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PAGE 1 1 ANAL YSIS OF ARTERIAL BIFURCATIONS IN THE HUMAN RETINA By RICHARD DEAN CLARK III A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE UNIVERSITY OF FLORIDA 2013 PAGE 2 2 2013 Richard Dean Clark III PAGE 3 3 To my family and friends, without whom I would have never gotten this far PAGE 4 4 ACKNOWLEDGMENTS I would like to thank my advisor Professor Greg Sawyer for allowing me to pursue this research and for sharing his resources, time and thoughts in helping me during the past two years. I would al so like to thank Dr. Dan Dickrell for helping me conquer the obstacles of this project over ma ny cups of coffee. I would also like to extend my appreciation to all the members of the University of Florida Tribology Lab for their support and encouragement throughout my graduate career PAGE 5 5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ............................ 4 LIST OF FIGURES ................................ ................................ ................................ .... 6 LIST OF SYMBOLS ................................ ................................ ................................ .. 9 ABSTRACT ................................ ................................ ................................ ............. 10 CHAPTER 1 INTRODUCTION ................................ ................................ .............................. 11 Anatomy of the Human Retina ................................ ................................ .......... 11 Fundus Photography ................................ ................................ ......................... 12 Diagnostic Methods and Morphology of the Retinal Vasculature ...................... 13 2 HISTORICAL REVIEW ................................ ................................ ..................... 17 Origins of Biological Design ................................ ................................ .............. 17 Murray and the Principal of Minimum Work ................................ ...................... 17 Principle of Minimum Volume ................................ ................................ ........... 21 Principle of Minimum Shear Stress ................................ ................................ ... 23 ................................ ...................... 25 Generalization of Optimization Theories for Varying Exponents ....................... 28 Relationship to Presented Work ................................ ................................ ....... 30 3 METHODS ................................ ................................ ................................ ........ 33 Image Dataset Description ................................ ................................ ................ 33 Network Separation ................................ ................................ .......................... 33 Network Skeletonization ................................ ................................ ................... 34 Arterial Source Selection ................................ ................................ .................. 35 Identification of Endpoints and Junctions ................................ .......................... 35 Determination of Connectivity and Junctions ................................ .................... 36 Calculation of Segment Geometry and Bifurcation Angles ............................... 37 Calculation of Junction Exponents ................................ ................................ .... 39 4 RESULTS AND DISCUSSION ................................ ................................ ......... 51 5 CONCLUSIONS ................................ ................................ ............................... 69 LIST OF REFERE NCES ................................ ................................ ......................... 71 BIOGRAPHICAL SKETCH ................................ ................................ ...................... 74 PAGE 6 6 LIST OF FIGURES Figure page 1 1 A cross sectional drawing of the typical anatomy of the human eye including a description of the various layers of retinal tissue. ................................ ...... 14 1 2 central retinal artery which pierces the optic nerve behind the eye. ............. 15 1 3 A non mydriotic Topcon digital fundus camera. ................................ ............ 16 1 4 Example red free fundus image with main components labeled. .................. 16 2 1 Drawing of an arterial bifurcation with labelled flow rates, radii and bifurcation angles. ................................ ................................ ................................ .......... 31 2 2 An arterial bifurcation positioned on a rectangular coordinate system along with its labeled lengths, endpoints and angles. ................................ ............. 3 2 3 1 An illustration of the vascular segmentation process. ................................ ... 40 3 2 Illustration depicting the separation of the original binary network into arterial and venous binary networks. ................................ ................................ ........ 41 3 3 Example of a vein next to an artery in a fundus image. ................................ 42 3 4 Illustration showing the separation process of arteries and veins. ................ 42 3 5 An example of neovascularization. ................................ ............................... 43 3 6 Illustration of the skeletonization process. ................................ .................... 44 3 7 Illustration of stub removal. ................................ ................................ ........... 45 3 8 Manual selection of the source of arterial flow. ................................ ............. 45 3 9 Skeletal network section with endpoints identified. ................................ ....... 46 3 10 Skeletal network section with junctions identified. ................................ ........ 46 3 11 Skeletal network section with a node and respective segments labeled. ..... 47 3 12 Skeletal network section with segment generation labels. ............................ 47 3 13 Example node with parent and children segments labeled. .......................... 48 3 14 Method of nearest non vessel pixel method of width calculation. ................. 48 PAGE 7 7 3 15 Sample skeletal segme nt with directional unit vectors labeled. .................... 49 3 16 Sample node with bifurcation angles labeled. ................................ ............... 49 3 17 Sample node with branch diameters labeled. ................................ ............... 50 4 1 Histogram of junction exponents. ................................ ................................ 53 4 2 Scattered data of area ratios vs. asymmetry ratios along with theoretical predictions. ................................ ................................ ................................ ... 54 4 3 Binned data of area ratios vs. asymmetry ratios along with theoretical predictions. ................................ ................................ ................................ ... 55 4 4 Measured area ratios versus asymmetry ratios along with theoretical predictions with a junction exponent of 1.5. ................................ .................. 56 4 5 Measured area ratios versus asymmetry ratios along with theoretical predictions with a junction exponent of 2.5. ................................ .................. 57 4 6 Measured area ratios versus asymmetry ratios along with theoretical predictions with a junction exponent of 3.0. ................................ .................. 58 4 7 Scattered larger diameter bifurcation angles versus asymmetry ratios along drag/surface area. ................................ ................................ ........................ 59 4 8 Binned larger diameter bifurcation angles versus asymmetry ratios along with imum power/volume and drag/surface area. ................................ ................................ ................................ ............. 60 4 9 Scattered smaller diameter bifurcation angles versus asymmetry ratios along with Z drag/surface area. ................................ ................................ ........................ 61 4 10 Binned smaller diameter bifurcation angles versus asymmetry ratios along drag/surface area. ................................ ................................ ........................ 62 4 11 Scatter ed larger diameter bifurcation angles versus asymmetry ratios along for a junction exponent of 1.5. ................................ ................................ ...... 63 4 12 Scattered smaller diameter bifurcation angles versus asymmetry ratios along for a junction exponent of 1.5. ................................ ................................ ...... 64 PAGE 8 8 4 13 Scattered larger diameter bifurcation angles versus asymmetry ratios along urface area for a junction exponent of 2.5. ................................ ................................ ...... 65 4 14 Scattered smaller diameter bifurcation angles versus asymmetry ratios along f or a junction exponent of 2.5. ................................ ................................ ...... 66 4 15 Scattered larger diameter bifurcation angles versus asymmetry ratios along etical predictions for minimum power, drag and surface area for a junction exponent of 3.0. ................................ ................................ ...... 67 4 16 Scattered smaller diameter bi furcation angles versus asymmetry ratios along for a junction exponent of 3.0. ................................ ................................ ...... 68 PAGE 9 9 LIST OF SYMBOLS asymmetry ratio area ratio fluid dynamic viscosity b ifurcation angle vessel shear stress segment diameter power loss through vessel length of segment drag force segment volume power loss per unit volume of blood volumetric flow rate dynamics viscosity pressure drop across a segment vessel radius drag force per unit len g th vessel directional unit vector junction exponent PAGE 10 10 Abstract of Th esis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Master of Science ANALYSIS OF ARTERIAL BIFURCATION S IN THE HUMAN RETINA By Richard Dean Clark III May 2013 Chair: Wallace Gregory Sawyer Major: Mechanical Engineering The bifurcating nature of the human vascular system has been of scientific and medical interest for over a century. The morphological growth of arteries has been attributed to various theories of optimization while changes in morphology and deviations from normal geometries have been suggested to be the result of pathological condition s Because of the noninvasive natu re of observing the retinal vasculature, fundus photographs present a means of easily viewing and recording an arterial network. If the morphology can then be quantified, it could be a useful tool for diagnosing ocular or systemic diseases. A program was developed which analyzed the bifurcation angles, area ratios, asymmetry ratios and junction exponents of 13 healthy retinal arterial networks and compared the results against various theories of optimization including the principles of minimum work, minimum drag force, minimum volume and minimum surface area. The results conformed to different theories to various degre es depending on the parameter measured, leading to the conclusion that not one theory alone can be assumed if an optimum model is to be developed. PAGE 11 11 CHAPTER 1 INTRODUCTION Anatomy of the Human Retina The human retina is a complex layer of tissue that lies on the anterior inner surface of the eye It is an extension of the central nervous system