Homogenized constrained mixture models for anisotropic volumetric growth and remodeling

Braeu, Fabian A.; Seitz, Alexander; Aydin, Roland C.; Cyron, Christian J.

doi:10.1007/s10237-016-0859-1

Cited by 97 publications

(160 citation statements)

References 52 publications

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“…We also modeled the passive behavior of SMCs by an anisotropic Fung‐type exponential function such as

{normalΨ}_{p a s}^{m} = \frac{C_{1}^{m}}{4 C_{2}^{m}} ((), \exp false(C_{2}^{m} {false(λ_{e}^{m}^{2} - 1 false)}^{2} false) - 1)

while we modeled its active behavior according to Braeu et al,

{normalΨ}_{a c t}^{m} = \frac{σ_{a c t m a x}}{ρ_{0}} ((), λ_{a c t} + \frac{{false(λ_{m a x}^{m} - λ_{a c t} false)}^{3}}{3 {false(λ_{m a x}^{m} - λ_{0}^{m} false)}^{2}})

where

C_{1}^{m}

and

C_{2}^{m}

are stress‐like and dimension‐less material parameters, respectively, σ actmax is the maximal active Cauchy stress, λ act is the active stretch in the fiber direction,

λ_{0}^{m}

and

λ_{m a x}^{m}

are the zero and maximum active stretches and ρ 0 denotes the total mixture density in the reference configuration;

λ_{e}^{m}

is the elastic contribution of SMCs calculated such as

λ_{e}^{m} = \frac{λ^{m}}{λ_{r}^{m}} with λ^{m} = G_{h}^{m} \sqrt{λ_{z}^{2} \cos^{2} α^{m} + λ_{θ}^{2} \sin^{2} α^{m} <...>}

…”

Section: Methodsmentioning

confidence: 99%

“…Kinematic growth theories commonly split multiplicatively the total deformation gradient into elastic and inelastic parts, where the inelastic one is related to growth . This theory has been widely used for single‐constituent solid continuum as well as for homogenized and nonhomogenized constrained mixture models (CMMs). For example, Valentín et al modelled arterial wall adaptation and maladaptation to different cases, such as loss of smooth muscle cells (SMCs), elastin degradation, and changes in fiber orientations and quantities.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A new finite‐element shell model for arterial growth and remodeling after stent implantation

Laubrie

Mousavi

Avril

2019

Numer Methods Biomed Eng

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The goal of this paper is to study computationally how blood vessels adapt when they are exposed to a mechanobiological insult, namely, a sudden change of their biomechanical conditions such as proteolytic injuries or implantation.Adaptation occurs through growth and remodeling (G&R), consisting of mass production or removal of structural proteins, such as collagen, until restoring the initial homeostatic biomechanical conditions. In some circumstances, the initial conditions can never be recovered, and arteries evolve towards unstable pathological conditions, such as aneurysms, which are responsible for significant morbidity and mortality. Therefore, computational predictions of G&R under different circumstances can be helpful in understanding fundamentally how arterial pathologies progress. For that, we have developed a low-cost open-source finite-element 2D axisymmetric shell model (FEM) of the arterial wall. The constitutive equations for static equilibrium used to model the stress-strain behavior and the G&R response are expressed within the homogenized constrained mixture theory. The originality is to integrate the layer-specific behavior of both arterial layers (media and adventitia) into the model. Considering different mechanobiological insults, our results show that the resulting arterial dilatation is strongly correlated with the media thickness. The adaptation to stent implantation is particularly interesting. For large stent oversizing ratios, the artery cannot recover from the mechanobiological insult and dilates forever, whereas dilatation stabilizes after a transient period for more moderate oversizing ratios. We also show that stent implantation induces a different response in an aneurysm or in a healthy artery, the latter yielding more unstable G&R. Finally, our G&R model can efficiently predict, with very low computational cost, fundamental aspects of arterial adaptation induced by clinical procedures. KEYWORDSarterial growth and remodeling, axisymmetric shell model, homogenized constrained mixture theory, layer-specific behavior, stent implantation Int J Numer Meth Biomed Engng. 2020;36:e3282.wileyonlinelibrary.com/journal/cnm

