Annotation of the results obtained in 2020
Research on the project in 2020 addressed a wide range of issues - from the analysis of raw data on the dynamics of viral infections to the construction, numerical implementation and study of various classes of mathematical models developed under the frameworks of deterministic, stochastic and hybrid approaches, including multiscale spatially distributed descriptions of immune processes. The research has resulted to a number of novel propositions related to (1) the possible mechanisms of complete elimination of HIV-1 infection from the human body, (2) the identification of promising targets for antiretroviral therapy of HIV-1 were identified, and (3) it was shown that the response of the type I interferon system at the early stages of cytopathic viral infections plays a key role in prevention of severe forms of disease. A computational approach to modeling and analysis of the human lymphatic system based on graph theory has been developed. It is based on two different methods of constructing the graphs of the lymphatic system: using the anatomical data and on the basis of a set of physiological rules characterizing the network organization of the system. The transformation of the anatomical data-based simple graph into a directed graph is implemented by quantifying the direction of lymph flows in the network of lymphatic vessels using the Poiseuille equation and the conservation law for the lymph flow at vascular branch points. Fundamental topological characteristics for the graph models of the human lymphatic system have been obtained. To conduct research on the calculation of the spatial fields of signaling molecular factors in the lymph node, an algorithm was developed for constructing a 3D full-scale geometric model of the structural organization of the lymph node. The algorithm enables the integration of the blood vascular network with the components of the lymph node: the fibroblast reticular cells network of, B-cell follicles, subcapsular sinus. The algorithm is based on the biophysical growth mechanism of the hypoxia-induced angiogenesis, phenomenologically represented by the oxygen level gradient. A simplified model of the drainage function of the lymph node has been developed, characterized by good consistency between the solution properties and experimental data. When passing through the lymph node, part of the lymph is absorbed into the blood vessels. A model of lymph absorption into the lymph node blood vessels has been formulated. It is assumed that the blood vessels are evenly distributed in the suction area. The flow of lymph into the blood is described by the Starling equation for unknown lymph velocity and pressure and known average blood vessel pressure in the node. Overall, this problem is treated as fluid filtration in a homogeneous porous medium with a smooth boundary, on which inhomogeneous boundary conditions are set. An algorithm for modeling the movement of cells in a 3D spatial setting is developed using Newton's second law of motion. For the first time, the methods of directional statistics, in particular, the Mises-Fischer distribution on a sphere in 3D, were used to simulate the active intrinsic cell motility force for spatial cell migration in a lymph node. The force is determined by realizations of correlated random walks, i.e. a discrete-time Markov process. The use of the Mises-Fischer and Gaussian distributions in determining the direction and jump magnitude of the motility force provides an inverse correlation of the angle between the trajectory of cell movement and the amplitude of the force of active mobility. A prototype of an environment for modeling immune processes has been implemented using the Qt Creator development software and the Qt programming language. In particular, the Composite Mathematical Immunology software package has been developed, which provides the user with the ability to computationally solve the following problem-oriented tasks: to deploy an interactive model of the lymphatic system graphs of arbitrary topology, represented in the form of connectivity matrices and perform various operations on them; simulate cell dynamics in a two-dimensional setting with the ability to flexibly modify the properties of elements of the computational domain; perform deterministic and stochastic simulations of the HIV intracellular replication with the ability to modify the model parameters within physiologically admissible ranges. Using a detailed model of HIV-1 replication in an infected T-cell, the biochemical processes that strongly affect the production of viruses were identified. To this end, the sensitivity analysis of the functional characterizing the total production of viruses by an infected cell was performed. Their impact on the total output of viral particles from the infected cell was ranked according to the calculated relative sensitivity indices. Note that for some of the identified processes there are still no effective drug inhibitors. The new, additional targets for antiviral therapy can be (1) HIV-1 