Growth description for vessel wall adaptation: a thick-walled mixture model of abdominal aortic aneurysm evolution

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Growth Description for Vessel Wall Adaptation:

a Thick-Walled Mixture Model of Abdominal Aortic Aneurysm Evolution

Andrii Grytsan¹, Thomas S. E. Eriksson², Paul N. Watton^3,4, T. Christian Gasser¹

1Department of Solid Mechanics, Royal Institute of Technology (KTH), Stockholm, Sweden

2Swedish Defence Research Agency, 164 90 Stockholm, Sweden

3Department of Computer Science, University of Sheffield, Sheffield, UK

4INSIGNEO Institute of in silico Medicine, University of Sheffield, Sheffield, UK

Abstract. Modeling the soft tissue volumetric growth has received considerable attention in the literature. However, due to the lack of experimental observations, the growth kinematics, that are reported in the literature, are based on a number of assumptions. The present study tested the plausibility of different growth descriptions when applied to the abdominal aortic aneurysm (AAA) evolution.

A structurally motivated material model and the multi-constituent tissue growth descriptions were utilized. The mass increment of the individual constituents preserved either the density or the volume. Four different growth descriptions were tested, namely isotropic (IVG), in-plane (PVG), in-thickness (TVG) growth and no volume growth (NVG) models.

Based on the model sensitivity to the increased collagen deposition, TVG and NVG models were found to be plausible scenarios, while IVG and PVG were found to be implausible. In addition, TVG and NVG models were less sensitive to the initial constituent volume fractions, than IVG and PVG models. In conclusion, the choice of the growth kinematics is of crucial importance when modeling the AAA growth and remodeling, and, probably, also for other soft biological tissues.

Keywords: soft tissue, mixture model, growth deformation, growth description, volume growth, vascular adaptation, abdominal aortic aneurysm, AAA

1 Introduction

Soft tissue growth and remodeling is inherent to many biological processes in the human body, such as cardiovascular diseases (stenosis, aneurysm evolution), cardiac growth, skin growth, cartilage growth. Such processes may alter the composition of the tissue by remodeling and changing the amount of the main structural proteins, collagen and elastin. In turn, the volume of the tissue and its individual components may change. However, there is little experimental knowledge about exact mechanisms of adding or subtracting volume to or from the tissue.

While such experiments remain technically challenging, various hypotheses regarding mechanisms of changing tissue volume can be tested by careful and thorough modeling.

In a normal arterial tissue, mechanically most important passive constituents are elastin, collagen, smooth muscle cells (SMC) and other extracellular matrix (proteoglycans, water).

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Aneurysm evolution is associated with up to 90% loss of elastin (Rizzo et al. [13]), SMC apop- tosis (Kazi et al. [6]), and possibly increased collagen content. Despite large increase in the diameter compared to the normal aorta, the literature reports no significant difference between AAA wall thickness and normal aorta thickness, implying a substantial tissue volume change.

Early efforts to model the volume change of the living tissue were of purely phenomenolog- ical nature, directly relating the stress-strained state to the volume change, see Rodriguez et al.

[14], Taber and Eggers [19]. Similar approach was also used in recent works by Rodriguez et al. [15], Tsamis et al. [21]. Recently, the mixture-model based approach has emerged in the literature, see Menzel [10], Wan et al. [23], Schmid et al. [17], Valentin et al. [22], Eriksson et al. [2]. This approach allows to compute the volumetric change of the whole tissue based on the mass density changes of the individual constituents. The constrained mixture modeling is also commonly used to model arterial growth and adaption, see Humphrey and Rajagopal [5], Watton et al. [25], Martufi and Gasser [8], Valentin et al. [22]. Therefore, the AAA evolution can be naturally modeled using a microstructure-motivated constitutive model, coupled to the constituent growth and remodeling (G&R) framework and volumetric growth description.

Due to the lack of experimental observations, it is not clear whether the volume change in arterial wall is isotropic or has a preferred direction. Isotropic volumetric growth (VG) is widely used in the literature (Kuhl et al. [7], Schmid et al. [17], Valentin et al. [22], Eriksson et al. [2]), this may be motivated by rather straightforward implementation. Given that the most of collagen fibers are distributed helically around the lumen, the volume change may result in different changes of linear dimensions in the thickness direction and in the tangential plane. Hence, variations of orthotropic VG models were also used in the literature. Transversely isotropic VG was employed by Menzel [10], principal stress-driven VG by Rodriguez et al.

[15], and orthotropic VG by Wan et al. [23]. To the best of our knowledge, there has been no study that has attempted to compare the influence of the different VG assumptions on the performance of the arterial (or aneurysm) growth and adaption framework, which could support some scenarios and disapprove others.

The initial volume fractions of the individual constituents of the tissue are difficult to obtain from experiments, especially in vivo. Therefore, it is important to investigate the influence of the initial volume fractions on the predictions of the G&R model.

Focus of this paper is on modeling abdominal aortic aneurysm (AAA) evolution, however the implications may be transferred to modeling other biological processes that involve volumetric changes. The aneurysm evolution is modeled by a mixture-theory-based growth and remodeling framework Grytsan et al. [4]. The constitutive model for volumetric growth is taken from Eriksson et al. [2] and developed further to consider different functional forms of the growth tensor.

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2 Methods

Aneurysm evolution model utilizes two different timescales. A short time scale, t in seconds, governs the mechanical equilibrium of the vascular wall tissue, at which the tissue is assumed to be nearly incompressible. A long time scale,τ in years, governs the growth and remodeling (G&R) of the vascular tissue constituents.

