Biaxial mechanical characterization and microstructure-driven modeling of elastic pulmonary artery walls of large mammals under hypertensive conditions by Philip Hsien-Lan Kao B.S., Purdue University, 2005 M.S., University of Colorado at Boulder, 2008 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirement for the degree of Doctor of Philosophy Department of Mechanical Engineering 2010 UMI Number: 3433301 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. UMI 3433301 Copyright 2011 by ProQuest LLC. All rights reserved. This edition of the work is protected against unauthorized copying under Title 17, United States Code. ProQuest LLC 789 East Eisenhower Parkway P.O. Box 1346 Ann Arbor, MI 48106-1346 This thesis entitled: Biaxial mechanical characterization and microstructure-driven modeling of elastic pulmonary artery walls of large mammals under hypertensive conditions written by Philip Hsien-Lan Kao has been approved for the Department of Mechanical Engineering H. Jerry Qi Kendall S. Hunter Date The final copy of this thesis has been examined by the signatories, and we Find that both the content and the form meet acceptable presentation standards Of scholarly work in the above mentioned discipline. Kao, Philip Hsien-Lan (Ph.D., Mechanical Engineering) Biaxial mechanical characterization and microstructure-driven modeling of elastic pulmonary artery walls of large mammals under hypertensive conditions Thesis directed by Professor H. Jerry Qi and Professor Robin Shandas Pulmonary Hypertension (PH) is a disease of the pulmonary vasculature which causes right heart failure. It is known that PH causes significant remodeling of the pulmonary arterial vasculature, but the effects of this remodeling are not well-understood. In addition, there is a dearth of research in large mammals for PH. Modeling of the arteries is also important in the simulation of deformation due to blood flow. Current models either do not reflect the microstructure, or are too complex for clinical use. This work presents mechanical characterization and analysis of the artery wall, in addition to a constitutive model driven by the microstructure of the artery. In this work, mechanical characterization of the artery wall is performed via multiaxial deformation using a custom-fabricated planar biaxial tester. This test device provides higher fidelity than the standard uniaxial tests. Using the data gathered from the biaxial tester, trends in aspects of the mechanical behavior due to PH can be elucidated. Specifically, in this work, the anisotropy of the elastin protein network has been quantified, with the circumferential direction being 1.4x stiffer than the longitudinal direction. In addition to this new finding, PH has been shown to slightly decrease the anisotropy of the pulmonary artery trunk. A new microstructurally-based constitutive model for the artery wall was developed to reflect this iii finding. This model uses decoupled anisotropy for the elastin and collagen networks, reflecting the true behavior of the artery wall. The model uses a sinusoidal elastic beam to model the collagen fibers, reflecting the microstructure. This microstructural basis is then verified through histology and correlation of material parameters to histological images. Using information from this data, prospective future analysis of mechanical behavior will be proposed. iv Acknowledgements I would like to thank Prof. H. Jerry Qi for his patience and guidance throughout my graduate school experience. I would like to thank the committee members, Prof. H. Jerry Qi, Prof. Robin Shandas, Prof. Kendall Hunter, Prof. Mark Rentschler and Prof. Kurt Stenmark for their participation in my doctoral process. I would like to thank Dr. Steven R. Lammers for his guidance and assistance for this work, Dr. Kevin N Long for his enlightening and invigorating discussions and Dr. Kristofer Westbrook for his perspectives and guidance. I would like to thank Al Zhang and Charles Procknow for their help in tracing the histological images. Funding was provided by NIH grants T32-HL072738, SCCOR-HL081506, K24-HL084923. v Table of Contents List of Figures ............................................................................................................................. vii Index of Tables ............................................................................................................................ xi 1 Introduction ............................................................................................................................... 1 1.1 Artery Mechanics ..................................................................................................... 1 1.2 Scope of Research .................................................................................................... 2 1.3 Artery Mechanics ..................................................................................................... 3 1.4 Pulmonary Arterial Hypertension ............................................................................ 6 1.5 Materials Testing ..................................................................................................... 7 1.6 Modeling .................................................................................................................. 9 2 The Crimped Fiber Model ..................................................................................................... 