Karieva V. V. The Problem of Finding an Upper Optimality Estimate for Liver Regeneration Strategies Using Adaptive Dynamic Programming Methods. – Qualification scientific work is as a manuscript.
А thesis on the degree of Doctor of Philosophy (PhD) in the Specialty 113 Applied Mathematics. – V.N.Karazin Kharkiv National University, Ministry of Education and Science of Ukraine, Kharkiv, 2025.
The thesis is devoted to the mathematical modeling of liver regeneration processes in the form of generalized Lotka-Volterra equations and determining optimality criteria for finding liver recovery strategies. It considers adaptive dynamic programming methods for solving the problem of finding an upper optimality estimate for liver regeneration strategies.
The first chapter is dedicated to the history of the study of the topics discussed in the dissertation. It provides a brief description of the state-of-the-art projects focused on liver research and the associated mathematical model development. Well-known methods of modeling biological processes are formulated, and examples from the aforementioned projects are provided. Possible approaches to the optimization problem that are used further are also discussed.
In the second chapter, a detailed analysis of the key processes that ensure liver regeneration under moderate toxic exposure and partial hepatectomy is provided. These processes include cell replication, hyperplasia, polyploidy, activation of anti-stress mechanisms, apoptosis, and necrosis. Each of these processes plays a crucial role in the restoration of liver tissue, ensuring adaptation to damage and maintaining the functionality of the organ.
Based on these processes, mathematical equations have been developed to describe the dynamics of various liver cell populations. Specifically, the dynamics of normal hepatocytes, polyploid cells, binuclear hepatocytes, hypertrophic cells, as well as hepatocytes in a state of anti-stress, apoptosis, and necrosis have been modeled. Additionally, mathematical models have been created to describe the detoxification processes in the organism, related to exogenous and endogenous toxins.
All of these equations were combined into a generalized mathematical model, which is presented as a system of equations. This system is an extension of well-known population dynamics models, such as Volterra integral-differential equations, generalized Lotka-Volterra equations, and their modifications with delayed arguments.
To verify the correctness of the proposed model, it was validated both for each process individually and for their combination, specifically in the case of partial hepatectomy. This allowed for the evaluation of the model's accuracy and its ability to reproduce the dynamics of liver regeneration under real conditions.
The third chapter is dedicated to adapting methods of adaptive dynamic programming to the problem of evaluating liver regeneration mechanisms.
In practice, adaptive dynamic programming tasks often employ stationary control strategies, where control actions depend on the state of the system. Such systems are a particular case of systems with nonstationary control.
Since the control strategy described in Chapter Two is nonstationary, the methods of adaptive dynamic programming have been adapted to this problem. To achieve this, several assumptions were made. First, the liver regeneration process is considered within a finite time interval, consistent with its biological context (a limited organism lifespan). However, formally, the system of equations describing liver regeneration processes can be extended to an infinite time axis. Second, the cost function is assumed to be bounded, which is natural for biological systems.
The fourth chapter is dedicated to numerical experiments aimed at studying the impact of various control parameters on liver regeneration processes after partial hepatectomy and in cases of alcoholic liver injury. The main focus is on analyzing key mechanisms such as hyperplasia, replication, the formation and division of binuclear hepatocytes, polyploidy, and controlled apoptosis.
Numerical experiments demonstrate that apoptosis of hypertrophic cells is critical for renewing the cellular composition of the liver and ensuring its adaptation to future stresses. Effective regeneration is only possible under conditions of dynamic regulation of polyploidy. A constant level of polyploidy promotes the stability of functioning, but reduces the organ’s adaptive reserve. The balance between the formation and division of binuclear cells is essential for maintaining the optimal cellular composition and functionality of the liver in the long term.
A numerical experiment for alcoholic liver injury was conducted for excessive ethanol consumption over two weeks. The obtained regeneration strategies confirm well-known biological data that, after the cessation of toxic exposure, even significantly damaged liver tissue can recover its functions.
