Vitalii Volpert
Doctor of Physical and Mathematical Sciences
Director of the interdisciplinary research center “Mathematical modeling in Biomedicine” of S.M. Nikol’skii Mathematical Institute of RUDN, s.M. Nikol’skii Mathematical Institute

Mathematics and modeling for the study of natural phenomena.


Graduated with honors from the Faculty of Mechanics and Mathematics of the Southern Federal University (SFU, Rostov–on-Don).


PhD. thesis “Autowave processes in chemically active media”.

1980 - 1991

Worked at the Institute of Chemical Physics of the USSR Academy of Sciences (Chernogolovka), worked his way up from engineer to head of the laboratory of Macrokinetics of polymerization processes.


Schelkin prize of the Soviet Academy of Sciences.


Visiting researcher at the Courant Institute Mathematical Sciences, New York University, USA. 

1991 - 1992

Research Fellow, Department of Materials Science, Northwestern University, USA.

1992 - present

Works at the French National Center for Scientific Research (Directeur de recherche, Centre National de la Recherche Scientifique) and at University Lyon 1, France. 


Habilitation, “Mathematical theory of reaction-diffusion equations and their application in chemical physics”, University Lyon 1.

2004 - 2006

Deputy Director of Camille Jordan Institute, Lyon, France.

2010 - 2012

Member of the board of Directors of the Institute for Systems Biology and Medicine, Lyon, France.

2012 - 2018

Member of the Council of the European Society for Theoretical and Mathematical Biology.

2018 - present

Director of the interdisciplinary research center “Mathematical modeling in Biomedicine” of S.M. Nikol’skii Mathematical Institute of RUDN.

Member of editorial boards of the following journals:

  • Mathematical modelling of natural phenomena (founder and editor-in-chief, 2006),
  • Complex variables and elliptic equations (2017),
  • Computation (2018),
  • Computer research and modelling (2018),
  • Pure and Applied Functional Analysis (2016),
  • Mathematics (2019).


  1. Gives the course of lectures “Reaction-diffusion equations and applications” for postgraduate students of the direction “Differential equations, dynamical systems and optimal control”.
  2. 2014 - 2018 gave courses on partial differential equations and mathematical modeling to students, postgraduate students at the Faculty of Mathematics at the Higher school of Kouba. (Ecole Normale de Kouba, Algeria) and other educational institutions in Algeria.
  3. 2017 gave courses on partial differential equations and mathematical modeling to students, postgraduate students at the Faculty of Mathematics of the National Institute of Technology Patna, India.
  4. 2012 - 2013 gave courses on partial differential equations and mathematical modeling to students at the Engineering Faculty of Ecole Centrale de Lyon France.
  5. Repeatedly was invited for scientific visits and lectures to England, Israel, India, Poland, USA, Chile and other countries. Last three years:
    • 2018 - University of Talca (Chili), University of Warsaw (Poland), University of Tlemcen (Algeria)
    • 2017 - King’s college London (UK), University of Tlemcen (Algeria), University of Talca (Chili), Baumann University Moscow (Russia), Indian Institute of Technology Patna (India)
    • 2016 - University of Leiden (Netherlands), University of Talca (Chili), Institute of numerical mathematics, Moscow (Russia)


  • Elliptic problems in unbounded domains were studied. Fredholm conditions for general elliptic operators in unbounded domains were obtained. Solvability conditions of linear problems, index, solvability conditions for non-Fredholm operators were studied. Property conditions of general nonlinear elliptic operators in unbounded domains were obtained and a topological degree was constructed.
  • Reaction-diffusion waves, in particular, existence and stability of waves for monotonic and locally monotonic systems, minimax representation of wave velocity were studied. Application to various problems of chemical kinetics, population dynamics, biomedicine. Generalized travelling waves.
  • Nonlocal reaction-diffusion equations and equations with delay were studied. Local and global stability of equilibrium states, existence of travelling waves and generalized travelling waves, bifurcations and nonlinear dynamics were studied. Applications to various problems of population dynamics and biomedicine.
  • Waves in combustion and chemical kinetics, the existence, stability, bifurcations, nonlinear dynamics were studied. Thermal explosion with convection, thermal explosion conditions, convection influence, space-time structures, vibrational thermal explosion, thermal explosion in porous medium were studied.
  • Existence, stability, propagation velocity, nonlinear dynamics of front polymerization waves were studied. Production technology of poly-epsilon-caprolactam based on frontal polymerization. Influence of convection on the propagation of frontal polymerization waves was studied. Low-temperature waves accompanied by the destruction of a solid.
  • Interphase and capillary phenomena in miscible liquids were studied, and experiments on the international space station were prepared.
  • Mathematical modeling of atherosclerosis. Models of atherosclerosis as a chronic inflammation of the artery walls were developed. Development of atherosclerosis depending on the cholesterol content was studied. Development of atherosclerosis as a reaction-diffusion wave. Existence and stability of waves in one-dimensional formulation and in two-dimensional formulation with nonlinear boundary conditions were investigated. Interaction of an atherosclerotic plaque with the flow.
  • Blood clotting and comorbidities. Different modes of blood clotting and conditions of their implementation. Blood clotting as a reaction-diffusion wave. Existence, stability and speed of propagation. Initial conditions for initiation of coagulation and existence of a solution in the form of a stationary pulse. Influence of various factors on clotting: blood flow, platelets, inflammation. Thrombosis and bleeding. Identification of patients with hemophilia.
  • Modeling of cancer: leukemia and multiple myeloma. Development of mathematical models, analysis and numerical modeling, characterization of myeloblastic leukemia by flow cytometry. Modeling of the development of multiple myeloma. Selection of optimal treatment protocols.
  • Mathematical modeling of erythropoiesis. Development of hybrid models of erythropoiesis taking into account the presence of cells of different types, intracellular and intercellular regulation. Functioning of erythroblastic islets and production of erythrocytes, reaction to hypoxia, regulation by erythropoietin.
  • Mathematical immunology. Development of viral infection as a reaction-diffusion phenomenon. Existence, resistance and modes of spread; influence of delayed immune response; virus mutations; antiviral therapy and emergence of resistant strains.
  • Electrical stimulation of the cerebral cortex for rehabilitation of patients after stroke. Modeling of electric potential waves in the cerebral cortex on the basis of integro-differential equations of the mean field theory. Different wave propagation modes, stability, bifurcations, nonlinear dynamics. Selection of stimulation modes to restore waves characteristics in damaged areas of the brain.
  • Development of hybrid models in biomedicine based on the combination of discrete cell models and continuous models for intracellular regulation (ODE) and intercellular regulation (PDE). Application to various problems of biomedicine (leukemia, lymphoma, myeloma, erythropoiesis). Hybrid models with dissipative particle dynamics to study blood flow clotting.
  • Evolution of biological species. Development of models of evolution based on nonlocal reaction-diffusion equations, taking into account competition for resources. Conditions of emergence of new species, description of different modes of evolution of species. Interrelation of different definitions of species (by Darwin and Mair). Nonlocal predator-prey models.
  • Morphogenesis and modeling of plant growth. Development of plant growth models, how to explain plant diversity, branching; vegetative hormones and nutrients. Two-dimensional models based on cell division, plant growth as self-similar structures. Different models of morphogenesis and wound healing.
  • Other issues: economic and demographic models, phase transitions in metal oxides, propagation of calcium waves, etc.