and is responsible for the nerve impulses that create vision through the collection of light. It lies between the vitreous, the clear liquid in the center of the eye, and the choroid, a vascular layer of the eye [ 1 ] The main components of the retina are the tissues layers, which are responsible for the chemical and electrical processes that interpret light and create vision; the optic disc, a small ov al portion through which the retinal blood flow enters and exits; the macula, a small yellow spot at the center of the b ack of the eye which contains a depression known as the fovea, responsible fo r sharp, high resolution vision; and the vasculature, responsible for supplyi ng oxygen and nutrients to the retina l tissue. The typical anatomy of the eye is seen in Figure 1 1. The retinal vasculature begins with the central retinal artery that branches from the ophthalmic artery (Figure 1 2) pierces the optic nerve behind the ey e and transports blood into th e retina through the optic disc. This artery then branches into smaller arterioles which ultimately branch down into capillaries through which oxygen and nutrients are dispersed into and collected from the retinal tissue. The collected blood is then funneled through venules which merge into larger veins and ultimately the central retinal vein which exits through the optic disc. PAGE 12 12 Fundus Photography Due to the requirement for light the retina contains the only vasculature of t he human body that can be observed noninvasively [ 2 ]. This fact has led to a wide variety of examinations and devices that focus on looking through the pupil and analyzing the retina in order to check for possible disease. Because these images generally on ly capture the fundus, or inner back surface, of the eye, this practice is known as fundus photography. Originally, fundus images were taken on film, but with the development of the charge coupled device for capturing light, the practice has moved into the digital realm. The ability for a clinician to accurately diagnose a fundus image has been determined to be equal for film and digital images [ 3 ]. A typical fun dus camera can be seen in Figure 1 3 Fundus cameras can vary in field of view and magnificatio n of the produced images. Some cameras also vary on their need of dilation ( n on mydriatic cameras do not need the patient to be dilated). Additionally, some cameras produce red free images as the red layer of an RGB image sometimes contributes to the ambig uity in discerning objects from the image [ 4 ]. Typically, fundus images (Figure 1 4) are fovea centered, contain a 30 to 50 field of view and display the optic disc as well as the arteries and veins at a magnification of 2.5. Initially the arteries and veins seem indistinguishable, but due to their function and anatomy they are generally able to be discerned from each other based on diameter, brightness, tortuosity and connectivity. PAGE 13 13 Diagnostic Methods and Morphology of the Retinal Vasculature The current method of practice involves a technician taking the image and then the image being surveyed by either a medical doctor or a specialized reader. This is a highly manual process that requires the viewer to pick out abnormalities or lesions in the ret ina, which has been found to be slow, laborious and prone to human errors [ 5 ]. The automation of diagnoses has mostly focuses on detection of microaneurisms, cotton wool spots, exudates, or hemorrhages [ 6,7 ], most of which are used as markers for diabetic retinopathy, a disease that has been rapidly growing across the world and results in an approximate 30 fold chance of blindness over someone without the disease [ 8 ]. In recent years interest has grown in the morphology of the retinal vasculature [ 9 ,10 ]. T he interesting shape has led many to try to classify the morphology for use of diagnosis [ 1 1 ]. While the resulting shapes are interesting but the underlying causes and principles of that geometry, and the geometry of the vascular system as a whole, have a lso interested scientists and doctors for almost a century. In particular, scientists and doctors have observed the diameters and angles of blood vessels at bifurcations to determine what underlying principles might contribute to the formation of, or devia tion from, an efficient method of blood transport [ 1 2 ]. This study collects the information from bifurcations in the arterial vasculature of a set of healthy eyes and compares this information to that of various proposed optimization theories. PAGE 14 14 Figure 1 1 A cross sectional drawing of the typical anatomy of the human eye including a description of the various layers of retinal tissue. (Source: http:/ /www.intechopen.com/source/html/26714/media/image1.png Last accessed March 201 3). PAGE 15 15 Figure 1 central retinal artery which pierces the optic nerve behind the eye. (Source: http://disac.co.uk/Ophthalmic%20Artery%20Branches%20Roof%20of%20Orb it%20Removed.jpg Last accessed March, 2013). PAGE 16 16 Figure 1 3. A non mydriotic Topcon dig ital fundus camera. (Source: http://upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Retinal_camera.j pg/1280px Retinal_camera.jpg ). Figure 1 4. Example red free fundus image with main components labeled. PAGE 17 17 CHAPTER 2 HISTORICAL REVIEW Origins of Biological Design Some of the first work on optimal vascular branching was done by during the [ 1 3 ] Murray claimed that physiological systems not only adapt underlying mathematical and scientific laws that govern the geometric growth of these systems. These laws are based in physiologically independent fields such as thermodynamics and fluid mechanics. is credited with the idea of the principle of similitude, in which he remarks upon the sc aling laws between the cross sectional area of bone and its