Ansys Hyperelastic Tissue Model

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An in silico-based review on anisotropic hyperelastic

Constitutive Models For Soft Biological Tissues

H¨usn¨u Dal∗1, Alp Ka˘gan A¸can1, Ciara Durcan2 and Mokarram Hossain2 Dumlupınar Bulvarı 1, 06800 C¸ankaya, Ankara , T¨urkiye. 2Zienkiewicz Centre for Computational Engineering, College of Engineering,

Abstract

We review nine invariant and dispersion-type anisotropic hyperelastic constitutive models for soft biological tissues based on their fitting performance to experimental data from three different human tissues. For this, we used a hybrid multi-objective optimization procedure.

A genetic algorithm is devised to generate the initial guesses followed by a gradient-based search algorithm. The constitutive models are then fit to a set of uniaxial and biaxial tension experiments conducted on tissues with differing fiber orientations. For the in silico inves- tigation, experiments conducted on aneurysmatic abdominal aorta, linea alba, and rectus sheath tissues are utilized. Accordingly, the models are ranked with respect to an objec- tive normalized quality of fit metric. Finally, a detailed discussion is carried out on the fitting performance of each model. This work provides a valuable quantitative comparison of various anisotropic hyperelastic models, the findings of which can aid those modeling the behavior of soft tissues in selecting the best constitutive model for their particular application.

Keywords: collagen fiber distribution, anisotropy, constitutive modeling, equibiaxial tension

1 Introduction

The physiological functions of many soft biological tissues are related to their material composition and mechanical properties. Examples of these functions include blood flow through the heart and blood vessels [12, 59, 68], peristalsis of the gastrointestinal (GI) tract [4, 52], and even the transmission of sound within the ear cavity . The mechanical properties of tissues can also be a predictor of their pathophysiology, such as the stiffness of arteries being a successful indicator of cardiovascular disease, or the elastic and viscoelastic parameters of skin being linked to the occurrence of autoimmune disorders [49, 51]. To successfully model the physiological functions of soft tissues and to investigate any pathophysiology, the characteristics of the material must first be established. Constitutive modeling can be used to analytically capture the mechanical behavior of these biological tissues. Such an ansatz forms the basis for developing computational models which consider the geometry of the tissue or organ to study how shape contributes to its function. Baseline hyperelasticity is a precursor in biomechanics to the modeling of more advanced phenomena such as viscoelasticity , electroelasticity and electro-viscoelasticity [9, 10], to

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mention a few. In addition to this, computational modeling can be used to study fluid-structure interaction scenarios, such as urine traveling down the urinary tract , or air filling up the alveoli of the lungs . However, before this is possible, the behavior of the solid structures must be determined and successfully simulated. A variety of applications exist for the computational models mentioned, including within the medical device industry, where the mathematical models can be used to aid in the design of devices that interact with the human body, such as stents and endoscopy devices [25, 81]. The use of the models in a finite element implementation can give greater insight into the devices’ mechanisms, highlight any potential risks for the patient, and conserve a range of resources such as time and manufacturing materials. Further to this, computational models of soft tissues can be used for surgical simulations; the training technique which allows surgeons to practice a procedure an infinite number of times without risking the health or lives of patients [1, 32, 61].

It has previously been well established that soft tissues respond elastically to a given force and past a certain point, i.e. under large strains, behave non-linearly [5, 26, 88]. For this, and when the time-dependent properties of the tissue are not considered, hyperelastic modeling, similar to that for polymers, is required [46, 79]. For polymers, isotropic, hyperelastic models are com- monly used as these substances often have a homogeneous material composition, unless specifically designed otherwise. Biological tissues, however, are generally observed through experimentation to be anisotropic. For fibrous tissues, this is often associated with different densities and/or orienta- tions of the tissue’s main constituent fibers, resulting in a differing response of the tissue depending on its loading direction. The anisotropy of biological tissues, hard or soft, can normally be linked with the physiological function of the organ when considering the mechanical characteristics of each different fiber. There are many types of tests to characterize the anisotropic hyperelastic behavior of biological tissues, and amongst the most common include uniaxial tension tests in two or more directions , biaxial tension tests which simultaneously load in two directions [29, 47], pure shear tension tests in two directions , inflation-extension tests for tubular tissues and compression tests in different directions . Experiments conducted on soft biological tissues can be difficult due to their delicate nature in addition to practical limitations including the need for highly sen- sitive data acquisition equipment to capture their comparatively low internal forces. This makes the experimental studies conducted on soft tissues extremely valuable in the field of biomechanics.

