We formulate a compartmental model of the murine lymphatic system with the transfer rate parameters derived from the data on the geometric characteristics of the lymphatic system (LS) graph structure and the Hagen-Poiseuille-based values of the lymph flows through the system components, i.e., vertices and edges. It is supplemented by the physics-based model of lymph node draining-related function which considers a paradigmatic view of its geometry with one- and three-afferent lymphatic vessels and one efferent vessel, and the lymph flow described by the Darcy-Starling equations. We discuss further modelling work needed to gain a predictive understanding of the LS function in response to various perturbations including infections and therapeutic treatments.

Animal models of diseases, particularly mice, are considered to be the cornerstone for translational research in immunology. The aim of the present study is to model the geometry and analyze the network structure of the murine lymphatic system (LS). The algorithm for building the graph model of the LS makes use of anatomical data. To identify the edge directions of the graph model, a mass balance approach to lymph dynamics based on the Hagen–Poiseuille equation is applied. It is the first study in which a geometric model of the murine LS has been developed and characterized in terms of its structural organization and the lymph transfer function. Our study meets the demand for quantitative mechanistic approaches in the growing field of immunoengineering to utilize or exploit the lymphatic system for immunotherapy.

The immune system is a complex distributed system consisting of cells, which circulate through the body, communicate and turnover in response to antigenic perturbations. We discuss new approaches to modelling the functioning of the immune system of humans and experimental animals with a focus on its 'complexity'. Emerging mathematical and computer models are reviewed to describe the immune system diversity, the cell/cytokine network communication structures, hierarchical regulation, and evolutionary dynamics of immune repertoires.

OBJECTIVE:Population screening for aneurysms in patients with risk factors and preventive surgical treatment are beneficial according to numerous studies. One of the most significant risk factors is heredity, namely, the presence of first-degree relatives (FDR) with aneurysmal subarachnoid hemorrhage (aSAH). Nevertheless, there are still no generally accepted approaches or evidence bases regarding the benefits of the aneurysm screening strategy. METHODS:Mathematical modeling of the dynamics of aneurysm development in the population was carried out using an algorithm implementing a discrete Markov's chain. To implement the model, all probabilities of events and distributions are taken from available literature sources. Three-dimensional time of flight noncontrast magnetic resonance angiography was chosen as a screening method. Patients underwent preventive surgical treatment if an aneurysm was detected. RESULTS:Screening and preventive treatment in the general population reduces the prevalence of aneurysms by 1.74% (3.44% in the FDR group) and the prevalence of aSAH by 14.36% (37.48% in the FDR group). Mortality due to aSAH was reduced by 14.44%. The number of disabilities also decreases. The occurrence of deep disability was reduced by 20.2% in the FDR group. Economic analysis of the part of the population consisting of FDRs showed annual savings of ies also decr CONCLUSIONS: The mathematical model demonstrated that screening and preventive treatment of cerebral aneurysms can reduce aSAH-associated morbidity and mortality. In the FDR group, there was decrease in the prevalence of aSAH and decrease in associated mortality. Screening for cerebral aneurysms is cost-effective.

A mathematical model of the human immunodeficiency virus Type 1 (HIV-1) life cycle in CD4 T cells was constructed and calibrated. It describes the activation of the intracellular Type I interferon (IFN-I) response and the IFN-induced suppression of viral replication. The model includes viral replication inhibition by interferon-induced antiviral factors and their inactivation by the viral proteins Vpu and Vif. Both deterministic and stochastic model formulations are presented. The stochastic model was used to predict efficiency of IFN-I-induced suppression of viral replication in different initial conditions for autocrine and paracrine effects. The probability of virion excretion for various MOIs and various amounts of IFN-I was evaluated and the statistical properties of the heterogeneity of HIV-1 and IFN-I production characterised.