strength as well as between the volume of bone and its weight. This gave insight into the animal sizes and a departure from the idea of randomness in biological structures. While also citing Darwin for the idea of macroscopic biological adaptation, Murray focused primarily on the concept of internal organization of physiological systems. As with the design of bones, intended to support animal weights and bear various other forces effect ively, the design of the vascular system must also perform its function effectively. Murray argued that this design is not only effective, but optimal. Murray and the Principal of Minimum Work Two competing factors contribute to the development of the vas cular system cost and function. It is the optimization of these two parameters, Murray states, that leads to an optimized overall design that should resemble the structure of our vascular PAGE 18 18 network. Discrepancies between a theoretical optimization and the observed network must be attributed to an unknown factor. The primary purpose of the arterial vascular system, Murray claimed, is to distribute oxygen to various organs in the body, and this is the focus of his work He attempts to balance the functionalit y of the system delivering oxygen via capillaries in tissue with the costs of the system. Murray i dentified two costs associated with blood transport the cost of friction and the cost of volume. In order to estimate the friction loss and volumetric loss, Murray assume d that a blood vessel is a cylinder with a constant radius r and length l. He also assume d that the volumetric flow rate is laminar at a rate of f and constant dynamic With these assumptions it was possible to use the Hag en Poiseuille equation ( Equation 2 1) to find the pressure drop p, along the length of the ve ssel due to friction. (2 1) By multiplying each side of Equation 2 1 by the volumetric flow rate, f, Equation 2 2 is found, which describes t he power loss due to friction. (2 2 ) In order to estimate the volumetric power loss of blood, Murray multiplied the volume of blood in a vessel by a constant b, representing the power cost per unit volume of blood as seen in Equation 2 3. PAGE 19 19 (2 3) This encompasses various contributing costs such as the metabolism of the blood and vessel walls, the cost of comprising elements of the blood such as hemoglobin and the actual weight of the blo od. For the sake of simplicity, Murray attributed these factors into a single parameter. The total power cost was thus estimated by the following equation: (2 4 ) Murray determined the optimized economy to exist when the total power cost is at a minimum. By this logic, he differentiated Equation 2 4 with regard to the radius and equated the result to zero ( Equation 2 5). (2 5 ) This allow ed b to be solved in terms of the flow rate, viscosity and radius: (2 6 ) The equation can then be solved for f to yield (2 7 ) used to encompass both terms and express the flow rate with a simple expression. PAGE 20 20 (2 8 ) (2 9 ) Equation 2 9 states that the flow rate through any vessel is proportional to the By applying the principle of conservation of mass, a relationship between the three vessels at a bi furcation can be described with Equation 2 12. (2 10 ) (2 1 1 ) (2 1 2 ) This relationship states that the sum of the cubes of the child radii r 1 and r 2 is equal to the cube of the parent radius r 0 at a bifurcation (Figure 2 1) This can also be stated using diameters: (2 13) Murray also investigated the effects that bifurcation angles have on minimum work [ 1 4 ] He demonstrated that a change in length of any of the three connecting vessels leads to known changes in bifurcation angles Using Equation 2 4 he determined the effects these length changes had on the power cost of each vessel, PAGE 21 21 which, for minimum work, must yield a total change of zero. This led to relationships for 1 and 2 the deviation angles of the first and sec ond child vessels, respectively: (2 1 4 ) (2 1 5 ) Substituting Equation 2 12 into Equation s 2 1 4 and 2 1 5 he yielded his final result of (2 1 6 ) (2 17 ) ` If the child branches are equal in diameter, each angle can be represented by Equation 2 18. (2 18 ) Principle of Minimum Volume Horsfield and Cumming [ 1 5 ] focused on the bronchial tree morphology to determine an optimum branching pattern, but the principles of fluid transport remain similar. The optimization problem of interest involved maximizing the al veolar surface area and cross sectional area of the air passageways in order to facilitate oxygen transport from the alveolar sacs to the pulmonary capillaries. A concurrent goal exists that strives for efficient use of volume within the lungs, which requi res small cross PAGE 22 22 sectional areas for the passageways, leading to investigations of how these parameters influence the morphology Assuming a symmetric dichotomous style of branching, Horsfield determined the optimum diameter of a child vessel r (w+1) to be a constant fraction of the parent vessel, r w This relationship is expressed by Equation 2 19 (2 19 ) 1 = r 2 He then focused on the principal of minimum volume, usin g a similar techni que to Murray. By changing the length of each vessel (L 1 L 2 and L 3 ), and assuming that the point of bifurcation is already at an optimum minimum volume position, a change in volume in the parent vessel is equal to the total change in volume of the child branches. (2 20 ) The changes in length were related to the deviation angles of the child branches, which ultimately led to a relationship between the area ratios and the deviation angles: (2 21 ) (2 22 ) (2 23 ) PAGE 23 23 This relationship is mathematically viable only when the total cross sectional area of the child vessels is not less than that of the parent vessel. For these special