Such studies conducted using animal soft tissues are more prevalent than with human tissue due to the former being more readily available and having fewer associated ethical constraints. To date, some non-exhaustive examples of the experimental data available on animal soft tissues includes the porcine aorta , rodent and porcine brain , rodent vagina , porcine liver , ovine oesophagi , swine tongue , equine urethra , porcine skeletal muscle , rabbit skin and porcine stomach .

Animal tissue may provide a good representation of how human tissue responds to certain loads or strains, but the data from such experiments cannot be used to accurately model human tissues, especially for use within medical applications. Despite being less prevalent than animal studies, there are also a wide variety of experimental studies on human soft tissues, with investigations being conducted both in vivo and ex vivo. In this review, we will focus purely on human data from ex vivo experiments as these allow for large deformations that clearly display the non-linear elastic behavior of the tissues. Experimental data for healthy human soft tissues can be found for the brain , spleen , skin , aorta [24, 84], bile duct , oesophagus [21, 22], GI tract , liver , cervical tissue , while also for diseased tissue such as aneurysmatic aortas and diabetic foot plantar tissues . As mentioned previously, and as can be witnessed from the time- independent stress-strain relationships of these experimental studies, at large strains soft biological tissues exhibit a non-linear hyperelastic material response, therefore their behavior cannot be simply captured by two constants of linear elasticity. A strain energy density function (SEDF) must be used to model the hyperelastic response of a soft tissue to certain loads or deformations, and SEDFs exist in many different forms. There are a variety of approaches including the very popular classical models by Fung et al. and Holzapfel et al. , as well as more recently developed models [14, 15], and those which describe certain time-independent phenomena such as the Mullins effect . Micromechanical models provide insight into the mechanical role of the different components that comprise a material. In these models, every parameter is linked to a physical phenomenon, with the aim to predict the behavior of the tissue based purely on its composition. Micromechanical dispersion models give further insight into the orientation and

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distribution of the tissues’ main constituent fibers. The parameter in these models associated with the dispersion of the fibers should ideally be identified using microstructural data obtained from the tissue in question . Phenomenological models, on the other hand, utilize material parameters that do not have a physical meaning, these, however, are often found to capture the experimental behavior very well and may be more relevant for a specific application or loading domain .

A number of comprehensive reviews have been carried out on the topic of hyperelastic con- stitutive models. Steinmann et al. , Hossain et al. [44, 45] and Dal et al. reviewed the range of hyperelastic models in literature. A review of the models used to simulate the behav- ior of anisotropic hyperelastic soft biological tissues was conducted by Chagnon et al. . This review presents an extensive and detailed list of many prominent models used in the field. Wex et al. provided a review of the most commonly used isotropic constitutive models for biolog- ical tissues. Mihai et al. reviewed models used for the brain and fat tissues, and successfully compared their performance ability under different loading conditions. Furthermore, Rosen and Jiang investigated current isotropic hyperelastic laws able to model tissue-mimicking materi- als characterized through shear wave elastography. Martins et al. carried out a concise review on isotropic models subjected to uniaxial loading. Wang et al. added a level of complexity and reviewed the state of modeling the compressibility of soft tissues. Holzapfel et al. looked at purely dispersion-type models. Bhattarai et al. reviewed the modeling of only one tissue, the large intestine. Recently, Cheng and Zhang provided a framework for the derivation of hyperelastic models, both isotropic and anisotropic, in the context of biomaterials.

The aforementioned reviews exclude a one-to-one quantitative comparison of models used for soft tissues. In this work, we provide a derivation of invariant- and dispersion-type models, an outline of the most prominent constitutive models in literature, and an in silico comparison of the models’ ability to simulate the anisotropic hyperelastic behavior of a variety of human tissues.

The data for this is obtained from experimental studies of aortic aneurysms and different regions of the healthy abdominal wall. This work provides a detailed framework of how to utilize the constitutive models, while also providing an in-depth critical analysis of each models’ performance ability according to different data sets and loading domains. The findings of this review can be used in research or industry to support and inform the selection of the best constitutive model for a specific soft tissue application.

We pursue our investigation based on the datasets proposed by Niestrawska et al. , Cooney et al. , and Martins et al. for abdominal aortic aneurysm (AAA) tissue, the linea alba and the rectus sheath, respectively. AAA is a pathological condition of the abdominal aorta which results in local bulging, while the linea alba and rectus sheath are sections of the abdominal wall.

