The Mechanics of Vasculogenesis in Collagen Tubes

MISSING IMAGE

Material Information

Title:
The Mechanics of Vasculogenesis in Collagen Tubes
Physical Description:
1 online resource (55 p.)
Language:
english
Creator:
Breaux, Jolie A
Publisher:
University of Florida
Place of Publication:
Gainesville, Fla.
Publication Date:

Thesis/Dissertation Information

Degree:
Master's ( M.S.)
Degree Grantor:
University of Florida
Degree Disciplines:
Mechanical Engineering, Mechanical and Aerospace Engineering
Committee Chair:
ANGELINI,THOMAS ETTOR
Committee Co-Chair:
HAHN,DAVID WORTHINGTON

Subjects

Subjects / Keywords:
angiogenesis -- blood -- buckling -- cell -- mechanics -- vasculogenesis -- vessel
Mechanical and Aerospace Engineering -- Dissertations, Academic -- UF
Genre:
Mechanical Engineering thesis, M.S.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract:
Cells in all types of tissue are sensitive to their mechanical environment. Understanding cell mechanics in tissue growth can lead to advancements in important medical applications, like technologies that enhance angiogenesis during wound healing. Great progress has been made in understanding the mechanics of angiogenesis with assays performed in flat bottomed culture dishes. Here we present results from an in vitro study of collective endothelial cell mechanics in a 3D culture system that mimics the geometry of a real endothelium. Human Aortic Endothelial Cells were grown inside of a collagen tube supported by a rigid cylindrical scaffold. We developed a time-lapse small angle light scattering method to directly measure the radial distribution of cells in the 3D matrix over time. Accompanying live-cell time-lapse microscopy was performed to monitor the cells collective movement and organization. We find that the cells generate sufficient contractile force to detach the collagen matrix from the support scaffold while maintaining a macroscopic cylindrical arrangement, creating a fiber. Image analysis coupled with dynamic small angle light scattering shows possible tube formation. Contractile forces of the cells cause the tube to buckle through time, in what is modeled as Euler buckling. The cells also cause a smaller undulation follows along with the model of buckling within an elastic medium
General Note:
In the series University of Florida Digital Collections.
General Note:
Includes vita.
Bibliography:
Includes bibliographical references.
Source of Description:
Description based on online resource; title from PDF title page.
Source of Description:
This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jolie A Breaux.
Thesis:
Thesis (M.S.)--University of Florida, 2014.
Local:
Adviser: ANGELINI,THOMAS ETTOR.
Local:
Co-adviser: HAHN,DAVID WORTHINGTON.

Record Information

Source Institution:
UFRGP
Rights Management:
Applicable rights reserved.
Classification:
lcc - LD1780 2014
System ID:
UFE0046841:00001


This item is only available as the following downloads:


Full Text

PAGE 1

THE MECHANICS OF VAS CULOGENESIS IN COLLAGEN TUBES By JOLIE ANNE BREAUX 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 2014

PAGE 2

2014 Jolie A nne Breaux

PAGE 3

To my parents, my sister and my husband for their support and encouragement

PAGE 4

4 ACKNOWLEDGMENTS First and foremost, I would like to express my great appreciation to Dr Thomas Angelini, for his teaching and guidance on this research work His willingness to help even at his busiest has been very much appreciated. I would also like to thank Steven Zehnder, Melanie Suaris, Ryan Nixon and all other members the members of the Bio and Soft Matter Lab for their valuable assistance. Finally, spec ial thanks to NSF grant CMMI 1161967 for the funding of this research project.

PAGE 5

5 TABLE OF CONTENTS page ACKNOWLEDGMENTS ................................ ................................ ................................ .. 4 LIST OF FIGURES ................................ ................................ ................................ .......... 6 ABSTRACT ................................ ................................ ................................ ..................... 8 CHAPTER 1 INTRODUCTION ................................ ................................ ................................ .... 10 Motivation ................................ ................................ ................................ ............... 10 Hypothesis ................................ ................................ ................................ .............. 10 Vasculogenesis and Angiogenesis ................................ ................................ ......... 11 Buckling in Blood Vessels ................................ ................................ ....................... 12 Cell Mechanics ................................ ................................ ................................ ....... 1 3 2D Cell Mechanics ................................ ................................ ........................... 13 3D Cell Mechanics ................................ ................................ ........................... 13 2 METHODS ................................ ................................ ................................ .............. 18 Cell Culture ................................ ................................ ................................ ............. 18 Sample Preparation ................................ ................................ ................................ 18 Time Lapse Microscopy and Image Analysis ................................ .......................... 19 Time Lapse Photography and Image Analysis ................................ ........................ 19 Multi Photon Confocal Imaging ................................ ................................ ............... 19 Dynamic Small Angle Light Scattering and Image Analysis ................................ .... 20 3 RESULTS ................................ ................................ ................................ ............... 23 Radial Collapse and Buckling of Endothelial Cell Fibers ................................ ........ 23 Mode Analysis of Fiber Shape ................................ ................................ ................ 25 Small Angle Light Scattering on Collapsing Fibers ................................ ................. 27 Buckling Mechanics and the shape of Endothelial Cell Tubes ................................ 28 Buckling in a Confined Container ................................ ................................ ..... 28 Buckling in an Elastic Medium ................................ ................................ .......... 31 Collagen Properties under Collapsing Conditions ................................ ............ 32 4 DISCUSSION ................................ ................................ ................................ ......... 49 5 FINAL REMARKS AND FUTURE PLANS ................................ .............................. 51 LIST OF REFERENCES ................................ ................................ ............................... 53 BIOGRAPHICAL SKETCH ................................ ................................ ............................ 55