Scientific interests

  • General theory of elliptic equations in unbounded domains;
  • Mathematical theory of reaction-diffusion waves with applications in chemical kinetics and combustion;
  • Nonlocal reaction-diffusion equations and equations with delay;
  • Vibrational thermal explosion;
  • Capillary phenomena in miscible fluids;
  • Mathematical theory of the origin and evolution of species;
  • Mathematical theory and modeling of biomedical processes: atherosclerosis and other chronic inflammations, blood clotting and thrombosis, cancer;
  • New methods of mathematical modeling in Biomedicine;
  • The study of mathematical models of various biological and ecological issues: plant growth, morphogenesis, etc.
Aneurysms of saccular shape are usually associated with a slow, almost stagnant blood flow, as well as a consequent emergence of blood clots. Despite the practical importance, there is a lack of computational models that could combine platelet aggregation, precise biorheology, and blood plasma coagulation into one efficient framework. In the present study, we address both the physical and biochemical effects during thrombosis in aneurysms and blood recirculation zones. We use continuum description of the system and partial differential equation-based model that account for fluid dynamics, platelet transport, adhesion and aggregation, and biochemical cascades of plasma coagulation. The study is focused on the role of transport and accumulation of blood cells, including contact interactions between platelets and red blood cells (RBCs), coagulation cascade triggered by activated platelets, and the hematocrit-dependent blood rheology. We validated the model against known experimental benchmarks for in vitro thrombosis. The numerical simulations indicate an important role of RBCs in spatial propagation and temporal dynamics of the aneurysmal thrombus growth. The local hematocrit determines the viscosity of the RBC-rich regions. As a result, a high hematocrit slows down flow circulation and increases the presence of RBCs in the aneurysm. The intensity of the flow in the blood vessel associated with the aneurysm also affects platelet distribution in the system, as well as the steady shape of the thrombus.
The paper is devoted to a compartmental epidemiological model of infection progression in a heterogeneous population which consists of two groups with high disease transmission (HT) and low disease transmission (LT) potentials. Final size and duration of epidemic, the total and current maximal number of infected individuals are estimated depending on the structure of the population. It is shown that with the same basic reproduction number R0 in the beginning of epidemic, its further progression depends on the ratio between the two groups. Therefore, fitting the data in the beginning of epidemic and the determination of R0 are not sufficient to predict its long time behaviour. Available data on the Covid-19 epidemic allows the estimation of the proportion of the HT and LT groups. Estimated structure of the population is used for the investigation of the influence of vaccination on further epidemic development. The result of vaccination strongly depends on the proportion of vaccinated individuals between the two groups. Vaccination of the HT group acts to stop the epidemic and essentially decreases the total number of infected individuals at the end of epidemic and the current maximal number of infected individuals while vaccination of the LT group only acts to protect vaccinated individuals from further infection.
Spatial distribution of the human population is distinctly heterogeneous, e.g. showing significant difference in the population density between urban and rural areas. In the historical perspective, i.e. on the timescale of centuries, the emergence of densely populated areas at their present locations is widely believed to be linked to more favourable environmental and climatic conditions. In this paper, we challenge this point of view. We first identify a few areas at different parts of the world where the environmental conditions (quantified by the temperature, precipitation and elevation) show a relatively small variation in space on the scale of thousands of kilometres. We then examine the population distribution across those areas to show that, in spite of the approximate homogeneity of the environment, it exhibits a significant variation revealing a nearly periodic spatial pattern. Based on this apparent disagreement, we hypothesize that there may exist an inherent mechanism that may lead to pattern formation even in a uniform environment. We consider a mathematical model of the coupled demographiceconomic dynamics and show that its spatially uniform, locally stable steady state can give rise to a periodic spatial pattern due to the Turing instability, the spatial scale of the emerging pattern being consistent with observations. Using numerical simulations, we show that, interestingly, the emergence of the Turing patterns may eventually lead to the system collapse.
In nature, different species compete among themselves for common resources and favorable habitat. Therefore, it becomes really important to determine the key factors in maintaining the bio-diversity. Also, some competing species follow cyclic competition in real world where the competitive dominance is characterized by a cyclic ordering. In this paper, we study the formation of a wide variety of spatiotemporal patterns including stationary, periodic, quasi-periodic and chaotic population distributions for a diffusive Lotka–Volterra type three-species cyclic competition model with two different types of cyclic ordering. For both types of cyclic ordering, the temporal dynamics of the corresponding non-spatial system show the extinction of two species through global bifurcations such as homoclinic and heteroclinic bifurcations. For the spatial system, we show that the existence of Turing patterns is possible for a particular cyclic ordering, while it is not the case for the other cyclic ordering through both the analytical and numerical methods. Further, we illustrate an interesting scenario of short-range invasion as opposed to the usual invasion phenomenon over the entire habitat. Also, our study reveals that both the stationary and dynamic population distributions can coexist in different parts of a habitat. Finally, we extend the spatial system by incorporating nonlocal intra-specific competition terms for all the three competing species. Our study shows that the introduction of nonlocality in intra-specific competitions stabilizes the system dynamics by transforming a dynamic population distribution to stationary. Surprisingly, this nonlocality-induced stationary pattern formation leads to the extinction of one species and hence, gives rise to the loss of bio-diversity for intermediate ranges of nonlocality. However, the bio-diversity can be restored for sufficiently large extent of nonlocality.
Blood coagulation represents one of the most studied processes in biomedical modelling. However, clinical applications of this modelling remain limited because of the complexity of this process and because of large inter-patient variation of the concentrations of blood factors, kinetic constants and physiological conditions. Determination of some of these patients-specific parameters is experimentally possible, but it would be related to excessive time and material costs impossible in clinical practice. We propose in this work a methodological approach to patient-specific modelling of blood coagulation. It begins with conventional thrombin generation tests allowing the determination of parameters of a reduced kinetic model. Next, this model is used to study spatial distributions of blood factors and blood coagulation in flow, and to evaluate the results of medical treatment of blood coagulation disorders.
A prey-predator model with a sexual reproduction in prey population and nonlocal consumption of resources by prey in two spatial dimensions is considered. Patterns produced by the model without nonlocal terms and periodic boundary conditions are studied first. Then, Turing patterns induced by the nonlocal interaction (see Banerjee et al. (2018) [1]) in the two dimensional space are explored along with the effects of the nonlocal interaction range on the resulting patterns under proper parametric restrictions. The Turing bifurcation conditions for the nonlocal model are derived analytically and bifurcation scenario of stationary hotspot pattern generated from the homogeneous steady-state are studied in detail, both analytically and numerically. Also, conversion of periodic and aperiodic solutions exhibited by the local model into stationary Turing pattern as an effect of the nonlocal interaction term is also explored. The resulting patterns are stationary when the range of nonlocal interactions are significantly large.
Damage to arterial vessel walls leads to the formation of platelet aggregate, which acts as a physical obstacle for bleeding. An arterial thrombus is heterogeneous; it has a dense inner part (core) and an unstable outer part (shell). The thrombus shell is very dynamic, being composed of loosely connected discoid platelets. The mechanisms underlying the observed mobility of the shell and its (patho)physiological implications are unclear. To investigate arterial thrombus mechanics, we developed a novel, to our knowledge, two-dimensional particle-based computational model of microvessel thrombosis. The model considers two types of interplatelet interactions: primary reversible (glycoprotein Ib (GPIb)-mediated) and stronger integrin-mediated interaction, which intensifies with platelet activation. At high shear rates, the former interaction leads to adhesion, and the latter is primarily responsible for stable platelet aggregation. Using a stochastic model of GPIb-mediated interaction, we initially reproduced experimental curves that characterize individual platelet interactions with a von Willebrand factor-coated surface. The addition of the second stabilizing interaction results in thrombus formation. The comparison of thrombus dynamics with experimental data allowed us to estimate the magnitude of critical interplatelet forces in the thrombus shell and the characteristic time of platelet activation. The model predicts moderate dependence of maximal thrombus height on the injury size in the absence of thrombin activity. We demonstrate that the developed stochastic model reproduces the observed highly dynamic behavior of the thrombus shell. The presence of primary stochastic interaction between platelets leads to the properties of thrombus consistent with in vivo findings; it does not grow upstream of the injury site and covers the whole injury from the first seconds of the formation. А simplified model, in which GPIb-mediated interaction is deterministic, does not reproduce these features. Thus, the stochasticity of platelet interactions is critical for thrombus plasticity, suggesting that interaction via a small number of bonds drives the dynamics of arterial thrombus shell.
We prove the existence of solutions for some semilinear elliptic equations in the appropriate H4 spaces using the fixed-point technique where the elliptic equation contains fourth-order differential operators with and without Fredholm property, generalizing the previous results.
The work is devoted to the analysis of cell population dynamics where cells make a choice between differentiation and apoptosis. This choice is based on the values of intracellular proteins whose concentrations are described by a system of ordinary differential equations with bistable dynamics. Intracellular regulation and cell fate are controlled by the extracellular regulation through the number of differentiated cells. It is shown that the total cell number necessarily oscillates if the initial condition in the intracellular regulation is fixed. These oscillations can be suppressed if the initial condition is a random variable with a sufficiently large variation. Thus, the result of the work suggests a possible answer to the question about the role of stochasticity in the intracellular regulation.
The paper is devoted to a reaction-diffusion problem describing diffusion and consumption of nutrients in a biological tissue consisting of small cells periodically arranged in an extracellular matrix. Cells consume nutrients with a rate proportional to cell area and to nutrient concentration. The dependence on the nutrient concentration can be linear or nonlinear. The cells are modeled by a potential approximating the Dirac’s delta-function. The potential has a periodically distributed support of small measure. The problem contains two small parameters: the diameter of a cell and the distance between the cells (in comparison with the characteristic macroscopic size). In the multi-dimensional formulation assuming some restriction on the relation of parameters, we prove convergence of solution of this problem to the solution of a limiting homogenized problem. We show that the problem is non-homogenizable in classical sense if this restriction fails.