One reason for selecting these tissues is the fact that they are often subjected to surgical treatment, therefore an accurate representation of their mechanical response leads to representative numerical analysis for surgical procedures and the design of proper medical devices. In addition, each tissue presents a unique stress-stretch response, thus the choice of these tissues allows the models to be assessed across different mechanical behaviors. The overall output of the paper is twofold. Firstly, the applicability of each constitutive model to a particular type of tissue will be assessed, specifically the aptness of various anisotropic fiber distribution functions to the three tissues mentioned above.

Secondly, the fitting performance of various dispersion-type anisotropic constitutive models will be

Compared. In This Study, We Will

• outline nine anisotropic hyperelastic models and derive the closed form stress expressions for the uniaxial tension (UT) and equibiaxial tension (ET) deformation states based on the classical

Incompressibility Assumption,

• implement the aforementioned constitutive models into a hybrid genetic-gradient search algo-

Rithm Outlined In Dal Et Al. ,

• identify the material parameters of each model (i) with respect to an ET dataset for AAA tissue , (ii) a UT dataset for the linea alba , and (iii) a UT dataset for the rectus sheath , • sort all the models with respect to an objective quality of fit metric according to their fitting performance to the AAA tissue dataset , the linea alba dataset , and the rectus sheath dataset .

• The identified parameters and the quality of fit values of each constitutive model will be presented in tabular form. Also, the stress-strain curves for each constitutive model will be provided in separate graphs.

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The paper is organized as follows: the mathematical preliminaries for the kinematics of incom- pressible anisotropic hyperelastic solids are presented in Section 2. In Section 3, nine anisotropic hyperelastic constitutive models for soft biological tissues are summarized. Section 4 outlines the experimental procedures for the characterization of the quasi-static hyperelastic response of the soft human tissues selected in this study. In Section 5, the parameter identification procedure and quality of fit metric are provided. Finally, a detailed discussion of the results is presented in Section 6.

2 Fundamentals Of Hyperelastic Deformable Solids

In this section, the kinematics, state variables and stress expressions for an anisotropic hyperelas- tic continuum will be introduced. The mathematical framework of invariant and dispersion-type anisotropic hyperelastic formulations will be briefly discussed.

2.1.1 Kinematics

Let the deformation map ϕ(X, t) represent the motion of a deformable solid body. It maps the ref- erence/Lagrangian configuration X ∈B0 of material point onto the current/Eulerian configuration of material points x = ϕt(X) at time t ∈T ⊂R+. The deformation gradient

(1)

maps the unit tangent of the reference configuration onto the spatial counterpart in the current configuration where TXB0 and TxB denote the tangent spaces in the reference and current config- urations, respectively. Additionally, the co-tangent spaces in the reference and current manifolds

Xb0 And T ∗

x B, respectively. The normal map between the unit normals in the undeformed and deformed configurations is defined as

(2)

where n is the unit normal vector in the reference configuration and N is the unit normal vector in the current configuration. In this sequence, let line, area, and volume elements in the Lagrangian configurations to be denoted as dX, dA, dV , respectively. The Eulerian counterparts of these elements are obtained through the deformation gradient F , its cofactor cof[F ] = det[F ]F −T and

(3)

see also Figure 1. J := det[F ] > 0 guarantees the nonpenetrable deformations ϕ. To define the

Dv

Fig. 1 Three fundamental maps of a continuum: (a) The deformation gradient F as a mapping of an infinitesimal line element, (b) its cofactor cof[F ] as an area map, and (c) and its determinant det[F ] as a volume map. mappings between the co- and contravariant objects in the Lagrangian and Eulerian manifolds, the undeformed B0 and deformed B configurations are locally equipped with covariant G and current g metric tensors in the neighborhoods NX of X and Nx of x, respectively. Here, the right Cauchy-Green tensor and the inverse of the left Cauchy-Green tensors

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are defined as the pull back of the current metric g and push-forward of the Lagrangian metric G, respectively. The inverse of the left Cauchy-Green tensor is commonly known as the Finger tensor and represented as b = c−1.

2.1.2 Principal Stretches And Invariants

The principle of material objectivity and the principle of frame indifference require that the energy stored in the hyperelastic material be a function of either principal stretches or invariants of the right Cauchy-Green tensor. The spectral decomposition of right Cauchy-Green tensor and its

(6)

are the principal areal stretches, see Figure 2. Moreover, the three isotropic invariants of the right

2 + Λ2

3 as the sum of squares of the principal stretches, (b) I2(C) =

2 + Ν2

3 as the sum of squares of principal areal stretches, where νi = J/λi, and (c) I3 = J2 = λ2

3 As The

square of the volumetric stretch.