PAGE 6

6 LIST OF FIGURES Figure page 1 1 Steps of vasculogenesis and angiogenesis 8 ................................ ....................... 15 1 2 Examples of tortuous blood vessels found in a tumor 14 ................................ ...... 16 1 3 Blood vessel growth modes ................................ ................................ ................ 17 2 1 Method of molding cylinders for endothelial cell growth ................................ ..... 21 2 2 Dynamic small angle light scattering set up ................................ ........................ 22 3 1 Macro time lapse imaging shows the collapse of the cell cylinder ...................... 34 3 2 The average radius of the cell cylinder through time. ................................ ......... 35 3 3 Time lapse microscopy images of the tube collapse ................................ .......... 35 3 4 Image enhancement shows the smaller wavelength undulations. ...................... 36 3 5 A bsorbance data gives in sight into the cross section of the cell fiber ................. 36 3 6 Confocal images reveal the cross section of the cell tube in time ....................... 37 3 7 F inding the backbone of t he collagen cell fiber ................................ ................... 38 3 8 Mode a nalysis of time lapse shows prominent wavlengths ................................ 39 3 9 Wavelength evolution over time during cell tube collapse ................................ .. 40 3 10 Scattering through a glass capillary tube ................................ ............................ 41 3 11 Scattering pattern through the cell filled cylinder ................................ ................ 41 3 12 The diffraction pattern in the q r direction ove rl aid with the numerical fits ........... 42 3 13 Numerical fit of the first side peak of the sc attering pattern ................................ 43 3 14 The density fluctuation spacing of the cell tub e ................................ .................. 44 3 15 The dens ity fluctuation spacing correlate s with the sma ll wavelength buckle .... 45 3 16 Rheology measu rements of the modulus of collagen. ................................ ........ 46 3 17 The storage modulus for different concentration s of collagen ............................ 47 3 18 Measured wavelengths and their corresponding models ................................ .... 48

PAGE 7

7 5 1 The contractility of the cells changes with drugging. ................................ ........... 52

PAGE 8

8 Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requir ements for the Degree of Master of Science THE MECHANICS OF VASCULO GENESIS IN COLLAGEN TUBES By Jolie A Breaux May 2014 Chair: Thomas E. Angelini Major: Mechanical Engineering Cells in all types of tissue are sensitive to their mechanical environment. Understanding cell mechanics in tissue growth can lead to advancements in important medical applications, like technologies that enhance angiogenesis during wound heal ing. Great progress has been made in understanding the mechanics of angiogenesis with assays performed in flat bottomed culture dishes. Here we present results from an in vitro study of collective endothelial cell mechanics in a 3D culture system that mimi cs the geometry of a real endothelium. Human Aortic Endothelial Cells were grown inside of a collagen cylinder supported by a rigid cylindrical scaffold. We developed a time lapse small angle light scattering method to measure the radial distribution of ce lls in the 3D matrix over time. Accompanying live cell time lapse We find that the cells generate sufficient contractile force to detach the collagen matrix from the suppo rt scaffold while maintaining a macroscopic cylindrical arrangement, creating a hollow fiber Image analysis coupled with dynamic small angle light scattering shows possible tube formation. Contractile forces of the cells cause the tube to buckle

PAGE 9

9 through t ime, in what is modeled as Euler buckling. The cells also cause a smaller undulation that follows along with the model of buckling within an elastic medium.

PAGE 10

10 CHAPTER 1 INTRODUCTION Motivation The past few decades have shown increased interest in the relationship between cells and their mechanical environment 1 3 Endothelial cells are the basis for the formation of blood vessels and are paramount in biological functions such as va sculogenesis, angiogenesis, and wound healing. When endothelial cells come together to form an endothelium within a blood vessel, they are subjected to many forces such as shear and tension due to the env ironment 4 Tortuous or twist ed blood vessels have been shown to cause medical issues. In this paper we investigate a mechanical instability created by endothelial cells that are grown in a tube structure. This is done with mac ro imaging, micro imaging, confocal imaging, and dynamic small angle light scattering. Hypothes is When endothelial cells are kept in a cylindrical environment, it was hypothesized that the cells could come together to form a tube like structure. We obser ved that after the cells were seeded within the cylinder that the method of creating a cylinder full of cells led to the non uniform distribution of cells within the cylinder. Through time the cells contracted the collagen within the tube and formed a tigh t strand. Using dynamic small angle light scattering, the composition of the cell cylinder was hypothesized to be evidence showed this to be a possibility, although more investigation is needed to confirm. As the tube evolved during the time lapse, two sets of undulations were seen at the micro and macro scale. The macro scale bending is hypothesized to follow Euler