assembly on the inner side of the cell membrane depending on Gag protein and (2) nonlinear Tat and Rev proteins-based regulation at the transcriptional and post-transcriptional levels. The probability of productive infection of cells as function of the number of bound virions were computed together with the histograms of the distribution of proviral DNA number integrated into the cell chromosomes for MOI ranging from 1 to 100. A non-Markovian stochastic model HIV-1 infection and its deterministic analogue were developed to examine relationships between the model parameters for which the probability of eradication of HIV-1 infection within a fixed period of time after initial infection is equal to one (corresponds to the case R0 < 1, where R0 is the basic reproductive number). For the case R0 > 1, an estimate of the probability of eradication of HIV-1 infection within a given period of time after infection was derived depending on the initial dose of infection (number of viral particles) and parameters characterizing the efficiency of the immune response. The values of the model parameters were estimated at which the results of the computational experiments reproduce the known empirical data on the dynamics of HIV-1 infection in the humans following the initial phase of infection. For the respective values of the model parameters, the probability of eradication of HIV-1 infection for a small virus dose during the first 8-12 days is equal to one. The computational results indicate a significant effect of a specific immune response on the eradication of HIV-1 population following infection with a small initial number of viral particles (5-10 virions) spreading to the nearest lymph node. The regularities of the spatiotemporal dynamics of viral infection in the infected host organism were studied, taking into account the distribution of viral quasi-species with respect to the genotype and the spatial location of infection in the tissue or organ. A mathematical model of this distributed parameter process was developed in the form of partial integro-differential equations. It is shown that the spread of infection is similar to the propagation of a reaction-diffusion wave, and that in a particular case a two-dimensional problem can be reduced to a one-dimensional problem using separation of variables. The 1D problem depending on the operator eigenvalue was considered as an approximation of 2D problem. In general, the proposed approach to the analysis allows an approximate investigation of the solutions to 2D problem based on the solution properties of a reduced 1D problem. The dependences of the viral virulence (wave speed) and the viral load (wave amplitude) on the width of the distribution of genotypes, the strength of the immune response, the level of immunity and the initial viral load (initial state) were determined. It was shown that an increase in the width of the distribution of genotypes increases both the virulence of the virus and the viral load. A systematic interdisciplinary analysis of a variety of data on the dynamics and structure of the HIV population during infection course before and following antiretroviral therapy, a new concept of a functional cure (elimination of HIV-reservoir) of HIV infection was formulated. It is based on the physiological mechanism of washing out a part of activated subpopulations of CD4+ T cells (including the latently-infected cells) with more mature phenotype as a result of the homeostatic inflow of less differentiated cells into the immune system and their competition for survival factors. It is proposed, through the use additional polyclonal activation, to intensify the process of renewal of the T-cell population (i.e., washing out subpopulations of memory CD4+ T-cells, including the latently infected CD4+ T-lymphocytes). This concept of HIV-1 infection treatment (‘rinse and replace’) opens up the possibility of implementing a new multimodal approach to the elimination of all latently infected cells by using inducers of activation of antigen-presenting cells in combination with antiretroviral therapy. Based on the analysis of the relationship between the kinetics of replication of cytopathic viruses, the site of localization of the infectious process, the reaction of the innate immune system (synthesis of type I interferon, IFN-I) and the severity of the infectious disease, a hypothesis has been formulated that protection against cytopathic viral infections developing in peripheral organs and tissues provided by the interferon reaction is extremely vulnerable to an increase in the rate of viral replication. It is proposed that an early treatment of the virus-exposed individualы with type I IFN in cytopathic viral infections (HIV-1, SARS-CoV-2) is necessary to prevent the development of severe forms of peripheral diseases. The following collection of works edited by the main project participants has been published: Bocharov, G., Ludewig, B., Meyerhans, A., Volpert, V., eds. (2020). Mathematical Modeling of the Immune System in Homeostasis, Infection and Disease. Lausanne: Frontiers Media SA. doi: 10.3389/978-2-88963-461-3 (276 pages). ISSN 1664-8714, ISBN 978-2-88963-461-3 The research results of the project were published in a number of specialized Q1 journals and some of them are discussed in the following media publications https://nauka.tass.ru/nauka/8912173 https://www.gazeta.ru/science/news/2020/07/08/n_14644921.shtml https://indicator.ru/medicine/novyi-podkhod-lecheniyu-vich-09-07-2020.htm https://iz.ru/1032939/2020-07-08/rossiiskie-uchenye-nashli-novyi-podkhod-klecheniiu-vich

 

