Gou, Kun

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Virginia Commonwealth University VCU Scholars Compass Biology and Medicine Through Mathematics Conference 2022 May 20th, 11:00 AM - 11:30 AM Mechanical Parameters Fitting for Layer-reduced Umbilical Arteries Used for Grafting Kun Gou Texas A&M University-San Antonio, kgou@tamusa.edu Follow this and additional works at: https://scholarscompass.vcu.edu/bamm  Part of the Life Sciences Commons, Medicine and Health Sciences Commons, and the Physical Sciences and Mathematics Commons https://scholarscompass.vcu.edu/bamm/2022/fri/10 This Event is brought to you for free and open access by the Dept. of Mathematics and Applied Mathematics at VCU Scholars Compass. It has been accepted for inclusion in Biology and Medicine Through Mathematics Conference by an authorized administrator of VCU Scholars Compass. For more information, please contact libcompass@vcu.edu. Mechanical Parameters Fitting for Layer-reduced UmbilicalArteries Used for GraftingRene Alvarado, Kun Gou∗AbstractThe umbilical artery (UA) functions to carry deoxygenated blood from the fetus to the placentaduring pregnancy. It is discovered that this artery can be used as an efficient by-pass graft to repairthe occluded or narrowed coronary artery for other patients to restore normal blood flow [1, 2]. Theouter layer of the UA is usually removed to get rid of the immunogenic cellular components in theUA to minimize the immune cells’ biological activities on the extracellular matrix for the UA’s betteradaptation to the patient’s coronary artery [3]. The stiffness of the layer-reduced UA is different fromthe original UA, and is difficult to measure directly by lab experiments due to its cylindrical shape.However, understanding stiffness of the bypass graft is important to predict how it can successfullyadapt to the patient’s coronary artery for normal functionality [4]. This study aims at establishingproper mathematical models and employing numerical optimization techniques to obtain the stiffnessparameters of the layer-reduced UA based on lab measurements. The artery wall is considered tobe of hyperelastic soft tissue [5] incorporated with two families of fibers. An objective function isestablished employing the difference between the theoretical and lab results. Minimization of theobjective function then provides the shear modulus of the isotropic matrix and the fiber stiffness andorientation parameters.1 Constitutive modelingThe reference configuration of the UA is unloaded with traction-free inner and outer boundary conditions.We also ignore the residual stress in this unloaded configuration due to experimental observation of noobvious opening angle when cutting the UA radially. Under the loaded, deformed configuration, the UAis subject to an inner pressure P in the inflating experiments (Fig. 1). The UA is nearly cylindrical. Wethus set up the model under the cylindrical coordinate system [6]. The coordinates in the radial, angular,and axial directions are expressed by (R,Θ, Z) in the reference configuration, and the correspondingcoordinates in the deformed configuration are expressed by the related lower case letters (r, θ, z). Thebasis vectors for the cylindrical coordinates in the reference and deformed configurations are, respectively,“eR, eΘ, eZ”, and “er, eθ, ez”.The mapping from the reference configuration to the deformed configuration is expressed as r =r(R), θ = Θ, z = λzZ, where r(R) is the radial function, and λz is the axial stretch ratio. The deformationgradient F is thus F = r′(R)er⊗eR+ rReθ⊗eΘ +λzez⊗eZ . The right Cauchy-Green tensor C satisfyingC = FTF, generating the first invariant I1 = trC. We further assume that the volume of the arterialsegment is preserved during the mapping from the reference configuration to the deformed configurationsatisfying detF = 1.Two symmetric families of fibers are oriented in the wall. The unit direction vectors for these fiberfamilies are N(1) = sinϕeΘ + cosϕeZ , and N(2) = −sinϕeΘ + cosϕeZ , where ϕ is the acute angle betweenthe fiber unit direction vector and the axial axis. These oriented fibers contribute strain energy to theUA. We take the fiber strain-energy density functions [7] to beWf =k12k22∑n=1[ek2E2n − 1]with En = κI1 + (1− 3κ)I(n)4 − 1, (1)∗Authors’ address: Texas A&M University-San Antonio, Department of Mathematical, Physical, and Engineering Sci-ences, One University Way, San Antonio, Texas 78224. Rene Alvarado was a junior undergraduate student in the mathe-matics major at the time of article publication. Corresponding author: Dr. Kun Gou; email: kgou@tamusa.edu.1where k1, k2, and κ in (1) are parameters for fibers in the wall. The pseudo-invariant I(n)4 (n =1, 2) for thetwo families of fibers is defined as I(n)4 = N(n) ·CN(n) [8]. The wall consists of elastic groundmatrix andthe two families of fibers. We take the elastin groundmatrix contributed strain-energy density function tobe in the form of a neo-Hookean model We =µ2 (I1−3), where µ is the shear modulus of the groundmatrix.The total strain-energy density function is W = We +Wf .For the incompressible hyperelastic material, the Cauchy stress tensor T is derived from the strainenergy density function W via the formula T = −pI+2F∂W∂C FT, where p is a constraint parameter needingdetermination, and I is the identity tensor. The Cauchy stress components in the radial, circumferential,and axial directions are expressed by Trr, Tθθ, and Tzz, respectively. The Cauchy stress tensor T in