2.1 Arterial Wall Modeling

The artery is modeled as a two-layered structure with layers γ = {M, A}, where M is the media-intima composite andA is the adventitia. The following main constituents of the tissue are considered: the isotropic network of elastin fibers (e), the isotropic ground substance (g) and two families of collagen fibers (c₁, c₂). The ground substance includes passive response of SMCs, and other extracellular matrix (e.g., proteoglycans). Adventitia is assumed to have no elastin. In summary, the media-intima composite has four constituents,ζ^M = {e, g, c1, c2} and the adventitia has three constituents,ζ^A= {g, c1, c2}.

2.2 Mixture theory

Vascular tissue is modeled as a constrained mixture, such that all tissue constituents are present at every material point. Affine deformation is assumed, such that the deformation of the constituents follows the deformation of the tissue. In addition, changes in the mass and volume of the tissue constituents sum up to the total change of tissue mass and volume during aneurysm evolution. Due to the lack of experimental evidence, mechanisms for adding and removing mass are unclear. In the present study, two extremal scenarios are tested for changing mass of a tissue constituent over time, namely constant constituent density (CCD) and constant constituent volume (CCV).

Figure 1(a) illustrates the constituent mass increment through CCD. At time τ = 0, Con- stituent 1 has massm1(0), volume V1(0) and density ̺1(0). Then, at time τ , the mass dm1with volumedV1and density̺1(0) is added. In result, the tissue mass and volume at current time τ increase to m1(τ ) = m1(0) + dm1 andV1(τ ) = V1(0) + dV1, respectively, while the density remains constant̺₁(0) = ̺₁(τ ). Normalized volume change reads ˆv₁ = V₁(τ )/V₁(0), mass changemˆ₁ = m₁(τ )/m₁(0) and density change ˆ̺₁ = ̺₁(τ )/̺₁(0) ≡ 1.

On the other hand, mass can be added through CCV, as illustrated in Figure 1(b). Con- stituent 2 with mass m2(0), volume V2(0) and density ̺2(0), accommodates additional mass dm2 with the same density̺2(0). In result, the tissue mass increases to m2(τ ) = m2(0) + dm2, while the volumeV₂(τ ) = V₂(0) remains constant, which leads to increased constituent density

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̺2(τ ) = ̺2(0) + d̺2 . Hence, the normalized volume change reads vˆ2 = V2(τ )/V2(0) ≡ 1, mass changemˆ2 = m2(τ )/m2(0) and density change ˆ̺2 = ̺2(τ )/̺2(0).

Different constituents of vascular tissue may follow different scenarios. Figure 1(c) illustrates addition of the mass to a tissue composed of two constituents, ζ = 1, 2, where Con- stituent 1 follows CCD and Constituent 2 follows CCV. At time τ = 0, the tissue has mass m(0), volume V (0), and density ρ(0), and it is composed of two constituents with mass m_ζ(0), volumeV_ζ(0), and initial volume fractions φ_ζ(0) = V_ζ(0)/V (0). In addition to constituent den- sities̺_ζ(0), partial densities are defined with respect to the tissue volume, ρ_ζ(0) = m_ζ(0)/V (0).

Then, massdm_ζ with volumedV_ζand density̺_ζ(0) is added to each constituent. At time τ , the mass, volume, and density of the individual constituents are changed as described above, while the change of partial densities of constituents is ρˆ_ζ(τ ) = ρ_ζ(τ )/ρ_ζ(0). The tissue’s volume change is now computed using the volume changes of constituents and their initial volume fractions,v(τ ) =ˆ PN

ζ=1ˆv_ζ(τ )φ_ζ(0). Taking into account that ˆv(τ ) ≡ ˆm_ζ(τ ) holds for a constituent following CCD, whileˆv(τ ) ≡ 1 holds for a constituent following CCV, the volume change of a multi-constituent tissue is given as

ˆ v(τ ) =

N

X

ζ=1

ˆ

v_ζ(τ )φ_ζ(0) = X

ζ∈CCD

ˆ

m_ζ(τ )φ_ζ(0) + X

ζ∈CCV

φ_ζ(0). (1)

Tissue volume changev is used to specify the growth kinematics in Section 2.4. The normalizedˆ mass changes mˆ_ζ and the partial density changes ρˆ_ζ of individual constituents ζ are used to connect the growth and remodeling of the vascular tissue to its elastic response in Section 2.5.

2.3 Kinematics

The growth and the elastic response of the artery are associated with different time scales (years and seconds, respectively). As shown in Figure 2, the two time scales can be separated by the multiplicative decomposition of the deformation gradient F(τ, t) = Fê(t) F^g(τ ), where the volumetric growth tensor F^g(τ ) accommodates the change of volume (det F^g = det F = J(τ )), while the elastic deformation gradient Fê(t) is restricted to volume preserving elastic deformation (det Fê = 1).

Tissue volume change J(τ ) ≡ ˆv(τ ) is defined by the sum of volume changes of all tissue constituents according to Eq. (1). However,limited experimental evidence on the mechanisms of the volume change motivates the study on different idealized growth kinematics, see Sec- tion 2.4.

Slightly compressible elastic response. The elastic deformation gradient Fê(t) is further split into the volumetric and isochoric parts, Fê(t) = (Jê)^1/3F withJê = detFêanddetF = 1.