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Annotation of the results obtained in 2018
Available experimental techniques of modern immunology for global screening and visualization of processes occurring in tissues and organs with resolution spanning multiple spatial- and temporal scales resulted in the accumulation of unprecedented level of detailed information on the structural organization and regulatory networks underlying the functioning of the immune system. To comprehend the systems dynamics of the immune processes and to understanding required for implementing a rational control of the immune system components, the efficient methodology of formulating high-resolution multiscale and hybrid mathematical models of hierarchically organized-, spatially structured-, and nonlinearly regulated physical, biochemical and physiological processes has to be developed. The multiscale and hybrid modeling approach to the immune processes is developed with the aim of specific applications in studies of the human immunodeficiency virus type 1 (HIV) infection and the experimental lymphocytic choriomeningitis virus infection (LCMV) in mice. During the first year, we critically reviewed ongoing world-wide studies on hybrid modelling of biomedical processes. We analyzed discrete-continuous models which implement nonlinear dependencies of the process rates on the concentration of interacting species and a multiscale description of the intracellular-, cell population-, tissue-, and the systems scales. The existing approaches to integrative modelling of the structure and organization of immune processes in lymphatic system, i.e., the lymph nodes (LN), are formulated using continuous deterministic, discrete-stochastic and hybrid approaches. We developed deterministic mathematical model of HIV infection with a detailed description of the immune response and the virus-target cell dynamics. The model is formulated as a system on integro-differential equations with time delays. A stochastic version of the model is developed using the branching process description for interacting particles. It takes into account the stages in maturation for cells, viruses and the proliferation of cytotoxic T lymphocytes. The global solvability and the positive invariance of the initial value problem for the model are proved. The analytical condition for asymptotic stability of the trivial steady-state, which corresponds to the infection-free human organism, is derived. Computational experiments were preformed to examine the possibility of a complete elimination of HIV within a given time interval. An estimate was obtained for the probability of falling of the stochastic model trajectory to the infection free steady state. We developed a family of embedded mathematical models describing the population dynamics of viruses and immune cells using two dimensional systems of nonlinear ODEs, delay differential equations, reaction-diffusion equations in one space dimension and the time-delay reaction-diffusion equations in 1D. For bistable time-delay dynamical systems modeling the dynamics of viral infections and the virus-induced immune response, an efficient approach is proposed for constructing optimal disturbances of steady states with a high viral load that transfer the system to a state with a low viral load. It is shown that the optimal disturbances found using the Sobolev norm are superior to those found using the Euclidian norm as applied to the development of adequate therapeutic strategies via estimating the amplification effect. For time-delay reaction-diffusion equations in 1D with specified initial and boundary condition efficient finite-difference methods were developed to solve numerically the initial-boundary value problems. The methods consider time-implicit approximation of nonstationary reaction- and diffusion terms, the central difference formula and tridiagonal matrix algorithm, as well as two algorithms for approximation of the delay variables. The methodology for qualitative analysis of 2 dimensional system of delay reaction-diffusion equations has been developed based on the reduction of the system to one equation of the same type. The numerical simulations revealed the existence of quasi-waves of spatiotemporal dynamics, propagating solutions without regular structure and often with complex aperiodic oscillations. We have formulated, calibrated and validated mathematical model of immune cell migration in LN in 2D spatial resolution. The model is formulated as a system of stochastic differential equations