Homogenized constrained mixture models for anisotropic volumetric growth and remodeling

Abstract

Constrained mixture models for soft tissue growth and remodeling have attracted increasing attention over the last decade. They can capture the effects of the simultaneous presence of multiple constituents that are continuously deposited and degraded at in general different rates, which is important to understand essential features of living soft tissues that cannot be captured by simple kinematic growth models. Recently the novel concept of homogenized constrained mixture models was introduced. It was shown that these models produce results which are very similar (and in certain limit cases even identical) to the ones of constrained mixture models based on multi-network theory. At the same time, the computational cost and complexity of homogenized constrained mixture models are much lower. This paper discusses the theory and implementation of homogenized constrained mixture models for anisotropic volumetric growth and remodeling in three dimensions. Previous constrained mixture models of volumetric growth in three dimensions were limited to the special case of isotropic growth. By numerical examples, comparison with experimental data and a theoretical discussion, we demonstrate that there is some evidence raising doubts whether isotropic growth models are appropriate to represent growth and remodeling of soft tissue in the vasculature. Anisotropic constrained mixture models, as introduced in this paper for the first time, may be required to avoid unphysiological results in simulations of vascular growth and remodeling.

Appendices

Appendix 1

In mechanobiological equilibrium, mechanical equilibrium is satisfied and at the same time no growth and remodeling occur because the stress of each constituent subject to growth and remodeling equals the homeostatic value (Cyron and Humphrey 2014). In two-dimensional membrane models of blood vessels, it is easy to define for a given vascular geometry an initial state in mechanobiological equilibrium. The reason is that in the balance of momentum only the membrane stress rather than the Cauchy stress appears (cf. Equation (2.2) in Cyron et al. (2014)). So, one can first solve this equation to ensure mechanical equilibrium. Subsequently, one can separately vary the remaining parameters (in particular wall thickness and mass fractions) to ensure that the Cauchy stress of each constituent equals the homeostatic value.

By contrast, in three dimensions the definition of an initial state in mechanobiological equilibrium is not straightforward. In Sect. 4, we applied, inspired by Gee et al. (2010), the following procedure to initialize the simulations in mechanobiological equilibrium:

• 1. First, the homeostatic stress (or equivalently stretch) of collagen and smooth muscle in fiber direction has to be defined. Herein, we adopt the values used already by Wilson et al. (2013) and presented in Table 1. The elastic stretches $\lambda _\mathrm{e}^{\mathrm{co}} \left( {t=0} \right)$ and $\lambda _\mathrm{e}^\mathrm{sm} \left( {t=0} \right)$ of collagen and smooth muscle in the initial configuration are then computed so that they correspond to these homeostatic stresses (or are equal to the chosen homeostatic stretches).

• 2. Second, the elastic in vivo stretches of elastin in axial and circumferential direction have to be defined. Herein, we adopt the values used already by Wilson et al. (2013) and presented in Table 1. These stretches are equal to the initial stretch of elastin in axial direction $\lambda _\mathrm{e}^{\mathrm{el}\left( {90^{\circ }} \right) } \left( {t=0} \right)$ and circumferential direction $\lambda _\mathrm{e}^{\mathrm{el}\left( {0^{\circ }} \right) } \left( {t=0} \right)$.

• 3. Third, the elastic stretch $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$ of elastin in wall thickness direction in the initial configuration at time $t=0$ is computed. The stretch depends on the distance r from the cylinder axis and is computed such that the Cauchy stress $\sigma ^{\bot }\left( r \right)$ of the whole constrained mixture in wall thickness direction linearly increases from the value $\sigma ^{\bot }\left( {r=R} \right) =-p$ at the inner radius of the cylinder to the $\sigma ^{\bot }\left( {r=R+H} \right) =0$ at the outer radius of the cylinder. That is, we use the condition

  $$\begin{aligned} p\left( {1-\frac{r-R}{H}} \right) +\sigma ^{\bot }\left( {\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right) } \right) =0 \end{aligned}$$