cases, a separate relationship was derived while holding th e parent vessel length constant (2 24 ) Kamiya and Togawa also attempted to optimize the branching of the vascular network by volume minimization, citing the inverse relationship between volume and transmission time of information as well as the increased metabolic needs maintaining greater amounts of blood unti l it is actually used at the capillary level. [ 1 6 ]. Principle of Minimum Shear Stress Zamir argued that the two principles discussed thus far minimum power and minimum volume which he im mediately discards, or that these patterns are genetically embedded. The latter, he argues, is improbable because of the natural variations witnessed in the vascular network. Through power and volume management, Zamir stated that there is no mechanism by which the vessels may try to optimize. That is, there is no feedback system governing the formation of these vessels. Instead of a global principle governing the geometry of vessels, he proposed that vessel s grow accor ding to local factors, which he believed to be local shear stresses [ 1 7 ]. Zamir defined the shear stress on the inner wall tangential to the vessel and parallel to flow as PAGE 24 24 (2 25 ) The viscosity, volumetric flow rate and radius are respectively. The total force in the vessel, T, is found by multiplying the shear stress by per unit length, t, is found by dividing the force by the length of the vessel. (2 26 ) (2 27 ) After asserting that the total tension in the three vessels involved at a bifurcation is equal to the sum of the ten sions of each individual vessel, Zamir attempted to minimize this total tension by moving the point of bifurcation in an x y coordinate system (Figure 2 2) T 0 T 1 and T 2 are the total drag force in the parent vessel, larger diameter child vessel and smaller diameter child vessel, respectively. This can be represented with drag per unit length. (2 28 ) PAGE 25 25 (2 29 ) When this equation is differentiated by x and y and set to zero, the bifurcation angles for minimum total drag are obtained, seen in Equation s 2 30 and 2 31. (2 3 0 ) (2 3 1 ) The solution is only dependent on the drag per length, which, as seen in Equation 2 27 is a function of vessel flow and radius. Optimization Theories Zamir later summarized four different optimization principles, all of which involve the minimization of a single factor [ 1 8 ] as seen in Equation s 2 32 through 2 40. For minimum lumen surface area: (2 3 2 ) (2 3 3 ) (2 3 4 ) PAGE 26 26 For minimum lumen volume: (2 35 ) (2 36 ) For minimum power: (2 37 ) (2 38 ) For minimum drag: (2 39 ) (2 4 0 ) Zamir then introduced two new terms to simplify these relationships [ 1 9 ] the asymmetry ratio, and the area ratio, seen in Equation s 2 4 1 and 2 4 2 PAGE 27 27 (2 4 1 ) (2 4 2 ) In these equations r 0 r 1 and r 2 are the parent vessel radius, larger child vessel radius and smaller child vessel radius, respectively. By comb in ing Equation s 2 4 1 and 2 4 2 with Equation 2 12 the following relationship was derived. (2 4 3 ) Combining Equation s 2 4 1 and 2 9 into Equation s 2 3 2 through 2 4 0 Zamir arrived at simple equations defining the bifurcation angles [ 20 ], seen in Equation s 2 44 through 2 47 The equations for minimum work mirrored those for minimum volume, while the equation for minimum drag mi rrored those for minimum surface area. For minimum power and lumen volume: (2 4 4 ) (2 45 ) For minimum drag and lumen surface area: (2 46 ) PAGE 28 2 8 (2 47 ) Generalization of Optimization Theories for Varying Exponents While the optimization for minimum lumen surface and lumen volume do not require the relationship stated by Murray, the equations for minimum power and drag rely on the fact that the flow is proportional to the cube of the radius. Uylings [ 2 1 ] argued that the possibility for turbulent flow exists in the vasculature, and derived an equation for the relationship of flow to the radius that encompasses both types of flow. (2 48 ) For laminar flow the value of j is 4.0 and for turbulent flow j is equal to 5.0. The power to which the radius is raised is represented by Equation 2 49 The principle of conservation of mass can be applied to determine the relationship between flow rates in Equation 2 5 0 in which the x is referred to as the junction exponent. According to Uylings, laminar flow optimization requires a junction exponent of 3.0 while turbulent flow requires a junction exponent 2.33. (2 49 ) (2 5 0 ) PAGE 29 29 3.0 with x, Roy and Woldenberg [ 2 2 ] derived more general versions of the relationships previously defined by Zamir seen in Equation s 2 5 1 through 2 59 (2 5 1 ) Minimum power: (2 5 2 ) (2 5 3 ) Minimum drag: (2 5 4 ) (2 55 ) Minimum volume: (2 56 ) (2 57 ) PAGE 30 30 Minimum surface area: (2 58 ) (2 59 ) Relationship to Presented Work In the following work, the distribution of junction exponents are analyzed to the theories presented by Zamir ( Equation s 2 4 3 through 2 47 ) and Roy ( Equation s 2 5 1 through 2 59 ) are tested against the measured data. PAGE 31 31 Figure 2 1. Drawing of an arterial bifurcation with labelled flow rates, radii and bifurcation angles. PAGE 32 32 Figure 2 2. An arterial bifurcation positioned on a rectangular coordinate system along with its labeled lengths, endpoints and angles. PAGE 33 33 CHAPTER 3 METHODS Image Dataset Description The images used in this study come from the Gold Standard Database for Evalu ation of Fundus Image Segmentation Algorithms. This database is maintained by the Department of Computer Science at University of Erlangen Nuremberg in Bavaria, Germany [ 2 3 ]. The database contains three categories: healthy, diabetic and glaucomatous. For e ach category there exist fifteen fundus images accompanied by a corresponding binary image of segmented