of the Newton’s second law of motion. The model considers a parameterized description of the cell-to-cell interaction-, viscous damping-, and intrinsic random cell motility forces. The parameters of the potential functions describing the intercellular interactions and intracellular motility are estimated. A hybrid mathematical model of spatiotemporal dynamics of antiviral immune response in lymph node tissue in 3D was computationally implemented. It considers a random-discrete description of cell populations, continuous-deterministic representation of the of virus and humoral factors fields in the T cell zone of LN, the fibroblastic reticular cell network and the destruction of the spatial structure of LN due to fibrosis in the course of HIV infection. To use the CUDA parallel computing platform based on GPUs, the algorithmic structure of the hybrid model was modified. This resulted in a more efficient computational treatment of the model via parallel implementation of some processes and reduction of the demands to dynamic memory allocation. The mathematical model of intracellular regulation of the interferon stimulated gene transcription and HIV replication was developed. It is formulated as system of nonlinear delay differential equations. The model was calibrated and is ready for embedding into the hybrid model of antiviral immune response in LN. The digital cell quantifier (DCQ) method was used to generate the time series of cell dynamics using the time-resolved mouse splenic transcriptomes in acute and chronic LCMV infections. It specifies the relative kinetics of 125 state space variables to be further used for multiscale modelling of LCMV infection. The approach is based on combined Weighted gene co-expression network analysis and DCQ. The kinetics of the CD8+ T cell, macrophage and monocyte subsets with the highest correlation scores to genes controlling the acute versus chronic infection fate was established. We formulated a mathematical model with a detailed description of the intracellular replication of HIV (24 stages), i.e., from binding to cell receptors to secretion of virus particles. The stages which are targeted by existing antiretroviral drugs are the reverse transcription, integration and maturation. The stochastic implementation of the model using the Gillespe algorithm was used to quatify the 95% uncertainty bundles of the individual model runs around the solution of the deterministic model. An initial study of the possibility to optimize the relative abundance of drugs in combined three-drug formulations under the constraint on the total amount of the drugs was conducted. The available model of mitochondrial AZT metabolism was analyzed for integration into the hybrid model to describe the drug toxicity. We examined the problem of the model reduction. To this end a reduced complexity model was elaborated. It allowed us to gain a better understanding of the type of nonlinear parameterizations and time delays which need to be considered to substitute the sequence of processes being collapsed in the reduce model. Mathematical model of HIV infection dynamics considering the division number-structured clonal expansion of T cells, the regulatory T cells and combined therapy based on antiretroviral treatment and blockade of the PD-L1 receptor has been developed. The antiretroviral treatment considers the reverse transcriptase- and protease inhibitors which act to reduce the cell infection and the maturation of the virus. Information-theoretic criterion was used to assess the parsimony of the set of mathematical models reflecting various scenarios of parameter combination which underlie the observed phenotypes of HIV infection and the effect of the PD-Li receptor blockade. This allowed one to specify a most likely model. The model was used to predict the effect of the blockade for five HIV infected patients under condition that the kinetic parameters are changed or the fraction of responsive CD8+ T cell is changed. To model the transport phenomena in lymphatic system, we developed mathematical model and numerical methods for convective flow and diffusion of lymph for 3D consideration of LN. The approach is based on application of the potential theory to the solution of Laplace's equation with Neumann boundary condition on spherical domains. The application of the potential theory allows one to reformulate the boundary value problem for the differential equations of fluid flow in equivalent form of integral equations on the boundary of the domain. We formulated mathematical model for a stationary viscous flow of lymph in porous medium in the Brinkman form of Darcy's law for a simple geometric model of LN represented as two embedded spheres with a common center. The space between the spheres corresponds to the subcapsular sinus of LN, whereas the internal sphere represents the T cell zone of LN. The two domains differ in terms of their hydraulic conductivity. The lymph flow was numerically simulated using a code programmed in FORTRAN 90 and implementing the boundary finite elements method.

