Quantifying uncertainties in lung dosimetry with application to plutonium oxide aerosols

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Quantifying uncertainties in lung dosimetry with application to plutonium oxide aerosols
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Huston, Thomas Edward, 1968-
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Thesis (Ph. D.)--University of Florida, 1995.
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Includes bibliographical references (leaves 410-429).
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by Thomas Edward Huston.
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Vita.

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QUANTIFYING UNCERTAINTIES IN LUNG DOSIMETRY WITH APPLICATION
TO PLUTONIUM OXIDE AEROSOLS













By

THOMAS EDWARD HUSTON


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


1995




























Dedicated to Rhonda, my wife and friend--

without her encouragement this work would have been impossible.













ACKNOWLEDGMENTS


This work has been supported in part by Grant Number R23/CCR409769-01 from

the Centers for Disease Control and Prevention (CDC); however, its contents are solely

the responsibility of the author and do not necessarily represent the official view of CDC.

Preliminary portions of the research were performed under appointment to the

Environmental Restoration and Waste Management Fellowship Program administered by

Oak Ridge Institute for Science and Engineering for the United States Department of

Energy. I am especially grateful to these agencies for providing the financial assistance

necessary to complete this work.

I wish to thank my supervisory committee members-Drs. William G. Vernetson

(chair), W. Emmett Bolch (cochair), Marc Jaeger, David E. Hintenlang, William S.

Properzio, and Wesley E. Bolch--for serving on the committee and for their guidance

during the course of this study. I am especially thankful to Drs. William Vernetson,

Emmett Bolch, and Marc Jaeger for their useful suggestions to improve content in early

draft versions of the manuscript. I offer a special note of thanks to Dr. Emmett Bolch,

who during the past two years provided me (through the resources of the CDC grant) with

the financial backing to complete this research; his care for students is indeed a credit to

his profession.








I am thankful to several others with whom I have had the privilege of associating

throughout various stages of this work. These persons include, among many others, Drs.

Gen Roessler, John Till, Helen Grogan, Tony James, and Alan Birchall. Dr. Gen Roessler

served as a point of contact for me during the early stages of this work. She introduced

me to Dr. Till, who (along with colleagues from Radiological Assessments Corporation)

expressed a need for quantitative information regarding uncertainties in lung dose

predictions for plutonium. Dr. Till assisted during the early stages of this work by helping

me obtain draft reports from the National Council on Radiation Protection and

Measurements (NCRP) and the International Commission on Radiological Protection

(ICRP). At that time, both agencies were in the process of releasing independent models

for respiratory tract dosimetry. I appreciate the willingness of the ICRP and NCRP to

release these draft reports to me before publication. Drs. Tony James and Alan Birchall

provided me with many useful materials and encouragement. Dr. Helen Grogan provided

several helpful comments throughout the work.

I would like to thank my wife, Rhonda, for her constant encouragement

throughout my "tenure" at the university. She kept me focused on my work, never

complaining about the long hours I kept. She also helped organize the reference section

of the manuscript. I have been truly blessed with a wonderful, caring wife and a best

friend. I certainly look forward to spending the rest of my life (yes, I do plan to get one

after graduation) with her. I want to thank our new son, Hayden Edward Huston (born








only days ago on July 24, 1995), for bringing me happiness and wonder while I have

prepared this manuscript.

I would like to thank some (past and present) fellow students, Kaiss Al-Ahmady,

Brian Birky, Thabet Tolaymat, George Harder, Louis Iselin, Susan Stanford, and Missy

Jones, among others, for their friendship and stimulating discussions regarding the

radiation sciences. I also appreciate the departmental staff in the Nuclear Engineering

Sciences office (Barbara, Joan, Beth, Cess, and Donna) and the Environmental

Engineering Sciences office (Sandy, Berdenia, and Shirley). Thanks also to Chuck, Al,

Gene, and Gary for providing me with an outlet to this insanity by way of the Tuesday

night trombone jazz sessions.

Finally, but certainly not least, I am very thankful to all of my family and friends

back home in Arkansas for giving me hope and encouragement throughout the many,

many years of my education. I miss them all greatly.














TABLE OF CONTENTS


pAge

ACKNOWLEDGMENTS ........................................ iii

LIST OF TABLES ............ .................................. x

LIST OF FIGURES ............................................... xiv

ABSTRACT ........................................... ........ xviii

CHAPTERS

1 INTRODUCTION ............................................ 1

1.1 Purpose of Study ....................................... 3
1.2. Background Information ................................ 4
1.2.1. Radiation Exposure Pathways ......................... 4
1.2.2. Internal Radiation Dosimetry, Reference Man, and Risk ...... 9
1.2.3. Advantages and Disadvantages of the Reference Man Approach 14
1.3. Previous Work ............ ... ................... 16
1.4. Description of Present Study .......................... 20
1.4.1. Assessment Problem Addressed .................. 20
1.4.2. Units of Exposure .................................. 22
1.4.3. Summary of the Research Approach .................... 24

2 METHODS FOR RESPIRATORY TRACT MODELING .............. 27

2.1. Introduction ............. ........................ ....... 27
2.2. Development of Respiratory Tract Dose Models ................. 31
2.3. Respiratory Tract Morphological Model ...................... 33
2.4. Respiratory Tract Deposition Model ........................ 35
2.5. Particle Size and Diffusion Characteristics ..................... 48
2.6. Respiratory Tract Clearance Model ........................... 51
2.7. Respiratory Tract Dose Model ............................. 63
2.8. Sum m ary ................................ ........... 70








page


3 METHODS FOR CONDUCTING PARAMETER UNCERTAINTY
ANALYSES .............. .......... ................ 72

3.1. Introduction ............................................. 72
3.2. Quantifying Parameter Correlations and Uncertainties ............. 75
3.2.1. Parameter Correlations .............................. 75
3.2.2. Parameter Uncertainties ............................ 78
3.2.3. Testing the Fit of an Assigned Distribution ............... 84
3.2.4. Subjective Probability Judgements and Expert Opinion ...... 86
3.3. Propagation of Parameter Uncertainty ......................... 88
3.3.1. Simple Random Sampling ............................ 89
3.3.2. Latin Hypercube Sampling .......................... 91
3.4. Presentation of Uncertainties in Model Predictions ................ 96
3.5. Parameter Sensitivity Techniques ........................... 97
3.6. Description of the Lung Dose Uncertainty Code (LUDUC) ........ 102
3.7. Summary .................. ............ .. ....... 109

4 UNCERTAINTIES IN RESPIRATORY TRACT MODEL PARAMETERS 110

4.1 Deposition Model Parameters .............................. 112
4.1.1. BodyHeight .......................... ......... 112
4.1.2. BodyMassIndex ........... ...................... 116
4.1.3. Airway Diameters and Lengths ................... .... 123
4.1.4. Anatomical Dead Space ................... ......... 131
4.1.5. Regional Volumes of the Respiratory Tract ............... 135
4.1.6. Functional Residual Capacity ......................... 137
4.1.7. Vital Capacity ................................... 139
4.1.8. Ventilation Rate and Volumetric Flow Rate ............... 141
4.1.9. Tidal Volume ................................... 160
4.1.10. Fraction of Air Breathed Through the Nose ............... 169
4.1.11. Uncertainty in Particle Inhalability .................... 174
4.1.12. Regional Deposition Efficiency Uncertainty ............... 176
4.2. Clearance Model Parameters .............................. 179
4.2.1. Introduction ..................................... 179
4.2.2. Partition of Deposition in Clearance Model Compartments ... 180
4.2.3. Fractional Clearance Rate Constants .................... 188
4.2.4. Absorption Rate Constants ........................ .. 190
4.3. Dose Model Parameters .................................. 200
4.3.1. Introduction ..................................... 200
4.3.2. Measurements of the Bronchial Epithelium and Critical
Cell Layers ............ ........... .......... 201








page


4.3.3. Extrathoracic Regions ................... ......... 207
4.3.4. Bronchial Region .................................. 207
4.3.5. Bronchiolar Region ............... ............... 214
4.3.6. Surface Area of Bronchial and Bronchiolar Regions ......... 219
4.3.7. Alveolar-Interstitial Region ......................... 222
4.3.8. Total Lung Mass ......... ...... ................ 223
4.3.9. Mass of Lymph Nodes .............................. 225
4.3.10. Regional Risk-Apportionment Factors .................. 226
4.3.11. Summary of Methods for Dose Model Parameters .......... 228

5 APPLICATION OF LUDUC TO INHALED PLUTONIUM OXIDE
AEROSOLS: RESULTS AND DISCUSSION .................. 231

5.1. Introduction ................. ....... .... .. ............ 231
5.2. Characteristics of P'PuO2 Aerosols ...................... 233
5.3. Dose Uncertainty Results for Inhaled "'PuO2 Aerosols ............. 237
5.3.1. Exposure Scenario and Notational Conventions ............ 237
5.3.2. Adult Males Exposed to "'PuO2 Aerosols ................ 239
5.3.3. Influence of Dose Integration Time ..................... 249
5.3.4. Influences of Age and Gender ........................ 252
5.3.5. Influences of Particle Size Dispersion .................. 256
5.4. Dose Sensitivity Results for Inhaled 9PuO, ..................... .264
5.4.1. Combined Lung Dose ............................ 266
5.4.2. Regional Tissue Doses ............ ............... 270
5.5. Deposition Sensitivity Results ..... .................... 279
5.5.1. Bronchial Region ............ .... .... ..... ....... 283
5.5.2. Bronchiolar Region .............................. 283
5.5.3. Alveolar-Interstital Region .......................... 284
5.6. Implications of Dose Results and Comparison to Reference Man Dose 285
5.7. Summary of Results ................ ................... 292

6 CONCLUSIONS AND RECOMMENDATIONS ..................... 296

6.1. Summary of Research ................................ 296
6.2. Conclusions Based on Results for Plutonium Oxides .............. 300
6.3. Recommendations for Future Research ...................... 305









APPENDICES

A COMPARISON OF ICRP AND NCRP MODELS .................... 311

B THE CLEARANCE MODEL: SYSTEM OF DIFFERENTIAL
EQUATIONS ......................................... 323

C DOSE CALCULATIONS IN TRACHEOBRONCHIAL AIRWAYS ...... 330

D TIDAL VOLUMES FOR CHILDREN ........................... 355

E DOSE DATA FOR INHALATION OF PLUTONIUM OXIDE .......... 364

REFERENCES ................. ........ .... ................. 410

BIOGRAPHICAL SKETCH ......................................... 429














LIST OF TABLES


Table page

1-1. Selected intake-to-dose conversion factors for inhalation of some plutonium
isotopes and other transuranics .............................. 11

1-2. Organ and total risk per inhalation intake of plutonium compounds ....... 12

1-3. Specific information related to the assessment question and scenarios
considered in this study ..................................... 21

2-1. Recommended parameters for use in model of regional deposition for the
fraction of intake inhaled and exhaled through the nose for any subject as
functions of respiratory variables and anatomical size. ............... 42

2-2. Recommended parameters for use in model of regional deposition for the
fraction of intake inhaled and exhaled through the mouth for any subject, as
functions of respiratory variables and anatomical size. ................ 43

2-3. Description of input parameters required by the revised ICRP deposition
model. ................ ... .................. .............. 44

2-4. Other parameters of use in modeling particle deposition in the respiratory
tract. .............. ................................. 46

2-5. Reference values for mechanical clearance rates in the particle transport
aspect of the ICRP human respiratory tract model .................. 56

2-6. Reference values for partition of deposit among compartments in region. ... 56

2-7. Assignment of compartments to source regions for short-range radiations in
the revised ICRP respiratory tract model. ........................ 66

2-8. Risk-apportionment factors recommended by the ICRP for weighting the
dose to various respiratory tract target tissues .................. .. 69

3-1. Main steps involved in conducting a parameter uncertainty analysis ....... 74








Table page

3-2. Intervals paired for the Latin hypercube sample shown in Figure 3-6 ...... 95

3-3. Example of a hypothetical data set and its rank-transformed data set ...... 101

4-1. Selected height data from the Second National Health and Nutrition
Examination Study (NHANES II) .............................. 113

4-2. Body mass index data from NHANES II ........................... 119

4-3. Correlations between height and BMI. .......................... 119

4-4. Dimensional model of the tracheobronchial tree in adult male ............ 126

4-5. Expressions relating airway lengths and diameters to body height ......... 127

4-6. Regression equations and related information for functional residual
capacity. ............................................. 138

4-7. Regression equations and related information for vital capacity .......... 140

4-8. Regression equations and related data characterizing basal metabolic rates
for given age and gender classes ............................... 152

4-9. Typical basal multiplier values for different activities .................. 155

4-10. Distributions assigned to basal multiplier for four physical exertion levels. .. 155

4-11. Summary of distributions and methods used to predict variabilities in VE. ... 157

4-12. Fraction ofventilatory airflow passing through the nose in normal
augmenters and in mouth breathers. ............................ 172

4-13. Scaling constants used to account for uncertainties in regional deposition
equations. ............................................. 178

4-14. Distributions and relationships describing partition of deposited material
into compartments. ................. ...................... 182

4-15. Distributions adopted to describe uncertainties in fractional clearance rate
constants used in mechanical clearance aspect. ..................... 189








Table aMe

4-16. Results of studies that examined long-term clearance of inhaled PuO2
aerosols from the lungs. ...................................... 191

4-17. Distributions and relationships adopted to describe the absorption clearance
aspect for plutonium oxides. ................................. 199

4-18. Range of alpha particles in water for given energies .................... 201

4-19. Summary data for bronchial epithelium based on study by Gastineau et al. .. 204

4-20. Summary data for bronchial epithelium measurements based on study by
Baldwin et al............................................ 205

4-21. Summary data for bronchial epithelium measurements based on study by
Mercer et al. ............. ...... ........................... 206

4-22. Summary of distributions used to model uncertainty in source and target
tissue dimensions in the BB and bb regions. ...................... 229

4-23. Summary of methods to model uncertainties in source and target tissue
volumes and masses .............. ..... ..................... 230

4-24. Summary of distributions and relationships used to model uncertainties in
risk apportionment factors. ................................... 230

5-1A. Combined equivalent dose to the lungs, HH/AE: Ranking of selected
variables ........... ..... ........................ 268

5-1B. Combined equivalent dose to the lungs, HH/AE: Results of step-wise rank
regression analysis ............... ................... .. 268

5-2A. Equivalent dose to BB secretary cell layer, H(BB,)/AE: Ranking of
variables ............................................ 271

5-2B. Equivalent dose to BB secretary cell layer, H(BB.)/AE: Results of
step-wise rank-regression analysis ............. .... .......... 271

5-3A. Equivalent dose to BB basal cell layer, H(BBb)/AE: Ranking of variables 272

5-3B. Equivalent dose to BB basal cell layer, H(BB )/AE: Results of step-wise
rank-regression analysis ................ ................... 272








Table page

5-4A. Equivalent dose to bb secretary cell nuclei layer, H(bb,)/AE: Ranking of
variables .................... ................. ........ 273

5-4B. Equivalent dose to bb secretary cell nuclei layer, H(bb,)/AE: Results of
step-wise rank-regression analysis ........................... 273

5-5A. Equivalent dose to AI tissue, H(AI)/AE: Ranking of variables ............ 274

5-5B. Equivalent dose to AI tissue, H(AI)/AE: Results of step-wise rank-regression
analysis ......................... ......................... 274

5-6A. Equivalent dose to thoracic lymph nodes (LNth), H(LNth)/AE: Ranking of
input variables ........................... ............ 275

5-6B. Equivalent dose to LNth tissue, H(LNth)/AE: Results of step-wise rank-
regression analysis ...................................... 275

5-7. Ranking of the top ten input variables as determined by standardized rank-
regression coefficients for deposition fractions in respiratory tract
regions ............... ...... ...................... 280

5-8A. Deposition fraction for BB region, DFBB: Results of step-wise rank-
regression analysis for the four top-ranked model parameters .......... 281