PAGE 11

11 bending, while the micro scale undulations could be caus ed by the encapsulation of cells within an elastic medium. Vasculogenesis and Angiogenesis The first step in creating a functioning vascular system starts with primitive mesodermal cells that differentiate into endothelial ce lls, known as angioblasts. 5 The vascular system is developed when angioblasts assemble into the main vessels of the vasculature in a process known as vasculogenesis. The main growth factor that leads to the differentiation of ang ioblasts is known as VEGF, which is also a catalyst for mitosis for endothelial cells 6 The creation of new blood capillaries after the main vascular development has occurred is referred to as angiogenesis. At this point the endothelial cells continue differentiating and grow from the existing vasculature I n sprouting angiogenesis the cells form branches off the main vasculature and uses cell proliferation to continue growing. Sprouting angiogenesis is most often seen in wound healing and other areas where there are distances to be bridged. 7 8 I ntussusceptive or splitting, angiogenesis is the formation of blood vessels by splitting already formed vasculature 7 The last phase of angiogenesis is pruning and remodeling, where the vasculature is pruned to look like the tree like structure seen in adult vasculature (Figure 1 1) 8 Research has shown that mechanical stimulation from the cells environment helps trigger angiogenesis 9 10 Because of the difficulties in recreating in vivo conditions, in order to fully understand the process of angiogenesis, better understanding of the mechanical boundary conditions is needed in vitro. 10 11

PAGE 12

12 Buckling in Blood Vessels Arteries and veins are normally straight pathways in the body that support life by delivering blood and nutrients, as well as removing waste products and carbon dioxide. Tortuous arteries and veins are blood vessels that have been twisted, looped or kinked (Figure 1 2) 12 14 These tortuous vessels are associated with diseases such as atherosclerosis, hypertension, and diabetes, and have been found to be related to aging, and genetics 15 There are several ways that blood vessels can become tortuous. One of these ways is from the normal day to day stress put on a blood vessel. Arteries are subject to mechanical stress due to blood flow and pressure tethering to surrounding tissue, and movement of the body during every day activit ies 4 T his pressure is sometimes enough to damage the lumen and cause mechanical instability within the blood vessel. This mechanical instability causes the blood vessel to buckle and become tortuous. 16 Another failure method comes from r educed elongation and tension of the blo od vessel which decreases the stability of the arteries The extra length of the blood vessel then causes this blood vessel to bend and kink. This can happen due to age, as well as growth of the blood vessels. 17 18 Finally, genetics and development can cause the malformation of blood vessels leading to tortuous arteries. Malformed blood vessels can be cause by too much VE G F, a blood vessel growth hormone, or by poor genetics in which the formation of the blood vessel causes it to be prone to buckling. 12 19 22 Different modes of blood vessel growth an d instabilities can be seen in F igure 1 3. Understanding the underlying mec hanics of tortuous vessels can help with developing treatment for kinked blood vessels. The exact mechanical failure type has been recently hypothesized to be related to cylindrical buckling. 16 M odels that simulate

PAGE 13

13 the blood vessel as a closed end pressure vessel have seen comparable results to the failure of blood vessels. This model ma ke s use of the linear elastic Euler buckling equation to estimate the internal pressure when the artery becomes unstable. Higher wavelength modes of buckling were seen due to the inclusion of a surrounding support matrix outsid e of the blood vessel. 20 23 This problem more closely relates to the failure of blood vessels while in use, but the tortuosity of blood vessels during development has not been studied as in depth. Cell M echanics 2D Cell M echanics Cell trac tion force microscopy is one of the methods to determine the forces generated by si ngle cells on their environment. Since cells need a surface to grow on, many researchers have spent time learning about the forces individual species of cells can exert on t heir surroundings, as well as the optimal surface stiffness for their growth. 3 24 25 Learning about the migration of cells helps in learning more about cancer metastasis, wound healing and development. 26 Cells have been shown to respond to forces such as shear stress, and the stiffness and pattern of the surface they are on. 3 27 28 Though much has been learned through 2D cell mechanics, there is need for a more natural microenvironment in order to fully understand the native behavior of the cells and tissue morphogenesis. 3D Cell M echanics It ha s been shown that cells in a 3D environment act th an a 2D environment 29 The stresses that come from endothelial cells moving within the extra cel lular matrix (ECM) are shown to regulate cell function and allow for capillary morphogenesis that is closer to that seen in vivo 30 The use of Collagen type 1 as the ECM has demonstrated

PAGE 14

14 endothelial cell growth toward angiogenesis. 31 Even though there are many positives associated with studying cells in their 3D environment, prob lems stem from being able to fully image the sample. Typically, microscopes have been found to be able to only image through thin samples (100 200 um). To be able to fully understand the mechanics involved with angiogenesis, the ability to quantify mechani cs and dynamics within tissue samples is necessary.