An extended SEIQR type model is considered in order to model the COVID-19 epidemic. It contains the classes of susceptible individuals, exposed, infected symptomatic and asymptomatic, quarantined, hospitalized and recovered. The basic reproduction number and the final size of epidemic are determined. The model is used to fit available data for some European countries. A more detailed model with two different subclasses of susceptible individuals is introduced in order to study the influence of social interaction on the disease progression. The coefficient of social interaction K characterizes the level of social contacts in comparison with complete lockdown (K=0) and the absence of lockdown (K=1). The fitting of data shows that the actual level of this coefficient in some European countries is about 0.1, characterizing a slow disease progression. A slight increase of this value in the autumn can lead to a strong epidemic burst.
A nonlocal reaction–diffusion equation arising in various applications is studied. The speed of traveling waves is determined by means of a minimax representation. It is used to obtain the wave speed estimates and asymptotic values.
This paper represents a literature review on traveling waves described by delayed reaction-diffusion (RD, for short) equations. It begins with the presentation of different types of equations arising in applications. The main results on wave existence and stability are presented for the equations satisfying the monotonicity condition that provides the applicability of the maximum and comparison principles. Other methods and results are described for the case where the monotonicity condition is not satisfied. The last two sections deal with delayed RD equations in mathematical immunology and in neuroscience. Existence, stability, and dynamics of wavefronts and of periodic waves are discussed.
Reaction-diffusion equation with a logistic production term and a delayed inhibition term is studied. Global stability of the homogeneous in space equilibrium is proved under some conditions on the delay term. In the case where these conditions are not satisfied, this solution can become unstable resulting in the emergence of spatiotemporal pattern formation studied in numerical simulations.
Various types of brain activity, including motor, visual, and language, are accompanied by the propagation of periodic waves of electric potential in the cortex, possibly providing the synchronization of the epicenters involved in these activities. One example is cortical electrical activity propagating during sleep and described as traveling waves [Massimini et al., J. Neurosci. 24, 6862-6870 (2004)]. These waves modulate cortical excitability as they progress. We consider the possible role of epicenters and explore a neural field model with two nonlinear integrodifferential equations for the distributions of activating and inhibiting signals. It is studied with symmetric connectivity functions characterizing signal exchange between two populations of neurons, excitatory and inhibitory. Bifurcation analysis is used to investigate the emergence of periodic traveling waves and of standing oscillations from the stationary, spatially homogeneous solutions, and the stability of these solutions. Both types of solutions can be started by local oscillations indicating a possible role of epicenters in the initiation of wave propagation.
A prey-predator model with a nonlocal or global consumption of resources by prey is studied. Linear stability analysis about the homogeneous in space stationary solution is carried out to determine the conditions of the bifurcation of stationary and moving pulses in the case of global consumption. Their existence is confirmed in numerical simulations. Periodic travelling waves and multiple pulses are observed for the nonlocal consumption.
This work deals with a reaction-diffusion model for prey-predator interaction with Bazykin's reaction kinetics and a nonlocal interaction term in prey growth. The kernel of the integral characterizes nonlocal consumption of resources and depends on space and time. Linear stability analysis determines the conditions of the emergence of Turing patterns without and with nonlocal term, while weakly nonlinear analysis allows the derivation of amplitude equations. The bifurcation analysis and numerical simulation carried out in this work reveal the existence of stationary and dynamic patterns appearing due to the loss of stability of the coexistence homogeneous steady-state.
Periodic traveling waves are observed in various brain activities, including visual, motor, language, sleep, and so on. There are several neural field models describing periodic waves assuming nonlocal interaction, and possibly, inhibition, time delay or some other properties. In this work we study the influences of asymmetric connectivity functions and of time delay for symmetric connectivity functions on the emergence of periodic waves and their properties. Nonlinear wave dynamics are studied, including modulated and aperiodic waves. Multiplicity of waves for the same values of parameters is observed. External stimulation in order to restore wave propagation in a damaged tissue is discussed.
Platelets upregulate the generation of thrombin and reinforce the fibrin clot which increases the incidence risk of venous thromboembolism (VTE). However, the role of platelets in the pathogenesis of venous cardiovascular diseases remains hard to quantify. An experimentally validated model of thrombin generation dynamics is formulated. The model predicts that a high platelet count increases the peak value of generated thrombin as well as the endogenous thrombin potential (ETP) as reported in experimental data. To investigate the effects of platelets density, shear rate, and wound size on the initiation of blood coagulation, we calibrate a previously developed model of venous thrombus formation and implement it in 3D using a novel cell-centered finite-volume solver. We conduct numerical simulations to reproduce in vitro experiments of blood coagulation in microfluidic capillaries. Then, we derive a reduced one-equation model of thrombin distribution from the previous model under simplifying hypotheses and we use it to determine the conditions of clotting initiation on the platelet count, the shear rate, and the plasma composition. The initiation of clotting also exhibits a threshold response to the size of the wounded region in good agreement with the reported experimental findings.
Existence of travelling waves is studied for a delay reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.
In this paper, we study complex dynamics of the interaction between natural convection and thermal explosion in porous media. This process is modeled with the nonlinear heat equation coupled with the nonstationary Darcy equation under the Boussinesq approximation for a fluid-saturated porous medium in a rectangular domain. Numerical simulations with the Radial Basis Functions Method (RBFM) reveal complex dynamics of solutions and transitions to chaos after a sequence of period doubling bifurcations. Several periodic windows alternate with chaotic regimes due to intermittence or crisis. After the last chaotic regime, a final periodic solution precedes transition to thermal explosion.
Existence of travelling waves is studied for a reaction–diffusion system of equations describing the distribution of viruses and immune cells in the tissue. The proof uses the Leray–Schauder method based on the topological degree for elliptic operators in unbounded domains and on a priori estimates of solutions in weighted spaces.
Reaction-diffusion equation with a bistable nonlocal nonlinearity is considered in the case where the reaction term is not quasi-monotone. For this equation, the existence of travelling waves is proved by the Leray-Schauder method based on the topological degree for elliptic operators in unbounded domains and a priori estimates of solutions in properly chosen weighted spaces.
The paper is devoted to the investigation of a reaction-diffusion system of equations describing the process of blood coagulation. Existence of pulse solutions, that is, positive stationary solutions with zero limit at infinity is studied. It is shown that such solutions exist if and only if the speed of the travelling wave described by the same system is positive. The proof is based on the Leray–Schauder method using topological degree for elliptic problems in unbounded domains and a priori estimates of solutions in some appropriate weighted spaces.
Correct interpretation of the data from integral laboratory tests, including Thrombin Generation Test (TGT), requires biochemistry-based mathematical models of blood coagulation. The purpose of this study is to describe the experimental TGT data from healthy donors and hemophilia A (HA) and B (HB) patients. We derive a simplified ODE model and apply it to analyze the TGT data from healthy donors and HA/HB patients with in vitro added tissue factor pathway inhibitor (TFPI) antibody. This model allows the characterization of hemophilia patients in the space of three most important model parameters. The proposed approach may provide a new quantitative tool for the analysis of experimental TGT. Also, it gives a reduced model of coagulation verified against clinical data to be used in future theoretical large-scale modeling of thrombosis in flow.
The aim of this paper is to integrate different bodies of research including brain traveling waves, brain neuromodulation, neural field modeling and post-stroke language disorders in order to explore the opportunity of implementing model-guided, cortical neuromodulation for the treatment of post-stroke aphasia. Worldwide according to WHO, strokes are the second leading cause of death and the third leading cause of disability. In ischemic stroke, there is not enough blood supply to provide enough oxygen and nutrients to parts of the brain, while in hemorrhagic stroke, there is bleeding within the enclosed cranial cavity. The present paper focuses on ischemic stroke. We first review accumulating observations of traveling waves occurring spontaneously or triggered by external stimuli in healthy subjects as well as in patients with brain disorders. We examine the putative functions of these waves and focus on post-stroke aphasia observed when brain language networks become fragmented and/or partly silent, thus perturbing the progression of traveling waves across perilesional areas. Secondly, we focus on a simplified model based on the current literature in the field and describe cortical traveling wave dynamics and their modulation. This model uses a biophysically realistic integro-differential equation describing spatially distributed and synaptically coupled neural networks producing traveling wave solutions. The model is used to calculate wave parameters (speed, amplitude and/or frequency) and to guide the reconstruction of the perturbed wave. A stimulation term is included in the model to restore wave propagation to a reasonably good level. Thirdly, we examine various issues related to the implementation model-guided neuromodulation in the treatment of post-stroke aphasia given that closed-loop invasive brain stimulation studies have recently produced encouraging results. Finally, we suggest that modulating traveling waves by acting selectively and dynamically across space and time to facilitate wave propagation is a promising therapeutic strategy.
In this paper, we consider the (3 + 1)-dimensional Burgers-like equation which arises in fluid mechanics, which constructed from Lax pair generating technique. The bilinear form for this model is obtained to construct the multiple-kink solutions.
The paper is devoted to the diffusion equation, with the Dirac-like periodic potential having different structure in two parts of the domain. The problem can be homogenized in one part of the domain while the standard homogenization does not work in the other. We introduce and test numerically the method of partial homogenization, combining in one multiscale model the homogenized and discrete description.
Epidemiological data on seasonal influenza show that the growth rate of the number of infected individuals can increase passing from one exponential growth rate to another one with a larger exponent. Such behavior is not described by conventional epidemiological models. In this work an immuno-epidemiological model is proposed in order to describe this two-stage growth. It takes into account that the growth in the number of infected individuals increases the initial viral load and provides a passage from the first stage of epidemic where only people with weak immune response are infected to the second stage where people with strong immune response are also infected. This scenario may be viewed as an increase of the effective number of susceptible increasing the effective growth rate of infected.
Attempts to curb the spread of coronavirus by introducing strict quarantine measures apparently have different effect in different countries: while the number of new cases has reportedly decreased in China and South Korea, it still exhibit significant growth in Italy and other countries across Europe. In this brief note, we endeavour to assess the efficiency of quarantine measures by means of mathematical modelling. Instead of the classical SIR model, we introduce a new model of infection progression under the assumption that all infected individual are isolated after the incubation period in such a way that they cannot infect other people.
