Mathematical modeling of collagen turnover in biological tissue

Abstract

We present a theoretical and computational model for collagen turnover in soft biological tissues. Driven by alterations in the mechanical environment, collagen fiber bundles may undergo important chronic changes, characterized primarily by alterations in collagen synthesis and degradation rates. In particular, hypertension triggers an increase in tropocollagen synthesis and a decrease in collagen degradation, which lead to the well-documented overall increase in collagen content. These changes are the result of a cascade of events, initiated mainly by the endothelial and smooth muscle cells. Here, we represent these events collectively in terms of two internal variables, the concentration of growth factor TGF-β and tissue inhibitors of metalloproteinases TIMP. While the upregulation of the former increases the collagen density, the upregulation of the latter increases the collagen density through decreasing matrix metalloproteinase MMP. We establish a mathematical theory for mechanically-induced collagen turnover and introduce a computational algorithm for its robust and efficient solution. We demonstrate that our model can accurately predict the experimentally observed collagen increase in response to hypertension reported in literature. Ultimately, the model can serve as a valuable tool to predict the chronic adaptation of collagen content to restore the homeostatic equilibrium state in vessels with arbitrary micro-structure and geometry.

Use it or lose it: multiscale skeletal muscle adaptation to mechanical stimuli

“…The following information could be included in a more advanced model of chronic muscle adaptation: vi) at the molecular scale: changes in titin isoform [129–131]; vii) at the subcellular scale: changes in number [140,143] and force [47] of actomyosin cross-bridges within the sarcomeres; viii) at the cellular scale: changes in desmin protein content, as desmin also affects passive myofibril stiffness [59–61]; ix) at the organ scale: changes in collagen fiber orientation and extracellular matrix composition [203, 204]; ix) at the organ scale: changes in tendon geometry and tendon stiffness, which affect passive mechanical properties of the muscle-tendon unit [205]. Beyond the organ scale, this approach does not yet address the wide variety of factors that influence muscle adaptation at larger, more integrative scales.…”

Computational modeling of acute myocardial infarction

“…We model changes in matrix metalloproteinase, c MMP , through the following rate equation [38], $${ċ}_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}={𝒬}_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}\hspace{1em}\text{with}\hspace{1em}{𝒬}_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}={true[\frac{c_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}}{c_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}^{*}}true]}^{-m}false[1-c_{{\mathrm{normalO}}_2}false],$$ where c O 2 is the normalized oxygen concentration, c MMP is the density of matrix metalloproteinases, $c_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}^{*}$ is its initial value, and the exponent m controls its evolution. We assume that the collagen concentration c col is correlated to the intensity of matrix metalloproteinase c MMP through the scaling coefficient γ MMP , $${ċ}_{\text{col}}={\gamma }_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}{ċ}_{\mathrm{normalM}\mathrm{normalM}\mathrm{normalP}}.$$ The continuing collagen turnover and increase in collagen degradation makes the ECM to lost its ability to serve as a scaffold of the cardiomyocytes.…”

Mathematical modelling of atheroma plaque formation and development in coronary arteries

“…where DC S,w is the variation of both contractile and synthetic SMCs with respect to the initial concentration of these species before atheroma plaque initiation, Vol foam cell ¼ 4=3pR 3 F and Vol SMC ¼ 4=3pR 2 SMC l SMC are the volume of one foam cell and one SMC, respectively, which can be estimated knowing the radius of a foam cell R F and a SMC R SMC , and r G the density of the collagen, taken as 1 g ml 21 [65]. The foam cell has been considered as spherical and the SMC shape as ellipsoidal [66].…”