  (40)

  With given strain energies from Sect. 3.3, given initial mass fractions (cf. Table 1), given initial (homeostatic) stretches of collagen and smooth muscle (cf. 1)) as well as given initial axial and circumferential stretch of elastin (cf. 2)), $\sigma ^{\bot }$ in (40) becomes a function of only the parameter $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$, which can thus be computed from (40). Note that the only other so far unknown parameter $\mu _\mathrm{2D}^\mathrm{el}$ does not appear in (40) because it affects only the axial and circumferential wall stress but not the radial one due to the two-dimensional elasticity of (32). Equation (40) is solved for $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$ by applying a Newton–Raphson method.

• 4. Fourth, the material parameter $\mu _\mathrm{2D}^\mathrm{el} \left( r \right)$ is defined such that it ensures a constant circumferential Cauchy stress of elastin despite the variation of $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$ in radial direction. This ensures mechanical equilibrium for the $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$ computed according to 3) and requires

  $$\begin{aligned} \mathop \sum \limits _{i=1}^n \sigma ^{i\left( {0^{\circ }} \right) }\left( {t=0} \right) =\frac{pR}{H} \end{aligned}$$

  (41)

  with $\sigma ^{i\left( {0^{\circ }} \right) }\left( {t=0} \right)$ the Cauchy stress of the ith constituent in circumferential direction in the initial configuration at time $t\,{=}\,0$. Equation (41) has to hold for any $r\in \left[ {R;R+H} \right]$, and it is solved analytically for the material parameter $\mu _\mathrm{2D}^\mathrm{el} \left( r \right)$. Note that all parameters that are required to compute the Cauchy stresses $\sigma ^{i\left( {0^{\circ }} \right) }$ from the strain energies defined in Sect. 3.3 are known at this point except for $\mu _\mathrm{2D}^\mathrm{el}$. Also note that, as the elastic radial prestretch of elastin $\lambda _\mathrm{e}^{\mathrm{el}\bot } \left( {t=0,r} \right)$ varies in wall thickness direction, also the material parameter $\mu _\mathrm{2D}^\mathrm{el} \left( r \right)$ has to.

An initially stress-free thick-walled homogeneous cylinder on which an internal pressure is imposed exhibits in general a parabolic stress profile both of the circumferential stress $\sigma ^{\left( {0^{\circ }} \right) }$ and radial stress $\sigma ^{\bot }$ as illustrated in Fig. 7a. By contrast, the prestressing procedure described in this appendix leads to a configuration with constant circumferential stress $\sigma ^{\left( {0^{\circ }} \right) }$ and linearly increasing radial stress $\sigma ^{\bot }$ (cf. Fig. 7b) in case of uniform mass fractions and fiber orientations throughout the wall. The real stress profile over the wall thickness in arteries is currently not exactly known so that the prestressing procedure used herein should be understood as a simple way to initialize our simulations in a state of mechanobiological equilibrium rather than as a physiologically accurate procedure.

Fig. 7

Circumferential (${{\sigma }}^{\left( {{0}^{\circ }} \right) })$ and radial (${{\sigma }}^{\bot })$ stress in a thick-walled cylinder from the inner radius R to the outer radius ${R+H}$ in an initially stress-free cylinder under internal pressure p (left) compared to the prestressed configuration produced by the procedure described in this appendix (right)

Appendix 2

In this appendix, we provide the first and second derivatives of the strain energy functions $W^{\mathrm{co}}$, $W^\mathrm{sm}$, and $W^\mathrm{el}$ in (27), (28), and (36) with respect to the total Green–Lagrange strain ${\varvec{E}}$. For each constituent, we assume a multiplicative split of the deformation gradient into an elastic part ${\varvec{F}}_\mathrm{e}$ and an inelastic part ${\varvec{F}}_\mathrm{gr}$ (cf. (13)) and the partial derivative with respect to the total strain is computed assuming that ${\varvec{F}}_\mathrm{gr}$ is constant so that a variation of ${\varvec{E}}$ translates into a variation of ${\varvec{F}}_\mathrm{e}$ only. Note that in this appendix we omit superscripts in ${\varvec{F}}_\mathrm{gr}$ and ${\varvec{F}}_\mathrm{e}$, given that in each equation it is evident to which constituent in the constrained mixture they refer.