Investigation of interacting populations is an active area of research, and various modeling approaches have been adopted to describe their dynamics. Mathematical models of such interactions using differential equations are capable to mimic the stationary and oscillating (regular or irregular) population distributions. Recently, some researchers have paid their attention to explain the consequences of transient dynamics of population density (especially the long transients) and able to capture such behaviors with simple models. Existence of multiple stationary patches and settlement to a stable distribution after a long quasi-stable transient dynamics can be explained by spatiotemporal models with nonlocal interaction terms. However, the studies of such interesting phenomena for three interacting species are not abundant in literature. Motivated by these facts here we have considered a three species prey-predator model where the predator is generalist in nature as it survives on two prey species. Nonlocalities are introduced in the intra-specific competition terms for the two prey species in order to model the accessibility of nearby resources. Using linear analysis, we have derived the Turing instability conditions for both the spatiotemporal models with and without nonlocal interactions. Validation of such conditions indicates the possibility of existence of stationary spatially heterogeneous distributions for all the three species. Existence of long transient dynamics has been presented under certain parametric domain. Exhaustive numerical simulations reveal various scenarios of stabilization of population distribution due to the presence of nonlocal intra-specific competition for the two prey species. Chaotic oscillation exhibited by the temporal model is significantly suppressed when the populations are allowed to move over their habitat and prey species can access the nearby resources.
A mathematical model describing viral dynamics in the presence of the latently infected cells and the cytotoxic T-lymphocytes cells (CTL), taking into consideration the spatial mobility of free viruses, is presented and studied. The model includes five nonlinear differential equations describing the interaction among the uninfected cells, the latently infected cells, the actively infected cells, the free viruses, and the cellular immune response. First, we establish the existence, positivity, and boundedness for the suggested diffusion model. Moreover, we prove the global stability of each steady state by constructing some suitable Lyapunov functionals. Finally, we validated our theoretical results by numerical simulations for each case.
In this opinion paper we make the statement that hybrid models in oncology are required as a mean for enhanced data integration. In the context of systems oncology, experimental and clinical data need to be at the heart of the models developments from conception to validation to ensure a relevant use of the models in the clinical context. The main applications pursued are to improve diagnosis and to optimize therapies.We first present the Successes achieved thanks to hybrid modelling approaches to advance knowledge, treatments or drug discovery. Then we present the Challenges that need to be addressed to allow for a better integration of the model parts and of the data into the models. And finally, the Hopes with a focus towards making personalised medicine a reality.
Drug resistance (DR) is a phenomenon characterized by the tolerance of a disease to pharmaceutical treatment. In cancer patients, DR is one of the main challenges that limit the therapeutic potential of the existing treatments. Therefore, overcoming DR by restoring the sensitivity of cancer cells would be greatly beneficial. In this context, mathematical modeling can be used to provide novel therapeutic strategies that maximize the efficiency of anti-cancer agents and potentially overcome DR. In this paper, we present a new multiscale model devoted to the interaction of potential treatments with multiple myeloma (MM) development. In this model, MM cells are represented as individual objects that move, divide, and die by apoptosis. The fate of each cell depends on intracellular and extracellular regulation, as well as the administered treatment.
Virus density distribution as a function of genotype considered as a continuous variable and of time is studied with a nonlocal reaction-diffusion equation taking into account virus competition for the host cells and its elimination by the immune response and by the genotype-dependent mortality. The existence of virus strains, that is, of positive stable stationary solutions decaying at infinity, is determined by the admissible intervals in the genotype space where the genotype-dependent mortality is less than the virus reproduction rate, and by the immune response under some appropriate assumptions on the immune response function characterizing virus elimination by immune cells. The competition of virus strains is studied, first, without immune response and then with the immune response. In the absence of immune response, the strain dynamics is different in a short time scale where they converge to some intermediate slowly evolving solutions depending on the initial conditions, and in a long time scale where their distribution converges to a stationary solution. Immune response can essentially influence the strain dynamics either stabilizing them or eliminating one of the strains. An antiviral treatment can also influence the competition of virus strains, and it can lead to the emergence of resistant strains, which were absent before the treatment because of the competition with susceptible strains.
This work is devoted to the investigation of virus quasi-species evolution and diversification due to mutations, competition for host cells, and cross-reactive immune responses. The model consists of a nonlocal reaction-diffusion equation for the virus density depending on the genotype considered to be a continuous variable and on time. This equation contains two integral terms corresponding to the nonlocal effects of virus interaction with host cells and with immune cells. In the model, a virus strain is represented by a localized solution concentrated around some given genotype. Emergence of new strains corresponds to a periodic wave propagating in the space of genotypes. The conditions of appearance of such waves and their dynamics are described.
Under voluntary vaccination, a critical role in shaping the level and trends of vaccine uptake is played by the type and structure of information that is received and used by parents of children eligible for vaccination. In this article we investigate the feedbacks of spatial mobility and the spatial structure of information on vaccination dynamics, by extending to a continuous spatially structured setting existing behavioral epidemiology models for the impact of vaccine adverse events (VAEs) on vaccination choices. We considered the simplest spatial setting, namely classical 'Fickian' diffusion, and focused on the noteworthy case where the infection is absent. This scenario mimics the important case of a population where a previously endemic vaccine preventable infection was successfully eliminated, but the re-emergence of the disease must be prevented. This is, for example, the case of poliomyelitis in most countries worldwide. In such a situation, the dynamics of VAEs and of the related information arguably become the key determinant of vaccination decision and of collective coverage. In relation to this 'information issue', we compared the effects of three main cases: (i) purely local information, where agents react only to locally occurred events; (ii) a mix of purely local and global, country-wide, information due e.g., to country-wide media and the internet; (iii) a mix of local and non-local information. By representing these different information options through a range of different spatial information kernels, we investigated: the presence and stability of space-homogeneous, nontrivial, behavior-induced equilibria; the existence of bifurcations; the existence of classical and generalized traveling waves; and the effects of awareness campaigns enacted by the Public Health System to sustain vaccine uptake. Finally, we analyzed some analogies and differences between our models and those of the Theory of Innovation Diffusion.
In December 2019, the first case of infection with a new virus COVID-19 (SARS-CoV-2), named coronavirus, was reported in the city of Wuhan, China. At that time, almost nobody paid any attention to it. The new pathogen, however, fast proved to be extremely infectious and dangerous, resulting in about 3-5% mortality. Over the few months that followed, coronavirus has spread over entire world. At the end of March, the total number of infections is fast approaching the psychological threshold of one million, resulting so far in tens of thousands of deaths. Due to the high number of lives already lost and the virus high potential for further spread, and due to its huge overall impact on the economies and societies, it is widely admitted that coronavirus poses the biggest challenge to the humanity after the second World war. The COVID-19 epidemic is provoking numerous questions at all levels. It also shows that modern society is extremely vulnerable and unprepared to such events. A wide scientific and public discussion becomes urgent. Some possible directions of this discussion are suggested in this article.
We study a two-phase S-E-I-R mathematical model, based on the current coronavirus epidemic. If contacts are reduced to zero from a certain time T close to the start of the epidemic, the final size of the epidemic is close to that obtained by multiplying the cumulative number of cases R(T) at that time by the reproduction number R0 of the epidemic. More generally, if contacts are divided at time T by q > 1 so that R0/q<1, then the final size of the epidemic is close to R(T) R0 (1-1/q)/(1-R0/q). The parameters of the model are roughly fitted to the coronavirus data in France.
We study variants of the SEIR model for interpreting some qualitative features of the statistics of the Covid-19 epidemic in France. Standard SEIR models distinguish essentially two regimes: either the disease is controlled and the number of infected people rapidly decreases, or the disease spreads and contaminates a significant fraction of the population until herd immunity is achieved. After lockdown, at first sight it seems that social distancing is not enough to control the outbreak. We discuss here a possible explanation, namely that the lockdown is creating social heterogeneity: even if a large majority of the population complies with the lockdown rules, a small fraction of the population still has to maintain a normal or high level of social interactions, such as health workers, providers of essential services, etc. This results in an apparent high level of epidemic propagation as measured through re-estimations of the basic reproduction ratio. However, these measures are limited to averages, while variance inside the population plays an essential role on the peak and the size of the epidemic outbreak and tends to lower these two indicators. We provide theoretical and numerical results to sustain such a view.
Cardiovascular diseases remain one of the largest causes of death worldwide. The lethal pathologies often occur due to atherosclerosis, clot formation and cardiac arrhythmia. It turns out that all these pathologies can be described using the generic framework of multi-component reaction–diffusion equations. These parabolic partial differential equations sustain waves and pattern formation that are relevant in the study of all those processes. Here we present the first review which combines the description of these fields: atherosclerosis, clot formation and cardiac arrhythmias, and includes classic and recent developments in these areas. We show how mathematical models for the underlying physiological processes have been constructed, and which generic properties follow from their analytical or numerical study. Finally, we discuss the possibility of integrative studies of cardiovascular disease which will include both clot formation and cardiac arrhythmias. Such approach will be highly relevant to atrial fibrillation, a common cardiac arrhythmia whose main complication is clot formation and stroke.
The paper is concerned with the existence of pulses for monotone reaction–diffusion systems of two equations. For a general class of systems we prove that pulses exist if and only if the wave solutions propagate at positive speed. This result is applied to investigate the existence of pulses for the system of competition of species.
A neural field model with different activation and inhibition connectivity and response functions is considered. Stability analysis of a homogeneous in space solution determines the conditions of the emergence of stationary periodic solutions and of periodic travelling waves. Various regimes of wave propagation are illustrated in numerical simulations. The influence of external stimulation on the wave properties is investigated.