5-8B. Deposition fraction for bb region, DF,: Results of step-wise rank-regression
analysis for the four top-ranked model parameters .................. 281

5-8C. Deposition fraction for AI region, DFu: Results of step-wise rank-regression
analysis for the four top-ranked model parameters ................. 281

5-9. Summary comparison of typical dose predictions for exposure of various
population groups ................... ................ 293

5-10. Summary list of the variables contributing the most to uncertainties in
equivalent doses to thoracic target tissues. ...................... 295

5-11. Summary list of variables contributing the most to uncertainties in deposition
fractions to thoracic regions. .................. ......... .. 295














LIST OF FIGURES


Figure page

1-1. Factors to consider in radiological risk assessment .................... 6

1-2. Ratio of upper to lower 95% confidence bounds for committed effective dose
equivalent per inhalation intake for adult male workers .............. 18

2-1. Components of a respiratory tract dosimetry model ................... 28

2-2. Anatomical representation of revised ICRP respiratory tract model ....... 34

2-3. Model of airway wall in the bronchial (BB) region .................... 36

2-4. Deposition model for the revised ICRP respiratory tract model .......... 37

2-5. Inhalability of particles as a function of the aerodynamic diameter for wind
speeds ofU= 1, 5,and 10 m/s. .............................. 39

2-6. Particle transport model adopted by the ICRP to represent mucociliary
and macrophage transport mechanisms to the gastrointestinal tract and
lymph nodes ............................................ 52

2-7. Compartment model recommended by ICRP to represent dissolution and
absorption to blood and lung tissues for materials deposited in the
respiratory tract. ............... .................. ....... 57

2-8. Alternate form for the dissolution/absorption model ................... 59

2-9. Schematic of the overall compartment model to represent clearance of
material from the respiratory tract by mechanical and absorptive processes. 61

3.1. Example of a uniform probability density function. ................ 80

3.2. Examples of triangular probability density functions ................... 81

3-3. Examples of two typical normal probability density functions. ........... 82








Figure page

3-4. Examples oflognormal probability density functions. .................. 83

3-5. Algorithm to generate a random number, on the interval (0,1) ......... 90

3-6. Example of a Latin hypercube sample. ............................. 94

3-7. Schematic diagram showing the flow of computations in the Lung Dose
Uncertainty Code (LUDUC). ................................ 103

4-1. Average body height as a function of age for males and females .......... 115

4-2. Ratio of predicted to observed fractiles in body height for males .......... 117

4-3. Average body mass index as a function of age for males and females ...... 118

4-4. Ratio of predicted to observed fractiles in BMI for males ............... 122

4-5. Average airway length versus airway generation at three body heights
and residual standard errors of the linear regression equations used to
predict lengths ................ .......................... 128

4-6. Average airway diameter versus airway generation at three body heights
and residual standard errors (RSE) of the linear regression equations used
to predict diameters ................. .... .................. 128

4-7. Comparison of the cumulative probability for the observed residuals of
ln(Vd) and the cumulative probability for Evd ...................... 133

4-8. Anatomical dead space as a function of body height ................... 134

4-9. Relationship between volumes at ambient temperature and pressure to
volumes reported at body temperature and pressure ................. 143

4-10. Predicted average values for VE as a function of age for males and females .. 158

4-11. Comparison of ventilation rates at rest for males and females ............ 160

4-12. Tidal volume at various physical exertion levels for males and females ..... 167

4-13. Comparison of average resting tidal volumes in present study with other
references ............ .................................... 169








Figr Rpage

4-14. Slow-clearing fraction of material deposited in the tracheobronchial
region as a function of the volume-equivalent particle diameter ........ 185

4-15. Normal probability plot of the residuals for the slow-clearing thoracic
deposit ................ .. ................ ............ 186

4-16. Total 239Pu disintegrations in the lung as a function of time for three values
ofs, for the scenario specified ................. ............. 197

4-17. Surface area of the bronchial region as a function of height ............. 221

4-18. Surface area of the bronchiolar region as a function of height ............ 221

4-19. Total (blood-filled) lung mass as a function of age for males and females ... 224

5-1. Fractiles predicted for the committed (50 year) equivalent dose per unit
activity exposure to the lungs for a population group of 25-34 year-old
males at light exertion ....................... ......... .. 240

5-2. Dose quantities versus particle diameter in all thoracic target tissues for adult
males exposed at light exertion ............................. .. 242

5-3. Equivalent dose to the lungs for adult males at four exertion levels: resting,
sitting, light, and heavy exertion ....... ...... ... ..... 245

5-4. Estimated values of median dose to all thoracic targets for adult males at four
exertion levels: resting, sitting, light, and heavy exertion ............. 247

5-5. Estimated values of the ratio of 95% to 5% dose fractiles for all thoracic
targets for adult males at four exertion levels: resting, sitting, light, and
heavy exertion .......... ............ .... ............ 248

5-6. Influence of integration time on dose uncertainties ................... 250

5-7. Influences of age and gender on dose uncertainties .................... 255

5-8. Estimated values of median dose to all thoracic targets for nine different
age/gender groups at light exertion .......................... .. 257

5-9. Estimated values of the ratio of 95% to 5% dose fractiles for all thoracic
targets for nine different age/gender groups at light exertion .......... 258









Figure page

5-10. Influence of activity-size dispersion on lung dose uncertainties ........... 260

5-11. Estimated values of median dose versus activity median diameter for thoracic
tissues in adult males at four activity-size dispersion values ........... 262

5-12. Estimated values of the ratio of 95% to 5% dose fractiles versus activity
median diameter for thoracic tissues in adult males at four activity-size
dispersion values ....................................... 263

5-13. Lognormal probability plots for various equivalent doses in the respiratory
tract .................................................. 286

5-14. Estimated values of the geometric standard deviation, GSD, of equivalent
dose per activity exposure in various target tissues for males at four broad
exertion levels ............. ....... ................... 288

5-15. Comparison of median dose values computed in this study to reference man
dose values as predicted by LUDEP .......................... 290













Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial
Fulfillment of the Requirements for the Degree of Doctor of Philosophy

QUANTIFYING UNCERTAINTIES IN LUNG DOSIMETRY WITH APPLICATION
TO PLUTONIUM OXIDE AEROSOLS

By

Thomas Edward Huston

August 1995


Chairperson: William G. Vernetson
Cochair: W. Emmett Bolch
Major Department: Nuclear Engineering Sciences

A complete respiratory tract model for predicting lung dosimetry of inhaled

radioactive aerosols involves several component models, including models for particle

deposition in airways, biokinetic clearance and radiological decay of deposited materials,

and radiological dose to critical target tissues. Each component depends on several

parameters which can vary among members of a population group. The traditional

approach has been to use reference values for parameters and to generate a single,

deterministic reference dose. Based on conducting parameter uncertainty analyses, a

methodology was developed in this study to incorporate parameter uncertainties into the

respiratory tract modeling process. The methodology allows lung dose predictions to be

determined as probability distributions, which better reflect the potential spread in doses

for members of population groups than a single reference dose. The study involved


xviii








compilation and critical evaluation of previous studies to recommend defensible

distributions representative of parameter uncertainties. Relationships were also identified

to account for correlations between many model parameters. An interactive computer

program, LUDUC (for LUng Dose Uncertainty Code), was developed to implement the

methodology. Doses resulting from inhalation of plutonium oxide aerosols (aerodynamic

diameters ranging from 0.1 to 50 microns) were investigated with LUDUC to demonstrate

the methodology. This specific application of the methodology developed dose data

which support an ongoing dose reconstruction study of plutonium released by the Rocky

Flats Plant in Colorado. Resulting dose distributions followed a lognormal distribution

shape for all scenarios examined. For many scenarios, the uncertainties in lung dose

predictions were substantial--with geometric standard deviations approaching values of

five. Uncertainties in doses increased by about a factor often from the smallest to the

largest particle sizes. Differences in predicted dose distributions were small when

comparing different age and gender groups from 2 to 35 years of age. Median doses for

plutonium oxide generally agreed with reference dose values, providing some level of

confidence in the reference-man approach. Parameter sensitivity analyses were conducted

for inhaled plutonium oxides and revealed that dose uncertainties are generally attributable

to only a few of the model parameters; parameter sensitivity depended on the inhaled

particle diameter.













CHAPTER 1
INTRODUCTION


As knowledge of particle deposition, clearance, and radiation dosimetry in the

human respiratory tract has increased, so have the sophistication and complexity of the

conceptual and computational models used to describe these processes [e.g., International

Commission on Radiological Protection (ICRP), 1960; Task Group on Lung Dynamics

(TGLD), 1966; ICRP, 1979 and 1994]. In spite of recent advances in respiratory tract

modeling, the reliability of model predictions should be addressed. A number of

publications have focused on the techniques of model reliability assessment [e.g., Morgan

and Henrion, 1990; International Atomic Energy Agency (IAEA), 1989; Organization for

Economic Cooperation and Development (OECD), 1987; Hoffman and Gardner, 1983].

Although the general techniques of model reliability assessment can be applied to any

mathematical model, past applications in the radiological dose assessment area (e.g.,

OECD, 1987; Desmet, 1988; Garten, 1980; Hoffman and Baes, 1979; Little and Miller,

1979) have primarily involved environmental transfer models (e.g., atmospheric, surface-

water, groundwater, and terrestrial/aquatic food-chain transport models). Similar

techniques have also been applied to light-water nuclear reactors to examine probabilities

associated with accident scenarios (Rasmussen, 1975; Lewis, 1978). The current study










extends the application of reliability assessment techniques to include anatomical and

physiological models adopted to simulate the human respiratory tract.

Several factors must be considered in a study of model reliability. According to a

report by the International Atomic Energy Agency, model reliability is affected by:

(1) uncertainty due to improper definition and conceptualization of the assessment
problem or scenario,
(2) uncertainty due to improper formulation of the conceptual model,
(3) uncertainty involved in the formulation of the computational model,
(4) uncertainty inherent within estimation of model parameter values, and
(5) calculational and documentation errors in the production of results.
(IAEA, 1989, p. 16)

The assessment problem and scope of this study, item (1), are discussed in detail in a

following section. Uncertainties in the formulation of the conceptual and computational

(or mathematical) models, items (2) and (3), were not examined here. In regard to items

(2) and (3), an underlying assumption in this research has been that the conceptual and

computational models adopted for this study appropriately represent the anatomy and

physiology of the human respiratory tract. This study has used the revised respiratory

tract model recently issued (ICRP, 1994) by the International Commission on Radiological

Protection (ICRP). The model is based on recommendations by an ICRP task group

composed of experts in the field of respiratory tract dosimetry. The effect of uncertainties

in respiratory tract model parameters on model predictions, item (4), has been the primary

focus of this research. These uncertainties were examined in detail by implementing

parameter uncertainty and parameter sensitivity analyses of the revised ICRP respiratory

tract model. Calculational errors, item (5), were examined to some level, since the

probabilistic computer code developed to implement parameter uncertainty analyses of the










model was verified by comparison to a simpler, deterministic code, LUDEP, developed

independently by the National Radiological Protection Board (Jarvis et al., 1993).


1.1 Purpose of Study

The primary purpose of this study was to develop a computational methodology

for quantifying uncertainties (including stochastic variabilities) in respiratory tract model

dose predictions for exposure of population groups to radioactive aerosols. The study is

based on parameter uncertainty analyses of recommended biological models describing the

fate of inhaled, radioactive aerosols in the human respiratory tract. A probabilistic

computer code, LUDUC (Lung Dose Uncertainty Code), was developed in this study to

perform these analyses. A secondary purpose of this study was to demonstrate this

computational methodology by application to various inhalation exposure scenarios for

plutonium oxide aerosols (while also providing guidance for application to other

radionuclides and chemical forms). An objective of this study was not to develop a new

respiratory tract model or to validate existing models, but rather to use a model currently

recommended by national and/or international scientific advisory bodies (NCRP, 1993;

ICRP, 1994) as the basis for conducting parameter uncertainty analyses. In addition,

parameter sensitivity analyses also were performed in the study to identify the parameters

that most influence model predictions.

The study was undertaken in an effort to improve current internal dose assessment

methods by including parameter uncertainties in the dose modeling process. The

methodology developed allows inhalation exposure-to-dose conversion factors (EDCFs),








4
among other assessment quantities, to be derived as probability distributions rather than as

single, deterministic values--which are based solely on a reference human and do not

account for parameter uncertainties arising from biological variability and lack of

knowledge. Probabilistic characterization of respiratory tract model predictions is

important for a many reasons.

First, probabilistic characterization helps to demonstrate the reliability of models

adopted to represent the anatomical and physiological characteristics of the human

respiratory tract. Second, it helps to improve the credibility of dose assessment

calculations by estimating the potential distribution or range of values (including

descriptive statistics such as medians, fractiles, means, and standard deviations) for

quantities of interest. Finally, probabilistic characterization, when performed in

conjunction with sensitivity analyses, helps to distinguish the more influential model

parameters (with respect to model predictions) from the less influential ones. This latter

sensitivity information can help guide future research efforts in respiratory tract modeling.


1.2. Background Information


1.2.1. Radiation Exposure Pathways

In scientific endeavors to quantify and predict human health risks from exposure to

ionizing radiations, mathematical models are both appropriate and necessary. Numerous

models exist for describing the release, environmental transport, food-chain propagation,

human intake/usage, internal and external dosimetry, and subsequent health risks of

radioactive materials (e.g., see Till and Meyer, 1983 and NCRP, 1984 for summary). An










illustration of the major factors considered in radiological risk assessment is shown in

Figure 1-1. The upper aspect of this figure (boxes 1-5) involves release conditions and

environmental transport factors, while the lower aspect (boxes 6-8) involves human

lifestyle and biological factors. As indicated by bold print in Figure 1-1, this study focuses

on components in the risk assessment chain that are associated with usage rates and dose

factors (boxes 6 and 7) for inhalation of radioactive aerosols.

Within the context of inhalation of radioactivity, usage rates refer to the amounts

of air and subsequently radioactivity inhaled per unit time by an individual. The amount of

radioactivity inhaled per unit time is termed the inhalation intake rate. The time integral

of the inhalation intake rate over the exposure duration is termed the inhalation intake

(i.e., the total activity inhaled during the exposure period). Using the International System

of units (SI), the inhalation intake is expressed in units ofbecquerels (Bq), where the

becquerel is the special SI unit for radioactivity; one becquerel is equal to one radioactive

disintegration (or transformation) per second.

A quantity of interest in this study, related to usage rates, is the activity exposure.

In this study activity exposure, denoted by AE and with units ofBq-hr/m3, is defined as the

time integral of the ambient activity concentration, C,(t) in Bq per m', over the exposure

duration, D, in hours. Thus,

D
A =f C.(t)dt = D*C. (1-1)
0
where E represents the time-averaged ambient activity concentration over the exposure

duration, D. The activity concentration represents the radioactivity per unit volume





























Figure 1-1. Factors to consider in radiological risk assessment. This study
focuses on components represented by boxes 6 and 7 for inhalation. After
NCRP 1984, Figure 1.



(ambient air) as a function of time for a specified radionuclide and a specified particle

activity-size distribution. The exposure duration, D, represents the time a group of

subjects is exposed to the specified activity concentration (i.e., the time over which the

inhalation intake occurs). The activity-size distribution characterizes the fraction of the

airborne activity associated with aerosol particles in various size ranges. Particle size

characteristics and activity-size distributions are discussed in section 2.6.

As defined above, AE is useful when the activity-size distribution and the physical

exertion level of the population group do not change significantly over the exposure

duration for the specified radionuclide. These conditions have been assumed in

conducting the present study. Throughout this study a simplifying assumption was to










model the deposition of inhaled material as an acute event. This assumption is justified

when the intake occurs over a relatively short exposure duration (during which ventilation

rates remain relatively constant for individuals within the population group). For a long

exposure duration (where ventilation and intake rates change), the mathematical model for

biokinetic clearance requires mathematical expressions for the intake rates for individuals.