PAGE 15

15 Figure 1 1. Steps of vasculogenesis and angiogenesis 8

PAGE 16

16 Figure 1 2 Examples of tortuous blood vessels foun d in a tumor 14

PAGE 17

17 Figure 1 3 Blood vessel growth modes 14 A) As the blood vessel grows, the blood vess el decreases in inner radius B) The undulations that cause varicose veins C) A buckled shape D) Helical

PAGE 18

18 CHAPTER 2 METHODS C ell Culture Human Aortic Endothelial Cells (HAEC) were cultured in Endothelial Basal Media that was modified with a purchased Endothelial Growth Media kit. The cells were grown to confluence then passaged and centrifuged down to a dense pellet that was about 20 micro liters This pellet of cells was thoroughly mixed and deposited into the sample chamber. Sample Preparation For imaging and light scattering, a vertical sample holder had to be created. This was made by drilling one centimeter diameter holes in the top co ver and bottom of a 35 mm petri dish. A small hole was also cut out of the sides of the petri dish to enable the sample chamber to be filled after gluing took place. The top cover and the bottom of the dish were glued together to form a water tight seal, and quartz glass was glued to the holes in the top and bottom cover to allow for better imaging through the sample. To create the cylinder, a mold was created out of a 1 mm diameter capillary tube and a micropipette tip that was cut to hold the capillary tube. This formed a negative of the hole, as well as created a funnel so that the cells could be easily pipetted into the newly formed column. The sample chamber was filled with a mixture of molte n agar and EGM, to create a 1.5 mg/ml solution While the m ixture is molten, the capillary tube mold is inserted into the top of the sample chamber to mold the cylinder. After the agar has solidified, the tube is filled with a 1.5 mg/ml collagen solution. The cells are added after the tube is filled with the colla

PAGE 19

19 allowed to fall down the tube from gravity (F igure 2 1) The sample is then incubated at 37C and 5% CO 2 for 30 minutes to allow for the collagen to gel, and for the cells to spread. T ime Lapse Microscopy and Image Analysis The cell cylinder was imaged in bright field, once every minute, for 24 hours in an environmental chamber at 37C and 5% CO 2 At each time point, a z stack of images was taken in order to image the cell density of the whole tube. Each z stack was then stack focused in MATLAB to make a focused image of the total tube, for each time point. To do this, images were processed with a median filter and the edges were detected using an edge detecting algorithm for each z slice. The in focus edges were saved to a stack, which were then processed using a minimum filter to create one in focus picture. T ime Lapse Photography and Image Ana lysis The cylinder was imaged by a macro lens and a Nikon D3X camera. The sample chamber was held by a 3D printed support and held vertically in an environmental chamber that kept the sample at 37C and 5% CO 2 An image was taken every 2.5 minutes until th e cylinder fully collapsed. The images were then made to 8 bit images, contrasted and background subtracted to create a sharper image. Multi Photon Confocal Imaging Human aortic endothelial cells were dyed with Cell Tracker Green, before centrifuging and bei ng placed within the tube. The sample was then incubated for the designated amount of time, from 1 hour to overnight, to simulate the experiment. After the time elapsed, samples were fixed using 3.7% formaldehyde and phosphate buffered

PAGE 20

20 saline (PBS) sol ution. After fixing took place the samples were rinsed and refrigerated until imaging took place. The sample was imaged using a multi photon confocal micr oscope. A z stack of images was taken every 4 um for the 1 hour sample, and ever y 2.5 um for the overn ight sample. Dynamic Small Angle Light Scattering and Image Analysis For dynamic small angle light scattering, the sample was placed in an environmental chamber at 37C and 5% CO 2 An 850 nm laser diode was used to scatter off the sample, and a computer controlled shutter system took images at three different exposure times in order to obtain the full range of data After 24 hours, the sample was removed and imaged in the microscope to make sure the cells were still viable. The images were bracketed in MATLAB, using the ratio of the exposure times to make the images seamless. This allowed for the major peak to be imaged as well as the fringe peaks without losing information from saturat ed pixels. The images were stitched together in MATLAB to make a comp lete picture.

PAGE 21

21 Figure 2 1. Method of molding cylinders for endothelial cell growth

PAGE 22

22 Figure 2 2. Dynamic small angle light scattering set up

PAGE 23

23 CHAPTER 3 RESULTS Radial Collapse and Buckling of Endothelial Cell Fibers The newly formed column of cells was observed for approximately 24 hours or until the column was fully collapsed. Time lapse photography, taken every two and a half minutes, was used to discern macro scale changes withi n the column, and to witness the larger scale dynamics that occurred within the system. Figure 3 1 shows an annotated version of the time lapse, where it is seen that the cell contractions causes the cylinder to collapse and form a dense cell thread by t he end of the time lapse. The average measured radius of the cylinder o ver time can be seen in Figure 3 2. The ed ges of the tube are highlighted to draw attention to the shape of the tube as the collapse takes place. It is seen that as the radius of the c olumn of cells decreases, the cylinder begins undulating The wavelength of this undulation increases as time goes on u ntil the end of the time lapse, where the cell cylinder becomes straight again. To gain further understanding of the cell fiber, and possibly discover the cause of the undulation, more experiments were completed. To elucidate the behavior of the cylinder collapse at small length scales, time lapse microscop y was used. Shown in Figure 3 3 the microscope images show the cells filling the entirety of the molded tube, and the tube collapsing as the cells contract the coll agen around them. Noticeable in these images are the appearance of another wavelength that is smaller than the wavelength shown in the macro scale images. In Figure 3 4 t he images are dilated to enhance the appearance this smaller wavelength