Spreading of viral infection in the tissues such as lymph nodes or spleen depends on virus multiplication in the host cells, their transport and on the immune response. Reaction–diffusion systems of equations with delays in proliferation and death terms of the immune cells represent an appropriate model to study this process. The properties of the immune response and the initial viral load determine the regimes of infection spreading. In the proposed model, the proliferation rate of the immune cells is represented by a bell-shaped function of the virus concentration which increases for small concentrations and decreases if the concentration is sufficiently high. Here we use such a model system to show that an infection can be completely eliminated or it can remain present together with a decreased concentration of immune cells. Finally, immune cells can be completely exhausted leading to a high virus concentration in the tissue. In addition, we predicted two novel regimes of infection dynamics not observed before. Infection propagation in the tissue can occur as a superposition of two travelling waves: first wave propagates as a low level infection front followed by a high level infection front with a smaller speed of propagation. Both of the travelling waves can have a positive or a negative speed corresponding to infection advancement or retreat. These regimes can be accompanied by instabilities and the emergence of complex spatiotemporal patterns.
We study the existence of monotone wavefronts for a general family of bistable reaction-diffusion equations with delayed reaction term g. Differently from previous works, we do not assume the monotonicity of with respect to the delayed variable that does not allow to apply the comparison techniques. Thus our proof is based on a variant of the Hale-Lin functional-analytic approach to heteroclinic solutions of functional differential equations where Lyapunov-Schmidt reduction is done in appropriate weighted spaces of C 2-smooth functions. This method requires a detailed analysis of associated linear differential operators and their formal adjoints. For two different types of -unimodal functions , we establish the existence of a maximal continuous family of bistable monotone wavefronts. Depending on the type of unimodality (equivalently, on the sign of the wave speed), two different scenarios can be observed for the obtained bistable waves: (1) independently on the size of delay, each bistable wavefront is monotone; (2) wavefronts are monotone for moderate values of delays and can oscillate for large delays.
Nonlinear dynamics of a reaction-diffusion equation with delay is studied with numerical simulations in 1D and 2D cases. Homogeneous in space solutions can manifest time oscillations with period doubling bifurcations and transition to chaos. Transition between two regions with homogeneous oscillations is provided by quasi-waves, propagating solutions without regular structure and often with complex aperiodic oscillations. Dynamics of space dependent solutions is described by a combination of various waves, e.g., bistable, monostable, periodic and quasi-waves.
Thrombosis is a life-threatening clinical condition characterized by the obstruction of blood flow in a vessel due to the formation of a large thrombus. The pathogenesis of thrombosis is complex because the type of formed clots depends on the location and function of the corresponding blood vessel. To explore this phenomenon, we develop a novel multiscale model of platelet-fibrin thrombus growth in the flow. In this model, the regulatory network of the coagulation cascade is described by partial differential equations. Blood flow is introduced using the Navier–Stokes equations and the clot is treated as a porous medium. Platelets are represented as discrete spheres that migrate with the flow. Each platelet can attach to the thrombus, aggregate, become activated, express proteins on its surface, detach, and/or become non-adhesive. The interaction of platelets with blood flow is captured using the Immersed Boundary Method (IBM). We use the model to investigate the role of flow conditions in shaping the dynamics of venous and arterial thrombi. We describe the formation of red and white thrombi under venous and arterial flow respectively and highlight the main characteristics of each type. We identify the different regimes of normal and pathological thrombus formation depending on flow conditions.
Existence of travelling waves is studied for a bistable reaction–diffusion system of equations with linear integral terms (dispersion) and with some conditions on the nonlinearity. The proof is based on the Leray–Schauder method using the topological degree theory for Fredholm and proper operators with the zero index and a priori estimates of solutions in properly chosen weighted spaces.
Travelling waves for nonlocal reaction–diffusion equations are studied. The minimax representation of the wave speed is obtained. It is used to obtain analytical estimates and asymptotic values of the speed. Two regimes of wave propagation are identified. One of them is dominated by diffusion and another one by the nonlocal interaction.
We study solvability of some linear nonhomogeneous elliptic problems and prove that under reasonable technical conditions the convergence in L 2 (ℝ d ) of their right sides implies the existence and the convergence in H 2 s (ℝ d ) of the solutions. The equations involve the second order non-Fredholm differential operators raised to certain fractional powers s and we use the methods of spectral and scattering theory for Schrödinger type operators developed in our preceding work (Volpert and Vougalter, Electron J Differ Equ 160:16 pp, 2013).
A reaction-diffuusion system describing blood coagulation in flow is studied. We prove the existence of stationary solutions provided that the speed of the travelling wave problem for the limiting value of the velocity is positive. The implications to the problem of clot growth are discussed.
The paper presents a review of recent developments of hybrid discrete-continuous models in cell population dynamics. Such models are widely used in the biological modelling. Cells are considered as individual objects which can divide, die by apoptosis, differentiate and move under external forces. In the simplest representation cells are considered as soft spheres, and their motion is described by Newton's second law for their centers. In a more complete representation, cell geometry and structure can be taken into account. Cell fate is determined by concentrations of intra-cellular substances and by various substances in the extracellular matrix, such as nutrients, hormones, growth factors. Intra-cellular regulatory networks are described by ordinary differential equations while extracellular species by partial differential equations. We illustrate the application of this approach with some examples including bacteria filament and tumor growth. These examples are followed by more detailed studies of erythropoiesis and immune response. Erythrocytes are produced in the bone marrow in small cellular units called erythroblastic islands. Each island is formed by a central macrophage surrounded by erythroid progenitors in different stages of maturity. Their choice between self-renewal, differentiation and apoptosis is determined by the ERK/Fas regulation and by a growth factor produced by the macrophage. Normal functioning of erythropoiesis can be compromised by the development of multiple myeloma, a malignant blood disorder which leads to a destruction of erythroblastic islands and to sever anemia. The last part of the work is devoted to the applications of hybrid models to study immune response and the development of viral infection. A two-scale model describing processes in a lymph node and other organs is presented.
In the advanced stages of cancers like melanoma, some of the malignant cells leave the primary tumor and infiltrate the neighboring lymph nodes (LNs). The interaction between secondary cancer and the immune response in the lymph node represents a complex process that needs to be fully understood in order to develop more effective immunotherapeutic strategies. In this process, antigen-presenting cells (APCs) approach the tumor and initiate the adaptive immune response for the corresponding antigen. They stimulate the naive CD4+ and CD8+ T lymphocytes which subsequently generate a population of helper and effector cells. On one hand, immune cells can eliminate tumor cells using cell-cell contact and by secreting apoptosis inducing cytokines. They are also able to induce their dormancy. On the other hand, the tumor cells are able to escape the immune surveillance using their immunosuppressive abilities. To study the interplay between tumor progression and the immune response, we develop two new models describing the interaction between cancer and immune cells in the lymph node. The first model consists of partial differential equations (PDEs) describing the populations of the different types of cells. The second one is a hybrid discrete-continuous model integrating the mechanical and biochemical mechanisms that define the tumor-immune interplay in the lymph node.
A numerical approach is developed to solve differential equations on an infinite domain, when the solution is known to possess a slowly decaying tail. An unorthodox boundary condition relying on the existence of an asymptotic relation for |y| 1 is implemented, followed by an optimisation procedure, allowing to obtain an accurate solution over a truncated finite domain. The method is applied to -(-Δ)γ/2u - u + up = 0 in , a non-linear integro-differential equation containing the fractional Laplacian, and is easily expanded to asymmetric boundary conditions or domains of a higher dimension.
In this paper, we are concerned with a cantilevered Timoshenko beam. The beam is viscoelastic and subject to a translational displacement. Consequently, the Timoshenko system is complemented by an ordinary differential equation describing the dynamic of the base to which the beam is attached to. We establish a control force capable of driving the system to the equilibrium state with a certain speed depending on the decay rate of the relaxation function.
The review presents the state of the art in the atherosclerosis modelling. It begins with the biological introduction describing the mechanisms of chronic inflammation of artery walls characterizing the development of atherosclerosis. In particular, we present in more detail models describing this chronic inflammation as a reaction-diffusion wave with regimes of propagation depending on the level of cholesterol (LDL) and models of rolling monocytes initializing the inflammation. Further development of this disease results in the formation of atherosclerotic plaque, vessel remodelling and possible plaque rupture due its interaction with blood flow. We review plaque-flow interaction models as well as reduced models (0D and 1D) of blood flow in atherosclerotic vasculature.
The surveillance of host body tissues by immune cells is central for mediating their defense function. In vivo imaging technologies have been used to quantitatively characterize target cell scanning and migration of lymphocytes within lymph nodes (LNs). The translation of these quantitative insights into a predictive understanding of immune system functioning in response to various perturbations critically depends on computational tools linking the individual immune cell properties with the emergent behavior of the immune system. By choosing the Newtonian second law for the governing equations, we developed a broadly applicable mathematical model linking individual and coordinated T-cell behaviors. The spatial cell dynamics is described by a superposition of autonomous locomotion, intercellular interaction, and viscous damping processes. The model is calibrated using in vivo data on T-cell motility metrics in LNs such as the translational speeds, turning angle speeds, and meandering indices. The model is applied to predict the impact of T-cell motility on protection against HIV infection, i.e., to estimate the threshold frequency of HIV-specific cytotoxic T cells (CTLs) that is required to detect productively infected cells before the release of viral particles starts. With this, it provides guidance for HIV vaccine studies allowing for the migration of cells in fibrotic LNs.
We study the existence of solutions of an integro-differential equation in the case of the anomalous diffusion with the negative Laplace operator in a fractional power in two dimensions. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.
Spreading of viral infection in the tissues such as lymph nodes or spleen depends on virus multiplication in the host cells, their transport and on the immune response. Reaction-diffusion systems of equations with delays in cell proliferation and death by apoptosis represent an appropriate model to study this process. The properties of the cells of the immune system and the initial viral load determine the spatiotemporal regimes of infection spreading. Infection can be completely eliminated or it can persist at some level together with a certain chronic immune response in a spatially uniform or oscillatory mode. Finally, the immune cells can be completely exhausted leading to a high viral load persistence in the tissue. It has been found experimentally, that virus proteins can affect the immune cell migration. Our study shows that both the motility of immune cells and the virus infection spreading represented by the diffusion rate coefficients are relevant control parameters determining the fate of virus-host interaction.