blood vessels (Figure 3 1) All fundus images are fovea centered with the optic disc to either the left or right side of the image, depending on whethe dexter) or left eye (oculus sinister) Each image has dimensions of 2336 x 3504 pixels with a red, green and blue color layer. The fundus can be observed in a trimmed circle with a radius of approximately 1600 pixels at the center of the image. The corresponding segmented image contains an array of logical values the same size as the original fundus images, with ones signifying the presence of vasculature These images were produced through the collaboration o f image processing experts, ophthalmologists, and ophthalmologic clinicians in order to create a segmentation was used for this dataset to ensure accuracy. Network Separation The seg mented images contain information on the locations of retinal vasculature, but no information is given on the classification of vasculature as arterial or venous. In order to analyze each network separately, two images were created from the PAGE 34 34 segmented image s: an arterial segmented image and a venous segmented image (Figure 3 2) Adobe Phot o shop (Adobe, 2012) was used to modify the segmented image to create the arterial and venous networks. In order to create the arterial network, the veins were identified a nd deleted from the image A set of vessel categorization rules was developed, most of which relate to the appearance of the vessel. Veins tend to be wider, darker and more tortuous than arteries at the same radial distance from the optic disc [ 2 4 ]. Through adherence to these criteria, it was possible to distinguish arteries from veins and delete the appropriate vessels (Figure 3 3 ) At crossover points, where arteries and veins meet, great care must be taken to ensure the right vessels are retained by observing the vessel properties such as diameter, orientation and intensity, and must also strive to retain a smooth vessel profile at points where a crossover vessel is deleted. This method introduces a small leve l of uncertainty in the geometry of the vessel through deletion errors (Figure 3 4 ) Due to the nature of the connectivity algorithms later implem ented, no loops can exist in the image, defined by areas of isolated black pixels These loops tend to form mo re often in diabetic eyes that experience neovascularization (Figure 3 5) If these Network S keletonization The separated image, whether arterial or venous, was imported into MATLAB ( The Mathworks, 201 2 ) and converted to binary using a native MATLAB threshold function. The image was then treated as a logical array and subjected to a MATLAB PAGE 35 35 native thinning algorithm [ 2 5 ] to create a single pixel wide network defining the skelet on of the network (Figure 3 6) In order to disregard any network fragments entering from the boundaries of the image as well as possible floating artifacts, each area of connected white pixels was labeled and its comprising pixels counted. The region con taining the most connected white pixels, i.e. the network to be analyzed, was retained while all other s were deleted. Often, sometimes due to the nature of separating the arterial and venous networks, (Figure 3 7 ) These stubs were deleted by isolating only endpoint segments and deleting any of those segments that fell below a certain pixel length filter. The skeletal image was then checked for loops by inverting the image and counting the number of 4 neighbor pixel connected regions. If this number exceeded one, loops were detected and analysis ceased. This required modification of the separated image file and starting the analysis again Arterial Source Selection After the skeletal network was defi ned, the source of arterial flow (or destination of venous flow) was manually chosen by drawing an ellipse around a set of pixels within the optic disc (Figure 3 8 ) This is the area inside which the central retinal artery and central retinal vein enter and exit the optic disc. This characteristic set of pixels was later used in determining the connectivity of the network. Identification of Endpoints and Junctions Endpoints are defined as s keletal pixels with only one neighbor (Figure 3 9) MATLAB contains a native function for finding these endpoints, but sometimes PAGE 36 36 attributes non white pixels as endpoints. Additionally, any endpoints within the source area are not to be considered. In order to identify only the endpoints that are wanted, the native MATLAB function was first used. Any endpoints that were black pixels were then neglected, as well as any endpoints lying within the source area. A junction is defined as a skeletal pixel that conn ects three separate lengths of pixels (Figure 3 10) Unlike intra segment pixels, these pixels have three neighbors not adjacent to each other. While the criterion of having three neighbors should suffice, an extra precaution is taken to avoid interpreting junctions. The process of thinning should inherently remove these kinks, but the additional criterion neighbor pixels not being adjacent d id not hinder the discovery of these points. Determination of Connectivity and J unctions With the endpoints, junctions and source area defined, the network connectivity all skeletal pixels between and including the two containing nodes (Figure 3 11) By defining which nodes connected to which segments, and vice versa, the connectivity of the entire network was able to be defined. In order to actually determine the connect ions between nodes and segments, a connected at that node was traced until another node was reached, at which point that length of pixels is defined as a segment with its respective bounding nodes. This PAGE 37 37 process is repeated until all pixels are accounted for, with every segment having two containing nodes. The generation