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(c) Mixture of two constituents

+

(a) Constituent 1: CCD

time τ = 0 time τ

m₁(0) V₁(0)

̺₁(0)

dm₁ dV₁

̺₁(0)

m₁(0) + dm₁ V₁(0) + dV₁

̺₁(0)

+

time τ = 0 time τ

m₂(0) V₂(0)

̺₂(0)

dm₂ dV₂

̺₂(0)

m₂(0) + dm₂ V₂(0)

̺₂(0) + d̺₂ (b) Constituent 2: CCV

+

m, V

m₂, V₂ dm₂, dV₂ m₂+ dm₂, V₂

m₁, V₁ dm₁, dV₁ m₁+ dm₁, V₁+ dV1

m+ dm, V + dV

Figure 1: Illustration of two assumed scenarios for adding mass to individual constituents and a mixture of two constituents. (a) Constituent 1, constant density (CCD): adding massdm₁with volumedV₁to the initial massm₁(0) with volume V₁(0) results in increased mass m₁+dm₁and volumeV1+ dV1, but the density̺1(τ ) = ̺1(0) remains constant. (b) Constituent 2, constant volume (CCV): adding mass dm₂ with density̺₂(τ ) = ̺₂(0) to the initial mass m₂(0) with density̺₂(0) results in increased mass m₂+ dm₂and unchanged volumeV₂(τ ) = V₂(0), hence the density̺2(τ ) = ̺2(0) + d̺2 increases. (c) Mixture of Constituent 1 (follows CCD) and Constituent 2 (follows CCV): tissue has massm and volume V , and is composed of constituents 1 (m₁, V₁) and 2 (m₂, V₂). Mass increment is added to each constituent (with massdm₁, dm₂ and volumedm1, dm2, respectively), which results in increased mass of both constituents, increased volume of Constituent 1 and unchanged volume of Constituent 2. Finally, the mass m + dm and the volume V + dV of the mixture have been changed.

Due to the volumetric-isochoric split, the elastic right and left Cauchy-Green tensors are Cê = FêTFê = (Jê)^2/3C and bê = FêFêT = (Jê)^2/3b, respectively. The three invariants of C and b are given by ¯I1 = trC = trb, ¯I2 = trC⁻¹ = trb⁻¹ and ¯I3 = det C ≡ 1.

Collagen structure and reference configuration. Collagen is assumed to be arranged into two fiber families with the referential directions a_0i, i = 1, 2, see Figure 2. Further considerations are valid for any fiber family, and hence the indexi is dropped for brevity. A structural tensor A₀ = a₀⊗ a₀(⊗ is the dyadic tensor product) is used to introduce an additional isochoric invariant of C and A₀, ¯I₄ = C : A₀ = ¯λ²₀, where ¯λ₀ is the isochoric stretch along the reference direction a₀. Finally, the isochoric spatial fiber direction reads as a= Fa₀, and the spatial structural tensor is A= a ⊗ a = Fa0⊗ Fa0.

It is assumed that the collagen fibers do not have compressive and bending stiffness. The natural reference configurations of the fibers and the surrounding tissue may be different, which

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F^g(τ ) F^e(t) F(τ, t)

Ω0 a₀₁

a₀₂ Φ

Ωg a₀₁

a₀₂ Φ

Ωτ

¯ a₁

¯ a₂

ϕ

Figure 2: Kinematics of growth. Deformation gradient F(τ, t) maps reference configuration Ω0

to the current configuration Ω_τ. On the other hand, the spatially homogeneous growth tensor F^g(τ ) (with det F^g = det F = ˆv) connects reference configuration Ω0 to the intermediate stress- free configurationΩg, which accounts for the volume change at the long time-scaleτ . Then, the elastic deformation tensor F^e(t) connects Ω_g to the current configurationΩ_τ. That is, the total deformation gradient F(τ ) is split into volumetric growth part F^g and elastic part F^e.

is enabled through an isochoric recruitment stretch variable ¯λr > 0 (see [25] for detailed discussion). This variable determines the tissue stretch in the direction ¯a, at which the collagen fiber starts to bear load, i.e., to store elastic energy. With all these assumptions, the square of the fiber stretch ¯λ²_c = ¯I_4creads

I¯_4c = max

I¯4

I¯4r

, 1

, (2)

where ¯I4r = ¯λ²r corresponds to the square of recruitment stretch. Note thatI4r is set to unity in Figure 2, for simplicity. Adaptation of the recruitment stretch to changes in the mechanical environment during G&R is defined in Section 2.5.2. The recruitment stretch can also be used to describe irreversible collagen fiber deformation [3].

2.4 Different Forms of Growth Tensor

With the tangential plane defined by the two collagen fiber families a_0i,i = 1, 2 (see Figure 2), or by the unit normal n= a01× a02, the general form for transversely anisotropic growth tensor F^g can be written as

F^g = α^gI+ β^gn⊗ n. (3)

Note that I is the second order identity tensor.

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Isotropic growth (IVG) is obtained by assumingα^g = ˆv^1/3,β^g = 0, such that

F^g = ˆv^1/3I. (4)

To distribute the new volume in the tangential plane, in-plane growth (PVG) is defined by the weight coefficientsα^g = ˆv^1/2,β^g = 1 − ˆv^1/2, such that

F^g = ˆv^1/2I+ (1 − ˆv^1/2)n ⊗ n. (5) On the other hand, to distribute the new volume perpendicular to the tangential plane, in- thickness growth (TVG) is defined by the weight coefficientsα^g = 1, β^g = ˆv − 1, such that

F^g = I + (ˆv − 1)n ⊗ n. (6)

Note that the weight coefficientsα^g, β^gare constrained bydet F^g = ˆv.