 

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Annotation of the results obtained in 2019
The contemporary stage of mathematical modelling of immune processes is directed by the need for a broad spectrum of mathematical models differing in their nature, the degree of detalisation of the state space, consideration of several regulation scales and levels and the consideration of spatiotemporal dynamics of immune responses. The research performed in 2019 dealt with the development, numerical implementation and analysis of the mathematical models at the single cell-, cell population-, tissue- and the systemic scales of the immune response consideration. We have developed a systematic review of the existing deterministic and stochastic models of the intracellular regulation of cell fate, including the division, differentiation, downregulation, death and spatial motility as controlled by cytokines, chemokines and viral products. Using the reaction-diffusion equations with time-delay (RDDE) we formulated the mathematical model of virus infection in 2D spatial consideration. The conditions for emergence of various spatiotemporal dynamics regimes in relation to the initial conditions and the geometry of the spatial domain were examined. These include the transition from a radially symmetric solution to irregular oscillations. We developed 1D model for the evolutionary dynamics of the virus density distribution (quasispecies) with respect to the genotype during infection formulated with reaction-diffusion integro-differential equation. The model enables to study the role of virus genotype-dependent non-local interactions between quasispecies related to their competition for target cells, cross-reactive immune responses and drug sensitivity in the intra-patient diversification and evolution of the viruses. An extended 2D distributed parameter model considering the dynamics of the viruses in 1D genetic space and in 1D physical space was formulated. The numerical simulations indicated that a non-local dependence of the magnitude of the immune response on the spatial distribution of the virus population can lead to a qualitative change of the infection dynamics. We developed the multiscale hybrid model of the spatiotemporal dynamics of HIV infection in lymphoid tissue (LT) that integrates 1) 2D model of cell motility governed by the Newtonian mechanics of isotropic particles, 2) the stochastic model of cell infection via cell-free infection and cell-to-cell transmission of viral genomes, 3) the deterministic low-dimensional model of HIV replication in productively infected cells, 4) the stochastic model for activation of the type I IFN response, 5) 2D reaction-diffusion model for the spatial spread of HIV and type I IFN in LT, 6) the stochastic description for homeostatic turnover of T cells in the lymphoid organ. The calibrated model was used to predict the short-term kinetics of the viral load in lymphoid organ following the arrival of productively infected antigen-presenting cell under various immune pathophysiological conditions. Transition to a 3D formulation for the description of immune cell motility in LT using the second Newtonian law for the governing equations was performed. Using the HPC programming technologies OpenMP and CUDA, the agent-based model of the immune cell dynamics in 3D space of the lymph node with the fibroblastic reticular cell (FRC) network included was developed. The model describes the T-cell migration, intercellular interactions, cell division and chemotaxis as well as the spreading of molecular signaling factors in the intercellular space displaying a complex geometry. The impact of the FRC integrity on migration of T cells towards antigen-presenting cells during the initial phase of response was examined. For the mathematical model of experimental infection of mice with lymphocytic choriomeningitis virus (LCMV) a comprehensive steady state analysis was performed using the algorithms developed by Dr. Yu. Nechepurenko. The domains in the parameter space enabling multi-stability of the model (up to four steady states) were identified. The transitions between the states were explored using the optimal perturbation methodology. For the Marchuk-Petrov model of antiviral immune response the existence of multiple steady states was examined. We examined the applicability of various model reduction approaches to deal with the formulation of low-dimensional models for intracellular HIV replication which are needed as building blocks for multiscale models. A set of parsimonious models was specified. For modelling the immune regulatory processes which determine the acute or chronic phenotype of the LCMV infection in mice the associated differences in the kinetics of type I IFN response were determined. The data were obtained experimentally (Prof. A. Meyerhans’s group) which characterize the subpopulations of dendritic cells (XCR1-cDC) and the antigen-induced exhausted CD8+ T cells in spleen of infected mice. Following the mechanism of chronic infection control discovered in the murine LCMV infection, the data sets were prepared for mathematical modelling. These data sets characterize the viral load, CD4+ T cell counts and the relative fraction of DC subsets (pDC, cCD1, cCD2) expressing CD45. We developed an extended mathematical model for the CTL-mediated control of