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“…We also modeled the passive behavior of SMCs by an anisotropic Fung‐type exponential function such as

{normalΨ}_{p a s}^{m} = \frac{C_{1}^{m}}{4 C_{2}^{m}} ((), \exp false(C_{2}^{m} {false(λ_{e}^{m}^{2} - 1 false)}^{2} false) - 1)

while we modeled its active behavior according to Braeu et al,

{normalΨ}_{a c t}^{m} = \frac{σ_{a c t m a x}}{ρ_{0}} ((), λ_{a c t} + \frac{{false(λ_{m a x}^{m} - λ_{a c t} false)}^{3}}{3 {false(λ_{m a x}^{m} - λ_{0}^{m} false)}^{2}})

where

C_{1}^{m}

and

C_{2}^{m}

are stress‐like and dimension‐less material parameters, respectively, σ actmax is the maximal active Cauchy stress, λ act is the active stretch in the fiber direction,

λ_{0}^{m}

and

λ_{m a x}^{m}

are the zero and maximum active stretches and ρ 0 denotes the total mixture density in the reference configuration;

λ_{e}^{m}

is the elastic contribution of SMCs calculated such as

λ_{e}^{m} = \frac{λ^{m}}{λ_{r}^{m}} with λ^{m} = G_{h}^{m} \sqrt{λ_{z}^{2} \cos^{2} α^{m} + λ_{θ}^{2} \sin^{2} α^{m} <...>}

…”

Section: Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A new finite‐element shell model for arterial growth and remodeling after stent implantation

Laubrie

Mousavi

Avril

2019

Numer Methods Biomed Eng

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show abstract

“…According to Braeu et al [24], the deformation caused by growth is regarded as an inelastic deformation, where the change of volume is related to a change in mass. Hence, the rate of inelastic deformation gradientḞ g is obtained as in Braeu et al [24] F g =ρ (t)…”

Section: Gradient-enhanced Healing Model Based On Gandrmentioning

confidence: 99%

Gradient-enhanced continuum models of healing in damaged soft tissues

Zuo

Hackl

et al. 2019

Biomech Model Mechanobiol

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Healing of soft biological tissue is the process of self-recovering or self-repairing the injured or damaged extracellular matrix (ECM). Healing is assumed to be stress-driven, with the objective of returning to a homeostatic stress metrics in the tissue after replacing the damaged ECM with new undamaged one. However, based on the existence of intrinsic length scale in soft tissues, it is thought that computational models of healing should be non-local. In the present study, we introduce for the first time two gradient-enhanced constitutive healing models for soft tissues including non-local variables. The first model combines a continuum damage model with a temporally homogenized growth model, where the growth direction is determined according to local principal stress directions. The second one is based on a gradient-enhanced healing model with continuously recoverable damage variable. Both models are implemented in the finite-element package Abaqus by means of a user subroutine UEL. Three two-dimensional situations simulating the healing process of soft tissues are modeled numerically with both models, and their application for simulation of balloon angioplasty is provided by illustrating the change of damage field and geometry in the media layer throughout the healing process.

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“…Alternative ways exist to account for deposition of more material, where it is advised to enforce volumetric growth in the vessel's thickness direction, keeping the density constant. [34,35] Future work will be directed at implementing the latter, since a more biofidelic representation of volumetric growth will become more important when modeling more realistic geometries.…”

Section: Limitations and Future Directionsmentioning

confidence: 99%

“…To incorporate these processes, so-called constrained mixture models have already been established by a number of authors, [17][18][19][20][21][22][23][24][25] including homogenized approaches. [26,27] They describe growth and remodeling as a superposition of continuous degradation and deposition processes of differential mass increments of multiple constituents in each differential volume element. Each constituent has a given deposition stretch in the configuration at which it is created.…”

Section: Introductionmentioning

confidence: 99%