Therefore, the methodology and results described herein are based on an acute activity

deposition in the lungs.

Dose factors for inhalation refer to the radiation dose delivered over a specified

time period to a tissue, organ, or the whole body per unit of inhaled activity. Dose

quantities of interest for an inhalation intake include the equivalent dose to various body

organs (including, but not limited to, the lungs) and the effective dose to the whole body.

These dose quantities are described in later sections of this dissertation. Generally, these

dose quantities represent the amount of radiation energy absorbed per unit mass (of the

target tissue) and include various weighting factors to account for the biological

effectiveness of different radiation types and for the differential radiation sensitivities of

organs in the body. The special SI unit for equivalent dose and effective dose is the

sievert, abbreviated Sv. The sievert has more fundamental units ofjoules per kilogram.

To compute radiation dose, the dose integration time must be specified. This

integration time refers to the period, following the intake, over which the radiation dose is

delivered. For radiation protection purposes, the ICRP (1991a) has recommended a dose

integration time of 50 years for adults (18 years or older) and 70 years for children when








8
deriving equivalent and effective doses. However, for epidemiological studies or radiation

litigation cases, other dose integration times might be desired.

It is important to distinguish clearly the exposure duration, D, from the dose

integration time. As described above, the exposure duration refers to the time over which

the intake occurs. The dose integration time refers to the time over which the dose rate is

integrated to compute radiation doses to various body tissues and organs. For

radionuclides that deposit in the body, dose continues to be delivered (in some cases over

many years) after the intake has occurred since material is retained by various tissues.

Of the possible pathways for intake of a radionuclide into the body (i.e., inhalation,

ingestion, absorption, and injection), inhalation is generally the most complex. Particles

that deposit in the respiratory tract can undergo a number of processes including: (1)

transport by mucociliary clearance to the gastro-intestinal (GI) tract, (2) phagocytosis by

alveolar macrophages accompanied by possible transport to the lung-associated lymph

nodes (where particles can be retained throughout the remainder of a person's life) or the

GI tract via the mucociliary escalator, and (3) dissolution accompanied by absorption from

lung tissues to the blood circulatory system. Therefore, in addition to being a region of

concern with respect to radiation-induced risk, the respiratory tract is also a portal for

inhaled radionuclides to other body tissues and organ systems. For example, animal

studies involving inhalation of a number of transuranic compounds have demonstrated

increased incidences of fatal cancer to the bones, liver, and kidneys, as well as the lungs

(ICRP 1980).










1.2.2. Internal Radiation Dosimetry, Reference Man, and Risk

Radiation dose calculations for a given inhalation intake have traditionally involved

deterministic techniques. In such techniques biological variabilities and uncertainties are

ignored in the dose prediction yielding a single quantitative result. The traditional

approach has been to define and apply anatomical and physiological models under the

auspices of a reference or standard man, where reference man represents an individual

comprising typical anatomical and physiological characteristics (e.g., ICRP, 1975, 1979).

Such characteristics have been chosen by scientific consensus in order to provide a

common (simplified and routine) basis for performing internal dose calculations.

Acknowledging differences attributable to age and gender, the reference-man approach

currently distinguishes between adult man, adult woman, and children of various ages

(ICRP, 1989). Although a useful concept for simplifying computational effort, relatively

few, if any, individuals in a given population group will match their reference man

counterpart identically.

Using the reference-man approach, tables of intake-to-dose conversion factors (or

DCFs) have been compiled for a number of specific intake conditions (e.g., Eckerman et

al., 1988; ICRP, 1979; ICRP, 1989; USNRC, 1992). Such tables are used extensively in

the nuclear industry and affect many decisions regarding past and future exposures of

both workers and the general public to potential internal radiation sources.

Generally for internal dosimetry, the DCF represents either the radiation equivalent

dose (to a tissue/organ) or the effective dose (to the whole body) per unit intake of

radioactivity and has units of sieverts per becquerel (Sv/Bq). These quantities are defined










in ICRP Publication 60 (1991 a). These dose quantities include a radiation weighting

factor, WR, to account for the different biological effectiveness of various radiation types

(e.g., photons, electrons, alpha particles, neutrons) per absorbed dose. When reflecting

the effective dose, the DCF also incorporates tissue weighting factors, wT, which account

for the different radiation sensitivities of organs/tissues to induction of detrimental

diseases.

Table 1-1 lists inhalation DCFs for some selected plutonium isotopes and

transuranics. Except for Pu-241, the dose from these radionuclides is due primarily to

energy deposited by alpha particles. Values in all but the last column are from Eckerman

et al. (1988) and are for a working, reference adult male. These values are based on the

dose methodology described in ICRP Publication 30 and Supplements (ICRP, 1979). An

aerosol with a 1 pm activity median aerodynamic diameter (AMAD, i.e., the diameter for

which half of the activity is associated with particles of smaller aerodynamic diameter)

with typical dispersion in size (i.e., geometric standard deviation or GSD < 3.5) has been

used in deriving these values. The last column, labeled effective dose (denoted H-f), was

derived using the organ equivalent doses from Eckerman et al. (1988) and the tissue

weighting factors adopted in ICRP Publication 60 (ICRP, 1991a). The clearance class

refers to the residence time of various chemical forms of the radionuclide in the lungs.

Clearance classes D, W, and Y (defined in ICRP, 1979) refer to materials having residence

times that are on the order of days, weeks, and years in the lungs, respectively. Generally,

plutonium oxides, which are of interest in this study, are assigned to class Y, while other

chemical forms of plutonium are assigned to class W. The term denoted f, corresponds to










the fraction of the activity in the small intestine that is absorbed to the blood. An

important point to note is that these tabulated DCFs have been listed without their

associated uncertainties; the reason is that uncertainties in these DCFs have not been

previously quantified.


Table 1-1. Selected intake-to-dose conversion factors (in Sv/Bq) for inhalation of some
plutonium isotopes and other transuranics based on an activity median aerodynamic
diameter, AMAD, of 1 um.
Inhaled Class"; Gonads' Lungs' Red Bone Bone Liver" Hfd
Nuclide fb Marrow' Surfaces'
Pu-238 W;1.0E-3 2.80E-05 1.84E-05 1.52E-04 1.90E-03 3.51E-04 6.26E-05
Y;1.0E-5 1.04E-05 3.20E-04 5.80E-05 7.25E-04 1.37E-04 6.15E-05
Pu-239 W;1.0E-3 3.18E-05 1.73E-05 1.69E-04 2.11E-03 3.78E-04 6.87E-05
Y;1.0E-5 1.20E-05 3.23E-04 6.57E-05 8.21E-04 1.51E-04 6.48E-05
Pu-240 W;1.0E-3 3.18E-05 1.73E-05 1.69E-04 2.11E-03 3.78E-04 6.87E-05
Y;1.0E-5 1.20E-05 3.23E-04 6.57E-05 8.21E-04 1.51E-04 6.48E-05
Pu-241 W;1.0E-3 6.82E-07 7.42E-09 3.36E-06 4.20E-05 6.57E-06 1.29E-06
Y;1.0E-5 2.76E-07 3.18E-06 1.43E-06 1.78E-05 3.01E-06 9.37E-07
Am-241 W;1.0E-3 3.25E-05 1.84E-05 1.74E-04 2.17E-03 3.91E-04 7.08E-05
Cm-244 W;1.0E-3 1.59E-05 1.93E-05 9.38E-05 1.17E-03 2.39E-04 4.04E-05
Source: All columns except last are from Eckerman et al. (1988). Last column is based on
a weighted sum of organ/tissue doses with tissue weighting factors from ICRP (1991a).
* "Class" refers to clearance class (D,W, or Y) as defined in ICRP Pub. 30 (ICRP 1979).
b The term f, represents the fraction of activity absorbed from the small intestines to the
blood.
SOrgan/tissue equivalent dose in sieverts per becquerel (Sv/Bq).
d Effective dose, HIf: sum of organ/tissue equivalent doses weighted by tissue weighting
factors as described by ICRP (1991 a).


For all of the transuranics listed in Table 1-1, the bone surfaces (endosteal tissues)

are predicted to receive the greatest equivalent dose (per inhalation intake). For class W

forms, the liver is predicted to receive the second highest equivalent dose followed by the

red bone marrow. For class Y forms, the lungs are predicted to receive the second highest










dose followed by the liver. In comparing organ DCFs, one should be aware that while

DCFs indicate the typical equivalent doses to organs (of reference man), they do not

necessarily reflect the radiation-induced health risks to these organs.

As a means to compare organs on a risk basis, the ICRP (1991a, Table B-20) has

recommended organ/tissue risk factors. These risk factors quantify the additional

probability (i.e., above the background or baseline probability) per organ equivalent dose

that a person will experience a fatal cancer or contribute a severe genetic effect to

offspring. The organ DCFs listed in Table 1-1 have been multiplied by respective organ

risk factors (from Table B-20 of ICRP, 1991a) to determine organ risks per intake. Table

1-2 lists results of these computations for some of the radionuclides in Table 1-1.

For all radionuclides listed in Table 1-1 with clearance class W, the three organs

having the greatest risk per unit intake (from highest to lowest risk) are: bone surfaces, red

marrow, and liver; the lungs rank fifth and account for only about 5% of the total risk

(based on data presented in Table 1-2). For clearance class Y, the order is lungs, bone

surfaces, and red marrow; and the risk to the lungs contributes over 70% to the total risk

from the intake. For both clearance classes, the order of organs as ranked by equivalent

dose does not match the order as ranked by organ risk. The discrepancy is due to the fact

that some organs are more sensitive to radiation than others.

As a practical example of what the numbers in Table 1-2 reflect, consider the total

risk per intake for class Y Pu-239; the value is 3.84x106 per Bq. The annual limit on

intake (ALI) for class Y Pu-239 is about 300 Bq (ICRP, 1991b). If one million (106)

people experienced a 300 Bq intake, then about 1000 out of the 106 people








13

[i.e., (300 Bq)*(3.84x106 Bq')*(106)] are predicted to develop a fatal cancer or pass on a

severe genetic effect to offspring. It is predicted that 70% of those people, based on Table

1-2, would develop fatal lung cancers. A number of assumptions are implied in this

methodology. The most important (and controversial) is that these risk factors, which are

based on exposures of persons at high dose/high dose rate, are appropriate for the low

dose/low dose rate exposures generally of interest in environmental exposure scenarios.


Table 1-2. Organ and total risk (of fatal cancer or severe genetic disorder) per inhalation
intake of plutonium compounds for working adult male.
Risk per Inhalation Intake (x 106 Bq'')
Pu-238 Pu-239 b
Organ Class W Class Y Class W Class Y
Lungs 0.156 (5.8%) 2.72 (73.7%) 0.147 (5.0%) 2.74 (71.4%)
Bone 0.950 (35.2%) 0.290 (7.9%) 1.05 (35.5%) 0.410 (10.7%)
Surfaces
Red 0.760 (28.1%) 0.362 (9.8%) 0.845 (28.5%) 0.328 (8.5%)
Marrow
Liver 0.526 (19.5%) 0.205 (5.6%) 0.567 (19.2%) 0.226 (5.9%)
Gonads' 0.280 (10.4%) 0.104 (2.8%) 0.318 (10.7%) 0.120 (3.1%)
Total Riskd 2.70 (99.0%) 3.69 (99.8%) 2.96 (98.9%) 3.84 (99.6%)
Notes: Values are based on product of dose factors from Eckerman et al. (1988) for
working adult male (presented in Table 1-1) and organ risk factors from ICRP Pub. 60
(1991a, Table B.20). Values in parentheses represent percent contribution of respective
organs to the total risk.
Values for Am-241 compounds are approximately equal to values for class W Pu-238.
b Values for Pu-240 are equal to values for Pu-239 (for respective classes).
Risk of severe genetic effects for gonads. Risk of fatal cancer for other organs.
d Total risk includes contributions from other organs (e.g., stomach, bladder, breast, etc.)
not listed in the table. Parenthetical values represent the percent contribution from
organs listed in the table.


In summary, the above discussion indicates that inhalation of radioactive aerosols

introduces risks to the lungs as well as to other body organs. Traditional methods provide








14

only point, deterministic predictions of organ dose/risk, not information on the uncertainty

in these quantities. Regardless of which organs acquire higher risks after an inhalation

intake, uncertainties in respiratory tract models are important and affect dose and dose-

based risk predictions. For certain inhaled radionuclides which pose the greatest risk to

lung tissues (e.g., the class Y transuranics listed in Table 1-1), uncertainties in all aspects

(i.e., deposition, clearance, and dosimetry) of the respiratory tract model are potentially

important. For inhaled radionuclides that pose greater risks to tissues other than the lungs

(e.g., class W transuranics listed in Table 1-1), uncertainties in the deposition and

clearance aspects of the respiratory tract model are nonetheless important because they

influence the amounts of radioactivity that reach these tissues. In the present study, the

focus is only on uncertainty in radiation dose to the lungs. However, the methodology

developed can also be useful for quantifying uncertainties in other organ doses (inasmuch

as they are influenced by the lung model).


1.2.3. Advantages and Disadvantages of the Reference-Man Approach

As discussed, the reference-man approach for dose assessment results in single,

quantitative deterministicc) dose estimates for a given intake scenario. In many

circumstances such an approach is adequate and useful. For example, the following

situations are well served by this approach: (1) if the purpose of the dose assessment were

simply to demonstrate compliance with radiation protection standards and regulatory

limits or, (2) if the purpose of the assessment were to perform screening calculations in

order to identify doses (and dose pathways) which might result in serious health effects.








15
However, if the dose assessment study were being performed to assess actual radiological

health risks either to an exposed individual or to an exposed population group (e.g., as in

radiation litigation cases or epidemiological studies), it is difficult to justify the reference-

man approach (depending on the magnitude of the predicted risks). Individual and

population-wide tissue doses might differ significantly from typical doses based on the

reference-man methodology. In the former case, an exposed individual would likely differ

from reference man with respect to anatomical and physiological characteristics. In the

latter case, the problem is that the reference-man approach does not address the biological

variabilities among individuals in a population group. Additionally, the reference-man

approach does not include uncertainties due to lack of knowledge regarding model

parameters.

A number of major studies are underway in the United States whose tasks are to

assess radiation doses and risks to population groups exposed to past releases of

radioactive and hazardous materials from nuclear weapons-related production and testing

sites. Facilities under study include the U.S. Department of Energy Hanford Site in

Washington, Rocky Flats Plant in Colorado, Savannah River Site in South Carolina, and

Oak Ridge National Laboratory in Tennessee, among others. A difficult issue that all of

these studies face is one of incorporating uncertainties in dose and risk estimates. Some of

these studies have attempted to include uncertainties in environmental transfer models

(e.g., PNL, 1991a, 1991b; 1991c); however, none have conducted a detailed examination

of the uncertainties in human biological models (describing the fate of the radionuclide

after it enters the body). It would be beneficial to know whether variabilities in predicted








16
dose attributable to uncertainties in the human biological models are negligible compared

to variabilities in dose attributable to uncertainties in environmental transfer models (i.e.,

uncertainties in source-term, atmospheric/hydrogeologic dispersion, and terrestrial/aquatic

food-chain models).


1.3. Previous Work

Upon conducting a literature search for previous work, three studies were found

which dealt with uncertainties in inhalation radiation doses attributable to human

respiratory tract models and associated parameters. Uncertainties have been examined

for: (1) the effective dose (to whole body) due to inhaled U-238, Th-230, Ra-226, and Pb-

210 (Wise, 1985), (2) dose to basal cells of the bronchial epithelium due to inhalation of

radon decay products (Hofmann and Daschil, 1986), and (3) dose to the lungs due to

inhalation of radon decay products based on the revised ICRP lung model (Birchall and

James, 1994).