13 2.1 Introduction ............................................................................................................ 13 2.2 Mechanics Preliminaries ........................................................................................ 15 2.3 Modeling An Individual Collagen Fiber Bundle ................................................... 18 2.4 Orthotropic Crimped Fiber Model ......................................................................... 24 2.5 The Orthotropy Of Arterial Elastin ........................................................................ 27 2.6 Complete Model Incorporating Volume Fractions ................................................ 28 2.7 Results .................................................................................................................... 29 2.8 Discussion .............................................................................................................. 34 2.9 Conclusion ............................................................................................................. 40 3 Experimental Methods and Materials Testing ..................................................................... 41 3.1 Uniaxial Tensile Tests............................................................................................ 41 3.2 Planar Biaxial Tensile Tester ................................................................................. 47 3.3 Uniaxial Data Fitting.............................................................................................. 56 3.4 Conclusions ............................................................................................................ 61 4 The Anisotropy of Arterial Elastin ........................................................................................ 63 4.1 Introduction ............................................................................................................ 63 4.2 Materials and Methods ........................................................................................... 66 4.3 Results .................................................................................................................... 69 4.4 Discussion .............................................................................................................. 74 5 Nonlinear Anisotropic Passive Stress-Strain Behaviors of Artery Tissues: Experiments and a Structure Based Constitutive Model ........................................................................ 77 5.1 Materials and Experiments .................................................................................... 82 5.2 Constitutive Model................................................................................................. 88 5.3 Results .................................................................................................................... 95 5.4 Discussion ............................................................................................................ 105 5.5 Conclusions .......................................................................................................... 107 6 References .............................................................................................................................. 109 vi List of Figures Figure 2-1: Schematic of crimped fiber model. The solid line is the undeformed fiber configuration and the dotted line is the deformed fiber configuration. ................................. 18 Figure 2-2: The behavior of a single fiber is plotted. The nominal fiber stress, P is normalized by R l the Young’s modulus to give the reduced fiber stress. A) As the bending stiffness, via 0 , is increased, the fiber behavior goes from a sharp engagement to a broad engagement. The q q shape of the fiber is held constant at 0=54°. B) As the shape, via 0, is increased, the fiber behavior goes from a sharp transition with some fully-developed stiffness to a broad transition with lower fully-developed stiffness. The stretch at which the transition occurs also q R l increases with increasing 0. The radius of gyration ratio is kept constant where 0 =0.05. ................................................................................................................................................ 22 Figure 2-3: The normalized force-extension behavior for three cases of fibers is shown. The fiber force, f is normalized by the Young’s modulus, E, to give the normalized fiber force. The solid line shows a typical force-extension behavior of a crimped fiber. The dashed line is shown with very small bending stiffness. The dotted line shows a crimped fiber where the bending stiffness is much greater than the extensional stiffness. In the limit of high bending stiffness to extensional stiffness, the fiber behaves linearly. ................................................. 24 Figure 2-4: The visualized ellipsoidal structure tensor. This is generated by observing how the structure tensor transforms a unit vector. A longer dimension indicates a higher concentration of fibers in that direction. Note here that a and g are not aligned with the global coordinate 0 0 system, but in the model, are fixed to the circumferential and longitudinal directions respectively. ........................................................................................................................... 