the chronic HIV infection. The model was used to make clinically relevant predictions on the therapeutic effect of the PD-L1 blockade which is used for revitalization of T- and B lymphocytes. The model predicts in a quantitative way the effect for various infection phenotypes in relation to the viral load and the immune status of the HIV-infected patient. A hybrid approach to modelling the stochastic dynamics of virus infection with the virus production delay was developed based on the Markov process formulation (joint work with Dr. I. Sazonov, Swansea University, UK). The method considers the differences in the population sizes and allows one to switch between the deterministic and stochastic modes of the solution trajectory computations. Three deterministic versions of the primary HIV infection model were formulated and examined. The models differ with respect to the number of state variables, the details of the target cell infection mode and the production of viruses as well as the development of the antiviral T cell immune response. The necessary and sufficient conditions for the stability of the virus-free steady state were deduced. Then, a two compartmental model of HIV infection was developed to describe the infection spreading between lymph nodes (LN). The HIV infection in each LN is described by a high-dimensional system of integral and differential equations whereas the fluxes of cells and viruses between the LNs are represented by time-delay terms. The numbers of cells and viruses moving between the LNs are described by integral equations of convolution type or equivalent special form delay-differential equations. The conditions on the parameters of the model characterizing the antigen-specific immune response required for a complete eradication of the HIV infection within a finite time interval were studied numerically. A stochastic version of the multi-compartmental model of HIV infection was developed. The models admits the description of the duration of the transfer time of cells and viruses between the compartments (correspond to LNs) in the form of rather arbitrary deterministic functions subject to some biologically-defined constrains. For the numerical solution of the model a Monte Carlo based method was developed. The method combines the algorithms for preserving the history of the random process in the form of the data lists for the point-distributions which represent the stochastic analogues of the integral convolution equations. The model was used to show that the probability of the HIV infection eradication depends on the virus dose, the specific antigen-cellular immune response parameters in each lymph node and the transfer rate of cell and virions between the LNs. For the 3D modelling of the lymph filtration in lymph node, a mathematical model based on the integral representation of the velocity and pressure fields was formulated, justified and numerically implemented (joint work with Dr. A.V. Setukha). The equivalence of the boundary integral equations and the boundary value problem for the original differential Darcy-Brinkman law was proved. The equivalence also holds true for a more general filtration problem for viscous fluid in a bounded domain with finitely many inclusions representing sources or sinks. It was proved that the velocity and pressure fields satisfying the boundary value filtration problem can be represented in the form of corresponding surface potentials, and the densities of these potentials must satisfy the formulated system of boundary integral equations. To treat the matching problem for the flows in the exterior and interior domains, the corresponding integral representation for the velocity- and pressure fields was developed. The quantification of 3D morphometric parameters and the topological properties representing the structural organization of the conduit network in LN of mice was performed (collaborative work with Prof. B. Ludewig, the Institute of Immunobiology, St. Gallen Hospital, Switzerland). The generated data include the frequency distribution of the edge lengths, the node degrees and the bifurcation angles. The obtained data provide a solid quantitative basis for developing a physiologically consistent multiscale LN model to describe the fluid balance in the lymphoid tissues. Multiscale mathematical model for the HIV infection dynamics has been developed which integrates the immune cell behavior across two physiological scales, the lymphoid tissue and blood. The model describes the lymphocyte turnover as a stochastic process. Some major results of the research in year 2019 were published in mainstream journals (Q1) and commented in press-releases: https://nauka.tass.ru/nauka/7109351; https://ria.ru/20191113/1560890750.html https://www.gazeta.ru/science/news/2019/11/13/n_13692140.shtml; https://www.kommersant.ru/doc/4157546; https://indicator.ru/mathematics/matematicheskaya-model-rasschitala-chto-12-tysyach-podvizhnyh-limfocitov-pobedyat-vich-19-06-2019.htm https://ria.ru/20190619/1555691649.html http://www.ras.ru/news/shownews.aspx?id=3d3b667e-5a2a-4eaf-ac01-0a2390fbe0a3&print=1 https://www.gazeta.ru/science/news/2019/06/19/n_13110013.shtmlhttps://indicator.ru/news/2019/06/19/matematicheskaya-model-rasschitala-chto-12-tysyach-podvizhnyh-limfocitov-pobedyat-vich/

 

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