Based on measured deposition data from human experiments by Chan and

Lippman (1980) and Yu et al. (1981), Wise (1985) examined uncertainties in the fitting

parameters of empirical equations used to predict particle deposition in the respiratory

tract. These uncertainties were propagated through the dose methodology recommended

in ICRP Publication 30 (1979) to obtain uncertainty bounds for committed-effective-dose

equivalents (i.e., dose integrated over 50 years post intake) for a number of selected

inhalation scenarios. Uncertainties in organ committed dose to the lung were not

presented; only the committed effective dose to the whole body. Selected statistics and










uncertainty limits were tabulated for a number of particle aerodynamic diameters,

radionuclides, and solubility classes. Wise did not directly account for stochastic

variabilities in the input parameters of the lung deposition model such as airway

dimensions, tidal volumes, ventilation rates, etc. Wise's study also neglected uncertainties

in the clearance aspect of the lung dose model. Furthermore, variability in parameters

involved in systemic and excretion models for anatomical and physiological systems

beyond the respiratory tract were not included. Thus, the extent of the uncertainty in

effective dose was under-estimated.

With the above limitations, Wise found that uncertainties in deposition model

predictions lead to committed effective dose equivalents whose lower and upper 95%

confidence limits differed by less than a factor of about 10. If a log-normal distribution is

assumed for dose predictions, this factor corresponds to a geometric standard deviation,

GSD, of about 1.8 (i.e., 101'4). Ratios of the upper 95% confidence bound to the lower

95% bound have been determined from results tabulated by Wise and are plotted in Figure

1-2. This plot represents a working adult male breathing (20 L/min) entirely through the

nose. The ratio for the two largest particle sizes (5 and 6 Prm) are not shown for U-238

since the lower 95% confidence limits were zero for these conditions. Wise also reported

data for mouth breathers; the ratios obtained were slightly less than for nose breathers.

While these results were important as a first attempt to quantify uncertainties in inhalation

dose coefficients, they are incomplete.

Hofmann and Daschil (1986) performed a parameter uncertainty analysis on the

respiratory tract models for radon progeny (Hofmann, 1982a, 1982b) in order to derive










15
S--- U-238(Y)

-40- Th-230 (Y)
S10- Ra-226 (W)

S .-- Pb-210 (D)




5 ---------------
--4T-



0 1 2 3 4 5 6
Particle Aerodynamic Diameter (microns)

Figure 1-2. Ratio of upper to lower 95% confidence bounds for committed effective dose
equivalent per intake for adult male workers. Based on results from Wise (1985). The
ratio is undefined for the largest two diameters (5 and 6 jpm) ofU-238 because the lower
bound was reported as zero.


distributions in radiation dose to the basal cells of various airway generations of the

bronchial epithelium. For the target cells considered (basal cells), Hofmann and Daschil

examined uncertainty in more detail than Wise (1985) by accounting for variabilities in a

number of model input parameters (including deposition fractions, total surface area of

bronchial generations, amount of inhaled activity, mucociliary clearance rates,

translocation rates to blood, and basal cell depths).

However, the authors did not examine uncertainties in radiation doses arising in

the pulmonary region of the lungs. Furthermore, doses were only examined for individual










basal cells as the targets. While dose at the cellular level is of interest, the approach

adopted in the revised ICRP lung model (ICRP, 1994) is to compute average doses for

critical tissues (composed of many critical cells). The authors found that doses to basal

cells of the bronchial epithelium under specified conditions were approximately

lognormally distributed and that the 50% and 99% fractiles differed by about a factor of

five depending on the tracheobronchial generation. This value corresponds to a GSD of

about two. This dispersion is similar to that which can be deduced from results of the

study by Wise (1985).

Birchall and James (1994) recently performed an uncertainty analysis of the

effective dose per unit exposure from radon progeny using the revised ICRP lung model

(ICRP, 1994). This model is also used to conduct the present study. The primary

purpose of their study was to reconcile dose factors derived from the revised lung model

with those derived from epidemiological estimates. The study was performed for a

working male with median ventilation rate of 20 L/min. Uncertainties were included for

aerosol size characteristics, ventilation rate, deposition and clearance factors, source and

target layer dimensions, and regional risk-apportionment factors (discussed later in

Chapter 2), among other variables. Here Birchall and James found that model predictions

were influenced largely by the choice of values for the regional risk-apportionment factors.

Results indicated that effective dose per unit radon progeny exposure varied over about

one order of magnitude based on parameter distributions selected.

While the study by Wise (1985) did produce results of a useful form for whole-

body dose assessment activities, the results are incomplete since uncertainties only








20
pertained to the lung deposition portion of the dose model. Hofmann and Daschil (1986)

produced results which accounted for more detail in the uncertainty and variability of lung

model parameters but did not examine uncertainties in doses to the pulmonary region of

the lungs. Consequently, it is difficult to apply results of that study to class Y materials

such as plutonium oxides where a relatively large dose is expected to be delivered to the

pulmonary region of the lungs. The study by Birchall and James (1994) included

parameter uncertainties in the revised ICRP lung model but focused strictly on radon

progeny. Furthermore, all of the above studies were performed for working reference

man--not for other variants of the population (i.e., other age and gender groups).


1.4. Description of Present Study


1.4.1. Assessment Problem Addressed

According to the IAEA (1989), the reliability of model predictions can only be

properly examined within the context of a well-defined assessment problem. The

assessment problem of interest throughout this study was the following:

Determine the assessed quantity (per unit activity exposure, i.e., per Bq-hr/m3) to
an unspecified individual in a population group exposed under the following
scenario:
(a) specified age/gender of the population group,
(b) specified exertion level (resting, sitting, light exertion, or heavy exertion) of
the population group,
(c) specified radionuclide/chemical form,
(d) specified aerosol activity-size distribution (including related parameters such as
particle mass density and particle shape factor, defined in section 2.5),
(e) specified dose integration time,
(f) specified ambient temperature and barometric pressure over exposure duration.










The term assessed quantity in the above assessment problem serves simply as a

placeholder for text describing specific quantities of interest in this study. Table 1-3 lists

the assessed quantities and scenario information which were investigated in the present

research. Quantities of interest included (1) the fraction of inhaled radioactivity that

deposits in various respiratory tract (RT) regions, (2) the total number ofradionuclide

transformations that occur in various RT source components as functions of time, and (3)

the radiation dose delivered to various RT target tissues for a specified dose integration

time. For item (3), the assessed quantity corresponds to a dose conversion factor,

expressed per unit activity exposure (i.e., an exposure-to-dose conversion factor, or

EDCF). These assessed quantities are discussed in more detail in Chapter 2.


Table 1-3. Specific information related to the assessment question and scenarios
considered in this study.
Radionuclide/ Exertion Level Population Assessed Quantityb
Chemical Form Age (years)"
239+20puO2 Resting (Sleeping) 2 Deposition Fraction
Sitting (Awake) 5 (activity deposited in RT
Light Exertion 10 regions)
Heavy Exertion 15
18-24 Number of Transformations
25-34 (in RT source components)
35-44
45-54 Equivalent Dose
55-64 (to RT target regions)

Equivalent Dose
(to lung: weighted sum)
a Both males and females have been considered at these ages.
b Quantities assessed in this study pertain to respiratory tract regions or to combinations of
regions; other organs/tissues are not examined. Assessed quantities and respiratory
tract regions are discussed in detail in Chapter 2.










Since this assessment problem addresses unspecified individuals within a

population group, it possesses a probabilistic rather than deterministic answer. In other

words, stochastic variability exists in the assessed quantity among unspecified individuals

in the population group. Therefore, the answer to the assessment question should be

represented by a probability density function (distribution) rather than a single number.

In performing demonstration calculations in this study, the scope of the assessment

problem has been narrowed to include only inhaled PuO2-laden aerosols with radionuclides

Pu-239 and Pu-240. These particular radionuclides/chemical form was selected to support

a dose reconstruction study underway by the Colorado Department of Public and

Environmental Health. One purpose of that study is to predict population risks arising

from historical releases of plutonium from the Rocky Flats Plant (a facility that produces

and stores nuclear weapons components and is located about 20 from downtown Denver,

Colorado). "2392PuO2 emits primarily alpha particles and is relatively insoluble in lung

fluids; also as shown by predictions in Table 1-2, this material (depending on the

magnitude of the intake) can result in potentially large risks to the lungs. As further

limitation on the work, aerosol particles were assumed to be non-hygroscopic (i.e.,

particles do not increase in size upon entering saturated conditions of the respiratory tract)


1.4.2. Units of Exposure

Traditional methods have predicted inhalation dose conversion factors per unit

intake, where the intake represents the amount of radioactivity inhaled. However, in the

assessment question defined above, the intake possesses stochastic variability among










individuals of the exposed population. In order to account for these variabilities, an

improved reference unit for dose predictions is the activity exposure, AE, defined earlier.

At least three reasons exist for expressing dose factors per unit activity exposure

rather than per unit activity intake. First, in assessing doses to humans from inspirable

radionuclides released to the environment, atmospheric dispersion models are usually

implemented to estimate airborne activity concentration, i.e., C,(t) in Bq m"3 at specific

locations. By expressing dose factors in terms of activity exposure rather than intake, a

more efficient coupling of the results of dispersion models and inhalation dose factors can

be achieved. If dose factors were per unit intake, one would still need to determine the

inhalation intake in order to use the dose factor.

Second, intake depends on the human ventilation rate which is influenced by

biological variability and lifestyle habits among members of a population. It is desirable in

this study to account for variability in ventilation ( and subsequently the intake) as a

contributor to uncertainties in dose factors. The activity exposure, AE, is not influenced

by the ventilation rate.

Third, the ventilation rate is a parameter in the deposition component of the lung

dose model (it is proportional to the average inspiratory flow rate); so the dose coefficient

(expressed per unit intake) is correlated with the intake since both are mutually dependent

on ventilation rate. Therefore, in this study inhalation dose coefficients were derived per

unit activity exposure and have units of Sv per Bqhrm"3.










1.4.3. Summary of the Research Approach

As discussed in the opening section of this chapter, the reliability of model

predictions is affected by several uncertainties introduced through the modeling process.

This study examined the uncertainty in model predictions attributable to parameter

uncertainties. The respiratory tract model issued recently by the ICRP (1994) was

selected to perform this study. The study was divided into three phases: I, II, and III.

Phase I involved development of a computer program to implement the adopted

respiratory tract model and to couple the model with existing numerical random sampling

programs. The program developed in this work accounts for all aspects of the RT model,

including particle deposition, clearance, and radiation dosimetry. An existing program that

performs Latin hypercube sampling (Iman and Shortencarier, 1984)--which is a form of

random number sampling used to select values for the model parameters from their

respective, assigned, probability distributions--was coupled with the program for solving

the respiratory tract model. Sampled model parameter values are combined to form input

vectors (or realizations) which are propagated through the model to produce numerical

distributions in model predictions. An overview of the RT model, and the techniques

implemented to solve it, are provided in Chapter 2 and related appendices. Techniques

employed in this study for generating random samples of the joint probability density

function for the combined ranges of all parameters for propagating parameter uncertainties

through the model structure are discussed in Chapter 3. That chapter also presents

techniques used to examine parameter sensitivities in the model predictions.










Phase H of the study involved quantifying uncertainties and variabilities in

respiratory tract model parameters. Some of the general concepts involved in this phase

of the study are discussed in Chapter 3. Based on an extensive literature review, Chapter

4 presents the specific parameter distributions adopted to perform uncertainty analyses in

this study. This phase of the work was necessarily tedious since approximately seventy

input parameters were assigned distributions in the RT model. Although many of these

parameters were not primary input parameters, they were needed in order to account for

correlations between and restrictions on the primary model parameters. Most of the

parameter distributions presented in Chapter 4 are independent ofradionuclide and

chemical form; so techniques developed in this study could be applied with minimal effort

to other radionuclides/forms. For example, except for hygroscopic materials, all

parameters related to the deposition component of the model are generally independent of

radionuclide and physico-chemical form. For the clearance model, only the parameters

related to dissolution and absorption clearance processes and the radioactive decay

constant depend on the radionuclide/form. For the dose model, all source and target

geometrical dimensions are independent of nuclide/form; however, energy deposition

computations do depend on the radionuclide decay scheme (radiation types, frequencies,

and energies).

Phase HI of the study involved application of the computational methodology

(developed in Phase I) and recommended parameter distributions (developed in Phase II)

to various assessment scenarios (i.e., substitution of entries in Table 1-3 into the

assessment problem model). These scenarios involved inhalation exposures to plutonium








26
oxide aerosols by various population groups specified by age and gender. Equivalent

doses to lung tissues and an overall, combined lung dose were examined for various

particle sizes. This final phase also included sensitivity analyses with the primary model

parameters to determine which parameters contributed most to the uncertainty in

equivalent dose predictions for the lungs. The results of these activities are presented in

Chapter 5. Chapter 6 presents a summary of the work, important conclusions and

recommendations, and suggests future research to expand the methodology and the body

of knowledge concerning uncertainties in human biological-radiological dosimetry models.













CHAPTER 2
METHODS FOR RESPIRATORY TRACT MODELING


2.1. Introduction

In this study, parameter uncertainty analyses have been conducted with the new

respiratory tract model recommended by the International Commission on Radiological

Protection (ICRP, 1994). The National Council on Radiation Protection and

Measurements (NCRP, 1993) has also been working to issue guidance on respiratory tract

modeling (independently of the ICRP). However, a comparison of the two models (see

Appendix A) revealed difficulties in implementing the NCRP model in the numerical

strategy employed by this study--which required solving the model repetitively and

efficiently. In this study, Monte Carlo techniques have been employed to treat model

parameters as random variables within the computational model recommended by the

ICRP (these techniques are discussed in Chapter 3). The purpose of the current chapter is

to provide an overview of the components of the new ICRP respiratory tract model and to

outline the computational techniques for solving these components. A complete

description of the model can be found in ICRP Publication 66 (ICRP, 1994).

Although generally cited as a single entity, a respiratory tract model for dose

assessment involves a number of component models arising from many specialized areas

of study. As illustrated in Figure 2-1, the overall model can be divided into three primary











components--deposition, clearance, and dosimetry models. Each component involves a

number of input parameters which can depend on the exposure scenario. Topics which

must be considered in developing the conceptual and computational models for the

respiratory tract and in quantifying associated input parameters include respiratory tract

morphology, radiation biology, and respiratory physiology. In this context, morphology

refers to the anatomy and structure of the airways and associated tissues; radiation biology

involves identification and sensitivities of various critical (target) cells in the respiratory

tract; and respiratory physiology involves the functions and dynamics of the airways and

associated tissues. Aerosol particle size characteristics must also be considered since they

affect deposition mechanisms.




Respiratory Tract Model

User Input Parameters Deposition Model:
(aerosol properties, particle
impaction, sedimentation,
and diffusion mehanisms)
(Deposition Fractions)

SClearance Model:
(biological removal/retention
radioactive decay processes)

(Total Transrmations/Rates per Source Region)

Dosimetric Model:
(radioactive decay,
Radiation Dose Estimates (e
type and energy of emissions,
source-target geometries)



Figure 2-1. Components of a respiratory tract dosimetry model.










Particles deposit on airway walls by several interacting mechanisms including

inertial impaction, gravitational sedimentation, Brownian diffusion, interception, and

electrostatic attraction. The efficiencies of these deposition processes depend on a number

of parameters including: particle physical characteristics (e.g., aerodynamic diameter, mass

density, shape factor, and diffusion coefficient), airway diameters and lengths, airway

branching and gravitational angles, tidal volume, functional residual capacity, and

ventilation rate. Due to particle physical characteristics and air flow patterns in the lung,

one or more of the above deposition mechanisms can dominate in various regions of the

respiratory tract. Deposition generally does not depend on the radionuclide or its

chemical form (except for hygroscopic materials). Predictions of the deposition

component of the respiratory tract model include the fractions of inhaled activity which

deposit in various respiratory tract regions. These deposition fractions are subsequently

inputs to the clearance model.