For collagen in (27), we have

$$\begin{aligned} \frac{\partial W^{\mathrm{co}}}{\partial {\varvec{E}}}=\frac{2\left[ {k_1^{\mathrm{co}} \left( {I_a -1} \right) e^{k_2^{\mathrm{co}} \left( {I_a -1} \right) ^{2}}} \right] }{\Vert {\varvec{F}}_\mathrm{gr} {\varvec{a}}_0\Vert ^{2}}{\varvec{a}}_0 \otimes {\varvec{a}}_0 \end{aligned}$$

(42)

where ${\varvec{a}}_0$ is the unit direction vector in reference configuration of the collagen fiber family. The second derivative of the strain energy function is

$$\begin{aligned} \frac{\partial ^{2}W^{\mathrm{co}}}{\partial {\varvec{E}}\partial {\varvec{E}}}= & {} \frac{4k_1^{\mathrm{co}} \left[ {1+2k_2^{\mathrm{co}} \left( {I_a -1} \right) ^{2}} \right] e^{k_2^{\mathrm{co}} \left( {I_a -1} \right) ^{2}}}{\Vert {\varvec{F}}_\mathrm{gr} {\varvec{a}}_0\Vert ^{4}}{\varvec{a}}_0\nonumber \\&\otimes \, {\varvec{a}}_0 \otimes {\varvec{a}}_0 \otimes {\varvec{a}}_0. \end{aligned}$$

(43)

The stress and elasticity of smooth muscle fiber families aligned with the unit vector ${\varvec{a}}_0$ in reference configuration are governed by

$$\begin{aligned} \frac{\partial W_\mathrm{pas}^\mathrm{sm} }{\partial {\varvec{E}}}= & {} \frac{2\left[ {k_1^\mathrm{sm} \left( {I_a -1} \right) e^{k_2^\mathrm{sm} \left( {I_a -1} \right) ^{2}}} \right] }{\Vert {\varvec{F}}_\mathrm{gr} {\varvec{a}}_0\Vert ^{2}}{\varvec{a}}_0 \otimes {\varvec{a}}_0, \end{aligned}$$

(44)

$$\begin{aligned} \frac{\partial ^{2}W_\mathrm{pas}^\mathrm{sm} }{\partial {\varvec{E}}\partial {\varvec{E}}}= & {} \frac{4k_1^\mathrm{sm} \left[ {1+2k_2^\mathrm{sm} \left( {I_a -1} \right) ^{2}} \right] e^{k_2^\mathrm{sm} \left( {I_a -1} \right) ^{2}}}{\Vert {\varvec{F}}_\mathrm{gr} {\varvec{a}}_0\Vert ^{4}}{\varvec{a}}_0 \otimes {\varvec{a}}_0 \nonumber \\&\otimes \,{\varvec{a}}_0 \otimes {\varvec{a}}_0, \end{aligned}$$

(45)

$$\begin{aligned} \frac{\partial W_\mathrm{act}^\mathrm{sm} }{\partial {\varvec{E}}}= & {} \frac{\sigma _\mathrm{actmax} }{\varrho _0 \left( 0 \right) \left[ {{\varvec{C}}:\left( {{\varvec{a}}_0 \otimes {\varvec{a}}_0 } \right) } \right] }\left( 1 -\frac{(\lambda _m -\lambda _\mathrm{act} )^{2}}{\left( {\lambda _m -\lambda _0 } \right) ^{2}} \right) \nonumber \\&{\varvec{a}}_0 \otimes {\varvec{a}}_0, \end{aligned}$$