And

I3 := det[C] .

(7)

The principal stretches and the invariants of the right Cauchy-Green stretch tensor are related as

2Λ2

3 .

(8)

For an infinitesimal cubic element, the three isotropic invariants of the right Cauchy-Green stretch tensor are associated with linear, areal, and volumetric stretches in the principal directions. Let the Lagrangian unit vectors M 1 and M 2 denote the orientation directions of two fiber families for the description of the anisotropic behavior of soft tissues. Four additional invariants that take into account the contribution of the preferential initial fiber alignment are defined as

1 = √I4 And Λf

2 = √I6 represent the stretches of the fibers in the orientation directions M 1 and M 2, respectively. Here, the unit orientation vectors of two families of fibers, symmetrically disposed in the e1 −e2 plane in current configuration are

And

m2 = F M 2 .

(10)

Moreover, the second order structure tensors can be defined as

6

2.2 Free energy function and the stress expressions Hyperelastic materials are governed by a potential function, the Helmholtz free energy function that describes the energy stored as a result of mechanical deformation. Polymeric materials and soft biological tissues exhibit a distinct response to bulk deformation and shear-type deformations. In this context, it is common practice to split the deformation into dilatational and volume-preserving

With

F vol := J1/31 .

(13)

Based on (12) and (13), the free energy function is expressed as

(14)

where U(J) and ¯Ψ( ¯F , A1, A2) represent the volumetric and isochoric response of the material, respectively. Further decomposition of the isochoric free energy function was suggested by Holzapfel and Weizsacker . They proposed decomposition of the free energy function into isotropic and

Anisotropic Parts,

¯Ψ( ¯F , A1, A2) = ¯Ψiso( ¯F ) + ¯Ψani( ¯F , A1, A2) .

(15)

The wavy collagen fibers do not store energy under contractile deformations. Hence, the isotropic ground matrix is active ¯Ψiso under small deformations. On the other hand, the collagen fibers dominate the overall behavior at high stretch levels governed by ¯Ψani . Following the arguments of G¨ultekin et al. [34, 35], we employ the specific form of the free energy function

Leading To

˜Ψ(F , A1, A2) = Ψiso(F ) + Ψani(F , A1, A2) .

(17)

Herein, the ¯F is replaced with F for the sake of convenience. 1 Recall that, in the incompressible limit, the formulations (14) and (16) lead to identical results. In this investigation, the volumetric part of the free energy function U(J) will be excluded and the pressure term will be obtained from boundary conditions by enforcing exactly the incompressible deformation state under uniaxial and biaxial deformations. A canonical relation between the stresses and the free energy function can

(18)

where S and τ are the second Piola-Kirchhoffand the Kirchhoffstresses, respectively. Thereafter, the Lagrangian moduli expression result from the second derivative of ˆΨ with respect to C,

(19)

2.2.1 Invariant-based anisotropic hyperelastic formulations The Lagrangian and Eulerian stresses for invariant-based formulations of anisotropic elasticity

∂Ii

∂Ii

∂g .

(20)

1In the susequent treatment, the biological tissues will be assumed to behave perfectly incompressible.

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The closed form expressions for the Lagrangian and Eulerian stress tensors (26) require the

(21)

Substitution of (21) into (26)1 gives the invariant-based representation for the second Piola-

Kirchhoffstress Tensor

S = 2(c1 + I1c2)1 −2c2C + 2c3A1 + 2c4(M 1 ⊗CM 1 + M 1C ⊗M 1) + 2c5(M 2 ⊗M 2) + 2c6(M 2 ⊗CM 2 + M 2C ⊗M 2) −pC−1 .

(22)

The derivatives with respect to the spatial metric g follow from the push-forward operation of (21)

∂Gi6 = (M2 ⊗M2) ,

∂gI7 = (m2 ⊗bm2 + m1b ⊗m2) .

(23)

Substitution of (23) into (26)2 leads to the invariant-based representation for the second Kirchhoff

Stress Expression

τ = 2(ψ1 + I1ψ2) b −2ψ2bgb + 2ψ3(m1 ⊗m1) + 2ψ4(m1 ⊗bm1 + m1b ⊗m1)

∂Ii

and p = −J∂JΨ. The stress expressions given in (22) and (24) can be expressed in terms of index notation for the principal directions

2Iλ2

i −p .

