A homogenized constrained mixture model of cardiac growth and remodeling: analyzing mechanobiological stability and reversal

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  Fig. 1. The different configurations during prestressing, and growth and remodeling. Each configuration consists of multiple constituents, namely cardiomyocytes, collagen fibers and elastin, which occupy every point in the domain at once. The reference configuration is obtained from imaging data of a heart in systole. The initial condition for the remodeling deformation gradients of cardiomyocytes and collagen fibers is chosen such that these constituents are in their preferred mechanical environment in the reference (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\widehat{=}$$\end{document}=^ imaged) configuration. Prestress, active stress and external loading cause the mixture to elastically deform. The prestress algorithm aims at an elastin prestretch tensor \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{G}^\text…

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  Fig. 2. The physical analog model of a homogenized constrained mixture model subjected to turnover as shown by Cyron et al. (2016). A Maxwell element (right) with a spring with stiffness \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^i$$\end{document}Ci in and dashpot with time constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\dot{\rho }_0^i}{\rho _0^i}+\frac{1}{T^i}$$\end{document}ρ˙0iρ0i+1Ti parallel to a motor unit (left) exerting the homeostatic (pre-)stress \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma _\text {h}^i$$\end{document}σhi

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  Fig. 3. The prolate spheroid used as a model left ventricle for the simulation of G&R of the myocardium. The base of the heart is fixed with omnidirectional springs and the pericardium is modeled with springs in reference normal direction to mimic the stiffness of the left atrium and the surrounding tissue. The ventricle is loaded with systolic pressure p

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  Fig. 4. The fiber distributions within the prolate spheroid. The helix angle \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\phi$$\end{document}ϕ of Cardiomyocytes (red) varies from \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+60^\circ$$\end{document}+60∘ at the endocardium to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-60^\circ$$\end{document}-60∘ at the epicardium. The four discrete collagen fiber families are aligned in myocyte fiber, cross-fiber and diagonal (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek}…

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  Fig. 5. Convergence of the prestress algorithm. After prestretching, G&R is activated with unchanged boundary conditions. a Maximum Euclidean norm of the nodal displacements between the reference configuration and the prestressed configuration for different convergence thresholds. After the subsequent G&R phase, the maximum Euclidean norm of the nodal displacements is around half of an order of magnitude higher than the chosen prestress convergence thresholds. b Maximum relative deviation of the myocyte fiber Cauchy stress from the homeostatic stress for different convergence thresholds

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  Fig. 6. Visualization of the elastin prestretch. a Principal direction and principal stretch of the largest Elastin expansion (first principal stretch) and b for the largest compression (third principal stretch) at every integration point, respectively

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  Fig. 7. Mechanobiological stability map of heart growth depending on the systolic pressure and the growth constant \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k^\text {i}\cdot T^i$$\end{document}ki·Ti. Blue circles mark simulations that result in a stable end configuration, whereas simulations with purple crosses mark cases with unbounded growth. The cases (1) and (2) are shown in more detail in Fig. 8

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  Fig. 8. Stage 1 and stage 2 hypertension of a left ventricle with a subsequent pressure normalization and resulting reversal of G&R. For each case, we plot (from top to bottom) the applied systolic pressure, the resulting Cauchy stress of cardiomyocytes with standard deviation, the mid cavity wall thickness, the endocardial diameter and the collagen mass fraction with the standard deviation in the myocardium. Stage 1 hypertension resulted in stable G&R. After a normalization of the systolic pressure, all quantities return back close to their pre-G&R reference values. Both configurations D and E are depicted in Fig. 9. Stage 2 hypertension results in unstable G&R. After the deviation from homeostasis of all constituents consistently increased, the simulation is stopped. This configuration B is depicted in Fig. 10. If hypertension is treated after \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$150\,{\textrm{days}}$$\end{document}150days (configuration A) the reverse G&R phase results in a stable configuration…

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  Fig. 9. The final configuration of each G&R phase of the simulated stage 1 hypertension. The reference configuration is depicted in gray. Configuration D has recovered homeostasis for an increased pressure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=140\,{\textrm{mmHg}}$$\end{document}p=140mmHg (stage 1 hypertension). The cavity volume of the ventricle increased and a slightly more spherical shape is observed. Configuration E has recovered homeostasis after returning to baseline pressure \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=120\,{\textrm{mmHg}}$$\end{document}p=120mmHg. An almost complete reversal of the growth is observed resulting in only a small deviation from the reference configuration

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  Fig. 10. Configuration B after stopping unstable G&R for a systolic pressure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=180\,{\textrm{mmHg}}$$\end{document}p=180mmHg (stage 2 hypertension). The cavity size increased and the shape renders spherically

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  Fig. 11. Stage 2 hypertension with an early intervention after \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$150\,{\textrm{days}}$$\end{document}150days. The ventricle was loaded with a systolic pressure of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$180\,{\textrm{mmHg}}$$\end{document}180mmHg. Configuration A after \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$150\,{\textrm{days}}$$\end{document}150days of G&R. The shown configuration is not in a homeostatic state. Configuration C after returning pressure to baseline \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb}…