Subarachnoid hemorrhages due to rupture of cerebral aneurysms have a high risk of disability and mortality. Screening of the population to detect aneurysms in patients with risk factors is currently not carried out in Russia. However, the detection of clinically silent aneurysms and their subsequent prophylactic surgical treatment are justified, according to numerous studies.BACKGROUND:Demonstrate the clinical and economic feasibility of screening the population (including first-line relatives) for cerebral aneurysms using an economic and mathematical model of the RF virtual population.MATERIAL AND METHODS:Mathematical modeling was carried out using an algorithm that implements a discrete Markov chain. The virtual population consisted of 145 million people (the population of the Russian Federation). Magnetic resonance angiography 3DTOF was chosen as a screening method. Virtual patients underwent preventive surgical treatment in case of detection of aneurysm during screening. The number of aneurysms in the population, the number of aneurysmal subarachnoid hemorrhage (aSAH), the cost and outcomes of treatment, and the risk of disability were calculated.RESULTS:In the case of screening and preventive surgical treatment of aneurysms, there is a decrease in the number of aSAH by 14.3% (37.5% in first-line relatives (RPLR), which affects the reduction in mortality due to aSAH by 14.4% (24.1% in The total number of disabled people is reduced by 1.5% (5.1% for the RPHR). A shift in the structure of disability towards greater labor and social adaptation of patients was noted. An economic analysis for the entire population showed that screening saves 7.7 billion annually rubles, including in the population consisting of RPLR - 4.9 billion rubles.CONCLUSION:The created mathematical model of the virtual population demonstrated that screening and prophylactic treatment of cerebral aneurysms makes it possible to reduce the number of aSAH and associated mortality among the entire population and in the RPLR group. The number of individuals with severe disabilities is decreasing. Thus, population screening for the detection of cerebral aneurysms may be clinically effective and cost-effective in the general population, especially in RPCR.

ФЕДЕРАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ ЦЕНТР ФУНДАМЕНТАЛЬНОЙ И ТРАНСЛЯЦИОННОЙ МЕДИЦИНЫ ФЕДЕРАЛЬНЫЙ ИССЛЕДОВАТЕЛЬСКИЙ ЦЕНТР «КРАСНОЯРСКИЙ НАУЧНЫЙ ЦЕНТР СИБИРСКОГО ОТДЕЛЕНИЯ РОССИЙСКОЙ АКАДЕМИИ НАУК» ИНСТИТУТ ВЫЧИСЛИТЕЛЬНОГО МОДЕЛИРОВАНИЯ СО

The immune system is a complex multiscale multiphysical object. Understanding its functioning in the frame of systemic analysis implies the use of mathematical modelling, formulation of data consistency criterion, estimation of parameters, uncertainty analysis, and optimal model selection. In this work, we present some promising approaches to modelling the multi-physics immune processes, i.e., cell migration in lymph nodes (LN), lymph flow, homeostatic regulation of immune responses in chronic infections. To describe the spatial-temporal dynamics of immune responses in lymph LN, we propose a model of lymphocyte migration, based on the second Newtons law and considering three kinds of forces. The empirical distributions of three lymphocytes motility characteristics were used for model calibration using the KolmogorovSmirnov metric. Prediction of lymph flow in a lymph node requires costly computations, due to diversity of sizes, forms, inner structure of LNs and boundary conditions. We proposed an approach to lymph flow modelling based on replacing the full-fledged computational physics-based model with an artificial neural network (ANN), trained on the set of pre-formed results computed using an initial mechanistic model. The ANN-based model reduces the computational time for some lymph flow characteristics by four orders of magnitude. Calibration of MarchukPetrov model of antiviral immune response for SARS-CoV-2 infection was performed. To this end, we used previously published data on the viral load kinetics in nasopharynx of volunteers, and data on the observed ranges of interferon, antibodies and CTLs in the blood. The parameters, which have the most significant impact at different stages of infection process, were identified. Inhibition of immune mechanisms, e.g., T cell exhaustion, is a distinctive feature of chronic viral infections and malignant diseases. We propose a mathematical model for the studies of regulation parameters of four exhausted T cell subsets in order to examine the balance of their proliferation and differentiation determined by interaction with SIRPa+ PD-L1+ and XCR+1 dendritic cells. The model parameters are evaluated, in order to study the reinvigoration effect of aPD-L1 therapy on the homeostasis of exhausted cells.

Pancreatic cancer represents one of the difficult problems of contemporary medicine. The development of the illness evolves very slowly, happens in a specific place (stroma), and manifests clinically close to a final stage. Another feature of this pathology is a coexistence (symbiotic) effect between cancer cells and normal cells inside stroma. All these aspects make it difficult to understand the pathogenesis of pancreatic cancer and develop a proper therapy. The emergence of pancreatic pre-cancer and cancer cells represents a branching stochastic process engaging populations of 64 cells differing in the number of acquired mutations. In this study, we formulate and calibrate the mathematical model of pancreatic cancer using the quasispecies framework. The mathematical model incorporates the mutation matrix, fineness landscape matrix, and the death rates. Each element of the mutation matrix presents the probability of appearing as a specific mutation in the branching sequence of cells representing the accumulation of mutations. The model incorporates the cancer cell elimination by effect CD8 T cells (CTL). The down-regulation of the effector function of CTLs and exhaustion are parameterized. The symbiotic effect of coexistence of normal and cancer cells is considered. The computational predictions obtained with the model are consistent with empirical data. The modeling approach can be used to investigate other types of cancers and examine various treatment procedures.