astatic state with gravity ignored for the light UA tissue satisfies the equation div T = 0, where “div”represents divergence of the related quantity. The inner boundary (R = Rin) of the UA is subjected tothe pressure in the vector form Per with P > 0, by which we obtain Trr|R=Rin = −P . Over the outerboundary (R = Rout), the radial Cauchy stress is 0, and we have Trr|R=Rout = 0.2 Parameters fittingThe inflating experimental design is illustrated in Fig. 1. The left end is subject to a pressure P . Wedenote the inner boundary radius of the inflated UA to be rin, and the outer boundary radius of theinflated UA to be rout. The right end is closed with a total force L imposed on the UA. The forcegenerated by Tzz to the left end of the UA, the force generated by the left pressure to the UA, and theforce L on the right end satisfy the force equilibrium equation2π∫ routrinTzzrdr − Pπr2in − L = 0. (2)Figure 1: Simple illustration of the inflated arterialwall. The wall is closed on the right end with a totalforce L imposed over the wall. The left end is subjectto the pressure P . The axial Cauchy stress on the leftend of the inflated wall is Tzz . The inflated inner radiusand outer radius of the wall are denoted by rin and rout,respectively.The experiments recorded values of pressure P , theinner radius of the pressurized UA rin, the outer radiusof the pressurized UA rout, the axial Cauchy stress valueTzz, the circumferential Cauchy stress value Tθθ, andthe force at the closed end L. We define the objectivefunction for minimization to bef =m∑k=1(ωL|Lkthe − Lklab|+ ωP |P kthe − P klab|), (3)where m is the number of equilibrium experiments per-formed, Lkthe and Lklab are the force values for L fromthe theoretical models and the lab experiments, respec-tively, and P kthe and Pklab are the pressure values Pfrom the theoretical models and the experiments, re-spectively. The quantities ωL(= 1) and ωP (= 200) areweights for the corresponding terms. The measuredradii of the inner and outer boundaries of the UA are used as input in the model. We use the opti-mization toolboxes of Matlab for the layer-reduced UA to obtain parameters µ, ϕ, κ, κ1, and κ2.2.1 Results and discussionsWe study one sample of the UA. Over the traction-free configuration (P = 0), the layer-reduced UAshows measure of rin = 1.02 mm and rout = 1.14 mm. Minimization of the objective function (3)generates the parameter values: µ = 36.24 kPa, κ1 = 142.41 kPa, κ2 = 81.82, κ = 0.30, ϕ = 1.45radians. These parameter results are subjected to change for different UA samples, different objectivefunctions, and even different initial input in the optimization [9]. In this short conference proceedingpaper, we skip the other fitting results. Average of all these different values for each parameter cangenerate a more accurate value for that parameter. Statistical methods can be used to find a confidenceinterval for each parameter under fitting with a specified confidence level. This confidence interval can2provide a fundamental understanding how broadly each parameter may vary. Determination of the layer-reduced UA stiffness critically facilitates prediction of compatibility and efficiency of the bypass graft asa substitute coronary artery section for regular blood transport [10]. Our study applies mathematicalmodels to fit the stiffness parameters based on experimental data, and resolves the difficulty of obtainingthe mechanical parameters experimentally in the lab.References[1] Gui L, Mutu A, Chan SA, Breuer CK, Niklason LE. Development of decellularized human umbilicalarteries as small-diameter vascular grafts. Tissue Eng Part A. 2009;15:2665–2676.[2] Victor RE, Brenda MG, Alejandro QG, Cynthia-Guadalupe RH, David FV, Oscar GZ, et al. Humanumbilical vessels: Choosing the optimal decellularization method. Asaio J. 2018;64:575–580.[3] Youngstrom DW, Barrett JG, Jose RR, Kaplan DL. Functional characterization of detergent-decellularized equine tendon extracellular matrix for tissue engineering applications. PloS One.2013;8:e64151.[4] Owens CD, Wake N, Jacot JG, Gerhard-Herman M, Gaccione P, Belkin M, et al. Early biomechanicalchanges in lower extremity vein grafts-distinct temporal phases of remodeling and wall stiffness. JVasc Surg. 2006;44:740–746.[5] Gou K, Baek S, Lutnesky MMF, Han HC. Growth-profile configuration for specific deformations oftubular organs: A study of growth-induced thinning and dilation of the human cervix. PLoS ONE.2021;16:e0255895.[6] Gou K, Muddamallappa M. An analytic study on nonlinear radius change for hyperelastic tubularorgans under volume expansion. Acta Mech. 2020;231:1503–1517.[7] Gasser TC, Ogden RW, Holzapfel GA. Hyperelastic modelling of arterial layers with distributedcollagen fibre orientations. J R Soc Interface. 2006;3:15–35.[8] Gou K, Fok PW, Fu Y. Nonlinear tubular organ modeling and analysis for tracheal angioedema byswelling-morphoelasticity. J Eng Math. 2018;12:95–117.[9] Hu JJ, Baek S, Humphrey JD. Stress–strain behavior of the passive basilar artery in normotensionand hypertension. J Biomech. 2007;40:2559–2563.[10] Hu JJ. Constitutive modeling of an electrospun tubular scaffold used for vascular tissue engineering.Biomech Model Mechanobiol. 2015;14:897–913.3

Mechanical Parameters Fitting for Layer-reduced Umbilical Arteries Used for Grafting

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Mechanical Parameters Fitting for Layer-reduced Umbilical Arteries Used for Grafting

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