Following a stroke, cortical networks in the penumbra area become fragmented and partly deactivated. We develop a model to study the propagation of waves of electric potential in the cortical tissue with integro-differential equations arising in neural field models. The wave speed is characterized by the tissue excitability and connectivity determined through parameters of the model. Post-stroke tissue damage in the penumbra area creates a hypoconnectivity and decreases the speed of wave propagation. It is proposed that external stimulation could restore the wave speed in the penumbra area under certain conditions of the parameters. Model guided cortical stimulation could be used to improve the functioning of cortical networks.
Hemostasis is a complex physiological mechanism that functions to maintain vascular integrity under any conditions. Its primary components are blood platelets and a coagulation network that interact to form the hemostatic plug, a combination of cell aggregate and gelatinous fibrin clot that stops bleeding upon vascular injury. Disorders of hemostasis result in bleeding or thrombosis, and are the major immediate cause of mortality and morbidity in the world. Regulation of hemostasis and thrombosis is immensely complex, as it depends on blood cell adhesion and mechanics, hydrodynamics and mass transport of various species, huge signal transduction networks in platelets, as well as spatiotemporal regulation of the blood coagulation network. Mathematical and computational modeling has been increasingly used to gain insight into this complexity over the last 30 years, but the limitations of the existing models remain profound. Here we review state-of-the-art-methods for computational modeling of thrombosis with the specific focus on the analysis of unresolved challenges. They include: a) fundamental issues related to physics of platelet aggregates and fibrin gels; b) computational challenges and limitations for solution of the models that combine cell adhesion, hydrodynamics and chemistry; c) biological mysteries and unknown parameters of processes; d) biophysical complexities of the spatiotemporal networks' regulation. Both relatively classical approaches and innovative computational techniques for their solution are considered; the subjects discussed with relation to thrombosis modeling include coarse-graining, continuum versus particle-based modeling, multiscale models, hybrid models, parameter estimation and others. Fundamental understanding gained from theoretical models are highlighted and a description of future prospects in the field and the nearest possible aims are given.
We prove the existence in the sense of sequences of stationary solutions for some systems of reaction–diffusion type equations in the appropriate H2 spaces. It is established that, under reasonable technical conditions, the convergence in L1 of the integral kernels yields the existence and the convergence in H2 of the solutions. The nonlocal elliptic problems contain the second-order differential operators with and without Fredholm property.
Formation of blood clot in response to the vessel damage is triggered by the complex network of biochemical reactions of the coagulation cascade. The process of clot growth can be modeled as a traveling wave solution of the bistable reaction–diffusion system. The critical value of the initial condition which leads to convergence of the solution to the traveling wave corresponds to the pulse solution of the corresponding stationary problem. In the current study we prove the existence of the pulse solution for the stationary problem in the model of the main reactions of the blood coagulation cascade using the Leray–Schauder method.
Darwin described biological species as groups of morphologically similar individuals. These groups of individuals can split into several subgroups due to natural selection, resulting in the emergence of new species. Some species can stay stable without the appearance of a new species, some others can disappear or evolve. Some of these evolutionary patterns were described in our previous works independently of each other. In this work we have developed a single model which allows us to reproduce the principal patterns in Darwin’s diagram. Some more complex evolutionary patterns are also observed. The relation between Darwin’s definition of species, stated above, and Mayr’s definition of species (group of individuals that can reproduce) is also discussed.
Spatiotemporal pattern formation in integro-differential equation models of interacting populations is an active area of research, which has emerged through the introduction of nonlocal intra- and inter-specific interactions. Stationary patterns are reported for nonlocal interactions in prey and predator populations for models with prey-dependent functional response, specialist predator and linear intrinsic death rate for predator species. The primary goal of our present work is to consider nonlocal consumption of resources in a spatiotemporal prey-predator model with bistable reaction kinetics for prey growth in the absence of predators. We derive the conditions of the Turing and of the spatial Hopf bifurcation around the coexisting homogeneous steady-state and verify the analytical results through extensive numerical simulations. Bifurcations of spatial patterns are also explored numerically.
We study solvability of some linear nonhomogeneous elliptic equations and show that under reasonable technical conditions the convergence in L2(Rd) of their right sides yields the existence and the convergence in H1(Rd) of the solutions. The problems involve the square roots of the second order non Fredholm differential operators and we use the methods of spectral and scattering theory for Schrödinger type operators similarly to our preceding work (Volpert and Vougalter in Electron J Differ Equ 160:16, 2013).
We prove the existence of stationary solutions for some reaction-diffusion equations with superdiffusion. The corresponding elliptic problem contains the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces is employed.
T lymphoblastic lymphoma (T-LBL) is a rare type of lymphoma with a good prognosis with a remission rate of 85%. Patients can be completely cured or can relapse during or after a 2-year treatment. Relapses usually occur early after the remission of the acute phase. The median time of relapse is equal to 1 year, after the occurrence of complete remission (range 0.2-5.9 years) (Uyttebroeck et al., 2008). It can be assumed that patients may be treated longer than necessary with undue toxicity. The aim of our model was to investigate whether the duration of the maintenance therapy could be reduced without increasing the risk of relapses and to determine the minimum treatment duration that could be tested in a future clinical trial. We developed a mathematical model of virtual patients with T-LBL in order to obtain a proportion of virtual relapses close to the one observed in the real population of patients from the EuroLB database. Our simulations reproduced a 2-year follow-up required to study the onset of the disease, the treatment of the acute phase and the maintenance treatment phase.
We establish the existence in the sense of sequences of stationary solutions for some reaction-diffusion type equations in appropriate H2 spaces. It is shown that, under reasonable technical conditions, the convergence in L1 of the integral kernels implies the existence and convergence in H2 of solutions. The nonlocal elliptic equations involve second order differential operators with and without the Fredholm property.
The paper is devoted to the recent works on reaction–diffusion models of virus infection dynamics in human and animal organisms. Various regimes of infection propagation in tissues are described. In particular, it is shown that infection can spread in tissues of organs as a reaction–diffusion wave. Methods for studying the conditions of the existence of wave modes of the time and space dynamics of infections is discussed.
Human population growth has been called the biggest issue the humanity faces in the 21st century, and although this statement is globally true, locally, many Western economies have been experiencing population decline. Europe is in fact homeland for population decline. By 2050 many large European economies are predicted to lose large parts of their population. In this work, we consider the dynamical system that corresponds to the model introduced by Volpert et al. [Nonlinear Anal. 159 (2017) 408-423]. With the help of this model, we illustrate scenarios that can lead, in the long-run, to sharp population decline and/or deterioration of the economy. We also illustrate that even when under certain conditions the population will go extinct, temporarily it might experience growth.
Under normal conditions, blood coagulation provides an effective protective mechanism preventing bleeding in case of vessel damage. Details of its functioning are of particular importance since any blood coagulation disorders lead to severe physiological aggravations. Multiple experimental and computational studies demonstrate the thrombin concentration distribution to determine the spatio-temporal dynamics of clot formation. Propagating from the injury site with constant speed, thrombin concentration profile can be modeled with a traveling wave solution of the system of partial differential equations describing main reactions of the coagulation cascade. In the current study, we derive conditions on the existence and stability of such solutions and provide an analytic approach of their wave speed estimation.
This paper is devoted to modelling tissue growth with a deformable cell model. Each cell represents a polygon with particles located at its vertices. Stretching, bending and pressure forces act on particles and determine their displacement. Pressure-dependent cell proliferation is considered. Various patterns of growing tissue are observed. An application of the model to tissue regeneration is illustrated. Approximate analytical models of tissue growth are developed.
Rheumatoid and psoriatic arthritis are chronic inflammatory diseases, with massive increase of cardiovascular events (CVE), and contribution of the cytokines TNF-α and IL-17. Chronic inflammation inside the joint membrane or synovium results from the activation of fibroblasts/synoviocytes, and leads to the release of cytokines from monocytes (Tumor Necrosis Factor or TNF) and from T lymphocytes (Interleukin-17 or IL-17). At the systemic level, the very same cytokines affect endothelial cells and vessel wall. We have previously shown that IL-17 and TNF-α specifically when combined, increase procoagulation, decrease anticoagulation and increase platelet aggregation, leading to thrombosis. These results are the basis for the models of interactions between IL-17 and TNF, and genes expressed by activated endothelial cells. This work is devoted to mathematical modeling and numerical simulations of blood coagulation and clot growth under the influence of IL-17 and TNF-α. We show that they can provoke thrombosis, leading to the complete or partial occlusion of blood vessels. The regimes of blood coagulation and conditions of occlusion are investigated in numerical simulations and in approximate analytical models. The results of mathematical modeling allow us to predict thrombosis development for an individual patient.
Reaction-diffusion equations with a space dependent nonlinearity are considered on the whole axis. Existence of pulses, stationary solutions which vanish at infinity, is studied by the Leray–Schauder method. It is based on the topological degree for Fredholm and proper operators with the zero index in some special weighted spaces and on a priori estimates of solutions in these spaces. Existence of solutions is related to the speed of travelling wave solutions for the corresponding autonomous equations with the limiting nonlinearity.
Vessel occlusion is a perturbation of blood flow inside a blood vessel because of the fibrin clot formation. As a result, blood circulation in the vessel can be slowed down or even stopped. This can provoke the risk of cardiovascular events. In order to explore this phenomenon, we used a previously developed mathematical model of blood clotting to describe the concentrations of blood factors with a reaction-diffusion system of equations. The Navier-Stokes equations were used to model blood flow, and we treated the clot as a porous medium. We identify the conditions of partial or complete occlusion in a small vessel depending on various physical and physiological parameters. In particular, we were interested in the conditions on blood flow and diameter of the wounded area. The existence of a critical flow velocity separating the regimes of partial and complete occlusion was demonstrated through the mathematical investigation of a simplified model of thrombin wave propagation in Poiseuille flow. We observed different regimes of vessel occlusion depending on the model parameters both for the numerical simulations and in the theoretical study. Then, we compared the rate of clot growth in flow obtained in the simulations with experimental data. Both of them showed the existence of different regimes of clot growth depending on the velocity of blood flow. Copyright