Different growth kinematics may result in different mechanical response of the adapted vessel wall. Since the vascular tissue in vivo is locally exposed to biaxial deformation, these effects can be investigated with an analytical model of a vessel patch under equibiaxial extension. The patch has referential dimensionsL × L × H, and the deformation tensor is F = diag [λ, λ, λ3].

The elastic deformation tensor is then F^e = F (F^g)⁻¹. For different growth tensors F^g, defined in Eqs. (4),(5),(6), the elastic deformation tensor is computed as

IVG : F^e = ˆv^−1/3diag [λ, λ, λ3] , (7) PVG: F^e = diagh

ˆ

v^−1/2λ, ˆv^−1/2λ, λ₃i

, (8)

TVG: F^e = diagλ, λ, ˆv⁻¹λ3 . (9)

Assuming single constituent tissue and neo-Hookean material model, the stress equilibrium equation is obtained

σ = −phI+ ˆρµC^e, (10)

Note that here ph is the hydrostatic pressure to enforce incompressibility analytically (not to be confused with the derivative of the volumetric function, as defined in Section 2.5). For the applied forceF0, the load can be expressed in terms of Cauchy stress as σ= F0/(λLλ3H)(I − e₃ ⊗ e₃), where e₃ is the current out-of-plane direction. Finally, Eq. (10) can be expanded for

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(a) IVG

0 1 2 3

Biaxial stretch λ 0

0.5 1 1.5 2 2.5 3

Non-dimensionalizedstressP/µ

ˆ v = 0.5 ˆ v = 1.0 ˆ v = 1.5

(b) PVG

0 1 2 3

0.5 1 1.5 2 2.5 3

Non-dimensionalizedstressP/µ

ˆ v = 0.5 ˆ v = 1.0 ˆ v = 1.5

(c) TVG

0 1 2 3

0.5 1 1.5 2 2.5 3

Non-dimensionalizedstressP/µ ˆv = 0.5 ˆ v = 1.0 ˆ v = 1.5

Figure 3: Effect of different growth descriptions on the stress-stretch response of the neo- Hookean material under equibiaxial extension. Response of the material with unchanged volumev = 1.0 (solid curves) is compared to the material response with ˆˆ v = 0.5 (dashed curves), and v = 1.5 (dash-dotted curves), according to (a) isotropic (IVG), (b) in-plane (PVG), andˆ in-thickness (TVG) growth descriptions.

different growth models. Thus, the equibiaxial stretchλ can be derived from

IVG : P = ˆρˆv^1/3µ(λ − ˆv²λ⁻⁵), (11)

PVG: P = ˆρµ(λ − ˆv³λ⁻⁵), (12)

TVG: P = ˆρˆvµ(λ − λ⁻⁵) = ˆmµ(λ − λ⁻⁵), (13)

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whereP = F0/LH is the applied load in terms of the first Piola-Kirchhoff stress. Noteworthy, for the TVG case, the densityρ and volume ˆˆ v change can be combined into the mass change

ˆ m = ˆρˆv.

The effect of the different growth descriptions on the mechanical characteristics of the vessel patch under equibiaxial extension is shown in Figure 3. In-thickness growth (TVG) affects only the material’s stiffness, while in-plane growth (PVG) affects only the material’s reference configuration. Finally, isotropic growth (IVG) modifies both the material’s stiffness and its reference configuration.

2.5 Constitutive Equations

2.5.1 Material Response

Following the introduced kinematics (Sec. 2.3), the vascular tissue is modeled as nearly incompressible material. Consequently, additive split of the strain-energy function into the volumetric and isochoric parts,

Ψ = U(Jê(t)) + Ψ( ¯I₁, ¯I_4ci), (14) is followed. HereU(Jê(t)) = µ_κ[Jê(t) − 1]²/2 is the penalty function used to enforce near incompressibility withµ_κbeing the penalty parameter equivalent to an artificial bulk modulus.

The isochoric response of the tissue is a sum of individual responses of each component, Ψ = ˆρ_eΨ_e I¯₁ + ˆρ_gΨ_g I¯₁ + X

i=1,2

ˆ

ρ_ciΨ_ci I¯_4ci , (15)

whereρˆ_e= 0 in the adventitia. The normalized partial density changes ˆρ_ζ(τ ), ζ = e, g, c1, c2, specify the growth of the constituents, according to Section 2.2. The neo-Hookean model is used to capture the isotropic response of ground substance and elastin,

Ψ_ζ = µ_ζ

2 I¯₁− 3 , (16)

whereµ_ζ, ζ = e, g are the material parameters defining the referential stiffness of the ground substance and elastin, respectively. The mechanical response of the collagen fiber aligned with the direction a_0i is modeled by an exponential strain energy

Ψ_ci = k₁ 2k2

nexph

k₂ I¯_4ci− 12i

− 1o

, i = 1, 2, (17)

wherek₁, k₂are the material parameters.