The lymph node (LN) represents a key structural component of the lymphatic system network responsible for the fluid balance in tissues and the immune system functioning. Playing an important role in providing the immune defense of the host organism, LNs can also contribute to the progression of pathological processes, e.g., the spreading of cancer cells. To gain a deeper understanding of the transport function of LNs, experimental approaches are used. Mathematical modeling of the fluid transport through the LN represents a complementary tool for studying the LN functioning under broadly varying physiological conditions. We developed an artificial neural network (NN) model to describe the lymph node drainage function. The NN model predicts the flow characteristics through the LN, including the exchange with the blood vascular systems in relation to the boundary and lymphodynamic conditions, such as the afferent lymph flow, Darcy’s law constants and Starling’s equation parameters. The model is formulated as a feedforward NN with one hidden layer. The NN complements the computational physics-based model of a stationary fluid flow through the LN and the fluid transport across the blood vessel system of the LN. The physical model is specified as a system of boundary integral equations (IEs) equivalent to the original partial differential equations (PDEs; Darcy’s Law and Starling’s equation) formulations. The IE model has been used to generate the training dataset for identifying the NN model architecture and parameters. The computation of the output LN drainage function characteristics (the fluid flow parameters and the exchange with blood) with the trained NN model required about 1000-fold less central processing unit (CPU) time than computationally tracing the flow characteristics of interest with the physics-based IE model. The use of the presented computational models will allow for a more realistic description and prediction of the immune cell circulation, cytokine distribution and drug pharmacokinetics in humans under various health and disease states as well as assisting in the development of artificial LN-on-a-chip technologies.

Pathophysiology of ischemic heart disease is a complex phenomenon determined by the interaction of multiple processes including the inflammatory, immunological, infectious, mechanical, biochemical and epigenetic ones. A predictive clinically relevant modelling of the entire trajectory of the human organism, from the initial alterations in lipid metabolism through to atherosclerotic plaque formation and finally to the pathologic state of the ischemic heart disease, is an open insufficiently explored problem. In the present review, we consider the existing mathematical frameworks which are used to describe, analyze and predict the dynamics of various processes related to cardiovascular diseases at the molecular, cellular, tissue, and holistic human organism level. The mechanistic, statistical and machine learning models are discussed in detail with special focus on the underlying assumptions and their clinical relevance. All together, they provide a solid computational platform for further expansion and tailoring for practical applications.

The human lymphatic system (HLS) is a complex network of lymphatic organs linked through the lymphatic vessels. We present a graph theory-based approach to model and analyze the human lymphatic network. Two different methods of building a graph are considered: the method using anatomical data directly and the method based on a system of rules derived from structural analysis of HLS. A simple anatomical data-based graph is converted to an oriented graph by quantifying the steady-state fluid balance in the lymphatic network with the use of the Poiseuille equation in vessels and the mass conservation at vessel junctions. A computational algorithm for the generation of the rule-based random graph is developed and implemented. Some fundamental characteristics of the two types of HLS graph models are analyzed using different metrics such as graph energy, clustering, robustness, etc.

This paper presents current knowledge about the structure and function of the lymphatic system. Mathematical models of lymph flow in the single lymphangion, the series of lymphangions, the lymph nodes, and the whole lymphatic system are considered. The main results and further perspectives are discussed.

Mathematical immunology is the branch of mathematics dealing with the application of mathematical methods and computational algorithms to explore the structure, dynamics, organization and regulation of the immune system in health and disease. We review the conceptual and mathematical foundation of modelling in immunology formulated by Guri I. Marchuk. The current frontier studies concerning the development of multiscale multiphysics integrative models of the immune system are presented.