Clearance of deposited materials refers to their biological and radiological removal

from the respiratory tract. Three primary mechanisms exist for biological clearance in the

respiratory tract: (1) mucociliary clearance of material to the gastrointestinal tract, (2)

phagocytosis by alveolar macrophages followed by subsequent translocation to lymph

nodes or to the mucociliary escalator, and (3) dissolution of material accompanied by

absorption to airway tissues and/or to the blood circulatory system. When the material is

radioactive a fourth removal mechanism exists--radioactive decay. Depending on the

region of deposition and the physico-chemical form of the deposited material, various

clearance mechanisms/patterns can occur. To model clearance, the respiratory tract is










compartmentalized and is traditionally described by a system of first-order, linear

differential equations with respect to time. Except in very unique circumstances,

deposition and clearance models are not intended solely for radioactive materials. From

both a biological and mathematical perspective, the only difference between a radioactive

and non-radioactive material with respect to deposition and clearance is the

accommodation of radioactive decay as one removal mechanism. Two predictions are

generally of interest for the clearance model as applied to inhaled radionuclides. One is

the total number of radioactive transformations (or disintegrations) that occur in individual

model compartments over a specified integration time; the other is the transformation rate

(or activity) of the radionuclide in individual model compartments at a specified time. The

latter model prediction is related to the radiation dose rate while the former is related to

the radiation dose (i.e., the time-integrated dose rate).

The dosimetry component of a respiratory tract model is unique to radioactive

materials. The radiation dose rate attributable to short-range radiation types (e.g., alpha

particles, beta particles, and electrons) is directly related to the amount of radioactivity

present in lungs; this activity is determined by solving the clearance model for the intake

scenario of concern. For gamma rays and other penetrating radiation types, activity in

other body organs can also contribute to the lung dose. The radiation dose is the integral

of the dose rate over time and is related to the total number of radioactive transformations

that occur in organs in the specified time interval. The dosimetry aspect depends on the

radionuclide since different decay schemes (radiation types, frequencies, and energies) are

characteristic of different radionuclides. Furthermore, for short-range radiations, the dose










to critical lung tissues is very sensitive to the source-target geometry. This geometry

refers to the spatial relationship between tissues that contain the radioactive material

(source regions) and tissues that contain the critical cells (target regions).


2.2. Development of Respiratory Tract Dose Models

A brief summary of the evolution of the model used in this study has been given by

the ICRP (1994). The first coherent model integrating respiratory tract deposition,

clearance, and dosimetry was issued in ICRP Publication 2 (ICRP, 1960). Prior to that

publication, discussions and recommendations concerning respiratory tract dosimetry were

made at various conferences, such as the Tripartite Conferences on Radiation Protection

held from 1949-1953 (Taylor, 1983). The model reported in ICRP Publication 2 assumed

that, lacking specific data, 75% of inhaled aerosol particles would deposit in the

respiratory tract (50% deposited in upper airways; 25% deposited in the lungs; and 25%

was exhaled). Concerning clearance, the model classified materials as either soluble or

insoluble. Soluble materials that deposited in the lungs were assumed to be absorbed

completely (by the blood) and translocated to other body tissues. For insoluble materials

that deposited in the lungs, half of the material was assumed to clear (to the GI tract) with

a biological half time of 24 hours (presumably to the GI tract); the other half was assumed

to clear with a biological half time of 120 days (radiological decay would affect the overall

clearance time).

In 1966, a revised lung model was published (TGLD, 1966) by the ICRP Task

Group on Lung Dynamics. A slightly modified version of this model was ultimately








32
published by the ICRP in Publication 30 (ICRP, 1979). Most of the current annual intake

limits ofradionuclides for workers (e.g., USNRC, 1992) are based upon the Publication

30 model, which is a considerable improvement over the Publication 2 model. The ICRP

Publication 30 model divides the thoracic region of the lungs into two regions

(tracheobronchial and pulmonary) and includes a deposition model that accounts for

particle size influences. The Publication 30 model incorporates an improved clearance

model which provides guidance for classifying materials into three (compared to only two

in the former model) clearance classes. Clearance to lymph nodes, blood, and the

gastrointestinal tract are all included in the model. The model is based on an adult lung

morphology for deposition and on both human and animal data for clearance of materials.

Dosimetry calculations are based on averaging energy deposition over the total lung mass.

In December of 1994, the ICRP issued a new respiratory tract model as ICRP

Publication 66 (ICRP, 1994). Among the reasons for revising the model were: (1) to

provide a means for calculating doses to the nasal and oral passages; (2) to compute more

biologically significant doses to critical target regions rather than simply averaging dose

over the total lung mass; (3) to provide more flexibility for selecting various clearance

rates so that predicted clearance patterns better match observed patterns; (4) to

incorporate new data on particle deposition below 0.1 Pm particle diameter; (5) to

incorporate new knowledge and data on particle retention and clearance in various regions

of the respiratory tract; and (6) to construct a model that applies to all members of the

world's population consistent with age, gender, and race. A principal feature of the








33
revised lung model is that the relative radio-sensitivities of various respiratory tract tissues

are taken into account when combining regional doses to obtain an overall lung dose.

As discussed in the opening paragraph, the NCRP (1993) has also been working,

independently of the ICRP, to issue new guidance on respiratory tract modeling. Many of

the items listed above for the ICRP Publication 66 model are also addressed in the model

adopted in principle, but not yet published, by the NCRP. The conceptual and

computational models recommended by the NCRP and the ICRP (1994) differ in several

ways. Differences exist in the deposition model resolution, specification of fractional

clearance rates, and specification of target regions in the dosimetry model. Some of these

differences between the models are discussed in more detail in Appendix A.


2.3. Respiratory Tract Morphological Model

The anatomical representation of the revised ICRP (1994) respiratory tract model

is shown schematically in Figure 2-2. Based on structure and function, the model divides

the respiratory tract into the following five primary regions:

(1) the extrathoracic region comprising the anterior nasal passages, ET,,

(2) the extrathoracic region comprising the posterior nasal passages, larynx, pharynx, and
mouth, ET2,

(3) the bronchial region, BB, comprising airway generations 0 through 8 (trachea through
the bronchi),

(4) the bronchiolar region, bb, comprising airway generations 9 through 15, and

(5) the alveolar-interstitial region, AI, comprising the first respiratory bronchioles through
the alveolar sacs and including interstitial connective tissues.























Thoracic


Bronchial


Bronchioles bb / Bronchiolar

\/ i Al Alveolar Interstitial


bb
\ Bronchioles
Terminal Bronchioles


AJ Respiratory Bronchioles


Alveolar Duct +
Alveoli






Figure 2-2. Anatomical representation of the revised ICRP respiratory tract model. The
respiratory tract is divided into five primary regions (ETI, ET2, BB, bb, and AI) and two
lymph node regions (not shown). Reproduced by permission from ICRP Publication 66
(ICRP, 1994, p. 9).










In addition two lymph node regions are identified to represent the lymph nodes (LNET)

which drain the extrathoracic regions (ET, and ET2) and the lymph nodes (LNTH) which

drain the thoracic regions (i.e., BB, bb, AI).

To account for the morphological variation of the respiratory tract with respect to

age, gender, and race, scaling factors are used in the deposition model. These factors

allow regional deposition to be computed for any subject by scaling certain aspects of the

model based on the ratio of subject airway dimensions to those of reference man. Scaling

factors are based on the diameters of the trachea and airway generations 8 and 15. The

thickness and cellular structure of surface and epithelial tissues in ET, BB, bb, AI regions

are taken to be invariant with age, gender, and body size; however the surface area of the

various airway regions does depend on body size and is accommodated in this study.

For ET, BB, and bb regions, the airways are modeled as cylinders. Source and

target tissues residing within the airway wall are considered to be cylindrical shells. Figure

2-3 is a schematic of the cross sectional view of a typical airway wall for the bronchial

region. Although there are some differences in thicknesses and tissue layer structure,

similar geometries are used in the model to represent bb and ET airway wall tissues.


2.4. Respiratory Tract Deposition Model

The clearance pattern and regional doses are influenced by the amounts of

radioactive material which deposit in the various respiratory tract regions. To compute

the fraction of inhaled radioactivity which deposits in the five broad respiratory tract

regions, the basic approach of the ICRP (1994) is to model the respiratory tract as a series








36

of filters, as illustrated in Figure 2-4. The first filter in the figure represents the inhalability

of the aerosol--discussed below. The other filters represent the respiratory tract regions

described above. Because deposition occurs during both inhalation and exhalation, some

respiratory regions appear more than once in the filter chain.




5 5n ~.. ".." ..+.Mucus (Gel Layer)
6pm Cilia + Sol Layer
10 am

Nuclei of
Secretory 35 pm
Cells 30 pm i
(Target)

I I Nuclei of
Basal Cells 5 Basal Cell
Basement (Target)
Membrane 5j m

Lamina
O10 Propria

Subepithelial
Layer <500 pm
of Tissue





/Alveolar Interstitiu




Figure 2-3. Model of airway wall in the bronchial (BB) region. Targets in this region
include secretary cell nuclei and basal cell nuclei. The region is separated from the
alveolar interstitium by a layer of subepithelial connective tissue. Reproduced with
permission from ICRP Publication 66 (ICRP, 1994, p. 15).













Region


Na
Pa



Extrathoracic


sal
thway
F


Environment
(ambient air)


Inhalability
Inhaled Air

,A


Bronchial

Bronchiolar

Alveolar-Interstitial


Bronchiolar

Bronchial


Extrathoracic E


Exhaled Air


hm(ET2), hin(ET2)


hil(BB)

hin(bb)


h(AI)

hex(bb)

he(BB)


hex(ET2), heET2m)


hexETl)


Exhaled Air


Figure 2-4. Representation of the deposition component of the revised ICRP respiratory
tract model. The respiratory tract is modeled as a series of filters with characteristic
volumes and deposition efficiencies. After Figure 8 in ICRP(1994).


Filfration


Filtration
Efficiency




h,(

NhTinT)










Two pathways are considered for air flow in and out of the thoracic region--the

nasal pathway (i.e., air inhaled via the nose) and the oral pathway (i.e., air inhaled via the

mouth). Both pathways can coexist during breathing. The fraction of air inhaled which

passes through the nose is denoted by F,. These pathways are distinguished because the

nasal pathway generally results in more filtration of particles than the oral.

Each deposition region (or filter) possesses two characteristic parameters--its

volume, V, and its deposition (or filtration) efficiency, h. The deposition efficiency

represents the overall efficiency of the region for removing aerosol particles (h = 1 -

output/input particles). Denoting the tidal volume of the flow by VT and the (dead space)

volume of thejth filter by Vj, the volumetric fraction of the tidal air which reaches filter

is given by:


1 for j = 0
Sj-1
S 1 Vi for 1 : j (N+1)/2 (2-1)
VT, iO

4N-j+ for (N+3)/2 j N

where a value of zero for j denotes the inhalability (a virtual filter with Vo = 0; discussed

in following paragraph); N is the total number of filters (not counting inhalability)

considered for the complete breathing cycle (inhalation and exhalation). Note that N = 9

for nose breathing, and N = 7 for mouth breathing. The middle component of Eq. 2-1

represents the fraction of air reaching filter during inhalation. The bottom component

represents the fraction of air reaching filter during exhalation and assumes that the

volume of air traversing a filter during inhalation equals the volume during exhalation.










Inhalability (the first "filter" in the chain) refers to the efficiency with which

particles of a given diameter are able to be inspired by the nose and mouth. Inhalability

effects arise due to the complex flow of particles around the human head and torso and is

important only for particles with aerodynamic diameters larger than about 0.5 pm. For

increased wind speeds and larger particles, the inhalability can be greater than one; this

effect is attributable to non-isokinetic sampling conditions of the human head. Based on

the model recommended by the ICRP (1994), the inhalability is plotted in Figure 2-5 as a

function of the particle aerodynamic diameter. In the deposition model, inhalability can be

regarded as a virtual filter which acts to remove particles from the air before they actually

enter the airways.

2-

1.8 h = 0.5 (1 [7.6x104 d ae + l] ) + 1.0xl0"5 U275 exp(0.055d a)
1.6-

1.2 -





0.6
0.4- U= 1 m/s
0.2-
0-
0 10 20 30 40 50 60 70 80 90 100
dae (microns)
Figure 2-5. Inhalability of particles as a function of the aerodynamic diameter for various
wind speeds ofU = 1, 5, and 10 m/s.










In this sequential filtration system, the index corresponds to a respiratory tract

region as it is encountered by air during the respiratory cycle (both inspiration and

expiration). The fraction of material inhaled which deposits in filter, DEj, is given by an

expression involving the filtration efficiencies, hi, of the preceding filters and the

volumetric correction factor,
j-I
DE = hjjH (1 h) (2-2)
i-O0
Deposition in each respiratory region results from both aerodynamic and

thermodynamic processes. Aerodynamic processes include inertial impaction and

gravitational settling; thermodynamic processes include particle diffusion and Brownian

motion. Aerodynamic deposition mechanisms generally dominate for particles with

aerodynamic diameter larger than about 0.2 pm while thermodynamic mechanisms

dominate for particles less than this size. The model assumes that particles in the aerosol

carry no net charge--i.e., electrostatic deposition mechanisms are not accommodated.

Denoting h, as the aerodynamic deposition efficiency and h, as the thermodynamic

deposition efficiency, James et al. (1994b) have represented the combined deposition

efficiency of region j by the quadratic expression:


hj = (h 2 + hh2)1 (2-3)


James et al. (1994b) provide a detailed discussion of the mathematical expressions

used to estimate h. and h. for the various respiratory tract regions/filters. The

expressions for extrathoracic regions are based upon regression analyses of experimental

data involving both persons (e.g., Stahlhofen et al., 1989; Rudolf et al., 1986) and hollow








41
casts (e.g., Swift et al., 1992). Expressions for thoracic regions are based upon regression

analyses of deposition data predicted by an underlying, more sophisticated, theoretical

model developed by Egan and Nixon (1985), Nixon and Egan, 1987, and Egan et al.

(1989). The equations adopted by the ICRP (1994) for regional deposition efficiencies are

listed in Tables 2-1 and 2-2. Expressions presented in Table 2-1 are for air inhaled and

exhaled through the nose; those in Table 2-2 are for air inhaled and exhaled through the

mouth.

The primary difference between nose and mouth breathing is that the anterior nasal

passages (region ET1) are not part of the flow pathway for mouth breathing. The

equations apply over a wide particle size range--from a 0.001 pm thermodynamic particle

diameter to a 100 pm aerodynamic particle diameter.

As shown in Tables 2-1 and 2-2, the deposition model depends on a number of

parameters. Parameters required as direct input to the deposition model are listed along

with brief descriptions in Table 2-3. Reference values for parameters have been presented

in ICRP Publication 66 (ICRP, 1994) and are specified for various age and gender

groups. Many of the parameters described in Table 2-3 depend on other, more

fundamental, parameters such as body height, weight, and age. Other parameters,

investigated during the course of this work to account for correlations between the direct

input parameters, are listed in Table 2-4. Uncertainties and variabilities in these

parameters are examined in detail in Chapter 4.