(25)

The Lagrangian moduli expression for invariant-based formulation can be written as

(26)

Furthermore, moduli expression can be additively decomposed into isotropic and anisotropic parts

As Follows

C = Ciso + Cani .

(28)

where α ∈[0, 2π].

2.2.2 Dispersion-Type Anisotropic Formulations

The tissue is considered a fiber-reinforced composite with the fibers distributed within an isotropic matrix in the dispersion-type anisotropic formulation; the models developed within this frame- work accurately describe the effect of the structural arrangement of the fibers on the mechanical

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response. Dispersion-type anisotropic approaches utilize density distribution functions to represent the distributed fiber architecture of tissues. Let unit fiber direction r on a unit-sphere be given in the undeformed configuration. The fiber density in direction r is expressed with ρ(r). The unit fiber orientation vector can be expressed in spherical coordinates as follows

(29)

in terms of Eulerian angles θ ∈[0, π] and φ ∈[0, 2π], see Figure 3. The Eulerian counterpart of r is derived as t = F r. In this part, we will summarize the two kinematic approaches for the modeling of dispersion-type anisotropy in soft biological tissues.

(B)

Fig. 3 (a) The unit micro-sphere and the orientation vector, (b) mean fiber directions of two families of fiber lie on e1-e2 plane. (i) Generalized structure tensor (GST) formulations: The generalized structure tensor is

(30)

and | S |= 4π for a unit sphere. Let h = F HF T be defined as the Eulerian counterpart of the generalized structure tensor. The Lagrangian and Eulerian stresses can be obtained by applying

(33)

is the one-dimensional average anisotropic fiber strain akin to the Green-Lagrangian strain. The closed form expressions for the Lagrangian and Eulerian stress tensors require the following

And

∂gE = h .

(34)

Substituting (21), (23) and (34) into (31) and (32), and by rearranging the terms, finally gives the generalized structure tensor-based representation of stresses for an anisotropic hyperelastic solid

(35)

with ψf = ∂EΨani. In the principal directions, (35) can be written in terms of the principal stretches,

Τi = 2Ψ1Λ2

i + 2ψfhi −p .

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The anisotropic part of the Lagrangian moduli expression can be written as

∂E2

.

(37)

(ii) Angular integration (AI) formulations: The total free energy of fibers is calculated by

(38)

where S represents a unit sphere with | S |= 4π. Starting with the free energy (38), the Eulerian stress tensor follows by the application of the chain rule

(40)

with ψf = ∂λF ψfib. The general form of the Kirchhoffstress based on the angular integration formulation is given in (40). In the principal directions, (40) can be written in terms of the principal

Ρ(R)Λ−1

f ψfcos2θ sinφ dA .

Λ

Λ

Λ

Λ

Fig. 4 Deformation modes: (a) uniaxial tension in e1, (b) uniaxial tension in in e2, and (c) equibiaxial tension.

2.3 Stresses Under Homogeneous Deformations

In this section, we briefly outline the deformation state of anisotropic biological tissues subjected to uniaxial and equibiaxial stresses. The deformation gradient tensor of anisotropic materials cannot be expressed in a straightforward manner as can be done for isotropic materials since the amount of contraction in the transverse direction and the shear deformations depend on the anisotropic fiber structure. Therefore, the deformation gradient tensor should be obtained iteratively to sat- isfy equilibrium conditions, e.g. in case of uniaxial tension σ22 = σ33 = σ12 = σ13 = σ23 = 0. To this end, we assume that the material is deformed in the principal directions corresponding to the principal axes of fiber orientations for AI- or GST-based models. For invariant formulations, the

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deformation axes are assumed to coincide with the symmetry axes for the two fiber family formu- lations, while in single fiber family formulations one of the principal deformation axes coincides with the fiber direction. Such an ansatz avoids a priori shear strains/stresses, leading to diagonal deformation and stress tensors. In this context, the deformation gradient and the nominal stress expression under homogeneous uniaxial, equibiaxial or pure shear deformations can be expressed



.

(42)

Uniaxial tension: For an incompressible anisotropic hyperelastic solid, the deformation and stress

(43)

see Figure 4(a),(b). The first two invariants under uniaxial deformation read

I2 = 2Λ + 1

λ2 .