of a segment is determined by how many bifurcations exist between the segment and the arterial origin. S egments emanating from the source were (Figure 3 12) By knowing the generation of each segment meeting at a junction, it was possible to determine which segment was the parent segment a nd which were the children segments of the bifurcation (Figure 3 13) Calculation of Segment Geometry and Bifurcation Angles The skeletal segments are an estimation of the centerline of a vessel segment. Using this information, it is possible to estimate the width based on the location of the nearest non segment pixel. Since the projection of a portion of a vessel resembled a rectangle, and the skeletal pixels represent a line that equally divides the area of the rectangle and runs parallel to the vessel w alls, it can be assumed that the shortest distance to a point outside of the vessel represents the approximate radius of a vessel. By inverting the segmented image, thus making vessel pixels black, the MATLAB zero value to all non vessel pixels and distance values to all vessel pixels. This distance value increases with distance from a vessel wall (Figure 3 14) By using the distance value from the points on which the skeletal pixels lie, a radius for the segm ent is determined and doubled to arrive at a diameter at that point. PAGE 38 38 This width has a fair amount of uncertainty due to two factors: the skeletal pixel may not lie exactly on the centerline of the vessel, and the line connecting the skeletal pixel to the nearest non vessel pixel may not be perpendicular to the vessel walls. In larger vessels, this is less worrisome because the uncertainty does not scale with vessel size. After widths were determined for every skeletal pixel of a segment, the mean was cal of a value immediately next to a bifurcation because of the difficulty in obtaining an accurate measurement at that point. In order to determine the bifurcation angles of the child segments it was required to attribute a directional unit vector to each segment. For each segment, two vectors were calculated, one at each end directed inwards along the segment (Figure 3 15) These were calculated by finding the unit vectors from one end of the segment to each of up to 50 pixels into the segment. These unit vectors were then averaged to find a unit vector for that end of the segment. This process was repeated for the other end of the segment to determine another unit vector. (3 1 ) (3 2 ) With the unit vectors of the parent segment and children segments defined at each bifurcation, the bifurcation angles 1 2 ) were calculated including the deviation angles for each child segment (Figure 3 16). PAGE 39 39 (3 3 ) (3 4 ) Calculation of Junction Exponents The junction exponent can be found through Equation 3 5 with the parent segment diameter for D 0, the smaller diameter of the children segments for D 1 and the larger for D 2 (Figure 3 17) Because there is not a closed form solution for x the solving capabilities If a solution coul d not be found numerically, the bifurcation was excluded from analysis. (3 5 ) PAGE 40 40 A B Figure 3 1. An illustration of the vascular segmentation process. A.) original fundus image, B.) binary segmented image PAGE 41 41 A B C D Figure 3 2. Illustration depicting the separation of the original binary network into arterial and venous binary networks. A.) Binary network, B.) Color coded network, C.) separated arterial network, D.) separated venous network PAGE 42 42 Figure 3 3 Example of a ve in next to an artery in a fundus image. Figure 3 4 Illustration showing the separation process of arteries and veins. PAGE 43 43 Figure 3 5 An example of neovascularization. PAGE 44 44 A B Figure 3 6. Illustration of the skeletonization process. A.) binary section, B.) skeletonized section PAGE 45 45 Figure 3 7. Illustration of stub removal. Figure 3 8. Manual selection of the source of arterial flow. PAGE 46 46 Figure 3 9. Skeletal network section with endpoints identified. Figure 3 10. Skeletal network section with junctions identified. PAGE 47 47 Figure 3 11. Skeletal network section with a node and respective segments labeled. Figure 3 12. Skeletal network section with segment generation labels. PAGE 48 48 Figure 3 13. Example node with parent and children segments labeled. Figure 3 14. Method of nearest non vessel pixel method of width calculation. PAGE 49 49 Figure 3 15. Sample skeletal segment with directional unit vectors labeled. Figure 3 16. Sample node with bifurcation angles labeled. PAGE 50 50 Figure 3 17. Sample node with branch diameters labeled. PAGE 51 51 CHAPTER 4 RESULTS AND DISCUSSION The methods described in the previous chapter were used to calc ulate various parameters from 13 healthy arterial retinal vascular networks. Before the data was analyzed and represented graphically, it was filtered through a set of criteria The first criterion was that the junction exponent was actually found through an iterative solving method. The second restriction was that the junction exponent was positive. The third [2 6 ] The fourth restriction was that both bifu rcation angles must be positive. The distribution of jun ction exponents was represented in a histogram (Figure 4 1) to With a distribution mean of 1.97 and a standard deviation of 0.59, the result did f junction exponents exists in the retinal vasculature. investigated. In Figure 4 the predictions of Zamir ( Equati on 2 43) and Roy ( Equation 2 51). The theoretical line junction exponents. In order to understand the distribution of the measured values better Figure 4 3 displays the same theories and data averaged over evenly spaced bins of the asymmetry ratio. The data seems very random and does not seem to conform to PAGE 52 52 Figures 4 4 thr ough 4 6 show a comparison of measured data and