The outputs of the deposition model are the fractions of activity that deposit in the

various respiratory tract regions--ET,, ET2, BB, bb, and AI. The deposition model does
















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Table 2-3. Description of input parameters required by the revised ICRP deposition model.
Quantity (units) Symbol Description/Comments
Aerodynamic Diameter d, Diameter of a unit density sphere with same settling
(pmn) velocity as the particle of interest. Used to quantify
aerodynamic deposition mechanisms (important for
particles larger than about 0.2 pm).
Thermodynamic d, Diameter of spherical particle with same diffusion
Diameter (ipm) coefficient as the particle of interest.
Diffusion coefficient D Physical constant for particle characterizing its
(cm2/s) diffusivity in specified medium (air in this case);
depends on particle size, air temperature, and
pressure, among other quantities. Used to quantify
thermodynamic deposition mechanisms (important for
particles smaller than about 0.2 pm).
Volumetric Flow Rate VF Total flow rate of air entering the respiratory tract
(mL/s) (nasal + oral pathways).
Nasal Volumetric VF. Flow rate for air entering by nose; V,=VFF;
Flow Rate (mL/s) F,= fraction of air inhaled by nasal pathway.
Oral Volumetric Flow VFm Flow rate for air entering by mouth; VFm=VF(1-F)
Rate (mL/s)
Tracheal Scaling SF, Scaling factor to relate regional deposition efficiencies
Factor in ET and BB regions of subject of interest to
reference man. SF, = 1.65/do (do = diameter of
trachea in subject, cm).
Bronchiolar Scaling SFb Scaling factor to relate regional deposition efficiencies
Factor in bb region of subject of interest to reference man.
SFb = 0.165/d, (d8 = diameter of airway generation 8
in subject, cm).
Alveolar Scaling SFA Scaling factor to relate regional deposition efficiencies
Factor in AI region of subject of interest to reference man.
SFA = 0.051/d16 (di6 = diameter of generation 16 in
subject, cm).
Residence time in tB Time constant for conduction of air through BB
BB Region (sec) region. The ICRP has recommended the following
expression to compute tB from other quantities:
_tB = VD(BB) (1 + 0.5 VT/FRC)/V











Table 2-3 -continued.
Quantity Symbol Description/Comments
Residence time in tb Time constant for conduction of air through bb
bb Region (sec) region. ICRP has recommended the following
expression to compute tb from other quantities:
tb = VD(bb) (1 + 0.5 VT/FRC)/VF.
Residence time in tA Time constant for residence of air in the alveolar
AI region (sec) region. ICRP has recommended the following
expression to compute tA from other quantities:
tA = [VT VD]i[1 + 0.5 VT/FRC]/VF
where VD= VD(ET) + VD(BB) + VD(bb).
Tidal Volume (mL) VT Volume of air inhaled or exhaled per breath; depends
on temperature and saturation conditions. Tidal
volume is specified at body temperature, ambient
pressure, and saturated with water vapor.
Volume of ET Region VD(ET) The (dead space) volume of the ET region airways
(mL) (anterior nose + oro/nasopharynx+larynx).
Volume of BB Region VD(BB) The (dead space) volume of the BB region airways
(mL) (trachea + airway generations 1-8).
Volume of bb Region VD(bb) The (dead space) volume of the bb region airways
(mL) (generations 9 to 15).
Functional Residual FRC Volume of air remaining in the lungs at the end of a
Capacity (mL) It normal exhalation.













Table 2-4. Other parameters of use in modeling particle deposition in the respiratory tract.
Quantity Symbol Description/Comments

Age (y) A Age of subject of interest in years.
Body Height (cm) Ht Height of subject of interest.
Body Mass (kg) Wt Body mass (or weight) of subject of interest.
Body Mass Index (kg/m2) BMI MI = (Wt)/(Ht2)
Anatomical Dead Space VD Total volume of the conducting airways including
(mL) extrathoracic region: VD = VD(ET) + VD(BB) + VD(bb)
Ventilation Rate (L/min) VE Amount of air inhaled or exhaled per unit time.
Oxygen Consumption Vo2 The amount of oxygen consumed by the body per unit
Rate (mL/min) time.
Ventilatory Equivalent VQ Ratio of the ventilation rate to the oxygen consumption
Ratio rate, VQ = V/V02.
Basal Metabolic Rate BMR Energy consumption rate needed to sustain basic life
(MJ/day) activities.
Basal Multiplier Bmt Multiplication factor to relate the metabolic rate at a
specified exertion level to the basal metabolic rate.
Oxygen Consumption Hoy Amount of oxygen needed to produce 1 kJ of energy.
Factor (L/kJ)
Vital Capacity (mL) VC The amount of air present in lungs at maximal
inhalation.









not depend on the total activity inhaled, only on the fraction of the total activity that is

associated with a given particle size. The fraction of activity associated with given

particle sizes are quantified by the activity-size distribution, where size in this case refers

to either the aerodynamic diameter or the thermodynamic particle diameter. These

distributions are probability density functions which express the fraction of activity

associated with particles in an incremental size range. Generally, measurements have

shown that particles are log-normally distributed with respect to size. Therefore, the

size distributions are usually characterized by a geometric mean aerodynamic (or

thermodynamic) diameter and a geometric standard deviation (GSD). The activity

median aerodynamic (or thermodynamic) diameter, denoted by AMAD (or AMTD) is

the diameter of the particle for which half of the activity is associated with particles less

than that diameter.

The ICRP (1994) has suggested four physical exertion levels be used to classify

the physical activities of persons. The levels are: resting (or sleeping), sitting awake,

light exertion, and heavy exertion. Based on these four exertion levels, the ICRP has

recommended age and gender-specific reference values characterizing the ventilation

rate, tidal volume, respiratory frequency, and fraction of time spent in these levels. The

present study considers all of these parameters except the fraction of time spent in

various levels; subsequently, computations developed in this work assume a subjects are

exposed while at only one of these four levels over the duration of the exposure. More

details concerning the types of activities assigned to exertion levels in this study are

given in Chapter 4.










2.5. Particle Size and Diffusion Characteristics

This section presents some of the relationships needed to compute parameters in

the deposition model which are related to particle size. Most of these relationships have

been discussed in more detail in Hinds (1982). The two parameters required by the

model that depend on particle size (and air temperature and pressure) are the

aerodynamic diameter, d,, and the particle diffusion coefficient, D. Other quantities that

are used to relate these two parameters are the equivalent volume diameter, d,, and the

thermodynamic diameter, dt, which are discussed below. For a dispersion-type aerosol

(e.g., windborne surface soil), the aerodynamic diameter (or more generally the

distribution of aerodynamic diameters) would likely be determined from measurements,

for example, employing cascade impactors. For an aerosol produced by a chemical

reaction (e.g., oxidation processes), the thermodynamic diameter (or more generally the

distribution of thermodynamic diameters) would likely be determined from

measurements, for example, employing graded screen arrays or diffusion batteries.

The aerodynamic diameter, d., of a particle (defined as the diameter of a unit

density sphere with the same settling velocity as the particle) is related to the equivalent

volume diameter, d, (defined as the diameter of a sphere with the same volume as the

particle), by the following expression:


d. d [ pC(d) /(2-4)
S X PoC(d) I








49

where X is the particle (dynamic) shape factor; p is the mass density of the particle; p, is

unit density (1 g/cm3); C(d,) and C(d.) are particle slip correction factors (Hinds, 1982).

If d. is given, d, can be computed from Equation 2-4.

The particle mass density is difficult to measure, especially for aggregates

(clusters of particles attached as a result of thermal coagulation). Generally, if the

source of the aerosol particles is known, the mass density of the particulates is assumed

equal to the mass density of the bulk material. James et al. (1994b) have suggested a

reference value of 3 g cm3 for environmental aerosols because it is typical of many

natural materials.

The slip correction factor generally depends on air temperature and pressure.

For particles in the respiratory tract (saturated air at 37C ) at atmospheric pressure P

(cm Hg), the slip factor has been reported by James et al. (1994b) as:

C(d ,P) = 1 +[13.57 +4.31 exp(-0.102 Pd.)]/[PdJ (2-5)


The slip factor for d., C(d.), is also computed with Equation 2-5 by replacing d, with

da.

The particle (dynamic) shape factor, X, is the ratio of the actual drag force on an

irregular particle, FD, to the drag force predicted by Stoke's law, Fs, using the particle's

equivalent volume diameter, d,;

F FD
x = (2-6)
Fs 3 arud








50
where tj is the viscosity of air in the respiratory tract and u is the particle velocity. For a

spherical particle X = 1; typical values of X range from 1 to 2 (Hinds, 1982). Shape

factors must generally be determined experimentally.

The diffusion coefficient, D (cm2/s), for a particle is estimated by the following

expression (Hinds, 1982):


C(d )kT
D =C(d (2-7)
3x t dh
where k is Boltzmann's constant, T is the absolute temperature (K), and d, is the

thermodynamic diameter (diameter of sphere with same thermal diffusivity as particle of

interest). Consistent units must be used for quantities in Equation. 2-7. Based on the

discussion in James et al. (1994b), the following expression determines the value of the

thermodynamic diameter to be used in Equation 2-7 (d4 and d, in ipm):


d. for d.> 0.005 pm
d, = (2-8)
d,[l+3 exp(-2.20x103d)] for d.0.005im


The lower portion of Equation (2-8) is based on recommendations, by Ramamurthi and

Hopke (1989), that a correction be made to Eq. 2-7 for particles less than about 2 nm.

A value of 5 nm was chosen as the application point for their results in this study, since

it can be shown that the error in assuming da equals d, is less than 0.01% for d, > 5 nm.

Solving Eqs. 2-4 and 2-5 for d, involves an iterative approach since the solution

cannot be obtained analytically. James et al. (1994b) have recommended an

approximate, initial solution by setting d, = d,(x/p)'"; convergence to the correct value

by iteration is reportedly rapid (usually requiring less than 10 iterations).











2.6. Respiratory Tract Clearance Model

The clearance model adopted by the ICRP (1994) is based on an approach

introduced by Cuddihy and colleagues (Cuddihy, 1976, 1984; Cuddihy et al., 1979;

Cuddihy and Yeh, 1988). In this approach clearance from various regions is based on

competitive clearance processes. In all regions but the anterior nasal passages, ET,,

clearance is achieved (1) by mechanical translocation processes (such as macrophage

uptake and transport in fluids over surfaces by mucociliary action to the gastrointestinal

tract), (2) by transport to lymph nodes, and (3) by dissolution and absorption to lung

tissues and/or to blood. Radioactive decay can also be considered a form of removal.

These processes compete to remove material from various regions. In the anterior

extrathoracic region (ETI) material is removed by external pathways (such as nose

blowing, sneezing, or coughing) and by radioactive decay.

Three major working assumptions have been made in the clearance model

(discussed by Bailey and Roy, 1994). These assumptions are as follows:

(1) clearance rates due to mechanical transport processes (mucociliary and macrophage
transport) and dissolution/absorption processes (to blood) are independent;
therefore, the overall fractional clearance rate from a model compartment is
represented by the sum of the rates due to individual processes;

(2) mechanical transport rates are the same for all materials; i.e., the chemical form of
the material does not influence the fractional clearance rate due to particle transport
processes; and

(3) the dissolution/absorption rate of a material to blood is the same for all regions of the
respiratory tract except the anterior nasal passages ET,, where no absorption is
assumed to occur.











The compartment model which has been adopted by the ICRP (1994) to

represent mechanical clearance processes (including particle transport by the mucociliary

mechanism and by alveolar macrophages) to the gastrointestinal tract and lymph nodes is

shown in Figure 2-6. This figure does not include absorption pathways. Arrows

represent routes (and directions) for particle transport in the clearance model.

According to the second assumption above, these routes and their associated fractional

clearance rates are generally independent of the chemical form of the material.




Anterior Extrathoraci Region
NasalT







Boar- B 2
13 12 11 GITract


Thoradic Region

Bronchi se2 BB !
9 87


Bronchioles LNh bb bb bb
6 AS 4

Alveolar-
Interstitim _Al AI A
10 3 3 2

I I [L
Squeatered in tissue Available for surface transport

Figure 2-6. Representation of particle transport model recommended by the ICRP
(1994) for mucociliary and macrophage transport mechanisms to the
gastrointestinal tract and lymph nodes (no absorption pathways shown).










Material that deposits in the respiratory tract is assigned to those compartments

which that are shaded in Figure 2-6. The deposition model predicts the overall amount

of material that deposits in a given broad region (i.e., ETi, ET,, BB, bb, and AI;

represented by the grouped boxes in Fig. 2-6). This regional deposition is then

partitioned among the shaded compartments within the region to account for observed

clearance patterns. An exception to the second assumption exists for the BB and bb

regions. For these regions the partition of the deposition to regional compartments is

based on particle size. This aspect of the model is discussed in more detail in Chapter 4.

Clearance of materials from respiratory tract regions does not always occur at a

constant rate with respect to time. To accommodate time-varying clearance rates in a

practical manner, the model designers have represented the broad respiratory regions

(ET, BB, bb, and AI) by a number of compartments (representing materials in different

clearance states) clearing at different rates. Uncertainties in the fraction of the initial

deposition assigned to these states (or compartments) and in clearance rates for these

compartments are discussed in section 4.2 and are included in parameter uncertainty

analyses. These compartments are discussed below.

The ET, region is modeled by a single compartment in which all material is

deposited and cleared at a constant rate. The ET2 region is modeled by two

compartments, ET2 and ET,. Based on animal data, the ICRP (1994) has suggested

that -0.05% of the regional deposition be associated with ET. and the rest (99.95%)

with the ET2 compartment. The ET, compartment represents material that is










sequestered by the airway walls in the ET2 region and can only be cleared to lymph

nodes or to blood (as do all following compartments subscripted by seq).

Mechanical clearance from the BB region is represented by three compartments,

BB,, BB2, and BBq. Based on animal data, the ICRP (1994) recommends that -0.7%

of the material deposited in BB be assigned to the BB, compartment. The rest is

partitioned between BBI (a fast-clearing compartment) and BB2 (a slow-clearing

compartment) based on particle size. The bb region is represented in an analogous

manner by compartments labeled bbl, bb2, bbq. These bb compartments are assumed to

receive the same partition of deposition as their BB region counterparts.

Mechanical clearance from the AI region is modeled by three compartments, AI,,

AI2, and Al3, representing compartments that clear fast, slow, and very slow,

respectively. The ICRP (1994) recommends partitioning the overall deposition in AI by

fractions of 0.3, 0.6, and 0.1 for compartments AI,, AI2, and Al3, respectively. These

values, along with reference values for all other compartments, are subject to

uncertainties and variabilities which are addressed in Chapter 4.

In the conceptual model structure, the mechanical clearance rates for

compartments in Figure 2-6 are taken to be independent of particle size and physico-

chemical form of the deposited material. For reference, the compartments in Fig. 2-6

have been numbered. Denoting the mechanical clearance rate from compartment i to

compartment in Fig. 2-6 as m,j, the ICRP (1994) reference values for mechanical










clearance rates are listed in Table 2-5. One interpretation of these rates is that they

represent the fraction of material in compartment i that passes from compartment i toj

per unit time by the specified pathway. Denoting the fraction of the regional deposition

assigned to a compartment as fd, Table 2-6 lists reference values for the partition of

deposition among compartments.

Unlike particle transport mechanisms represented by the compartment scheme in

Fig. 2-6, absorptive processes do depend on the physico-chemical properties of the

radionuclide. Such properties include solubility of material within, and transportability

through, lung tissues. The model adopted by the task group to describe these processes

is shown in Figure 2-7. According to the third assumption listed in the beginning of this

section, absorption rates are taken to be the same from all regions (except ETi where no

absorption is assumed) and depend on the chemical form of the radionuclide. Similar to

the processes represented in Figure 2-6, experimental evidence suggests that for some

materials the fractional absorption rate of materials to blood is time-varying. Rather

than using time-varying clearance rates (which would lead to a system of differential

equations with time-varying coefficients), the ICRP (1994) has modeled absorption to

blood by multiple compartments with constant fractional absorption rates.