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Partial derivatives of Eqs. (16) and (17) are introduced to facilitate stress computation

Ψ_ζ,1 = ∂Ψ_ζ

∂ ¯I1

= µ_ζ

2 , with ζ = e, g, (18)

Ψ_ci,4 = ∂Ψ_ci

∂ ¯I_4i = k₁I¯_4ri⁻¹ I¯_4ci− 1 exph

k₂ I¯_4ci− 12i

. (19)

The second Piola-Kirchhoff tensor S^e = 2∂Ψ/∂C^eis defined at the intermediate configuration Ω_gand calculated using the chain rule,

Sê= Jêp_h(Cê)⁻¹+ 2(Jê)^−2/3 (

ˆρ_eΨ_e,1+ ˆρ_gΨ_g,1 Dev (I) + X

i=1,2

ˆ

ρ_ciΨ_ci,4Dev A_0,i )

, (20)

where Dev (•) = (•) − (1/3) [• : C^e] (C^e)⁻¹ is the material deviatoric operator, and ph =

∂U/∂Jê = µ_κ(Jê− 1). The Cauchy stress tensor is computed by push-forward operation σ = (Jê)⁻¹FêSêFêT, i.e.,

σ = p_hI+ 2(J^e)⁻¹ (

ˆρ_eΨ_e,1+ ˆρ_gΨ_g,1 dev ¯b + X

i=1,2

ˆ

ρ_ciΨ_ci,4dev A_i )

, (21)

wheredev (•) = (•) − (1/3) [• : I] I is the spatial deviatoric operator.

2.5.2 Collagen Growth and Remodeling

The collagen fabric of arterial walls is continuously remodeled via a process of cell-based (SMC, fibroblast, etc.) fiber deposition and matrix metalloproteinase (MMP) based degradation, at a half-life time of approximately two months. The new collagen fibers are assumed to be configured to the matrix in a state of stretch, which is referred to as the attachment stretch ¯λ_ai [25]; note terminology includes deposition stretch or collagen pre-stretch. The invariant form I¯_4ai = ¯λ²_aiis used, and indexi is omitted in the following considerations, for convenience.

Collagen fiber properties remain unchanged during the growth and remodeling. Collagen remodeling is simulated by a rate equation that evolves the recruitment stretch field throughout the artery. Specifically, the square of collagen stretch ¯I4c remodels towards the square of the predefined attachment stretch ¯I4a,

∂¯λ_r

∂τ = αI¯_4c− ¯I_4a

I¯4a− 1 , (22)

whereα is a remodeling time scale parameter related to the collagen fiber half-life.

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Collagen fiber deposition and degradation are balanced at homeostasis. However, in response to perturbations to the mechanical environment, vascular cells can respond by up (down)- regulating synthesis, while MPP respond by down (up)-regulating degradation leading to a net increase (decrease) in collagen mass. The rate equation for adapting the net mass change of the collagenous constituents is formulated as

∂ ˆmc

∂τ = β ˆmc

I¯4c− ¯I4a

I¯4a− 1 , (23)

where the net growth parameter β relates to the imbalance between collagen production and degradation caused by altered mechanical environment.

2.5.3 Elastin Degradation

Aneurysm development is associated with up to 90% loss of elastin [13], the exact cause of which is not fully understood. While the initial elastin degradation is hypothetically caused by altered hemodynamic conditions (e.g., low levels of wall shear stress) or a vessel wall injury, the progression of the degradation may be linked to the formation and growth of intraluminal thrombus. For simplicity, such coupling effects are neglected and the elastin degradation is prescribed by a simple exponential decaymˆ_e = ˆm_e(X, τ ) (see Watton et al. [25]). This allows to keep the focus on explicit implications of the volumetric growth model.

2.6 Axisymmetric Model

An axisymmetric model of aneurysm evolution has been used, starting from normal artery (modeled as a two-layered cylinder, as described in Section 2.1). Due to axisymmetry and distal-proximal symmetry, only one-eighth of a cylinder was modeled. Finite element mesh consisted of 1920 elements (30 elements in axial direction, 8 in circumferential, 4 through each layer thickness). The simulation workflow is divided into three phases: loading, homeostasis, and the aneurysm growth.

Loading. Symmetry boundary conditions are applied to 1/8 of a cylinder, which is then pre-stretched byλ_zin the axial direction and loaded by internal pressurepi.

Homeostasis. Collagen is remodeled following Eq. (22) until its stretch λ_c is spatially uniform and equal to the attachment stretchλ_a, i.e. the homeostasis is reached. Then, the time is set toτ = 0 and the aneurysm growth starts.

Aneurysm growth. At the timeτ , the elastin mass is prescribed by ˆ

me(z, τ ) = 1 −

1 − c^{τ /T}_min

exp−m1(2z/L − 1)² , (24)

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wherez is the axial Lagrangian coordinate, and L is the Lagrangian length of the artery; Elastin degradation is prescribed such that after timeT , cmin is the amount of elastin left atz = L/2.

The width of elastin degradation is controlled by the parameterm1. This degradation function has been previously introduced and discussed in [25]. In response to altered biomechanical environment, collagen grows and remodels, following Eqs. (22,23). Then, the normalized volume change is computed using Eq. (1). Finally, the new mechanical equilibrium is computed, and the next time step starts.

The reference parameters that were used to set up the axisymmetric model is presented in the Table 1.

Table 1: Reference set of parameters for axisymmetric model of aneurysm growth.