(44)

For symmetrically orthotropic fibers, the equalities I4 = I6 and I5 = I7 hold leading to

(45)

where ϕ is the angle between the fiber and the symmetry axis e1. The components of the nominal stresses under uniaxial loading in the symmetry axes for the invariant-, generalized structure tensor-

(46)

Equibiaxial tension: For an incompressible hyperelastic anisotropic solid, the deformation and



.

(47)

see Figure 4(c). The first two invariants under equibiaxial deformation read

I2 = Λ4 + 2

λ2 .

(48)

The invariants associated with the structural tensors are

And

I5 = I7 = λ4cos2ϕ + λ4sin2ϕ .

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The components of the nominal stresses under equibiaxial loading in the symmetry axes for invariant-, generalized structure tensor- and angular integration-based formulations read

F Ψfsin2Φsin3Θ Da + 2Ψ1Λ −1

λp .

3 Hyperelastic Material Models

In this section, we will review a common isotropic model and nine anisotropic hyperelastic models under two main categories: (i) strain invariant-based models, and (ii) fiber dispersion-based models. The free energy functions and necessary derivatives for the stress expressions are outlined. In order to emphasize the degree of anisotropy and the anisotropy distribution obtained from paramater identification process from each tissue is presented in terms of polar plots for each constitutive model investigated. The polar plots consider the density distribution (DD) function for the plane of interest along with the directional stiffness (DS) computed from equation (28). For the invariant based formulations, the polar plots of the directional stiffness obtained merely from the anisotropic part of the initial elastic moduli tensor is used for visualization of the plane of interest.

3.1.1 The Neo-Hookean Model

The invariant-based models assume perfect alignment of fibers embedded into an isotropic ground matrix. Models have been presented for two families of fibers symmetrically disposed. The neo-Hookean model is the most fundamental model for hyperelastic constitutive models.

Many researchers represent the isotropic ground matrix of soft tissues with the neo-Hookean model [2, 3, 36, 41, 88]. Therefore, it deserves a separate description. Based on Wall’s treatment of elasticity of a molecular network, Treloar proposed the following free energy function

(51)

with µ = nkBθ. Herein, n is the volume specific chain density, kB is the Boltzmann constant and θ is the absolute temperature, and µ = nkBθ is the shear modulus. The nonzero derivative of ψ

Ψ1 = Μ

2 .

3.1.2 Newman-Yin (Ny) Model

Newman and Yin proposed a exponential free energy form analogous to the one proposed by Fung et al. to describe hyperelastic behavior of mitral valve tissue. They assumed that the material is transversely isotropic and that the free energy function depends on the first and the fourth invariants. They observed that for a constant I4, both ψ1 and ψ4 increase nonlinearly.

330◦

330◦

330◦

Fig. 5 The polar plots of the directional stiffness (DS) for the NY model ; (a) two families of fibers distribution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT datset for the rectus sheath .

Q is the quadratic function of the invariants. In order to have a strain energy increase with increas- ing I4, k0 should be positive. Also, if the tissue is not able to support compressive load, k2 should be positive. The original model was proposed for tissues with a single family of fibers. In this work, we have extended the formulation considering two families of fibers as given in (53)2 by incorporating the latter term depending on I6. The derivatives of Ψani with respect to the invariants are

(54)

The polar plots of the directional stiffness for the NY model obtained with the identified parameters for each tissue are depicted in Figure 5.

3.1.3 Holzapfel-Gasser-Ogden (Hgo) Model

Holzapfel et al. proposed a constitutive model for arteries. They additively decomposed the isochoric free energy function into isotropic and anisotropic parts, Ψ = Ψiso(I1) + Ψani(I4, I6) .

(55)

They utilized the neo-Hookean model (51) for the isotropic part since collagen fibers are thought to not contribute to the mechanical behavior of the tissue at low pressures. The free energy stored

(56)

where k1 > 0 is a stress-like parameter and k2 > 0 is a dimensionless parameter. The collagen fibers do not support compressive stresses due to their wavy nature. Therefore, the Macauley brackets

(57)

are utilized in order to filter out the tensile stretches. The derivatives of ψi with respect to the ith

.

(58)

The polar plots of the directional stiffness for the HGO model obtained with the identified parameters for each tissue are depicted in Figure 6.

330◦

330◦

330◦

Fig. 6 The polar plots of the directional stiffness (DS) for the HGO model ; (a) two families of fibers distribution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT datset for the rectus sheath .