In the model, particles are deposited into "initial state" compartments. These

initial-state compartments correspond to the shaded compartments in Fig. 2-6. After

deposition, competing mechanisms act either to remove material from a location or to

change the state of the material. Material deposited in the initial state can be








56

Table 2-5. Reference values for (fractional) mechanical clearance rates in the particle
transport aspect of the ICRP human respiratory tract model.
Pathway From To Rate (d-') Half Time"
m,4 AI, bb, 0.02 35 d
m2, AI2 bb, 0.001 700 d
m3,4 Al3 bb, 0.0001 7000 d
...n ............ I ............. ..........0.00002.........
mn4, bb, BBI 2 8h
ms,7 bb2 BBI 0.03 23 d
... 0.......01 ...................0.01
... ,o....................b.b ..,....... L.N ....................... ......................................................................................
mi7, BBi ET, 10 100 min
ms,11 BB2 ET2 0.03 23 d
.. L N0 .0 1 7 0.................... .......... ....................... 1 .................... 7.. ...........................................
ma1i, ET2 GI tract 100 10 min
m1,13 ET, LN, 0.001 700 d
m,I ,1 ET, Environment 1 17 h
Source: ICRP (1994, Table 17A).
'Half times are approximate since they are based on rounded values for the rates
(half-time = ln(2)/rate). No half-time is reported for m3,0; the rate for this route
was chosen simply to direct the desired amount of material to the lymph nodes.
Units: d = day; h = hour; min = minutes.


Table 2-6. Reference values for partition of deposition among compartments in region.
Respiratory Tract Clearance Fraction, fd, of Regional Deposition
Region Compartment Assigned to Compartment"
ET2 ET2 fd(ET2) = 0.9995
ET f(Tq=............ f~ ). = 0.0005
........................................................ .. T ..................................... fdf .T.. ....... = O...0005.................................
BB BBI fd(BB,) =0.993 -f,
BB2 fd(BB2) =f,
....................................................... .. ........ .........................f () ......... = ..0 7 ...................................
bb bb, fd(bbl) = 0.993 f
bb2 fd(bb2) =
................. .. ......... (b .......... ...................................
AI All fd(AIi) = 0.3
AI2 fd(A12) = 0.6
AI, f&(AI,) = 0.1
Source: ICRP (1994, Table 17B).
' The term f, represents the slow-clearing fraction of particles deposited in the
tracheobronchial airways; it depends on particle size (discussed in section 4.2.2.1).










PARTICLES IN INITIAL STATE MATERIAL IN TRANSFORMED STATE
(Depoited Particle) ----.
(ComparmnUts: 1, 2, 3,..., 14) pt (Compartmnat: IT, 2T, 3T, .13T)


fbp xbst

(l-fbi
(fb P MATERIAL IN BOUND STATE (f t

(Compartment: AI(b), bb(b), BB(b) ET(b),
LNet(b), LNth(b))


V tb
BLOOD

Figure 2-7. Compartment model recommended by ICRP (1994) to represent dissolution
and absorption to blood and lung tissues for materials deposited in the respiratory tract.


(1) transported mechanically between initial state compartments by the pathways

indicated in Figure 2-6, (2) transformed to a different clearance state (i.e., "transformed

state") represented by a different compartment although in the same spatial location, or

(3) absorbed from the initial state to either a "bound state" or to the blood. The bound

state represents material that chemically binds to the tissues in the airway wall after

dissolving in lung fluids. The total fractional clearance rate for materials from initial

state compartments to the blood and to the bound state is denoted by sp. The fraction of

material that clears from the initial state to the bound state is given by fb so that the net

fractional clearance rates to the bound state and the blood are fbSp and (1-fb)sp,

respectively.










The transformed state represents material that is absorbed to the blood and

bound state at a different rate, s, than initial state material. The fractional clearance rate

from initial state to transformed state is represented by s,. According to the clearance

model adopted by the ICRP (1994), material in the transformed state can either be (1)

transported mechanically among transformed state compartments by assuming the same

pathways apply as in Figure 2-6 or (2) absorbed to the blood and bound state. The total

fractional clearance rate for materials from the transformed state to the blood and the

bound state is denoted by s,. The fraction of material that clears from the transformed

state to the bound state is given by fb (assumed to equal the fraction to bound state from

initial state) so that the net fractional clearance rates to the bound state and the blood are

fbs, and (l-fb)s, respectively.

The ICRP (1994) has also examined an alternate form for the absorption model

which has parameters that are generally easier to determine from experimental data than

the model shown in Figure 2-7. This alternate model is depicted in Figure 2-8. In this

alternate form, material is assumed to be deposited in either a rapid dissolution state (or

rapid absorption phase) or a slow dissolution state (or slow absorption phase). The

fraction of material in the rapid phase is denoted by f (so that the fraction in the slow

phase is 1-). Fractional clearance rates from the rapid and slow states to the blood and

bound states are given by s, and s,, respectively. As before, fb represents the fraction of

cleared material that goes to the bound state. Concerning the amount of material that is

absorbed to blood over a specified time period, this alternate form provides the same






















(1-fY Sr
Bound Material





Blood



Figure 2-8. Alternate form for the dissolution/absorption model. This model form is
related to the form in Figure 2-7 through Equation 2-9.


numerical results as the form in Figure 2-7. The following expressions can be derived

which relate the two models:


sp = s. + f,, s.)
s, = (1 f)( Sr s,) (2-9)
s, = s,


When reliable human data exist, the absorption aspect of the model can use

observed rates of absorption to blood; when data is not available, the absorption rate is

classified as slow, moderate andfast (denoted S, M, and F) and is based on reference

values recommended by the ICRP. In the present study, literature related to absorption










of plutonium oxides was investigated (section 4.2.4) to develop distributions for the

absorption rate constants for that specific material.

Upon coupling the absorption model of Figure 2-7 with the particle transport

model of Figure 2-6, the overall compartmental clearance model is obtained; it is

represented schematically in Figure 2-9. The overall clearance model is presented in its

complete mathematical form (i.e., as a system of first-order differential equations) in

Appendix B, where the mathematical form of the older ICRP Publication 30 lung model

(ICRP, 1979) is also presented for comparison. The revised lung model possesses 33

compartments compared to only 10 for the older model.

Since the mathematical form of the clearance model involves linear, first-order

differential equations with constant coefficients, the model can be solved analytically for

simple exposures of interest. Due to mathematical complexity in accounting for changes

in activity intake rates over time (e.g., due to changes in ventilation rate with level of

physical exertion), this study has focused on an acute intake scenario. For an acute

activity intake, material is deposited instantaneously at time to in the respective initial

state compartments (shaded regions in Fig. 2-6). This assumption simplifies the solution

to the differential equations presented in Appendix B for the clearance model. This

simplifying assumption is appropriate when ventilation rates remain relatively constant

over the exposure duration (which is less at most a few hours) and when the exposure

duration is much less than the dose integration time. A relatively simple solution to the

differential equations also exists for relatively constant, non-acute intakes however, such

scenarios are reserved for future study.






























N

fin

vim


o\ e


15

0










I-
I

c4







0



E
t-









0
0

o
4-

8
4-
o




I-
SCO



o C

Sc&
.0


~I.









Several methods are available for solving clearance models represented by

systems of linear first-order differential equations in this exposure situation. In this

study, a computational algorithm described by Birchall (1986) has been employed. This

algorithm is based on analytical equations (so-called Bateman equations) presented by

Skrable et al. (1974), which originate from equations first suggested by Bateman (1910).

The algorithm is restricted to first-order, non-recycling compartment models, such as the

clearance model described here. Non-recycling models are those which do not allow

material to re-enter clearance compartments (i.e., routes do not allow material to return

to previous compartments of residence). The algorithm can solve the clearance model

for both acute and constant, chronic activity intake scenarios. Given the initial number

of atoms deposited into initial state compartments, the algorithm can solve the clearance

model for two quantities: (1) the number of atoms remaining in compartments at time 1,

and (2) the total number of radioactive transformations occurring in compartments over

an integration time r. For a compartment of interest, the algorithm operates by first

finding all pathways that lead to that compartment. For each pathway leading to the

compartment the algorithm uses the Bateman equations to compute the contribution

made to (1) the activity in compartment at time, I and (2) to the total transformations in

compartment over time r. Contributions from all pathways are then summed.

Provisions are included to avoid duplicating contributions from overlapping pathways.

A complete description of the algorithm and a listing of the source code for computer

implementation can be found in the original paper by Birchall (1986).










2.7. Respiratory Tract Dose Model

Dose calculations in the revised ICRP respiratory tract model are based on the

methodology presented in ICRP Publication 30 (ICRP, 1979) for radiation workers and

later expanded for exposures to the general population in ICRP Publication 56 (ICRP,

1989). In that methodology, the committed equivalent dose to a tissue T, denoted HT(r)

with special SI unit of Sv, is the time-integrated equivalent dose rate for that tissue,

where r is the integration time. For radiation protection purposes the ICRP (1989 and

1991a) has recommended an integration time of r = 50 years for adults and r = 70 years

for children. However, other integration times can be specified if desired. The

equivalent dose rate at time t for target tissue Tis given by:


HT(t) = c q,(t) SEE (T-S;t)j (2-10)
j

where c is a constant accounting for conversion of units, qj(t) is the activity of

radionuclidej in source region S for a subject at time t; SEE(T- S; t) represents the

specific effective energy for tissue T irradiated by radionuclidej in source S and is

defined below. Due to source-target geometry effects, the SEE generally depends on

the age of the subject at time t. The summation over s is over all source regions, S, that

irradiate the target. The summation over is over all radionuclides in source region S.

The double summation over s andj then accounts for all radionuclides in all source

regions that affect the target tissue.

The specific effective energy, SEE(T+- S)j for radionuclidej is expressed by:










w,E,Y,YAF(T-S;t),
SEE (T-S;t) = E wEYAF(T-S;t) (2-11)
R MT(t)

where w. is the radiation weighting factor for radiation type R emitted from radionuclide

j; ER is the energy of radiation R; YR is the yield of radiation type R per transformation

of radionuclidej; AF(T-S; t)R is the fraction of energy of radiation R emitted by source

S that is absorbed in target T and depends, among other things, on the age of subject at

time t; and MT(t) is the mass of target tissue T for the subject at time t.

Assuming an acute intake at time to, the committed equivalent dose at time t due

to the intake is computed as:



HT(t) = fH(tdt' = c f q(t )SEE (T-S;t j (2-12)
to to
If the SEE is assumed to be independent of time t (e.g., for adults where growth is

negligible), the term enclosed in brackets becomes:



j j SEE (T-S;t Idt = U(r) SEE (T-S) (2-13)
to
where

U j() = f q(t)dt' (2-14)
to

and where U,(zr) equals the total number of transformations of radionuclidej that occur

in source region S over an integration time of r = t to When a respiratory tract region

is the source, Uf(t) depends on both the deposition and clearance models described










above and is determined by integrating, with respect to time, the solutions to the

differential equations that are presented in Appendix B.

The present study focuses on doses to various lung tissues as the targets. The

scope of the present study has been limited to include only source regions within the

respiratory tract. The scope is further limited to include only those doses which are

contributed by alpha particles. For an inhalation of primarily alpha-emitting

radionuclides with negligible photon, electron, and beta emissions--such as Pu-238, Pu-

239, Pu-240--the dose to lung tissues is due almost exclusively to alpha particles which

originate within the lung tissues. To justify this statement, the computer program

LUDEP (Jarvis et al., 1993) has been run for a class S (insoluble) "Pu-laden aerosol

having an AMAD of 5 pmr and a GSD of 2.5; the scenario involved a reference adult

male at light exertion. The lung model described here and the ICRP (1979) metabolic

models for other organ systems were used with reference values for all parameters.

Results indicated negligible contributions to the lung dose from source organs other than

the lungs (<<0.1%) and from radiations other than alpha particles that originated within

the lungs (<<0.1%). Methods developed in this study are applied to Pu-239 and Pu-240

oxides in Chapter Five; for these radionuclides doses contributed by radiations other

than alpha particles are negligible.

For short-range alpha, beta, and electron emitters, source regions in the lungs are

identified in Table 2-7. In total the ICRP (1994) has identified 17 source components to

be considered for short-range radiation emitters. These regions have been modeled to

correspond with anatomical regions in the airway wall as shown earlier in Fig. 2-3. The








66

thicknesses and depths of various source and target layers are important determinants of

the fraction of energy that can reach the target layers. These dimensions are examined in

more detail in Chapter 4.


Table 2-7. Assignment of compartments to source regions for short-range radiations in
the revised ICRP respiratory tract model.
Target" Source Region Clearance Compartments Comprising
Sources in the Respiratory Tractb
ETSurface d it .................... .. 14
ET2 Surface fluid 11 + 1T
Particles sequestered in airway wall 12 + 12T
Bound material ET(b)
LN All transformations in LN ........................... 13. +1..3T..+..LN.(bnd)......................................
BB Mucous gel layer 7 + 7T
Mucous sol layer 8 + 8T
Particles sequestered in airway wall 9 + 9T
Bound material BB(b)
Alveolar-interstitium 1 + 2 + 3 + IT + 2T + 3T + AI(bnd)
bb Mucous gel 4 + 4T
Mucous sol 5 + 5T
Particles sequestered in airway wall 6 + 6T
Bound material bb(bnd)
Alveolar-interstitium 1 + 2 + 3 + IT + 2T + 3T + AI(bnd)
AI All transformations in AI and LNth 1 + 2 + 3 + IT + 2T + 3T + AI(bnd) +
........................................................................................................l..... 1..... ....n.. ... ..................................
LNth All transformations in LNth 10 + 10T + LN,(bnd)
Source: ICRP (1994)
Target cell nuclei layers in respective respiratory tract regions are discussed in text.
b See Figs. 2-6 and 2-7 and Appendix B for additional clarification of model
compartments.


Target tissues in the revised model include the following tissue and cell layers.

For the extrathoracic tissues the target regions are: (1) the basal cell nuclei layer in the

epithelium of region ET,; (2) the basal cell nuclei layers in the epithelium of the naso-









oropharynx and larynx in region ET2; and (3) the total mass of the extrathoracic lymph

nodes, LNT. For the thoracic tissues (or lungs) the target tissues are: (1) the secretary

cell nuclei layer in the epithelium of the BB region; (2) the basal cell nuclei layer in the

epithelium of the BB region; (3) the secretary cell nuclei layer in the epithelium of the bb

region; (4) the total mass of the alveolar-interstitial tissues, AI; and (5) the total mass of

the thoracic lymph nodes, LNa. This aspect of the revised model (computing doses to

target regions within the lungs) differs from the ICRP Publication 30 model (ICRP,

1979) in which a single dose was computed to the lung by averaging over the total lung

mass.

Based on a review of human epidemiological studies, Masse and Cross (1989)

have suggested that target tissues in the respiratory tract be assigned varying degrees of

radiation sensitivity in the new model. To account for varying sensitivities, the revised

respiratory tract model (ICRP, 1994) has incorporated regional weighting factors to

partition the risk from irradiation of lung tissues. The approach adopted in the model

specifies that combined equivalent doses to extrathoracic and thoracic (lung) tissues be

computed by the following expressions:

HT(ET) = HT,ET(I) AET(I) + HT,ET(2) AET(2) + HT,LN(ET) ALN(ET) (2-15)

HT(TH) = HBB ABB + HT,b Ab + HT, AA + HT,LN ALN (2-16)

where HT(ET) and HT(TH) are the detriment-adjusted equivalent doses to the

extrathoracic and thoracic (lung) tissues, respectively; HT,ET(1),..., etc., are the equivalent

doses to targets in the extrathoracic regions; HT BB,..., etc., are equivalent doses to

targets in the thoracic regions; AET(I),..., etc., are detriment apportionmentfactors for









the partition of risk in ET tissues; and ABB,..., etc., are detriment apportionment factors

for the partition of risk in the thoracic (lung) tissues.

The detriment apportionment factors are used to weight the doses to the

individual regions and reflect the relative sensitivity of the target regions to radiation-

induced effects. The values recommended by the ICRP in Publication 66 are listed

below in Table 2-8; it is important to note that these values are quite uncertain.