Aims: The problem of effective treatment of HIV-infected patients is an important task of clinical virology and immunology due to the high cost of drugs, the presence of side effects and the need for strict adherence to the schedule of drug intake for patients. Therefore, the urgent task is to develop new approaches to optimize the use of antiretroviral therapy to reduce the cost of treatment and to improve the quality of life for patients. The tasks are addressed to test the hypothesis that the system of therapeutic interruptions in the treatment of HIV infection can give better results (both the duration and comfort of the patient’s life, and the need for fewer drugs) compared with regular medication in standard doses. Methods: In this work, an extended version of the mathematical model of the immune response in HIV infection (proposed in Hadjiandreou et al., 2009) was constructed to take into account the hormonal regulation of the immune response and the impact of antiretroviral drugs on the course of the disease, the calibration of the parameters of the resulting model to match the actual trends of the disease and the search for an optimal treatment strategy. The model is formulated as a system of ordinary differential equations. The therapy optimization is modeled following the structured treatment interruptionapproach using the methods of simulated annealing and the simplex method. The mathematical model and optimization methods are implemented in C ++. Results: It has been shown that in treating HIV-infected patients, it is possible to significantly (up to 3 times) reduce the total amount of required medications simultaneously with an increase in the duration of the period with a high quality of life (due to reducing the intensity of side effects) during antiretroviral therapy. Conclusion: The use of mathematical models and optimization methods opens up the possibility for the implementation of personalized approaches to the treatment of HIV infection, taking into account the side effects, the hormonal status of patients and the cost of drugs.

The lymphatic system is a body-wide network of lymphatic vessels and lymphoid organs. The complexity of the structural and functional organization of the lymphatic system implies the necessity of using computational modeling approaches to unravel the mechanisms of its regulation in quantitative terms. Although it is a vital part of the circulatory and immune systems, the lymphatic system remains poorly investigated as a mathematical modeling object. Modeling of the lymphatic vessel network needs to be established using a systematic approach in order to advance the model-driven research of this important physiological system. In our study, we elucidate key general features underlying the 3D structural organization of the lymphatic system in order to develop computational geometry and network graph models of the human lymphatic system based on available anatomical data (from the PlasticBoy project), which provides an estimate of the structure of the lymphatic system, and to analyze the topological properties of the resulting models.

In this study, we discuss critical issues in modelling the structure and function of lymph nodes (LNs), with emphasis on how LN physiology is related to its multi-scale structural organization. In addition to macroscopic domains such as B-cell follicles and the T cell zone, there are vascular networks which play a key role in the delivery of information to the inner parts of the LN, i.e., the conduit and blood microvascular networks. We propose object-oriented computational algorithms to model the 3D geometry of the fibroblastic reticular cell (FRC) network and the microvasculature. Assuming that a conduit cylinder is densely packed with collagen fibers, the computational flow study predicted that the diffusion should be a dominating process in mass transport than convective flow. The geometry models are used to analyze the lymph flow properties through the conduit network in unperturbed- and damaged states of the LN. The analysis predicts that elimination of up to 60%–90% of edges is required to stop the lymph flux. This result suggests a high degree of functional robustness of the network.

The human immune system is characterized by enormous cellular and anatomical complexity. Lymph nodes are key centers of immune reactivity, organized into distinct structural and functional modules including the T-cell zone, fibroblastic reticular cell (FRC) network and the conduit system. A thorough understanding of the modular organization is a prerequisite for lymphoid organ tissue-engineering. Due to the biological complexity of lymphoid organs, the development of mathematical models capable of elaborating the lymph node architecture and functional organization, has remained a major challenge in computational biology. Here, we present a computational method to model the geometry of the FRC network and fluid flow in the conduit system. It differs from the blood vascular network image-based reconstruction approaches as it develops the parameterized geometric model using the real statistics of the node degree and the edge length distributions. The FRC network model is then used to analyze the fluid flow through the underlying conduit system. A first observation is that the pressure gradient is approximately linear, which suggests homogeneity of the network. Furthermore, calculated permeability values (≈0.0033μm2) show the generated network is isotropic, while investigating random variations of pipe radii (with a given mean and standard deviation) shows a significant effect on the permeability. This framework can now be further explored to systematically correlate fundamental characteristics of the FRC conduit system to more global material properties such as permeability.

This work presents an idealized geometric model of the lymph node (LN) consisting of a set of 'elementary' macroscopic structures such as subcapsular sinus, trabecular sinuses with a special focus on fibroblastic reticular cells (FRC) network. An experimental data driven computational algorithm has been developed to reconstruct the macroscopic structure of FRC network based on high-resolution confocal microscopy data of a murine LN. The FRC geometric model can be further discretized for use in computational studies of lymph and cytokines transport through the LN.