3.1.4 Holzapfel-Sommer-Gasser-Regitnig (HSGR) model Holzapfel et al. proposed a free energy function of the form Ψ(I1, I4) = Ψiso(I1) + Ψani(I1, I4). The model uses the neo-Hookean model (51) as the isotropic part of the free energy function. The anisotropic part of the free energy function has a mixed representation

330◦

330◦

330◦

Fig. 7 The polar plots of the directional stiffness (DS) for the HSGR model ; (a) two families of fibers distri- bution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT datset for the rectus sheath .

where k1 > 0 is a stress-like parameter and k2 > is a dimensionless parameter. The measure of anisotropy parameter p ∈[0, 1] interpolates between the contributions of the first I1 and fourth invariants I4. It mimics the degree of fiber dispersion phenomenologically. The switch function sgn⟨I4 −1⟩enforces the tension-only condition and is activated for I4 −1 > 0. The model recovers the HGO model for p = 1. The fiber related terms drop and the model reduces to an exponential isotropic model for p = 0. The nonzero derivatives of the free energy function are

Ψ1 = Μ

2 + 2k1(1 −p)(I1 −3)sgn⟨I4 −1⟩exp{k2[(1 −p)(I1 −3)2 + p⟨I4 −1⟩2]} ,

K2

p⟨I4 −1⟩exp{k2[(1 −p)(I1 −3)2 + p⟨I4 −1⟩2]} .

330◦

330◦

330◦

Fig. 8 The polar plots of the directional stiffness (DS) for the OS model ; (a) two families of fibers distribution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT datset for the rectus sheath .

3.1.5 Ogden-Saccomandi (Os) Model

Ogden and Saccomandi , following the work of Horgan and Saccomandi , proposed a log- arithmic constitutive law for arterial tissue with two fiber families in which the fiber extension is limited. They additively decomposed the free energy function into isotropic and anisotropic parts, Ψ = Ψiso(I1) + Ψani(I4, I6) .

(61)

To model the isotropic behavior of the tissue, they adapted the well-known rubber elasticity model of Gent . The free energy function of the Gent model reads

(62)

where µ is the shear modulus and Jm is the parameter that controls the chain extensibility limit for the matrix material. The deformation limit for the first invariant is I1 < 3+Jm and the stresses tend to infinity asymptotically at this limit. As Jm →∞, the isotropic part of the free energy function (62) recovers the neo-Hookean model. A similar model of Gent was proposed by Horgan and Saccomandi for transversely isotropic materials. Instead of limiting the polymer chain extensibility, the model of Horgan and Saccomandi limits the extensibility of the fibers. The

(B)

Fig. 9 (a) Graphical representation of the von-Mises distribution of collagen fibers and (b) relation between the dispersion parameter κ and the concentration parameter of the von-Mises distribution.

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where k1 is a stress-like parameter, and Jf is the limiting parameter of extensibility of collagen

Jf + 1 ,

α = 4, 6 .

(64)

confines the stretches beyond the fiber extensibility limit. The derivatives of the free energy function



.

(65)

The polar plots of the directional stiffness for the OS model obtained with the identified parameters for each tissue are depicted in Figure 8. 3.2 Dispersion-type anisotropic constitutive models It has been determined that not only do the mean orientation of fibers within soft tissues affect their mechanical properties, but also the amount by which they are dispersed around this mean .

For instance, if the majority of fibers are orientated along the direction of the mean, the behav- ior will be very different to that if the fibers are distributed in a cone-like span around the mean orientation . For this reason, a number of models have been developed to incorporate this dispersion into the constitutive law describing the stress-strain relation of the tissue. These fiber dispersion-based models utilize a probability distribution function to model the histological struc- ture of tissues. The models which have an angular integration approach and a generalized structure tensor approach are outlined in this section.

3.2.1 GST-based Gasser-Ogden-Holzapfel (GOH) model Gasser et al. assumed that the tissue’s fibers are distributed rotationally symmetric around a mean fiber orientation direction M. They utilized a planar π-periodic von-Mises distribution as a fiber density distribution around M. The von-Mises distribution function is a one-dimensional probability distribution which is a function of Θ and concentration parameter b. The standard

(66)

where I0(b) is a modified Bessel function of the first kind of order zero. Applying the normalization

I0(B)

.

2B

.

(68)

By inserting the von-Mises type density distribution function (68) into (30), the generalized

(69)

is the fiber dispersion parameter. A one-to-one relation exists between the dispersion parameter κ and the concentration parameter b, see Figure 9. Hence, κ ∈[0, 1/3] enters the constitutive model as an additional material parameter responsible for the degree of dispersion. The lower limit κ = 0 recovers the invariant-based anisotropy and the upper limit κ = 1/3 leads to an isotropic constitutive response. The polar plots of the density distribution and directional stiffness for the

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Fig. 10 The polar plots of the density distribution (DD) and the directional stiffnesses (DS) for the GOH model ; (a) two families of fibers distribution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT dataset for the rectus sheath .