25 Figure 2-5: The data from four sets of uniaxial tests and their corresponding model fits. Squares and triangles denote circumferential and axial data respectively, while Solid and dashed lines denote circumferential and axial data fits. Here it is seen that the model’s versatility in different situations of material behaviors (A-D). ................................................................... 31 Figure 2-6: The data and corresponding fits for two selected sets of data. The data is shown with markers; triangles denote axial data while squares denote circumferential data. The Bischoff- Arruda model fits are presented with dotted lines, while the model presented here uses solid lines for the circumferential data and dashed lines for the axial data. In (A) the Bischoff- Arruda model can follow the circumferential data well, but sacrifices fidelity in the longitudinal direction. In (B) the Bischoff-Arruda model follows the longitudinal data, but circumferential data is not fit well. ........................................................................................ 33 Figure 2-7: A) A parametric study varying γ, with κ = 0.8. It is seen that when κ = γ, (shown in circles), the behavior is transversely isotropic; the uniaxial stress-stretch curves lay on top of one another. As γ is increased in relation to κ, shown in squares (γ = 0.95) and triangles (γ = 0.9), the degree of anisotropy is increased. The circumferential directions are in solid lines and the longitudinal directions are in dotted lines. The parameters held constant are q R l KA=8x10-4, E=10GPa, 0=45°, 0 =0.1. B) Results for the crimped fiber model only, q q showing the effect of changing crimped fiber parameter 0. As 0 increases, it pushes the vii engagement strain of the collagen further out. It also decreases the stiffness, as the contour R l length is increased. The parameters held constant are KA=8x10-4, E=10GPa, 0 =0.1, R l κ=0.90 and γ=0.95. C) The effect of changing crimped fiber parameter 0 . As the radius of gyration is changed, it causes the transition to broaden and become more gradual. The q parameters held constant are KA=8x10-4, E=10GPa, 0=36°, κ=0.90 and γ=0.95. .............. 36 q Figure 2-8: The engagement stretch as a function of (a) R/l and (b) 0. The parameters used to 0 m m q calculate these data are µ = a= g=5kPa, KA=12x10-4, E=10GPa, 0=45°, κ=0.90 and γ=0.95. The dotted lines denote typical parameter values. .................................................... 39 Figure 3-1: Schematic of strips taken from excised pulmonary arteries. .................................... 42 Figure 3-2: Uniaxial test setup showing the Insight 2 with ancillary environmental chamber and temperature controller. ........................................................................................................... 43 Figure 3-3: The carbon fiber composite extension rod. The mounting points are hollow stainless steel. ....................................................................................................................................... 44 Figure 3-4: Custom-machined tissue grips. The prominent grooves in the moveable side hold the tissue firmly. .......................................................................................................................... 45 Figure 3-5: The force sensing assembly is seen here, showing the adjustable leverage load cell mounting, the force balancing pulleys, and the load cell ....................................................... 48 Figure 3-6: Hooks used to transmit the load to the specimen. These hooks are made of 304 stainless steel wire, hand-bent and sharpened. The string used to tie them together is nylon, and the knots are sealed with cyanoacrylate adhesive. .......................................................... 49 Figure 3-7: The miniature elastin tissue clamps. These are made of 304 stainless steel. ............ 50 Figure 3-8: The pulleys and pivots which transmit the tensile forces to the specimen. .............. 50 Figure 3-9: A sample used to demonstrate the strain markers and method of attaching hooks. This is a typical size for a sample, at around 15mm on a side. .............................................. 51 Figure 3-10: The loading paths of the seven biaxial tests. The circumferential-axial first Piola- Kircchof stress plane is shown, and the loading paths are shown in dotted red lines. The bounds of the stress are shown in shaded blue. ..................................................................... 53 Figure 3-11: A set of uniaxial test data taken on the biaxial tensile tester. Note that the opposing stretches are recorded, but not plotted here for clarity. The data is from calf lot number 184, left pulmonary artery.............................................................................................................. 55 Figure 3-12: The full set of biaxial data from one tissue. Data from the same tests are in the same color, with dotted lines representing the circumferential behavior, while the solid lines represent the longitudinal behavior. The data is from calf lot number 184, left pulmonary artery. ..................................................................................................................................... 56 Figure 3-13: The isotropic shear modulus of the uniaxial data fits. ............................................ 58 Figure 3-14: The circumferential shear modulus of the uniaxial data fits. .................................. 59 Figure 3-15: The anisotropic shear modulus ratio of the uniaxial data fits. ................................ 