Reaction – diffusion model of multicomponent myocardial injury based on Dirichlet concentration parameters

Shchetinin E.Y., Shevchuk A.A.

A spatially distributed mathematical model is proposed for the evolution of five myocardial tissue states — healthy myocardium, inflammatory infiltrate, necrosis, replacement fibrosis, and interstitial fibrosis — in the form of a multicomponent parabolic reaction – diffusion system. At each spatial point the tissue state is described by a vector of Dirichlet concentration parameters, which allows one to encode both the expected tissue composition and the uncertainty of classification associated with transition zones of damage. For a truncated system that coincides with the original one inside an invariant region, local Lipschitz continuity and quasi-positivity of the extended reaction operator are established. Global existence of a weak solution, non-negativity, an explicit a priori upper bound, and a quantitative exponential lower bound are proved. These results imply invariance of the admissible region and hence existence of a global weak solution to the original model. Uniqueness is established in a strengthened class of solutions characterized by additional gradient regularity. For an auxiliary reversible transition matrix, a formal entropy balance is derived for smooth positive solutions; for the clinically motivated irreversible transition matrix, a modal spectral stability condition is formulated for the linearized problem. On the numerical side, a positivity condition is derived for the reaction substep of a splitting scheme relative to an adaptive upper trajectory bound, and numerical experiments are conducted for a reproducible calibrated computational variant. In the one-dimensional scenario, by day $60$ the replacement-fibrosis fraction at the lesion center increases from $0.091$ to $0.855$, while the total concentration parameter $\alpha_0$ rises from $11.0$ to $15.9$; in the two-dimensional scenario, on day $14$ the mean value of $\overline p_4^{}$ is $0.431$ in the lesion core versus $0.097$ in the healthy region. These results confirm the mathematical well-posedness of the proposed model and its applicability for the quantitative description of post-infarction scar formation and heterogeneous transition zones.

Keywords: reaction – diffusion system, Dirichlet distribution, myocardial fibrosis, global existence, invariant region, uniqueness in a strengthened class, entropy balance, Markov chain
Citation in English: Shchetinin E.Y., Shevchuk A.A. Reaction – diffusion model of multicomponent myocardial injury based on Dirichlet concentration parameters // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 929-972
Citation in English: Shchetinin E.Y., Shevchuk A.A. Reaction – diffusion model of multicomponent myocardial injury based on Dirichlet concentration parameters // Computer Research and Modeling, 2026, vol. 18, no. 4, pp. 929-972
DOI: 10.20537/2076-7633-2026-18-4-929-972
URL: http://crm-en.ics.org.ru/journal/article/3753/
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Copyright © 2026 Shchetinin E.Y., Shevchuk A.A.

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