PAGE 24

24 undulation that is visible. This smaller wavelength undulation decreases as time goes on. relates the amount of light that passes through the object to the concentration of the object by measuring the how much light has been absorbed by the object, compared to the orig inal amount of light. Equation 3 1, describes this relationship where P is the transmitted intensi ty, P 0 is the original intensity, a is the molar absorbivity, b is the path length and c is the concentration. (3 1) If you define A, the absorbance to be equal to the path length times the molar absorbivity and the concentration, you can simplify Equation 3 1 using Equation 3 2, and can find Equation 3 3. (3 2) Equation 3 3 shows the relationship between initial inten sity and the measured intensity and the absorbance of the sample. (3 3) Using Equation 3 3, the projection of the cell density within the cylinder could be determined. The r esults of this can be found in F igure 3 5 From the results, we concluded that the cell cylinder could be forming a tube as the collapse occurs To help ch eck the results gathered from the cell tube, another trial was run, using a cylinder filled with 1um particles. The profile completed using images gathered from that experiment did show an absorbance profile that closely matches that of a cylinder. The

PAGE 25

25 res ults from this analysis provide insight to what could be happen ing inside the tube, but the results warrant further investigation. Confocal imaging of the cylinder helped to provide some more insight as to the cross section of the tube (Figure 3 6) After an hour, the cells are seen creating pockets of dense cells while leaving many spaces clear of cells. This is from the cells falling through the collagen before gelation of the collagen occurs. The cells coalesce and disperse throughout the tube, attaching to walls and the collagen. Over time these bunches of cells contract in the collagen, and the collagen and cells come together to form a thread of cells. The appearance of the cross section in the 12 hour case appears to show a sparse area of cells in the middle of the tube, possibly from the cells forming a rough tubular shape while the col lapse occurs. Due to distance imaging could only occur through part of the cylinder. Mode Analysis of Fiber Shape To quantitatively analyze the wavelength of the buckle, the backbone of the cell cylinder was found by fitting a Gaussian profile to each row of pixels in the images (Figure 3 7 ). From here we used mode analysis to determine the wavelengths of the buckle. To do this, the backbone coordinates were transla ted into the tangent angle as a function of contour position and then a discrete cosine transform was taken of the measurements. To do this, first we get the backbone in x and y coordinates. This we transla te into s space, which is related to the length along the backbone using Equation 3 4 and Equation 3 5 (3 4 )

PAGE 26

26 (3 5 ) To find the tangent angle along the backbone, Equation 3 6 was used, and allowed us to find each and S for each X and Y value. (3 6) At this point, a discrete cosine transform was taken of as seen in Equation 3 7 (3 7) The results of the analysis provided a frequency spectrum, where the peaks were fit with log normal functions to determine the location of the peaks in q space (Figure 3 8 ). These peaks then corresponded to the wavelength of the buckle with the use of E quation 3 2 (3 8 ) Figure 3 9 illustrates the results from the mode analysis. T he two peak movements corresponding to the long wavelength undulation seen in the macro imaging and the short wavelength undulation seen in the microscopy images are shown over time. As seen from the images, and in the mode analysis, the long wavelength increases over time as the tube becomes straighter after encountering the buckle. The short wavelength goes the opposite d irection in time, and decreases in wavelength slightly.

PAGE 27

27 Small Angle Light Scattering on Collapsing Fibers Dynamic small angle light scattering was used to determine the w hole profile of the tube. Scattering through a cylinder leads to a scattering patt ern that follows the following E quation 3 9 and can b e seen in Figure 3 10. (3 9 ) The addition of cells to the cylinder adds a bit more diffuse scattering to the diffraction pattern as seen in Figure 3 11. To get the full range of inform ation out of the images from the experiment, image bracketing was used. This allowed for lower intensity portion of the images to be brightened, while still retaining the information from the higher intensity center region of the scattering pattern. (3 10 ) To get information about the radial direction of the cylinder over time, a slice was taken along the q r direction and th e peaks were fit using Equation 3 10 (Figure 3 12) Here G is a Gaussian wit c is the radius of the initial cylinder (500 um), J 0 and J 1 are Bessel functions, and q is the wave vector. The peak width is inversely proportional to the width of the cylinder, so as the cylinder collapses the peaks co alesce as can be seen in Figure 3 12 B. The fitting function parameters matched up to the inner and outer radius of the cylinder, allowing insight to the formation of the tube, as well as the outer radius of the tube as it collapsed These results seem to c orroborate the results from the absorbance measurements as well as the measurements from the macro imaging The rate of collapse differs between the two

PAGE 28

28 experiments because of the discrepancy between the cell densities in any given experiment. To get info rmation about the axial direction of the cylinder, slices of data were taken in the q z direction and numerically fit. By taking the width of the first fringe peak from the center of the scattering pattern, the persistence length of the tube can be found (Figure 3 13 ). The persistence length is the length scale over which the cylinder remain s straight, and is therefore a measurement of how stiff the tube is. A smaller persistence length means the cylinder is more flexible, whereas a high persistence length is a stiffer tube. From the results of the persistence length measurements the persist ence length of the cell tube is mostly constant around 1 mm (Figure 3 15 ). The density fluctuation spacing is another material property that can be found by analyzing the diffraction pattern in the q z direction. Seen in Figure 3 14 the profile of the cent er peak is plotted, and then numerically fit to determine the width of the shoulder. The density fluctuation spacing gives the length between oscillations in the axial direction. By comparing the re sults of the time lapse to the results from the short wave length buckle from the microscopy data, the two numbers seem to correspond, and reinforce that what the scattering data is portraying is reasonable. Buckling Mechanics and the shape of Endothelial Cell Tubes Buckling in a Confined C ontainer The appearance of a n undulation with a changing wavelength, leads to the conclusion that as the cell cylinder is collapsing, it is also succumbing to the contraction yielded by the cells themselves. Since the cells are not just contracting radially but axially the colla gen is unable to withstand the force, and consequentially the cylinder buckles. The long wavelength buckle seen in the macroscopic images harken to the