Concerning the values reported in the table, the following statement was made:

It was concluded that there was no basis for deriving factors to represent
regional differences in radiation sensitivity with any acceptable degree of
confidence. Therefore, the weighting factors given in Table 31 [Table 2-8] are
recommended. These factors may be revised by the Commissions as better
information becomes available. Revisions can be made without changing the
respiratory tract dosimetry model itself. (ICRP, 1994, p. 114)

Based on this statement, the focus of this study has been more toward regional doses

than an overall, combined lung dose. Uncertainties in the apportionment factors do not

influence regional doses (only their interpretation on a radiological risk basis).

Nonetheless, the methodology developed in this study (1) predicts uncertainties in the

combined lung dose and (2) provides a means to change the detriment-apportionment

(actually their assigned uncertainties) for the thoracic region with little effort. It is noted

that although methods are provided for predicting doses to the extrathoracic tissues,

these tissues are not included in the list of organs that receive tissue weighting factors in

the ICRP (1991a) system of radiological protection. Consequently, these tissues do not

receive as much emphasis in this study.










Table 2-8. Risk-apportionment factors recommended by the ICRP for weighting the
dose to various respiratory tract target tissues.


Extrathoracic Detriment Thoracic Detriment
Regions Apportionment Regions Apportionment
Factors (A)" Factors (A)
ET, 0.001 BB 0.333
ET2 1 bb 0.333
LNET 0.001 AI 0.333
LN, 0.001
Total = 1.002 Total = 1.000


Source: ICRP (1994).
* Values are as listed in the reference; do not sum to one.


Masses for target layers associated with ET, BB, and bb regions can be

estimated from the products of the regional surface area and the target layer thickness.

The regional surface areas are both age- and sex-dependent and must generally be

estimated from airway dimensions, which depend on body size. Absorbed fractions for

short-range radiations in ET, BB, and bb tissues depend primarily on the source and

target layer thicknesses and depths. Measurements (e.g., Gehr, 1987) suggest that

dimensions for source and target layers in the airways are independent of age or body

size. Furthermore, computations have shown that the airway diameter is of secondary

importance to absorbed fractions. Based on these findings, the ICRP (1994) has

suggested that absorbed fractions for short-range radiations in the lung are independent

of age.

The computational methods used to compute energy absorption in BB and bb

target tissues in this study are discussed in detail in Appendix C. The approach for ET

regions has been to use the simplified expressions for the absorbed fractions, AF, as








70

presented by James et al. (1994a). For ET regions, the uncertainties in source and target

dimensions are not included in the present study since these regions are relatively

resistant to radiation effects and for reasons discussed above. For AI and lymph node

tissues, the absorbed fractions suggested by the ICRP(1994) for short-range radiations

are used to compute doses. These absorbed fractions (AF) in these regions for alpha

particles may be summarized as:

AF(AI- BB) = AF(AI- bb) = AF(AI- ET)= 0 (2-17)

AF(AI- AI) = AF (LNr.- LNm) = AF(LNET- LNET) = 1. (2-18)

Equation 2-17 simply states that alpha particles emitted from sources in the BB, bb, or

ET regions deposit a negligible amount of energy in the AI region. Equation 2-17 has

been suggested by the ICRP since the additional energy deposited in the AI region from

sources in other respiratory tract regions is negligible compared to the energy deposited

in the AI region due to sources in the AI region (due primarily to the thickness of tissues

separating these regions and to large surface area and mass of the AI region). Equation

2-18 states that alpha particles emitted by the AI and lymph node tissues (as source

regions) are completely absorbed by the those regions (as target tissues).


2.8. Summary

The purpose of this chapter has been to provide an overview of the new ICRP

respiratory tract model (ICRP, 1994). In this chapter, the deposition, clearance, and

dosimetry components of the respiratory tract model have been presented, and model

parameters have been introduced. The following chapter presents the methods








71
employed to treat model parameters and associated quantities as random variables within

the (computational) model framework. Chapter 4 examines the respiratory tract model

parameters in greater detail by providing results of a formidable literature review--

undertaken to assign probability distributions to parameters and to account for

relationships (or correlations) among parameters.













CHAPTER 3
METHODS FOR CONDUCTING PARAMETER UNCERTAINTY ANALYSES


3.1. Introduction

A computational methodology has been developed to quantify uncertainties in lung

dose quantities for population groups exposed to radioactive aerosols. This methodology

is based on conducting parameter uncertainty analyses using the revised ICRP respiratory

tract model described in Chapter 2 within the scope of the assessment problem defined in

Chapter 1. The purpose of a parameter uncertainty analysis is to quantify variabilities in

model predictions attributable to uncertainties (including stochastic variability and lack of

knowledge) in model parameters. Additionally, parameter sensitivity analyses of the

model are desired in order to determine the relative importance of parameters in affecting

model predictions (i.e., which parameters contribute most to variability of the predictions).

Specific methods involved in performing such analyses for complex mathematical models

have been the subject of many studies and reports (IAEA, 1989; Morgan and Henrion,

1990; OECD, 1987; Hoffman and Gardner, 1983; Iman and Helton, 1985; Iman and

Conover, 1982; Hamby, 1995; Iman and Helton, 1988; Nowak and Hofer, 1987; Paschoa

and Wrenn, 1987). The purpose of this chapter is to provide a description of methods

used in the current study to perform such analyses with the selected respiratory tract

model.








73

The International Atomic Energy Agency (IAEA, 1989) has listed the main steps

involved in conducting a parameter uncertainty/sensitivity analysis. These steps are

repeated in Table 3-1 and have been followed in the current study. To examine the

concepts and techniques associated with these steps, this chapter is divided into five

additional sections. In section 3.2 parameter uncertainties and correlations are addressed.

Section 3.3 involves a discussion of the numerical methods used to sample the joint

probability density function (i.e., the probability density for combined parameter values)

and to propagate this joint density function through the model. In section 3.4, some

measures of model prediction uncertainty and formats for presenting results are discussed.

Section 3.5 provides a discussion of techniques employed to determine parameter

sensitivities in the model (i.e., methods to determine dominant parameters and their

contributions to model prediction uncertainty). Section 3.6 provides a description of the

computer code developed to implement the respiratory tract model and to integrate it with

the methods described in this chapter.

Due to the mathematical nature of the respiratory tract model and the large number

of model parameters involved, a numerical approach has been adopted for propagating

parameter uncertainties through the model. A Latin hypercube sampling technique

(McKay et al., 1979; described in section 3.3) has been used in this study to generate a

numerical representation of the joint probability density function for the combined range of

parameter values. This technique involves numerical generation ofn values from each of

the m hypothesized parameter distributions. A special pairing process is then implemented

to combine values randomly for the m model parameters into n input vectors which can be










used to solve the model for n output values for each desired model prediction. Various

aspects of the technique are discussed in more detail in the following sections of this

chapter.


Table 3-1. Main steps involved in conducting a parameter uncertainty analysis.
(1) List all of the parameters that are potentially important contributors to uncertainty in
the final model prediction.
(2) For each parameter listed, specify the maximum reasonable range of applicable
alternative values.

(3) Specify the degree of belief (in percentage) that the appropriate parameter value is
not larger than specific values selected from the range established in Step 2 above and
select a probability distribution that best fits the quoted degrees of belief.
(4) Account for dependencies among model parameters by introducing suitable
restrictions, by incorporating appropriate conditional degrees of belief, or by specifying
suitable measures of degree of association.
(5) Set up a subjective probability density function (pdf) for the combined range of
parameter values. This will subsequently be referred to as a joint pdf. Propagate this
joint pdf through the model to generate a subjective probability distribution of predicted
values.

(6) Derive quantitative statements about the effects of parameter uncertainties on the
model prediction.
(7) Rank the parameters with respect to their contribution to the uncertainty in the
model prediction [sensitivity of parameters].
(8) Present and interpret the results of the analysis.
Source: IAEA (1989, pp. 31-32) with minor alterations.











3.2. Quantifying Parameter Correlations and Uncertainties

The first four steps listed in Table 3-1 involve identifying model input parameters

and quantifying their correlations and uncertainties. Primary input parameters for the

respiratory tract model have been identified in Chapter 2. In this section, the techniques

used to account for parameter correlations and uncertainties are discussed. Chapter 4

contains a detailed discussion of individual respiratory tract model parameters, focusing on

the distributions and relationships adopted in this study to accommodate parameter

uncertainties and correlations.


3.2.1. Parameter Correlations

Two approaches for accommodating parameter correlations/dependencies have

been considered. One approach involves specifying and using a rank correlation matrix

and is implemented during the pairing process for the m parameter values. The other

approach involves using regression equations along with random error terms. The latter

approach generally accounts for correlations directly by incorporating detailed

relationships between parameters into the uncertainty analysis; consequently, this approach

is used here and is more heavily weighted in the following discussion. However, the

methodology accommodates the rank correlation matrix approach (as discussed below and

in section 3.6).

The correlation matrix approach uses a restricted pairing technique (Iman and

Conover, 1982) to induce correlation between the ranks of two numerically sampled input

variables. Iman and Shortencarier (1984) have developed a computer code which










implements this approach within the Latin hypercube sampling scheme (discussed in

section 3.3). Use of the code requires the user to specify rank correlation coefficients for

pairs of parameters. If an appropriately sampled data set exists for two parameters, for

example parameters X and Y, then the rank correlation coefficient (also known as

Spearman's correlation coefficient) can be estimated by the sample correlation coefficient

of the rank-transformed data. The rank correlation coefficient expresses the degree of

linear relationship between the ranks of X and Y (i.e., it is a measure of the monotonic

relationship between parameters). For example, if an increase in X always results in an

increase in Y, the rank correlation coefficient would be (positive) one. If an increase in X

always resulted in a decrease in Y, the rank correlation coefficient would be negative one.

This approach is discussed in more detail by Iman and Conover (1982) and by Liebetrau

and Doctor (1987).

A second approach for accommodating relationships between parameters (and the

one used in this study) is to use reported regression equations which predict the value of

one variable (or input parameter) as a function of another variable (or input parameter).

Consider a hypothetical example involving two variables, X and Y. Assume a literature

survey has revealed that Y can be predicted (to some degree) by a regression equation

relating Y to X; for example, Y = aX (i.e., a power function with fitting parameters a and

b determined, for example, by the least squares method). This expression is equivalent to

InY = In(a) + b*ln(X), and the associated coefficient of determination, R2, provides a

measure of the variance in Y that is unexplained by the variance in X. For example, a

value ofR2 = 0.90 means that 90% of the variance in Y is explained by the regression. A








77
residual, or random, error term can be introduced to explain the remaining uncertainty in

Y. For this example, the expression above could be multiplied by an error term, Eyx so

that Y = (aXb)*Ex; equivalently, InY = In(a) + b*ln(X) + In(Ey,). A distribution for Y,

that accounts for its correlation with X, is then obtained by sampling values from a

distribution for X and from a distribution for Ex. The parameter Y does not, in this case,

need to be assigned a distribution of its own since it is determined by sampled values for X

and Ex.

The choice of the distribution shape for the residual error term depends on the type

of regression equation used. For linear, quadratic, and higher-order polynomial regression

models the error term is customarily added to the regression expression and assigned a

normal distribution with mean value of zero and standard deviation estimated by the

standard deviation of the residual errors. The residual errors are given by the differences

between the observed values of Y and the values predicted by the regression equation

without the random error term. For exponential and power-function regression models,

the error term is customarily multiplied into the regression expression and assigned a

lognormal distribution with geometric mean of one and geometric standard deviation

estimated by the sample standard deviation of the log residuals, where log residuals refer

to the differences between In(observed Y) and In(predicted Y). The term In refers to the

natural logarithm. Given the data set for the residuals or log residuals, the assigned

distribution for the residuals can be subjected to various statistical tests (section 3.2.3) to

determine whether the assigned (or hypothesized) distribution is accepted or rejected

based on the data.










Throughout this study, parameter correlations have been handled directly (when

possible) by using reported regression equations and accompanying residual error terms.

This approach allows all random variables to be sampled independently in the Latin

hypercube sampling scheme, improving computational efficiency. More details are given

for specific model parameters in Chapter 4, where literature is reviewed, parameter

uncertainties and regression relationships are discussed, and distributions are assigned.

Although a mechanism exists in the methodology for its inclusion (see section 3.6), the

rank correlation pairing approach for inducing parameter correlations has not been

employed in this study.


3.2.2. Parameter Uncertainties

Uncertainties in respiratory tract model parameters are accommodated by

introducing continuous random variables. Such variables can attain any real value within

their allowed range. Associated with each random variable is a probability density

function, orpdf. The pdfcharacterizes uncertainties (and variabilities) of the random

variable by describing the probability associated with a specified range of values for the

random variable. For a continuous random variable, a continuous probability density

function is used. All of the parameters investigated in this study possess uncertainties due

both to lack of knowledge and to stochastic variability. Since lack of knowledge exists,

thepdfs generally incorporate some subjective information. One interpretation of the

distributions assigned in Chapter 4 is that all represent subjective prior distributions in

which distribution parameters (e.g., mean and standard deviation) are Bayesian estimators










(Morgan and Henrion, 1990, p. 83). Future information and data for model parameters

can be used to update these prior distributions by employing Bayesian statistics (DeGroot,

1970) to compute the posterior distributions. Notwithstanding these issues, one must

work in the present with limited data sets and attempt to assign, on a case by case basis,

distributions to model parameters that realistically reflect the current state of knowledge

(as perceived by the assessor). If measured data exist for model parameters, graphical and

statistical tests can be used to determine whether to reject an assigned (hypothesized) pdf,

however, no tests exist which can unequivocally prove that an assigned distribution is the

correct one.

Many distribution shapes can be used to represent parameter uncertainties.

Among thepdfs considered in this study are the uniform (or rectangular), triangular,

normal, and lognormal probability density functions. The choice of distribution depends

on the information available for the specific parameter being considered. A description of

each of these distribution shapes follows.

The uniform pdf represents the simplest distribution shape for continuous random

variables. For this pdf the random variable has an equal probability of assuming any value

between a specified minimum and maximum value. If the minimum value is represented by

a and the maximum by b, thepdffor the uniform distribution is given as:

1
f(x) = asx b (3-1)
(b -a)

where x refers to a specific value of the random variable, denoted as X. The integral of

the pdf over a specified range of values Ax is the probability of X taking on a value in that










range. A typical uniform distribution is shown in Figure 3-1. This distribution shape is

useful when the only information known about the parameter is the range of possible

values and when no other sufficiently detailed information exists to justify one of the other

Spriori pdfs.


0.15





0


a= ,b= 10





-5 0 5 10 1


x


Figure 3-1. Example of a uniform probability density function with a = 0 and b = 10.


The triangular probability density function has three parameters, the minimum

value a, the mode (or most probable value) b, and the maximum value c. Three cases can

be identified for triangular distributions. For case one, a < b < c; for case two, a = b < c;

and for case three, a < b = c. Typical distribution shapes associated with these cases are

shown in Figure 3-2. Functions representing the pdf are given by the following equations

for these three cases:


Case one (a < b < c):


2(x a)
(c a)(b a)
f(x) = *
2(c x)
(c a)(c b)


b sxe


(3-2)











Case two (a = b < c):



Case three (a < b = c):


f(x) = 2(c x) axc
(c a)2


f(x) = 2(x a) axc
(c -a)2


(3-3)



(3-4)


This distribution shape is useful for model parameters with values that tend toward a

single most likely value (i.e., stochastic variability is not expected to be large), but the

paucity of data precludes that value from being specified exactly.


a) Case 1: a a=0, b = 3, c= 10




0 5 10 1


b) Case 2: a = b < c
a = b =0, c = 10




0 5 10 15
x


c) Case 3: a < b = c
a=0, b = c = 10




0 5 10 15
x


Figure 3-2. Examples of triangular probability densities for the three possible cases
discussed in text: (a) Case one: a

The normal (or Gaussian) probability density function is specified by two

parameters, p and a, where p represents the mean value and a represents the standard

deviation. The pdf for the normal distribution is:


1 ^ ( _
f(x) = 1 (xI- )
,(2; o 2o )


(3-5)