# Patient-specific predictions of aneurysm growth and remodeling in the ascending thoracic aorta using the homogenized constrained mixture model

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## Abstract

In its permanent quest of mechanobiological homeostasis, our vasculature significantly adapts across multiple length and timescales in various physiological and pathological conditions. Computational modeling of vascular growth and remodeling (G&R) has significantly improved our insights into the mechanobiological processes of diseases such as hypertension or aneurysms. However, patient-specific computational modeling of ascending thoracic aortic aneurysm (ATAA) evolution, based on finite element models (FEM), remains a challenging scientific problem with rare contributions, despite the major significance of this topic of research. Challenges are related to complex boundary conditions and geometries combined with layer-specific G&R responses. To address these challenges, in the current paper, we employed the constrained mixture model (CMM) to model the arterial wall as a mixture of different constituents such as elastin, collagen fiber families and smooth muscle cells. Implemented in Abaqus as a UMAT, this first patient-specific CMM-based FEM of G&R in human ATAA was first validated for canonical problems such as single-layer thick-wall cylindrical and bilayer thick-wall toric arterial geometries. Then it was used to predict ATAA evolution for a patient-specific aortic geometry, showing that the typical shape of an ATAA can be simply produced by elastin proteolysis localized in regions of deranged hemodymanics. The results indicate a transfer of stress to the adventitia by elastin loss and continuous adaptation of the stress distribution due to change in ATAA shape. Moreover, stress redistribution leads to collagen deposition where the maximum elastin mass is lost, which in turn leads to stiffening of the arterial wall. As future work, the predictions of this G&R framework will be validated on datasets of patient-specific ATAA geometries followed up over a significant number of years.

## Keywords

Finite elements Constrained mixture theory Growth and remodeling Anisotropic behavior Zero-pressure configuration Residual stresses## List of symbols

- \(\mathbf{a}_0^k\)
The unit vector pointing direction of the

*k*th fiber- \(\mathbf{C}^i_{{\text {el}}}\)
Elastic right Cauchy–Green deformation tensor of the

*i*th constituent- \(\overline{\mathbf{C}}^i_{{\text {el}}}\)
Modified elastic right Cauchy–Green deformation tensor of the

*i*th constituent- \(D_{\text {max}}\)
Maximum damage of elastin

- \(\dot{D}^i_{\mathrm{g}}\)
Generic rate function of

*i*th constituent- \(\mathbf{F}\)
Total deformation gradient of the mixture

- \(\mathbf{F}^i_{\text {tot}}\)
Total deformation gradient of the

*i*th constituent- \(\mathbf{F}^i_{\text {el}}\)
Elastic deformation gradient of the

*i*th constituent- \(\mathbf{F}^i_{\text {gr}}\)
Total inelastic (G&R) deformation gradient of the

*i*th constituent- \(\mathbf{F}^i_{\text {g}}\)
Deformation gradient of the

*i*th constituent due to growth- \(\mathbf{F}^i_{\text {r}}\)
Deformation gradient of the

*i*th constituent due to remodeling- \(\mathbf{G}^i_{\mathrm{h}}\)
Deposition stretch tensor of the

*i*th constituent*J*Jacobian of the mixture

- \(\overline{I}_1^i\)
First invariant of the right Cauchy–Green deformation tensor for the

*i*th constituent- \({I}_4^i\)
Fourth invariant of the right Cauchy–Green deformation tensor for the

*i*th constituent- \(k^{{\text {c}}_j}_\sigma\)
Gain or growth parameter of collagen fiber families

- \(k^k_1\)
Fung-type material coefficient the

*k*th constituent- \(k^k_2\)
Fung-type material coefficient the

*k*th constituent- \(L_{\text {dam}}\)
Spatial damage spread parameter of elastin

- \(\mathbf{S}\)
Second Piola–Kirchhoff stress

- \(T^i\)
Average turnover time of the

*i*th constituent- \(t_{\text {dam}}\)
Temporal damage spread parameter of elastin

*W*The specific strain energy density function of the mixture

- \({W}^i\)
Strain energy of the

*i*th individual constituents- \(\mathbf{X}\)
Material point in a reference configuration

- \(\mathbf{x}\)
Material point in a deformed or current configuration

- \(\alpha ^{c_{j}}\)
Each direction of collagen fiber families

- \(\mu ^{{\text {e}}}\)
Neo-Hookean material coefficient of elastin

- \(\kappa\)
Bulk modulus of elastin

- \(\sigma ^i\)
Current stress of extant

*i*th constituent- \(\sigma ^{{\text {c}}_j}_{\mathrm{h}}\)
Average stress of

*i*th constituent at homeostasis- \(\lambda ^{{\text {e}}}_z\)
Axial elastin deposition stretch value

- \(\lambda ^{{\text {e}}}_\theta\)
Circumferential elastin deposition stretch value

- \(\lambda ^k\)
Deposition stretch value of

*k*th constituent in fiber direction- \({\varvec{\Omega }}_0\)
Reference configuration

- \({\varvec{\Omega }}(t)\)
Deformed or current configuration

- \(\varrho ^i_0\)
Mass densities of the

*i*th constituent before G&R- \(\varrho ^i_t\)
Mass densities of the

*i*th constituent at time*t*- \(\dot{\varrho }^{{{\text {e}}}}(t)\)
Rate of mass degradation of the elastin

- \(\dot{\varrho }^{{\text {c}}_j}_{\text {adv}}(t)\)
Rate of mass degradation or deposition in the adventitia for collagen fibers

- \(\dot{\varrho }^{{\text {c}}_j}_{\text {med}}(t)\)
Rate of mass degradation or deposition in the media for collagen fibers

## Notes

### Acknowledgements

The authors are grateful to the European Research Council for Grant ERC-2014-CoG BIOLOCHANICS. The authors would also like to thank Nele Famaey (KU Leuven, Belgium), Christian J. Cyron (TU Hamburg, Germany) and Fabian A. Braeu (TU München, Germany) for inspiring discussions related to this work.

### Compliance with ethical standards

### Conflict of interest

The authors declare that they have no conflict of interest.

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