(46)

$$\begin{aligned} \frac{\partial ^{2}W_\mathrm{act}^\mathrm{sm} }{\partial {\varvec{E}}\partial {\varvec{E}}}= & {} -2\frac{\sigma _\mathrm{actmax} }{\varrho _0 \left( 0 \right) \left[ {{\varvec{C}}:\left( {{\varvec{a}}_0 \otimes {\varvec{a}}_0 } \right) } \right] ^{\mathbf{2}}}\nonumber \\&\left( 1-\frac{(\lambda _m -\lambda _\mathrm{act} )^{2}}{\left( {\lambda _m -\lambda _0 } \right) ^{2}} \right) {\varvec{a}}_0 \otimes {\varvec{a}}_0 \otimes {\varvec{a}}_0 \otimes {\varvec{a}}_0. \end{aligned}$$

(47)

For elastin we have, according to (36), $W^\mathrm{el}=W_\mathrm{2D}^\mathrm{el} +W_\mathrm{3D}^\mathrm{el} +W_\mathrm{vol}^\mathrm{el}$ with

$$\begin{aligned} \frac{\partial W_\mathrm{2D}^\mathrm{el} }{\partial {\varvec{E}}}= & {} \mu _\mathrm{2D}^\mathrm{el} \left( {{\varvec{F}}_\mathrm{gr}^{-1} {\varvec{A}}_\mathrm{gr}^\parallel {\varvec{F}}_\mathrm{gr}^{-\mathrm{T}} -\frac{1}{\left| {{\varvec{A}}_\mathrm{gr}^\parallel {\varvec{C}}_\mathrm{e} {\varvec{A}}_\mathrm{gr}^\parallel +{\varvec{A}}_\mathrm{gr}^\bot } \right| }{\varvec{A}}_0^\parallel } \right) , \nonumber \\\end{aligned}$$

(48)

$$\begin{aligned} \frac{\partial ^{2}W_\mathrm{2D}^\mathrm{el} }{\partial {\varvec{E}}\partial {\varvec{E}}}= & {} \frac{2\mu _\mathrm{2D}^\mathrm{el} }{\left| {{\varvec{A}}_\mathrm{gr}^\parallel {\varvec{C}}_\mathrm{e} {\varvec{A}}_\mathrm{gr}^\parallel +{\varvec{A}}_\mathrm{gr}^\bot } \right| }\left( {{\varvec{A}}_0^\parallel \otimes {\varvec{A}}_0^\parallel +{\varvec{A}}_0^\parallel \odot {\varvec{A}}_0^\parallel } \right) , \end{aligned}$$

(49)

where ${\varvec{C}}_\mathrm{e}$ is the elastic Cauchy–Green deformation tensor of elastin, ${\varvec{F}}_\mathrm{gr}$ its inelastic deformation gradient, ${\varvec{A}}_0^\parallel ={\varvec{F}}^{-1}{\varvec{A}}_\mathrm{gr}^\parallel {\varvec{F}}^{-\mathrm{T}}$ and the special tensor product $\odot$ is defined such that in index notation $\left( {{\varvec{A}} \odot {\varvec{B}}} \right) _{ijkl} =\left( {{\varvec{A}}_{ik} {\varvec{B}}_{jl} +{\varvec{A}}_{jk} {\varvec{B}}_{il} } \right) /2$. The three-dimensional neo-Hookean contribution of elastin leads to

$$\begin{aligned} \frac{\partial W_\mathrm{3D}^\mathrm{el} }{\partial {\varvec{E}}}=\mu _\mathrm{3D}^\mathrm{el} \left| {{\varvec{C}}_\mathrm{e} } \right| ^{-1/3}\left( {{\varvec{C}}_\mathrm{gr}^{-1} -\frac{1}{3}\hbox {tr}\left( {{\varvec{C}}_\mathrm{e} } \right) {\varvec{C}}^{-1}} \right) \end{aligned}$$