theoretical prediction of Roy based on a junction exponent of 1.5, 2.5 and 3.0, respectively. These plots give insight into the exponent seems to follow thes Figures 4 7 and 4 1 2 against the ( Equation s 2 44 through 2 47) are also shown. The data seems to have a fairly wide means of the bins fit fairly well between the two theories. Figures 4 11 through 4 16 display theoretical relationships between the measured bifurcation angles and the asymmetry ratio as predicted by Roy ( Equation s 2 52 through 2 59). Each figure is isolated to a junction exponent of either 1.5, 2.5 or 3.0 and within each figure various theories of optimization are compared. For some p lots the theoretical relationship was not applicable (e.g. a junction exponent of 1.5 in Equation 2 56 yields imaginary results, therefore the relationship for volume is not shown in Figures 4 11 and 4 12). In each figure the data follows at least one tre nd that aligns with a theory. For a junction exponent of 1.5 the data only seems to follow the principle of minimizing surface area, while with other junction exponents it could be argued that the data could follow any of the four optimization principle pr edictions. PAGE 53 53 Figure 4 1. Histogram of junction exponents PAGE 54 54 Figure 4 2. Scattered data of area ratios vs. asymmetry ratios along with theoretical predictions. PAGE 55 55 Figure 4 3. Binned data of area ratio s vs. asymmetry ratio s along with theoretical predictions. PAGE 56 56 Figure 4 4. Measured area ratio s versus asymmetry ratio s along with theoretical predictions with a junction exponent of 1.5. PAGE 57 57 Figure 4 5. Measured area ratio s versus asymmetry ratio s along with theoretical predictions with a junction exponent of 2.5. PAGE 58 58 Figure 4 6. Measured area ratio s versus asymmetry ratio s along with theoretical predictions with a junction exponent of 3.0. PAGE 59 59 Figure 4 7. Scattered larger diameter bifurcation a ngle s versus asymmetry ratio s along drag/surface area. PAGE 60 60 Figure 4 8. Binned larger diameter bifurcation angles versus asymmetry ratios along inimum power/volume and drag/surface area. PAGE 61 61 Figure 4 9. Scattered smaller diameter bifurcation angles versus asymmetry ratios drag/surface area. PAGE 62 62 Figure 4 10. Binned smaller diameter bifurcation angle s versus asymmetry ratio s along drag/surface area. PAGE 63 63 F igure 4 11. Scattered larger diameter bifurcation angle s versus asymmetry ratio s theoretical predictions for minimum power, drag and surface area for a junction exponent of 1.5. PAGE 64 64 Figure 4 12. Scattered smaller diameter bifurcation angle s versus asymmetry ratio s urface area for a junction exponent of 1.5. PAGE 65 65 Figure 4 13. Scattered larger diameter bifurcation angles versus asymmetry ratios area for a junction exponent of 2.5. PAGE 66 66 Figure 4 14. Scattered smaller diameter bifurcation angles versus asymmetry ratios area for a junction exponent of 2.5. PAGE 67 67 Figure 4 15. Scattered larger diameter bifurcation angles ve rsus asymmetry ratios area for a junction exponent of 3.0. PAGE 68 68 Figure 4 16. Scattered smaller diameter bifurcation angles versus asymmetry ratios ictions for minimum power, drag and surface area for a junction exponent of 3.0. PAGE 69 69 CHAPTER 5 CONCLUSION S Overall the program seemed proficient at collecting information about retinal arterial bifurcations. Occasionally the junction exponent could not be defined or angles could not be calculated, but this is probably due to either the methods of segmentation, methods of separation, or errors in the calculation of segment diamete rs or directional unit vectors. For the use of comparison to theory, it also provided some additional insight. The i.e. that the junction exponent is always equal to 3.0, was not a valid assumption to make. This could be due to a multit ude of factors ranging from the assumptions of laminar flow the possibilities of other principles of optimization outside of minimum work. In predicting the relationship between area ratio and the asymmetry ratio, the inclusion of a junction exponent seeme d to follow theory more closely than a junction used, there seemed to be no discernable conformity to the predicted values. When predicting bifurcation angles, however, the the ories supposed by Zamir seemed to encompass the data much more closely than those of Roy with various exponents. The inconsistencies between various theory predictions may suggest that vascular growth is based on multiple theories, or relies on factors out side of this analysis. Future work should investigate possible additional relationships based on flow based factors, such as flow rate, pressure gradients or fluid resistances. As Zamir suggested, vascular growth may rely on local factors instead of a full network PAGE 70 70 optimization principle. 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JB Lippincott, 1908 PAGE 74 74 BIOGRAPHICAL SKETCH Richard Clark was born in 1988 in Orlando, Florida. He attended Winter Springs High School in Winter Springs, FL and continued his education at the University of Florida, majoring in m echanical e ngineering. During his undergraduate career he held various leadership positions including president of the Sigma Omicron branch of Pi Tau Sigma, an honor society for mechanical engineers. After graduating magna cum laude in 20 06 with his Bachelor of Scien ce degree he continued his studies under Professor W. Gregory Sawyer in the University of Florida Tribology Laboratory. While researching the vasculature of retinal fundus images, he also held a mentor ing position in the Engineering Freshman Transition Program and led weekly lectures in the undergraduate statics course. He graduated in the spring of 2013 with his Master of Science degree in mechanical engineering. 