Reference geometry and loading Value

Inner radius R_i 8.4 mm

Artery length L 147.4 mm

Media-intima thickness H_M 1.18 mm

Adventitia thickness H_A 0.59 mm

Inner pressure pi 16 kPa

Axial pre-stretch λ_z 1.2

Material model Media Adventitia

Elastin shear modulus µ_e 133.81 kPa 0 kPa

Elastin initial volume fraction φe 0.12 0

Ground matrix shear modulus µg 33.45 kPa 33.45 kPa Ground matrix initial volume fraction φ_g 0.73 0.86

Collagen parameter kc1 3.52 kPa 3.52 kPa

Collagen parameter kc2 40 40

Collagen family initial volume fraction φⁱ_c 0.075 0.075

Fictitious bulk modulus κ 100 MPa 100 MPa

Aneurysm growth Equation Value

Collagen remodeling rate α Eq. (22) 0.6 yrs⁻¹

Collagen net growth rate β Eq. (23) 1.0 yrs⁻¹

Collagen attachment stretch λa Eqs. (22,23) 1.093 Initial value collagen recruitment stretch λrec Eq. (22) 1.13

Target amount of elastin c_min Eq. (24) 0.2

Degradation time T Eq. (24) 10 yrs

Shape parameter m1 Eq. (24) 20

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2.7 Parameter Study

In addition to the growth descriptions (IVG, PVG, TVG), specified in Section 2.4, also a no volume growth (NVG) case was considered. The tissue volume was assumed to be constant in NVG model, and hence, both elastin and collagen mass increments preserved constant constituent volume (CCV) for this model. Whereas the tissue volume may change for IVG, PVG, and TVG, the mass increments of one or both the constituents (elastin, collagen) preserved constant density, i.e., followed CCD assumption.

Study 1: collagen net growth. For all volume-changing growth descriptions (IVG, PVG, TVG), the collagen mass increment preserved CCD, while the elastin mass increment preserved CCV. Model sensitivity to the collagen net growth parameterβ = [0.75, 1.00, 1.25] was tested for all growth descriptions, and other model parameters are listed in Table 1.

Study 2: elastin growth and initial volume fractions. For all volume-changing growth descriptions (IVG, PVG, TVG), the effect of elastin mass change through CCD was tested and compared to NVG model predictions. Model sensitivity to the initial volume fraction of elastin φe = [0.12, 0.18] was tested for all growth descriptions. Collagen mass increment preserved CCD and other model parameters are listed in Table 1.

Study 3: collagen growth and initial volume fractions. For all volume-changing growth descriptions (IVG, PVG, TVG), the effect of collagen mass change through CCD was tested and compared to NVG model predictions. Model sensitivity to the initial volume fraction of collagen φⁱ_c = [0.075, 0.15] was tested for all growth descriptions. Elastin mass increment preserved CCD and other model parameters are listed in Table 1.

3 Results

Figure 4 illustrates the predicted evolution of the aneurysm shape. The maximum inner diameter d_i is measured between points A^′ − A. The normal aorta at the homeostatic configuration is shown in Figure 4(a). At this configuration, the inner diameter was d_i = d₀ = 23 mm, and the wall thickness was h0 = 0.98 mm. Along with the collagen G&R, the aneurysm diameter and the sac length increased over time, compare Figure 4(b,c). Longitudinally growing aneurysm sac compressed the non-aneurysmal part of the vessel, such that it finally folded and buckled, see Figure 4(d). In addition, when the finite elements in the folded region became highly distorted, the simulation was terminated.

Study 1: collagen net growth. Influence of the collagen net growth parameter β on the aneurysm growth framework is shown in Figure 5. The key predicted quantities are plotted at the maximum inner diameter location, i.e., at the pointA specified in Figure 4. The AAA expansion

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d₀= 23 mm max(di) = 2d0 max(di) = 3d0 max(di) = 4d0

(a) (b) (c) (d)

A^′ A A^′ A A^′ A A^′ A

Figure 4: Shape change during abdominal aortic aneurysm (AAA) expansion. Normal aorta at the homeostatic configuration with inner diameterd0 = 23 mm. Maximum inner diameter diis measured between pointsA^′− A. AAAs with maximum inner diameter diof2d0 (b), 3d0 (c), 4d₀ (d) are shown, respectively.

over time was characterized by the normalized change of the maximum inner diameter ˆdi(τ ) = di(τ )/d0, with respect to the homeostatic configuration, and by the growth rate of maximum inner diameter d_i per year, incm, see Figure 5(a) and (b), respectively. The growth models can be separated into two groups with the different growth predictions. On the one hand, the AAA expanded slower for higher collagen net growthβ for NVG and TVG, i.e., AAA reached the same diameter later for larger β. On the other hand, predictions were opposite for IVG and PVG, i.e., the AAA expanded faster for higher collagen net growthβ. After 5 to 7 years of growth, IVG and PVG models reach the clinical repair indication of a maximum diameter of 55 mm [1, 11] and an expansion rate of 1 cm/year. On the other hand, TVG and NVG stayed below this indication. Only withβ = 0.75, TVG and NVG models reached 55 mm after 9.5 years.

Figure 5(c) shows the evolution of the collagen stretchλ_cat the maximum diameter location, (pointA, see Figure 4). For NVG and TVG, the results were very similar both quantitatively and qualitatively, an increased β led to lower collagen stretch at the given time. In contrast, for the case of PVG, increased β led to increased collagen stretch at the given time, which is counterintuitive. Finally, IVG case showed crossover response to increasedβ.