(50)

and

$$\begin{aligned}&\frac{\partial ^{2}W_\mathrm{3D}^\mathrm{el} }{\partial {\varvec{E}}\partial {\varvec{E}}}=\frac{2}{3}\mu _\mathrm{3D}^\mathrm{el} \left| {{\varvec{C}}_\mathrm{e} } \right| ^{-\frac{1}{3}}\left[ -\left( {{\varvec{C}}_\mathrm{gr}^{-1} \otimes {\varvec{C}}^{-1}+{\varvec{C}}^{-1} \otimes {\varvec{C}}_\mathrm{gr}^{-1} } \right) \right. \nonumber \\&\quad \left. +\frac{1}{3}\hbox {tr}\left( {{\varvec{C}}_\mathrm{e} } \right) {\varvec{C}}^{-1} \otimes {\varvec{C}}^{-1}+\hbox {tr}\left( {{\varvec{C}}_\mathrm{e} } \right) {\varvec{C}}^{-1} \odot {\varvec{C}}^{-1} \right] \end{aligned}$$

(51)

with ${\varvec{C}}_\mathrm{gr} ={\varvec{F}}_\mathrm{gr}^\mathrm{T} {\varvec{F}}_\mathrm{gr}$ and the standard Cauchy–Green deformation tensor ${\varvec{C}}={\varvec{F}}^\mathrm{T}{\varvec{F}}$. The volumetric penalty term ensuring an isochoric elastic deformation of elastin leads to

$$\begin{aligned} \frac{\partial W_\mathrm{vol}^\mathrm{el} }{\partial {\varvec{E}}}=2\varepsilon \left| {{\varvec{C}}_\mathrm{e} } \right| ^{1/2}\left( {\left| {{\varvec{C}}_\mathrm{e} } \right| ^{1/2}-1} \right) {\varvec{C}}^{-1} \end{aligned}$$

(52)

and

$$\begin{aligned}&\frac{\partial ^{2}W_\mathrm{vol}^\mathrm{el} }{\partial {\varvec{E}}\partial {\varvec{E}}}=4\varepsilon \left| {{\varvec{C}}_\mathrm{e} } \right| ^{1/2}\left[ \left( {\left| {{\varvec{C}}_\mathrm{e} } \right| ^{1/2}-\frac{1}{2}} \right) {\varvec{C}}^{-1}\otimes {\varvec{C}}^{-1}\right. \nonumber \\&\quad \left. -\left( {\left| {{\varvec{C}}_\mathrm{e} } \right| ^{1/2}-1} \right) {\varvec{C}}^{-1} \odot {\varvec{C}}^{-1} \right] . \end{aligned}$$

(53)

The above equations (42), (44), (46), (48), (50), and (52) can directly be used in (9) to compute for given reference mass densities of the different constituents the second Piola–Kirchhoff stress at each point. The elasticity tensor of the constrained mixture that is required to compute the tangent stiffness matrix is, as usual, given by

$$\begin{aligned} {\mathbb {C}}=\frac{\partial {\varvec{S}}}{\partial {\varvec{E}}}=\frac{\partial ^{2}\varPsi }{\partial {\varvec{E}}\partial {\varvec{E}}}=\mathop \sum \limits _{i=1}^n \varrho _0^i \frac{\partial ^{2}W^{i}}{\partial {\varvec{E}}\partial {\varvec{E}}}. \end{aligned}$$

(54)

where the second derivatives of the strain energies $W^{i}$ can be computed using (43), (45), (47), (49), (51), and (53).

Appendix 3

Time integration of growth and remodeling in a finite element scheme requires updating in each time step both the inelastic deformation gradients ${\varvec{F}}_\mathrm{gr}^i$ at each Gauss point and the total displacements at each node. For our homogenized constrained mixture model, we tested two time integration schemes. In both we discretized time by a series of n discrete points in time $t=t^{1},\ldots ,t^{n}$ with $t^{1}=0$. The distance between two subsequent points in time was ${\Delta }t=t^{k+1}-t^{k}$. The total deformation gradient and inelastic deformation gradients at time $t^{k}$ are ${\varvec{F}}\left( {t^{k}} \right)$ and ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k}} \right)$, respectively.