PAGE 29

29 bucklin g of a cylinder in compression. We treat the fiber as an elastic tube in a confined container and explore the relations between the dominate short wavelength mode and material properties, and the tension generated by endothelial cells. Here we develop beam theory in a container, to demonstrate why higher modes of buckling are being seen. To do this, f irst the bending moment of a beam is found. ( 3 11 ) Where M is the bending moment, P is load applied and v is the deflection along the axial direc tion. By substituting Equation 3 11 into E quation 3 12 which describes beam bending, we find Equation 3 13 ( 3 12 ) ( 3 13 ) By assuming a solution of the form found in E quation 3 14 and making use of the end conditions, it is determined that therefore c 2 =0. B y substituting v(L) = 0 into Equation 3 14 the previous equation can be further simplified into Equation 3 15 ( 3 14 ) ( 3 15 ) c 1 =0 and therefore the equa tion can then be simplified to E quation 3 16 ( 3 16 ) Which is satisfied when

PAGE 30

30 ( 3 17 ) The P from E quation 3 17 is the load which the column maintains its deflected shape, and is termed the critical load ( 3 18 ) In the case of a tube, where T is thickness, and R is the radius of the tube which when plugged into E q uation 3 18, yields Equation 3 19 ( 3 19 ) Defining yields Equation 3 20 ( 3 20 ) To relate this to yield stress in a cylinder, the stress in the cylinder at t his point in compression can be found in Equation 3 21. ( 3 21 ) Substituting E quation 3 20 into E quation 3 2 1 and solving, the mode of buckling can be found for a cylinder where horizontal constraints prevents the fundamental mode from occurring ( 3 22 ) The buckling mode can then be tran slated into a wavelength by E quation 3 23 ( 3 23 ) The results of this derivation can be found in Figure 3 18 and matches well with the data gathered experimentally. As the radius of the cylinder decreases, the model

PAGE 31

31 suggests that the wavelength decreases over time, due to the change in bending stiffness, and the strengthening of material properties within the cylinder due to the compression. Buckling in an Elastic M edium To explain the shorter wavelength buckle, we looked into buckling within an elastic medium. The problem at hand could be explained by t he collagen and the cells buckling due to localized cell forces. To derive this we begin with the equilibrium equation found in Equation 3 24 where E is the elastic modulus, I is the moment of inertia, P is the force, is a restoring modulus of the surro unding matrix, and v is the deflection of the cylinder (3 24) By starting with a deflection that follows a sine curve as in Equation 3 25, and substituting it in Equation 3 24, we find the calculation for the critical force in Equation 3 26. (3 25) (3 26) The critical force in Equation 3 26 holds terms for both the bending force and the restoring force caused by the elastic medium. By assuming that the restoring force is not negligible we then estimate that the restoring force is about equal to the bending force, seen in Equation 3 27

PAGE 32

32 (3 27) Also, the period of the sine wave from Equation 3 25, leads to Equation 3 28 (3 28) Equation 3 27 is then rearranged to find Equation 3 29. (3 29) When Equation 3 28 is then substituted into Equation 3 29, we find a relationship between the wavelength of the buckling mode and the material properties, shown in Equation 3 30. (3 30 ) Where (3 31 ) In Equation 3 31 G is the elastic modulus of the support material, l is the characteristic length scale of a buckle, K is the bending rigidity of the cylinder and a is the cylinder radius. Collagen Properties under Collapsing Conditions For insight into the mechanics of the cell tube, the mechanical properties of the collagen mixture needed to be found. To do this, a strain sweep with constant frequency at 0.5 hz rh eology experiment was performed on three separate concentrations of collagen, 1.5 mg/ml, 3 mg/ml and 5 mg/ml. The range was deemed necessary in order to estimate the change in material properties as the cells collapsed the cylinder, which consequently incr eases the concentration of the collagen within the tube. This causes

PAGE 33

33 an increase in the modulus of the tube itself. The loss and storage modulus from these experiments can be found in Figure 3 16. From these measurements the average storage modulus was tak en from each of the concentrations and the increase in strength was fit with a power law to determine the approximate increase in strength that occurred from an increase in concentration (Figure 3 17). The modulus of the collagen was found to increase by t he 1.5 power as the concentration increased, as shown in Equation 3 32. (3 32) To determine the increase of concentration of the collagen within the collapsing cell cylinder, we assumed that the concentration increase is proportional to the ratio of the volumes, as shown in Equation 3 33. (3 33) Because of Equations 3 32 and 3 33, we assume that the modulus of the collagen increases as the ratio of the initial volume to the volume at a given time, raised to the 1.5 power, shown in Equation 3 34. (3 34) For the models used, the initial three times the storage modulus found from the rheology experiments. 32 By makin g use of the material properties found in the rheology experiments and the estimated cross sectional make up of the cylinder, the modeled wavelength was found (Figure 3 18). The modelled wavelength showed good correspondence with the measured results and f ollowed the same general trend as the results we found.