0

0

0

Fig. 11 (a) Relation between the dispersion parameter κop and the out-of-plane concentration parameter b of von- Mises distribution and (b) relation between the dispersion parameter κip and the in-plane concentration parameter a of von-Mises distribution.

GOH model obtained with the identified parameters for each tissue are depicted in Figure 10. Gasser et al. can be considered as the GST counterpart of the GHO model . The anisotropic

(70)

where Ei = Hi : C −1 replaces the fourth and sixth invariants in the GHO model. The classical neo-Hookean model (51) is utilized for the isotropic part of the free energy function. The derivatives

Ψf = K1Eiexp(K2E2

i ) .

(71)

3.2.2 GST-based Holzapfel-Niestrawska-Ogden-Reinisch-Schriefl(HNORS)

Model

Holzapfel et al. take into account both the in- and out-of-plane dispersion of fibers based on the observations of Schrieflet al. where they recorded that the fibers are dispersed both in-plane and out-of-plane. Their observations reveal that no correlation exists between in- and out- of-plane dispersions. Based on these arguments, the probability density function is multiplicatively

Decomposed As

ρ(r) = ρip(Φ)ρop(Θ).

(72)

For in-plane distribution, they considered a basic von-Mises distribution

(73)

where a is the concentration parameter and I0(a) is the modified Bessel function of the first kind of order zero. The out-of-plane distribution is in the form

2B)

.

(74)

The measures of dispersion in the in-plane and the out-of-plane directions read

The Structure Tensor H Has The Form

H = 2κipκop1 + 2κop(1 −2κip)M i ⊗M i + (1 −2κop −2κipκop)M n ⊗M n

(76)

where M i is the in-plane mean fiber direction, whereas Mn is the out-of-plane vector. The Eulerian counterpart of the generalized structure tensor reads h = 2κipκop1 + 2κop(1 −2κip)mi ⊗mi + (1 −2κop −2κipκop)mn ⊗mn

(77)

where mi = F M i and mn = F M n, respectively. The lower and upper bounds for the in-plane and out-of-plane dispersion parameters are, respectively,

And

0 ≤κip ≤1 .

(78)

The polar plots of the density distribution and directional stiffness for the HNORS model obtained with the identified parameters for each tissue are depicted in Figure 12. The free energy functions are identical to the GOH model as depicted in (70).

3.2.3 AI-based Alastru´e -Martinez-Doblar´e -Menzel (AMDM) model The angular integration-based anisotropic model of Alastru´e et al. takes into account rotationally symmetric fiber dispersion based on the micro-sphere model. The model utilizes a planar π-periodic von-Mises distribution for the fiber density distribution ρ(r, M) around a mean direction M, in the same sense as Gasser et al. . r is the unit orientation vector of a micro-fiber and t := F r is the Eulerian counterpart of the Lagrangian fiber vector. The affine-stretch of a single fiber in the

Where

t♭:= gt .

(79)

The macroscopic free energy corresponding to one family of fibers with the mean direction M and

(80)

where ψfib is the free energy function associated with the orientation direction r and | S |= 4π for a unit-sphere. For the isotropic ground matrix, they utilized the neo-Hookean free energy function. The anisotropic free energy function is as follows

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Fig. 12 The polar plots of density distribution (DD) and the directional stiffness (DS) for the HNORS model ; (a) in-plane distribution of two families of fibers based on ET dataset for AAA tissue , (b) out-of-plane distri- bution of one families of fibers based on ET dataset for AAA tissue (c) in-plane distribution of one family of fibers based on UT dataset for the linea alba , (d) out-of-plane distribution of one family of fibers based on UT dataset for the linea alba , (e) in-plane distribution of one family of fibers based on UT dataset for the rectus sheath , (f) out-of-plane distribution of one family of fibers based on UT dataset for the rectus sheath .

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Fig. 13 The polar plots of the density distribution (DD) and the directional stiffness (DS) for the AMDM model ; (a) two families of fibers distribution based on ET dataset for AAA tissue , (b) one family of fibers distribution based on UT dataset for the linea alba , (c) one family of fibers distribution based on UT dataset for the rectus sheath .

F −1]2) −1] ,

for λf ≥1 .

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