The collagen mass change mˆc (see Figure 5(d)) and the local tissue volume changev (seeˆ Figure 5(e)) were consistently higher for higher values ofβ, for all growth cases. Similar trends are observed in Figure 5(f), showing the evolution of the total vessel volume change relative to the volume of the aorta in homeostasis. Despite rather high local volume growth values (up to

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(a) (b)

0 2 4 6 8 10

Time τ , [years]

1 1.5 2 2.5 3 3.5 4 4.5

AAAexpansion

ˆ d(τ),[-]ⁱ

β=0.75 β=0.75

β= 0.75 β=1.00

β=1.00

β= 1.00 β=1.25

β=1.25

β= 1.25 IVG

PVG TVG NVG

0 2 4 6 8 10

Time τ , [years]

0 0.5 1 1.5 2 2.5 3

AAAgrowthperyear,[cm/year]

β=0.75

β= 0.75 β=1.00

β=1.00

β= 1.00 β=1.25

β= 1.25

β= 1.25 IVG

PVG TVG NVG

(c) (d)

0 2 4 6 8 10

Time τ , [years]

1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Collagenstretchλc(τ),[-]

β =0.75 β=0.75

β= 0.75 β =1.00 β=1.00

β= 1.00 β =1.25

β=1.25

β= 1.25 IVG

PVG TVG NVG

0 2 4 6 8 10

Time τ , [years]

0 2 4 6 8 10 12 14 16 18 20

Collagenmasschangeˆmc(τ),[-]

β= 0.75 β=

0.75

β=0.75 β=

1.00

β= 1.00

β=1.00 β=

1.25

β= 1.25

β=1.25 IVG

PVG TVG NVG

(e) (f)

0 2 4 6 8 10

Time τ , [years]

1 1.5 2 2.5 3 3.5 4

Localvolumechangeˆv(τ),[-]

β=0.75 β=0.75

β=0.75 β=1.00

β=1.00

β=1.00 β=1.25

β= 1.25

β=1.25 IVG

PVG TVG

0 2 4 6 8 10

Time τ , [years]

0 5 10 15 20 25 30 35 40

Totalvesselvolumechange,[%]

β=0.75

β= 0.75

β=0.75 β=

1.00

β= 1.00

β=1.00 β=

1.25

β= 1.25

β=1.25 IVG

PVG TVG

Figure 5: Effect of varying the collagen net growth (parameter β) on the predictions of the different growth models, namely isotropic (IVG), in-plane (PVG), in-thickness (TVG) growth, and no volumetric growth (NVG) models. At the location of maximum inner diameterd_i, four predicted quantities are plotted versus timeτ : (a) AAA expansion ˆdi= di/d0, (b) AAA growth per year, (c) collagen stretchλ_c, (d) collagen mass changemˆ_c, (e) local tissue volume changeˆv.

The total tissue volume change, in%, is plotted versus time τ in (f). Clinically used AAA repair indication of55 mm [11] and 1 cm/year is plotted (dotted curve) in (a) and (b), respectively.

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400%), the total vessel volume changed only up to 40%.

Figure 6 shows the effect of the different growth models on the AAAs at twofold expansion, which corresponds to the configuration shown in Figure 4(b). Only the impact of the collagen net growth parameterβ = 1.25 is presented. Transmural variation of the collagen stretch λ_cis shown in Figure 6(a). The stretch values were slightly elevated towards the lumen, with a more significant gradient in TVG and NVG cases. The collagen stretch was highest for PVG, and slightly smaller for IVG. Figure 6(b) presents transmural plot of the normalized tissue volume changev. Noteworthy, the order of the curves was inversed, compared to the collagen stretchˆ plots. The largest volume change was observed in TVG case, while PVG resulted in the smallest volume change. There was a more significant gradient of the volume change, compared to the collagen stretch plots (see Figure 6(a)). The transmural plots of Cauchy hoop stress are shown in Figure 6(c). The normalized change of the vessel wall thickness (compared to the thickness in homeostasish₀) is shown in Figure 6(d). The stress magnitude correlated inversely to the thickness measure at the same location. The largest stress values and smallest thickness were predicted by the NVG model. On the other hand, the nearly twice lower stress values and twice larger thickness were predicted by the TVG model. Intermediate values were predicted by the IVG and PVG models.

Study 2: elastin growth and initial volume fractions. Figure 7(a) shows the effect of the different elastin growth descriptions on the aneurysm growth (solid curves, φ_e = 0.12).

Elastin degradation using TVG model resulted in fastest aneurysm expansion, while IVG model predicted slower aneurysm expansion. PVG and NVG models predicted slowest aneurysm expansion, and the two predicted curves were crossing over. Similar effects were observed in the tissue patch model in Section 2.4.

Effect of the increased IVF of elastinφ_e = 0.18 on the different growth model predictions is shown using asterisks in Figure 7(a). For PVG, IVG and NVG models, increasedφ_eresulted in slower aneurysm expansion. Largest effect of the increasedφe was in PVG model, and the smallest in NVG model. Interestingly, the increased φe had no apparent effect on the TVG model. In fact, the aneurysm expanded slightly more (by0.06%) after 10 years. However, the local normalized volume change at the maximum diameter (pointA in Figure 4) after 10 years wasv = 0.9047 for φˆ _e = 0.12 and ˆv = 0.8571 for φ_e= 0.18, i.e., higher IVF of elastin resulted in more elastin remaining in the wall, but the wall became thinner.