PAGE 34

34 Figure 3 1. Macro time lapse imaging shows the collapse of the collagen and cell tube through time. Higher mode buckling is apparent due to the constrained nature of the tube

PAGE 35

35 Figure 3 2. The aver age radius of the cell cylinder through time found by analyzing the cylinder edges. Figure 3 3 Time lapse microscopy images of the tube collapse in time illustrates small wavelength bumps

PAGE 36

36 Figure 3 4 Image enhancement shows the smaller wavelength undulations in the tube. Figure 3 5 By utilizing Be from microscopy images and gave insight into the cross section of the cell fiber. A tube containing 1 um sized particles was also imaged to see the effects of light scattering through a cylinder.

PAGE 37

37 Figure 3 6 Confocal images (in green) reveal the cross section of the cell tube in time At 12 hours, the tube has fully collapsed and the center appears to have a lesser cell density than the outer edge.

PAGE 38

38 Figure 3 7 By finding the backbone of the collagen cell fiber, mode analysis can be performed to determine the wavelengths of the buckling

PAGE 39

3 9 Figure 3 8 Mode analysis of time lapse shows one prominent wavelength during the beginning of the experiment that moves to two prominent wavelengths as time increases

PAGE 40

40 Figure 3 9 Wavelength evolution over time during cell tube collapse

PAGE 41

41 Figure 3 1 0 Scattering through a glass capillary tube Figure 3 1 1 Scattering pattern through the cell filled cylinder

PAGE 42

42 A B Figure 3 1 2 The diffraction pattern in the q r direction overlaid with the numerical fits. A) 380 minutes into the experiment, th e peaks are shown to be many sharp, defined peaks corresponding to a large cylinder. A s time progresses in B) the peaks coalesce and become wider, which correlates to a smaller cylinder radius

PAGE 43

43 Figure 3 1 3 Numerical fit of the first side peak of the scattering pattern yields the persistence length of the cell fiber

PAGE 44

44 Figure 3 1 4 The density fluctuation spacing of the cell tube is found by taking the width of the fitted should of the center peak of the scattering pattern

PAGE 45

45 Figure 3 1 5 The density fluctuation spacing seems to correlate with the small wavelength buckle previously noted from the microscope images.

PAGE 46

46 Figure 3 16. Rheology measurements of the storage and loss modulus of different concentrations of collagen determine the relationship between concentration increase and modulus increase.

PAGE 47

47 Figure 3 17. The storage modulus was found by rheology for differen t concentrations of collagen to determine the power law increase.

PAGE 48

48 Figure 3 18 Measured wavelengths and their corresponding models

PAGE 49

49 CHAPTER 4 DISCUSSION Endothelial cells within a collagen cylinder through time collapse the cylinder and form a tight thr ead of cells. This fiber of cells was observed in time lapse microscopy, dynamic small angle light scattering and through confocal imaging. Through this the radial distribution of cells in time is hypothesized to go from an heterogeneous mix of cells, that migrate outward forming a tube of cells, which then contracts and collapses due to the collective cell forces exerted. During the time lapse it was observed that the cell tube went through an evolution of buckling shapes in the macro scale as well as the micro scale. It is thought that c ontraction by the endothelial cells caused the tube to buckle. The buckling on the macro scale was seen to encompass a wavelength around 1 centimeter which then increased in wavelength. This la rge wavelength buckling agreed with an Euler buckling model of a tube in compression The deviance from this model could be due to the inclusion of collagen within the void of the tube. Further analysis would need to take place to verify this. The smaller wavelength buckle observ ed was measured around 500 um and resembled the results found when explained using theory for buckling within an elastic medium. The cells themselves could be creating small buckles within the material as t he cells contract, causing a ru ching effect along the length of the tube. The implications of these results could show that the development of a tube shape is susceptible to mechanical instabilities When this data was compared to the data collected from small angle light scattering, the persistence leng th was on the same length scales as the small wavelength buckle observed in time lapse. This correlates the results seen and helps to

PAGE 50

50 verify that dynamic small angle light scattering could be used to analyze collective cell mechanics in 3D.

PAGE 51

51 CHAPTER 5 FIN AL REMARKS AND FUTURE PLANS Inherently, blood vessels have to support many forces, stemming from blood pressure, as well as external forces on the body, and from movement itself. Because of this, many papers model the bucklin g of blood vessels due to the blood pressure causing buckling after the degradation of the external matrix. However, this research has found that capillaries and blood vessels can buckle without the flow of blood, the formation of a lumen and without the s upporting smooth muscle cell, giving insight to the forces that endothelial cells themselves can exert in a blood vessel. Future work should be done in order to verify the radial distribution of the cells within the cylinder. This would allow for serious consideration of the implications to angiogenesis as well as further elucidate the collective behavior of endothelial cells in 3D. Also of interest would be the creation of cell tubes within other mediums as well as other configurations in order to underst and better the way cells respond to tissue scaffolding. Also of interest is the effect of drugging the endothelial cells in order to increase or decrease the amount of contractility of the cells. Initial experiments with endothelial cells drugged with bleb bistatin, a drug that reduces the contractility of the cells, and thrombin, a drug that increases the contractility, has shown promise in changing the final wavelength of the cylinder after collapse (Figure 5 1) A smaller force exerted by the cells is exp ected to increase the wavelength of the buckle because of the reduction of force, whereas the highly contractile cells were expected to decrease the buckling wavelength.