59 Figure 3-16: The fiber density, KA, of the uniaxial data fits. ...................................................... 60 Figure 3-17: The anisotropy of the ellipsoid for the uniaxial data fits. ....................................... 60 Figure 3-18: The normalized radius of gyration for the crimped fibers of the uniaxial data fits. 61 Figure 3-19: The crimped fiber shape parameter, q , of the uniaxial data fits. .......................... 61 0 Figure 4-1: Fresh and digested test data from one sample is shown. Tension is shown rather than stress in order to ignore the thickness change due to digestion. The elastin data is very close viii to the fresh data, showing minimal change in low stretch behavior due to the digestion method.................................................................................................................................... 69 Figure 4-2: A representative plot of one set of equibiaxial test data. A) Equibiaxial stress test data from one digested elastin tissue shows large anisotropy. B) Circumferential data from the same tissue is shown with the tangent and secant methods to emphasize the difference between them. ........................................................................................................................ 71 Figure 4-3: Histograms showing the distribution of the anisotropy ratios as calculated by A) the tangent method and B) the secant method. The data show a clear right-tailed distribution with a mode greater than unity. B) Compared with the tangent method, the data is concentrated more toward unity due to the inclusion of the toe region. ..................................................... 72 Figure 4-4: Histograms of the pooled (PH and Control) data are shown. The MPA group (A) has a high anisotropy ratio, while the RPA group (B) has a low anisotropy ratio. The LPA group (C) has a large range. ............................................................................................................. 73 Figure 5-1: A set of biaxial experiments on the artery tissue specimen. The dotted lines are circumferential data and the solid lines are the longitudinal data. The loading ratios are denoted by symbols. Here, it is seen that the material shows pronounced anisotropy. From this plot, it can be seen that the addition of transverse stretch affects the behavior of the material. From the 100:0 to the 100:25 loading ratios, it is seen that adding longitudinal stress will cause a small, but noticeable change in the high-stretch behavior. ................................ 84 Figure 5-2: A) A histological image for tracing. The red channel is the two-photon emission of collagen, while the green channel is the autofluorescence of elastin. The collagen is seen as wavy fiber bundles which have distributed orientation. B) The traced lines for collagen fibers. ..................................................................................................................................... 85 Figure 5-3: (a) A single representative fiber tracing showing the orientation angle, θ, with respect to the circumferential direction. The projections of the unit length in the circumferential and longitudinal directions are shown as cos θ and sin θ respectively. (b) An idealized fiber with sinusoidal shape. The arc length and end-to-end lengths are both shown. The included angle of the sinusoid determines the tortuosity. .............................................................................. 86 Figure 5-4: The fiber stretch as a function of the applied stretches. The shape of the contours is circular, and the coupling of the stretch cannot be tuned. ..................................................... 93 Figure 5-5: A histogram of the tortuosity from histological images. The x-axis is logarithmically scaled for clarity. The distribution shows most fibers having a tortuosity close to 1, however the distribution is very long-tailed. ........................................................................................ 97 Figure 5-6: A rose histogram of the orientation data from histology. The distribution shows a strong preference for the circumferential direction, with a slight dip near the longitudinal direction. ................................................................................................................................ 98 Figure 5-7: The biaxial test data and corresponding fits. The model parameters of the sinusoid included angle and fiber distribution anisotropy were fixed from histology, while the other parameters were fit parameters. ............................................................................................. 99 Figure 5-8: Contour plots of the fiber stretch in the stretch space for a) c =-2 and b) c =-4. The 1 1 stress coupling decreases with decreasing c . ...................................................................... 101 1 Figure 5-9: With decreasing stretch coupling parameter c , a smaller effective anisotropy ratio is 1 observed. .............................................................................................................................. 102 Figure 5-10: The effect of c on the stretch coupling is shown here. Different values of c are 1 1 shown by symbols, and loading ratios are shown by line styles. For each value of c , two 1 stretch ratios are plotted, an equibiaxial case (dotted lines) and a near-uniaxial case with the ix