Study 3: collagen growth and initial volume fractions. Figure 7(b) shows the effect of the increased IVF of collagenφⁱc = [0.075, 0.15] (solid curves and asterisks, respectively) in the media on the aneurysm growth. IVG and PVG models responded by predicting faster aneurysm growth. On the other hand, TVG and NVG predicted slower aneurysm growth for increasedφⁱc. Same qualitative effects, with much lower magnitudes, were observed for changing the IVF of

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(a) (b)

0 0.2 0.4 0.6 0.8 1

Normalized thickness ˆh, [-]

1.09 1.1 1.11 1.12 1.13 1.14 1.15 1.16 1.17 1.18

Collagenstretchλc,[-]

Media-Intima Adventitia

Homeo IVG PVG TVG NVG

0 0.2 0.4 0.6 0.8 1

1 1.2 1.4 1.6 1.8 2 2.2

Localtissuevolumechangeˆv,[-]

IVG PVG TVG

(c) (d)

0 0.2 0.4 0.6 0.8 1

0 200 400 600 800 1000 1200 1400

Cauchyhoopstressσϕϕ,[kPa]

0 0.2 0.4 0.6 0.8 1

Normalized length ˆz, [-]

0 0.2 0.4 0.6 0.8 1 1.2

Vesselwallthicknesschange,[-]

Figure 6: Properties of the AAAs at twofold expansion for the different growth models, namely isotropic (IVG), in-plane (PVG), in-thickness (TVG) growth, and no volumetric growth (NVG) models. The net growth of collagen is specified byβ = 1.25. Transmural plots are shown for (a) collagen stretchλ_c, (b) local tissue volume changev, (c) Cauchy hoop stress σˆ _ϕϕ. Thickness change (relative to the thickness of aorta in homeostasis) along AAA length is shown in (d).

collagen in the adventitia. Therefore, these results were not presented separately.

4 Discussion

Presented results highlight the influence of the different growth kinematics on the predictions of the aneurysm evolution model. Despite the soft tissue growth kinematics have not been observed experimentally to date, our computational study helped to judge the plausibility of the model predictions for different assumed growth kinematics. Increased collagen net growthβ at no tissue volume change (NVG) decreased the aneurysm growth rate in Watton and Hill [24], using a membrane implementation. The in-thickness growth (TVG) model produced qualitatively and quantitatively similar results, and can be regarded as the most plausible kinematic description for the aneurysm growth. On the other hand, the isotropic (IVG) and in-plane (PVG)

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(a) (b)

0 2 4 6 8 10

Time τ , [years]

1 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8

AAAexpansion

ˆ d(τ),[-]ⁱ

IVG PVG TVG NVG

0 2 4 6 8 10

Time τ , [years]

1 1.5 2 2.5 3 3.5 4 4.5

AAAexpansion

ˆ d(τ),[-]ⁱ

IVG PVG TVG NVG

Figure 7: Effect of the increased initial volume fraction of elastin φe and collagen φⁱc on the predictions of different growth models, namely isotropic (IVG), in-plane (PVG), in- thickness (TVG) growth, and no volumetric growth (NVG) models. AAA expansion ˆd_i = (di)/d0 is plotted versus time τ for (a) different elastin growth models with φe = 0.12 (solid curves) andφe = 0.18 (asterisks), (b) different collagen growth models with φⁱc = 0.075 (solid curves) andφⁱ_c= 0.15 (asterisks).

growth models predicted faster aneurysm expansion at increased collagen net growth. This is a non-intuitive and nonplausible, such that IVG and PVG kinematics should be used with caution to model aneurysm growth.

Due to its definition, the stress in aneurysm wall is largely influenced by the wall thickness. The literature reports a population mean abdominal aorta wall thickness of about1.9 mm (Rosero et al. [16]) and a mean AAA wall thickness of2.09 mm (anterior region) to 2.73 mm (posterior region) (Thubrikar et al. [20]),2.24 mm (Shang et al. [18]). Similarly, other references reported a median AAA wall thickness of1.48 mm (Raghavan et al. [12]) and 1.75 mm (Martufi et al. [9]), to name a few. In summary, the data suggest at most 25% reduction in the AAA wall thickness compared to normal aortic wall thickness. In addition, there is no statis- tically significant difference between the wall thickness in small and large AAAs (Raghavan et al. [12]). On the other hand, our model predicts the wall thickness of 0.28 mm (NVG) to 0.54 mm (TVG) at twofold AAA expansion and collagen net growth of β = 1.25. This is 75% and 50% decrease, respectively, when compared to the thickness of normal aortic wall in homeostasis of0.98 mm. Hence, even the TVG model overestimated wall-thinning due to the AAA expansion. This discrepancy could be due to the fact that only the extreme scenarios for the mass change of collagen were considered, i.e., constant density or constant volume. It is known that collagen is well organized and tightly packed in the normal aorta, leading to high density. However, collagen in the AAA is more dispersed and hence loosely packed, which could indicate lower density. To model such an effect, both collagen density and volume should

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be allowed to change. Despite the TVG assumption seemed to be the most appropriate, it still requires experimental validation.

Accurate experimental estimate of the initial volume fractions (VF) of the individual constituents in the vascular wall is difficult, especially if only non-invasive methods should be used.

Therefore, it is important to investigate the sensitivity of the growth model to the variations in the VF of individual constituents. Interestingly, the NVG and TVG models have shown very little sensitivity to changes of initial elastin VF between12% and 18%, as well as to changes of the initial collagen VF in both the media-intima composite and the adventitia between7.5% and 15% per collagen fiber family. These results suggest that the crude estimate of the individual VF may affect the model predictions only slightly, and therefore population-based data may be used. On the other hand, the IVG and PVG models exhibit high sensitivity to the initial VF of collagen in the media-intima composite. If these models are used, the VF of collagen has to be identified accurately.

Presented model does not consider ILT presence, its growth over time, as well as its influence on the metabolic activity in the vascular wall. The active response of the SMCs was omitted, for simplicity. In addition, elastin degradation was simply prescribed.

Although this work illustrated the effects of the different growth kinematics on the aneurysm growth, same considerations and conclusions should apply to modeling other growth of the blood vessels (e.g., aging, stenosis), as well as the growth of other soft tissues (skin, tendon, cartilage).

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