Dynamics of human populations depends on various economical and social factors. Their migration is partially determined by the economical conditions and it can also influence these conditions. This work is devoted to the analysis of the interaction of human migration and wealth distribution. The model consists of a system of equations for the population density and for the wealth distribution with conventional diffusion terms and with cross diffusion terms describing human migration determined by the wealth gradient and wealth flux determined by human migration. Wealth production and consumption depend on the population density while the natality and mortality rates depend on the level of wealth. In the absence of cross diffusion terms, dynamics of solutions is described by travelling wave solutions of the corresponding reaction–diffusion systems of equations. We show persistence of such solutions for sufficiently small cross diffusion coefficients. This result is based on the perturbation methods and on the spectral properties of the linearized operators.
Moving from the molecular and cellular level to a multi-scale systems understanding of immune responses requires the development of novel approaches to integrate knowledge and data from different biological levels into mechanism-based integrative mathematical models. The aim of our study is to present a methodology for a hybrid modelling of immunological processes in their spatial context.
Spatio-temporal pattern formation in reaction–diffusion models of interacting populations is an active area of research due to various ecological aspects. Instability of homogeneous steady-states can lead to various types of patterns, which can be classified as stationary, periodic, quasi-periodic, chaotic, etc. The reaction–diffusion model with Rosenzweig–MacArthur type reaction kinetics for prey–predator type interaction is unable to produce Turing patterns but some non-Turing patterns can be observed for it. This scenario changes if we incorporate non-local interactions in the model. The main objective of the present work is to reveal possible patterns generated by the reaction–diffusion model with Rosenzweig–MacArthur type prey–predator interaction and non-local consumption of resources by the prey species. We are interested in the existence of Turing patterns in this model and in the effect of the non-local interaction on the periodic travelling wave and spatio-temporal chaotic patterns. Global bifurcation diagrams are constructed to describe the transition from one pattern to another one.
One of the main characteristics of blood coagulation is the speed of clot growth. In the current work we consider a mathematical model of the coagulation cascade and study existence, stability and speed of propagation of the reaction-diffusion waves of blood coagulation. We also develop a simplified one-equation model that reflects the main features of the thrombin wave propagation. For this equation we estimate the wave speed analytically. The resulting formulas provide a good approximation for the speed of wave propagation in a more complex model as well as for the experimental data.
Human Immunodeficiency Virus (HIV) infection of humans represents a complex biological system and a great challenge to public health. Novel approaches for the analysis and prediction of the infection dynamics based on a multi-scale integration of virus ontogeny and immune reactions are needed to deal with the systems' complexity. The aim of our study is: (1) to formulate a multi-scale mathematical model of HIV infection; (2) to implement the model computationally following a hybrid approach; and (3) to calibrate the model by estimating the parameter values enabling one to reproduce the "standard" observed dynamics of HIV infection in blood during the acute phase of primary infection. The modeling approach integrates the processes of infection spread and immune responses in Lymph Nodes (LN) to that observed in blood. The spatio-temporal population dynamics of T lymphocytes in LN in response to HIV infection is governed by equations linking an intracellular regulation of the lymphocyte fate by intercellular cytokine fields. We describe the balance of proliferation, differentiation and death at a single cell level as a consequence of gene activation via multiple signaling pathways activated by IL-2, IFNa and FasL. Distinct activation thresholds are used in the model to relate different modes of cellular responses to the hierarchy of the relative levels of the cytokines. We specify a reference set of model parameter values for the fundamental processes in lymph nodes that ensures a reasonable agreement with viral load and CD4+ T cell dynamics in blood.
The interaction between natural convection and the heat explosion in porous media is studied. The model consists of a nonlinear heat equation coupled with the Darcy equation for the motion of an incompressible fluid in a porous medium. Numerical simulations are performed using the alternate direction finite difference method and the fast Fourier transform method. A complex behavior of solutions is observed, including periodic and aperiodic oscillations and an oscillating heat explosion. It is shown that convection can decrease the risk of the explosion due to additional mixing and heat loss, but it can also facilitate the explosion due to temperature oscillations arising as a result of instability of stationary convective regimes.
Multiple myeloma (MM) is a genetically complex hematological cancer that is characterized by proliferation of malignant plasma cells in the bone marrow. MM evolves from the clonal premalignant disorder monoclonal gammopathy of unknown significance (MGUS) by sequential genetic changes involving many different genes, resulting in dysregulated growth of multiple clones of plasma cells. The migration, survival, and proliferation of these clones require the direct and indirect interactions with the non-hematopoietic cells of the bone marrow. We develop a hybrid discrete-continuous model of MM development from the MGUS stage. The discrete aspect of the model is observed at the cellular level: cells are represented as individual objects which move, interact, divide, and die by apoptosis. Each of these actions is regulated by intracellular and extracellular processes as described by continuous models. The hybrid model consists of the following submodels that have been simplified from the much more complex state of evolving MM: cell motion due to chemotaxis, intracellular regulation of plasma cells, extracellular regulation in the bone marrow, and acquisition of mutations upon cell division. By extending a previous, simpler model in which the extracellular matrix was considered to be uniformly distributed, the new hybrid model provides a more accurate description in which cytokines are produced by the marrow microenvironment and consumed by the myeloma cells. The complex multiple genetic changes in MM cells and the numerous cell-cell and cytokine-mediated interactions between myeloma cells and their marrow microenviroment are simplified in the model such that four related but evolving MM clones can be studied as they compete for dominance in the setting of intraclonal heterogeneity.
The paper is devoted to a reaction–diffusion equation with doubly nonlocal nonlinearity arising in various applications in population dynamics. One of the integral terms corresponds to the nonlocal consumption of resources while another one describes reproduction with different phenotypes. Linear stability analysis of the homogeneous in space stationary solution is carried out. Existence of traveling waves is proved in the case of narrow kernels of the integrals. Periodic traveling waves are observed in numerical simulations. Existence of stationary solutions in the form of pulses is shown, and transition from periodic waves to pulses is studied. In the applications to the speciation theory, the results of this work signify that new species can emerge only if they do not have common offsprings. Thus, it is shown how Darwin's definition of species as groups of morphologically similar individuals is related to Mayr's definition as groups of individuals that can breed only among themselves.
In case of vessel wall damage or contact of blood plasma with a foreign surface, the chain of chemical reactions called coagulation cascade is launched that leading to the formation of a fibrin clot. A key enzyme of the coagulation cascade is thrombin, which catalyzes formation of fibrin from fibrinogen. The distribution of thrombin concentration in blood plasma determines spatio-temporal dynamics of clot formation. Contact pathway of blood coagulation triggers the production of thrombin in response to the contact with a negatively charged surface. If the concentration of thrombin generated at this stage is large enough, further production of thrombin takes place due to positive feedback loops of the coagulation cascade. As a result, thrombin propagates in plasma cleaving fibrinogen that results in the clot formation. The concentration profile and the speed of propagation of thrombin are constant and do not depend on the type of the initial activator. Such behavior of the coagulation system is well described by the traveling wave solutions in a system of "reaction - diffusion" equations on the concentration of blood factors involved in the coagulation cascade. In this study, we carried out detailed analysis of the mathematical model describing the main reaction of the intrinsic pathway of coagulation cascade. We formulate necessary and sufficient conditions of the existence of the traveling wave solutions. For the considered model the existence of such solutions is equivalent to the existence of the wave solutions in the simplified one-equation model describing the dynamics of thrombin concentration derived under the quasi-stationary approximation. Simplified model also allows us to obtain analytical estimate of the thrombin propagation rate in the considered model. The speed of the traveling wave for one equation is estimated using the narrow reaction zone method and piecewise linear approximation. The resulting formulas give a good approximation of the velocity of propagation of thrombin in the simplified, as well as in the original model.
The article deals with the existence of solutions of a system of integro-differential equations in the case of anomalous diffusion with the Laplacian in a fractional power. The proof of existence of solutions is based on a fixed point technique. Solvability conditions for non Fredholm elliptic operators in unbounded domains are used.
In this article, we establish the existence of solutions of a system of integro-differential equations arising in population dynamics in the case of anomalous diffusion. The proof of the existence of solutions is based on a fixed point technique. Solvability conditions for elliptic operators without the Fredholm property in unbounded domains are used.
We establish the existence of stationary solutions for certain systems of reaction-diffusion equations with superdiffusion. The corresponding elliptic problem involves the operators with or without Fredholm property. The fixed point technique in appropriate H2 spaces of vector functions is employed.
The work deals with the existence of solutions of an integro-differential equation arising in population dynamics in the case of anomalous diffusion. The proof of existence of solutions relies on a fixed point technique. Solvability conditions for non-Fredholm elliptic operators in unbounded domains are being used.
The purpose of this study is to model a lymphedema following a mastectomy and its management (compression therapy). During surgery for breast cancer, an axillary node dissection can be done and cause damages to the lymphatic system leading to a secondary lymphedema located in upper limb. Limb lymphedema is an incurable disease associated with chronic and progressive limb swelling condition. The main clinical consequence of lymphedema is the limb edema, clinically resulting in pain, discomfort, strength reduction and musculoskeletal complications due to limb excessive heaviness. Some devices for lymphedema (e.g. bandaging and garments) could be more personalized, taking into account both characteristics of compressions and patients. Before the evaluation of these therapeutic strategies in humans, an ''in silico'' approach could be used to investigate the interest of gradual or intermittent compression testing in virtual patients. For that purpose, we developed a simplified model of the lymph flow through the lymphatic system in a whole upper limb including the corresponding interstitial fluid exchanges.
Blood coagulation is regulated through a complex network of biochemical reactions of blood factors. The main acting enzyme is thrombin whose propagation in blood plasma leads to fibrin clot formation. Spontaneous clot formation is normally controlled through the action of different plasma inhibitors, in particular, through the thrombin binding by antithrombin. In the current study we develop a mathematical model of clot formation both in quiescent plasma and in blood flow and determine the analytical conditions on the antithrombin concentration corresponding to different regimes of blood coagulation.