In the first, explicit time integration scheme we compute at time point $t^{k}$ from a given total deformation gradient ${\varvec{F}}\left( {t^{k}} \right)$ and inelastic deformation gradient ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k}} \right)$ via the evolution equations (16)–(18) and (25) the rate $\dot{{\varvec{F}}}_\mathrm{gr}^i$ and approximate ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right) ={\varvec{F}}_\mathrm{gr}^i \left( {t^{k}} \right) +\dot{{\varvec{F}}}_\mathrm{gr}^i\left( {\varvec{F}}_\mathrm{gr}^i \left( {t^{k}} \right) , {\varvec{F}}\left( t^k\right) \right) \Delta t$. Then we compute from this ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right)$, using the balance of linear momentum (4) and the equations provided in Appendix 2, iteratively the mechanical equilibrium configuration at $t^{k+1}$, that is, ${\varvec{F}}\left( {t^{k+1}} \right)$.

In the second, implicit time integration scheme we first compute an estimate of ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right)$ by solving (iteratively) the implicit equation ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right) ={\varvec{F}}_\mathrm{gr}^i \left( {t^{k}} \right) +\dot{{\varvec{F}}}_\mathrm{gr}^i\left( {\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right) , {\varvec{F}}\left( t^k\right) \right) \Delta t$, where $\dot{{\varvec{F}}}_\mathrm{gr}^i$ is again computed using the evolution equations (16)–(18) and (25). Subsequently, we compute an estimate of the mechanical equilibrium configuration at time $t^{k+1}$, that is, ${\varvec{F}}\left( {t^{k+1}} \right)$, by solving (iteratively) the balance of linear momentum (4), using the equations provided in Appendix 2. Then we compute an updated estimate of ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right)$ based on our estimate of ${\varvec{F}}\left( {t^{k+1}} \right)$ instead of ${\varvec{F}}\left( {t^{k}} \right)$. This updated estimate of ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right)$ is used to update also the estimate of ${\varvec{F}}\left( {t^{k+1}} \right)$, and these updates of ${\varvec{F}}_\mathrm{gr}^i \left( {t^{k+1}} \right)$ and ${\varvec{F}}\left( {t^{k+1}} \right)$ are iterated until ${\varvec{F}}$ has converged.

Both in the explicit and implicit time integration scheme, the balance of linear momentum (4) has to be solved iteratively to compute ${\varvec{F}}\left( {t^{k+1}} \right)$. To this end, Newton–Raphson iterations can be used. Note that the tangent stiffness matrix used in these iterations can be directly based on (54) in Appendix 2 in an explicit time integration scheme. In an implicit time integration scheme, however, to obtain quadratic convergence of the Newton–Raphson iterations, one has to add to the second derivatives of the strain energies in (54) an additional fourth-order tensor accounting for expected changes of the inelastic deformation gradient through the iterations of the displacement field. This additional fourth-order tensor can be computed for the ith constituent as

$$\begin{aligned} {\mathbb {C}}_\mathrm{implicit}^i =\left[ {\frac{\partial }{\partial {\varvec{F}}_\mathrm{gr}^i }\left( {\frac{\partial W^{i}}{\partial {\varvec{E}}}} \right) } \right] :\frac{\partial {\varvec{F}}_\mathrm{gr}^i }{\partial {\varvec{E}}}. \end{aligned}$$

(55)

The computational results shown in Sect. 4 were all computed using the above-described implicit time integration scheme. Note that explicit and implicit refer in the above discussion to the update of the inelastic deformation gradient only. The balance of linear momentum is always obtained via the solution of an implicit system of equations.