PAGE 52

52 Figure 5 1. The contractility of the cells changes with drugging, affecting the wav elength at which the cylinder of cells buckle.

PAGE 53

53 LIST OF REFERENCES 1. G. Bao and S. Suresh, Nat Mater 2003, 2 715 725. 2. M. Dembo and Y. L. Wang, Biophysical Journal 1999, 76 2307 2316. 3. D. E. Discher, P. Janmey and Y. l. Wang, Science 2005, 310 1139 1143. 4. S. Lehoux and A. Tedgui, Journal of Biomechanics 2003, 36 631 643. 5. W. Risau and I. Flamme, Annual Review of Cell and Developmental Biology 1995, 11 73 91. 6. L. Wolpert, R. Beddington, J. Brockes, T. Jessell, P. Lawrence and E. Meyerowitz, Current Biology Ltd London 7. P. H. Burri, R. Hlushchuk and V. Djonov, Developmental Dynamics 2004, 231 474 488. 8. W. Risau, Nature 1997, 386 671 674. 9. A. L. Sieminski, R. P. Hebbel and K. J. Gooch, Experimental Cell Research 2004, 297 574 584. 10. M. D. Brown and O. Hudlicka, Angiogenesis 2003, 6 1 14. 11. G. Kasper, N. Dankert, J. Tuischer, M. Hoeft, T. Gaber, J. D. Glaeser, D. Zander, M. Tschirschmann, M. Thompson, G. Matziolis and G. N. Duda, STEM CELLS 2007 25 903 910. 12. G. Schep, D. Kaandorp, M. Bender, H. Weerdenburg, S. van Engeland and P. Wijn, Physiological measurement 2001, 22 475. 13. J. Weibel and W. S. Fields, Neurology 1965, 15 462 468. 14. A. M. Waxman, Microvascular research 1981, 22 32 42. 15. H. C. Han, Journal of Vascular Research 2012, 49 185 197. 16. H. C. Han, J. W. Chesnutt, J. Garcia, Q. Liu and Q. Wen, Ann Biomed Eng 2013, 41 1399 1410. 17. B. M. Learoyd and M. G. Taylor, Circulation Research 1966, 18 278 292. 18. J Sugawara, K. Hayashi, T. Yokoi and H. Tanaka, JACC: Cardiovascular Imaging 2008, 1 739 748.

PAGE 54

54 19. S. Hughes, H. Yang and T. Chan Ling, Investigative Ophthalmology & Visual Science 2000, 41 1217 1228. 20. H. C. Han, Journal of Vascular Research 2012, 49 185 197. 21. F. Hafeez, R. L. Levine and D. A. Dulli, Journal of Stroke and Cerebrovascular Diseases 1999, 8 217 223. 22. N. Ferrara, H. P. Gerber and J. LeCouter, Nat Med 2003, 9 669 676. 23. H. C. Han, Journal of Biomechanics 2009, 42 2797 2801. 24. J. C. Wang and J. S. Lin, Biomech Model Mechanobiol 2007, 6 361 371. 25. C. A. Reinhart King, M. Dembo and D. A. Hammer, Langmuir 2003, 19 1573 1579. 26. P. Friedl and D. Gilmour, Nat Rev Mol Cell Biol 2009, 10 445 457. 27. C. S. Chen, M. Mrksich, S. Huang, G. M. Whitesides and D. E. Ingber, Science 1997, 276 1425 1428. 28. R. G. Wells, Hepatology 2008, 47 1394 1400. 29. O. W. Petersen, L. Ronnov Jessen, A. R. Howlett and M. J. Bissell, Proceedings of the National Acade my of Sciences of the United States of America 1992, 89 9064 9068. 30. S. E. Bell, A. Mavila, R. Salazar, K. J. Bayless, S. Kanagala, S. A. Maxwell and G. E. Davis, Journal of cell science 2001, 114 2755 2773. 31. R. B. Vernon and E. H. Sage, Microva scular research 1999, 57 118 133. 32. C. P. Brangwynne, F. C. MacKintosh, S. Kumar, N. A. Geisse, J. Talbot, L. Mahadevan, K. K. Parker, D. E. Ingber and D. A. Weitz, The Journal of Cell Biology 2006, 173 733 741.

PAGE 55

55 BIOGRAPHICAL SKETCH Jolie Br eaux received her Bachelor of Science degree in mechanical engineering from the Florida State University in spring of 2011. While at FSU, she worked as an undergraduate research assistant at the High Performance Materials Institute under the tutelage of Dr Okenwa Okoli and Dr. Tarik Dickens, and completed an undergraduate honors in the m ajor thesis titled After graduatin g, Jolie matriculated into the Department of M echa nical and Aerospace E ngineering at the University of Florida, and conducted research in the Soft Matter and Bio Laboratory under the direction of Dr. Thomas Angelini. Jolie graduated with her m continue her studies to obt ain her PhD in engineering.