Virus spreading in tissues is determined by virus transport, virus multiplication in host cells and the virus-induced immune response. Cytotoxic T cells remove infected cells with a rate determined by the infection level. The intensity of the immune response has a bell-shaped dependence on the concentration of virus, i.e., it increases at low and decays at high infection levels. A combination of these effects and a time delay in the immune response determine the development of virus infection in tissues like spleen or lymph nodes. The mathematical model described in this work consists of reaction-diffusion equations with a delay. It shows that the different regimes of infection spreading like the establishment of a low level infection, a high level infection or a transition between both are determined by the initial virus load and by the intensity of the immune response. The dynamics of the model solutions include simple and composed waves, and periodic and aperiodic oscillations. The results of analytical and numerical studies of the model provide a systematic basis for a quantitative understanding and interpretation of the determinants of the infection process in target organs and tissues from the image-derived data as well as of the spatiotemporal mechanisms of viral disease pathogenesis, and have direct implications for a biopsy-based medical testing of the chronic infection processes caused by viruses, e.g. HIV, HCV and HBV.
We propose to study the wound healing in Zebrafish by using firstly a differential approach for modelling morphogens diffusion and cell chemotactic motion, and secondly a hybrid model of tissue regeneration, where cells are considered as individual objects and molecular concentrations are described by partial differential equations.
The paper is devoted to the numerical investigation of the interaction between natural convection and heat explosion in a fluid-saturated porous media in a rectangular domain. The model consists of Darcy equations for an incompressible fluid in a porous medium coupled with the nonlinear heat equation. Numerical simulations are performed using the radial basis functions method (RBFs). We study the bifurcation of the periodic oscillation of the response born by Hopf bifurcation. First, a symmetry-breaking bifurcations observed; then is followed by successive period-doubling bifurcations leading to chaos.
The paper is devoted to multi-scale modelling of erythropoiesis and hemoglobin production. Red blood cells, which carry oxygen from the lungs to the other body tissues, are produced in the bone marrow of adult humans in cell units called erythroblastic islands. Erythroblastic islands are composed by a central macrophage surrounded by erythroid cells in different stages of maturation. Immature cells, the colony-forming units-erythroid, make a choice between self-renewal, differentiation and apoptosis determined by the intracellular proteins and extracellular substances. Moreover, this choice is regulated by erythropoietin and other hormones. Erythropoietin is produced in the kidney in response to hypoxia from decreased numbers of red blood cells, and it is delivered in the plasma to the bone marrow. Erythropoietin stimulates differentiation of erythroid cells and increases their proliferation by downregulating apoptosis. The rate of erythropoietin production depends on the level of hemoglobin in blood which is function of the number of circulating red blood cells. Hemoglobin is produced in the erythroid cells within the bone marrow in the process of their terminal differentiation. Thus, there is a feedback between production of red blood cells by the bone marrow, the level of hemoglobin contained in these cells and the level of erythropoietin. The multi-scale model developed in this work includes erythroid cells in the bone marrow, their intracellular and extracellular regulations, hemoglobin production, and the feedback by erythropoietin. This model describes normal functioning of erythropoiesis and its response to anemia resulting from the loss of red blood cells.
Collective behavior of a group of individuals is studied. Each individual adopts one of two alternative decisions on the basis of a neural network bistable dynamical system. The parameters of this system are regulated by collective behavior of the group with the purpose to control the number of individuals with certain decision. It is shown how behavior of the group depends on the distribution of initial states of individuals before they begin the process of decision making. If this distribution is narrow, then it can be impossible to achieve a stable coexistence of two decisions, and oscillations in the number of individuals with given decisions are observed. Various implications of this theory are discussed.
The prey-predator model with nonlocal consumption of prey introduced in this work extends previous studies of local reaction-diffusion models. Linear stability analysis of the homogeneous in space stationary solution and numerical simulations of nonhomogeneous solutions allow us to analyze bifurcations and dynamics of stationary solutions and of travelling waves. These solutions present some new properties in comparison with the local models. They correspond to different feeding strategies of predators observed in ecology.
The paper is devoted to the diffusion equation with discrete absorption described by a sum of Dirac (Formula presented.) -functions. Their supports are located at the nodes of some regular grid with the distance between them determined by the integral of solution. This model describes contraction of biological tissue when cells consume some substance influencing their interaction. In the one-dimensional formulation we prove existence of solutions of the discrete problem and their convergence to the solution of the limiting homogenized problem.
Multiple myeloma (MM) infiltrates bone marrow and causes anemia by disrupting erythropoiesis, but the effects of marrow infiltration on anemia are difficult to quantify. Marrow biopsies of newly diagnosed MM patients were analyzed before and after four 28-day cycles of nonerythrotoxic remission induction chemotherapy. Complete blood cell counts and serum paraprotein concentrations were measured at diagnosis and before each chemotherapy cycle. At diagnosis, marrow area infiltrated by myeloma correlated negatively with hemoglobin, erythrocytes, and marrow erythroid cells. After successful chemotherapy, patients with less than 30% myeloma infiltration at diagnosis had no change in these parameters, whereas patients with more than 30% myeloma infiltration at diagnosis increased all three parameters. Clinical data were used to develop mathematical models of the effects of myeloma infiltration on the marrow niches of terminal erythropoiesis, the erythroblastic islands (EBIs). A hybrid discrete-continuous model of erythropoiesis based on EBI structure/function was extended to sections of marrow containing multiple EBIs. In the model, myeloma cells can kill erythroid cells by physically destroying EBIs and by producing proapoptotic cytokines. Following chemotherapy, changes in serum paraproteins as measures of myeloma cells and changes in erythrocyte numbers as measures of marrow erythroid cells allowed modeling of myeloma cell death and erythroid cell recovery, respectively. Simulations of marrow infiltration by myeloma and treatment with nonerythrotoxic chemotherapy demonstrate that myeloma-mediated destruction and subsequent reestablishment of EBIs and expansion of erythroid cell populations in EBIs following chemotherapy provide explanations for anemia development and its therapy-mediated recovery in MM patients.
The article is devoted to the proof of the existence of solutions of a system of integro-differential equations appearing in the case of anomalous diffu sion when the negative Laplacian is raised to some fractional power. The argument relies on a fixed point technique. Solvability conditions for ellip tic operators without Fredholm property in unbounded domains along with the Sobolev inequality for a fractional Laplace operator are being used.
The article deals with the existence of solutions of a system of nonlocal reaction-diffusion equations which appears in population dynamics. The proof relies on a fixed point technique. Solvability conditions for elliptic operators in unbounded domains which fail to satisfy the Fredholm prop- erty are being used.
This paper presents a general review on hybrid modelling which is about to become ubiquitous in biological and medical modelling. Hybrid modelling is classically defined as the coupling of a continuous approach with a discrete one, in order to model a complex phenomenon that cannot be described in a standard homogeneous way mainly due to its inherent multiscale nature. In fact, hybrid modelling can be more than that since any types of coupled formalisms qualify as being hybrid. This review first presents the evolution and current context of this modelling approach. It then proposes a classification of the models through three different types that relate to the nature and level of coupling of the formalisms used.
We study a spatio-temporal prey-predator model with nonlocal interaction terms. Nonlocal interactions are considered for prey and predator species to describe the nonlocal intra-specific competition for limited resources. We show that the region of pattern formation increases with the increase of the range of nonlocal interaction. Numerical continuation technique is used to determine the existence of multiple stationary patterns.
In the preface we present a short overview of articles included in the issue "Bifurcations and pattern formation in biological applications" of the journal Mathematical Modelling of Natural Phenomena.
Abstract Reaction-diffusion system of equations describing blood clotting is studied. Different regimes of clot growth are identified in a quiescent plasma and in blood flow depending on the relative strength of initiation, propagation and inhibition of the thrombin production.
A short review on pharmacokinetics-pharmacodynamics (PK-PD) presented below aims to show the evolution of some concepts and ideas in this field. Some of them are developed in more detail in the papers of this issue. The key question for a practical application of PK-PD models is the ability to estimate the model parameters using patients data.
This review is devoted to recent developments in blood flow modelling. It begins with the discussion of blood rheology and its non-Newtonian properties. After that we will present some modelling methods where blood is considered as a heterogeneous fluid composed of plasma and blood cells. Namely, we will describe the method of Dissipative Particle Dynamics and will present some results of blood flow modelling. The last part of this paper deals with one-dimensional global models of blood circulation. We will explain the main ideas of this approach and will present some examples of its application.
Deep venous thrombosis (DVT) is characterized by formation of blood clot within a deep vein. The resulting thrombus can partially or completely block blood circulation. It can also detach and migrate with the flow resulting in pulmonary embolism. Anticoagulant drugs such as warfarin are usually prescribed to prevent recurrent thrombosis. The action of warfarin is monitored using a blood test for the International Normalized Ratio (INR) which is based on prothrombin time measurement. A high INR indicates a predisposition of the patient to bleeding, while a low INR shows that the warfarin dose is insufficient to prevent thromboembolic events. The therapeutic target of INR varies from case to case depending on clinical indications. It tends to be in the range 2.0-3.0 in most conditions. In this work we develop a model describing blood clotting during warfarin treatment. The action of warfarin is introduced by a Pharmacokinetics-Pharmacodynamics (PK-PD) sub-model. It describes the inhibition of synthesis of the vitamin K dependent factors by warfarin in the liver. We generate a population of patients with individual characteristics and assess their response to warfarin treatment by comparing the simulated INR and the corresponding developed clot height. Using this approach, we determine the underlying causes behind thrombosis and bleeding persistence even for an INR in the normal range. Thus, we suggest a novel methodology to predict the targeted INR depending on individual patient characteristics.
The main objective of our work was to compare different randomized clinical trial (RCT) experimental designs in terms of power, accuracy of the estimation of treatment effect, and number of patients receiving active treatment using in silico simulations. Study Design and Setting A virtual population of patients was simulated and randomized in potential clinical trials. Treatment effect was modeled using a dose-effect relation for quantitative or qualitative outcomes. Different experimental designs were considered, and performances between designs were compared. One thousand clinical trials were simulated for each design based on an example of modeled disease. According to simulation results, the number of patients needed to reach 80% power was 50 for crossover, 60 for parallel or randomized withdrawal, 65 for drop the loser (DL), and 70 for early escape or play the winner (PW). For a given sample size, each design had its own advantage: low duration (parallel, early escape), high statistical power and precision (crossover), and higher number of patients receiving the active treatment (PW and DL). Conclusion Our approach can help to identify the best experimental design, population, and outcome for future RCTs. This may be particularly useful for drug development in rare diseases, theragnostic approaches, or personalized medicine.