Research

30 pages, 4252 KiB

Open AccessArticle

Efficient Methods for Parameter Estimation of Ordinary and Partial Differential Equation Models of Viral Hepatitis Kinetics

by Alexander Churkin, Stephanie Lewkiewicz, Vladimir Reinharz, Harel Dahari and Danny Barash

Mathematics 2020, 8(9), 1483; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091483 - 02 Sep 2020

Cited by 2 | Viewed by 3322

Abstract

Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge [...] Read more.

Parameter estimation in mathematical models that are based on differential equations is known to be of fundamental importance. For sophisticated models such as age-structured models that simulate biological agents, parameter estimation that addresses all cases of data points available presents a formidable challenge and efficiency considerations need to be employed in order for the method to become practical. In the case of age-structured models of viral hepatitis dynamics under antiviral treatment that deal with partial differential equations, a fully numerical parameter estimation method was developed that does not require an analytical approximation of the solution to the multiscale model equations, avoiding the necessity to derive the long-term approximation for each model. However, the method is considerably slow because of precision problems in estimating derivatives with respect to the parameters near their boundary values, making it almost impractical for general use. In order to overcome this limitation, two steps have been taken that significantly reduce the running time by orders of magnitude and thereby lead to a practical method. First, constrained optimization is used, letting the user add constraints relating to the boundary values of each parameter before the method is executed. Second, optimization is performed by derivative-free methods, eliminating the need to evaluate expensive numerical derivative approximations. The newly efficient methods that were developed as a result of the above approach are described for hepatitis C virus kinetic models during antiviral therapy. Illustrations are provided using a user-friendly simulator that incorporates the efficient methods for both the ordinary and partial differential equation models. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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18 pages, 1522 KiB

Open AccessArticle

Mathematical Modelling of the Structure and Function of the Lymphatic System

by Anastasia Mozokhina and Rostislav Savinkov

Mathematics 2020, 8(9), 1467; https://0-doi-org.brum.beds.ac.uk/10.3390/math8091467 - 01 Sep 2020

Cited by 14 | Viewed by 4427

Abstract

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

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

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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16 pages, 7511 KiB

Open AccessArticle

Myocardial Fibrosis in a 3D Model: Effect of Texture on Wave Propagation

by Arsenii Dokuchaev, Alexander V. Panfilov and Olga Solovyova

Mathematics 2020, 8(8), 1352; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081352 - 12 Aug 2020

Cited by 6 | Viewed by 2278

Abstract

Non-linear electrical waves propagate through the heart and control cardiac contraction. Abnormal wave propagation causes various forms of the heart disease and can be lethal. One of the main causes of abnormality is a condition of cardiac fibrosis, which, from mathematical point of [...] Read more.

Non-linear electrical waves propagate through the heart and control cardiac contraction. Abnormal wave propagation causes various forms of the heart disease and can be lethal. One of the main causes of abnormality is a condition of cardiac fibrosis, which, from mathematical point of view, is the presence of multiple non-conducting obstacles for wave propagation. The fibrosis can have different texture which varies from diffuse (e.g., small randomly distributed obstacles), patchy (e.g., elongated interstitional stria), and focal (e.g., post-infarct scars) forms. Recently, Nezlobinsky et al. (2020) used 2D biophysical models to quantify the effects of elongation of obstacles (fibrosis texture) and showed that longitudinal and transversal propagation differently depends on the obstacle length resulting in anisotropy for wave propagation. In this paper, we extend these studies to 3D tissue models. We show that 3D consideration brings essential new effects; for the same obstacle length in 3D systems, anisotropy is about two times smaller compared to 2D, however, wave propagation is more stable with percolation threshold of about 60% (compared to 35% in 2D). The percolation threshold increases with the obstacle length for the longitudinal propagation, while it decreases for the transversal propagation. Further, in 3D, the dependency of velocity on the obstacle length for the transversal propagation disappears. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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18 pages, 3841 KiB

Open AccessArticle

Analysis of Operating Modes for Left Ventricle Assist Devices via Integrated Models of Blood Circulation

by Sergey Simakov, Alexander Timofeev, Timur Gamilov, Philip Kopylov, Dmitry Telyshev and Yuri Vassilevski

Mathematics 2020, 8(8), 1331; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081331 - 10 Aug 2020

Cited by 2 | Viewed by 2262

Abstract

Left ventricular assist devices provide circulatory support to patients with end-stage heart failure. The standard operating conditions of the pump imply limitations on the rotation speed of the rotor. In this work we validate a model for three pumps (Sputnik 1, Sputnik 2, [...] Read more.

Left ventricular assist devices provide circulatory support to patients with end-stage heart failure. The standard operating conditions of the pump imply limitations on the rotation speed of the rotor. In this work we validate a model for three pumps (Sputnik 1, Sputnik 2, Sputnik D) using a mock circulation facility and known data for the pump HeartMate II. We combine this model with a 1D model of haemodynamics in the aorta and a lumped model of the left heart with valves dynamics. The model without pump is validated with known data in normal conditions. Simulations of left ventricular dilated cardiomyopathy show that none of the pumps are capable of reproducing the normal stroke volume in their operating ranges while complying with all criteria of physiologically feasible operation. We also observe that the paediatric pump Sputnik D can operate in the conditions of adult circulation with the same efficiency as the adult LVADs. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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16 pages, 3672 KiB

Open AccessArticle

Computational Analysis of Coronary Blood Flow: The Role of Asynchronous Pacing and Arrhythmias

by Timur Gamilov, Philipp Kopylov, Maria Serova, Roman Syunyaev, Andrey Pikunov, Sofya Belova, Fuyou Liang, Jordi Alastruey and Sergey Simakov

Mathematics 2020, 8(8), 1205; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081205 - 22 Jul 2020

Cited by 13 | Viewed by 2888

Abstract

In this work we present a one-dimensional (1D) mathematical model of the coronary circulation and use it to study the effects of arrhythmias on coronary blood flow (CBF). Hydrodynamical models are rarely used to study arrhythmias’ effects on CBF. Our model accounts for [...] Read more.

In this work we present a one-dimensional (1D) mathematical model of the coronary circulation and use it to study the effects of arrhythmias on coronary blood flow (CBF). Hydrodynamical models are rarely used to study arrhythmias’ effects on CBF. Our model accounts for action potential duration, which updates the length of systole depending on the heart rate. It also includes dependency of stroke volume on heart rate, which is based on clinical data. We apply the new methodology to the computational evaluation of CBF during interventricular asynchrony due to cardiac pacing and some types of arrhythmias including tachycardia, bradycardia, long QT syndrome and premature ventricular contraction (bigeminy, trigeminy, quadrigeminy). We find that CBF can be significantly affected by arrhythmias. CBF at rest (60 bpm) is 26% lower in LCA and 22% lower in RCA for long QT syndrome. During bigeminy, trigeminy and quadrigeminy, respectively, CBF decreases by 28%, 19% and 14% with respect to a healthy case. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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20 pages, 829 KiB

Open AccessArticle

Optimization of Dose Fractionation for Radiotherapy of a Solid Tumor with Account of Oxygen Effect and Proliferative Heterogeneity

by Maxim Kuznetsov and Andrey Kolobov

Mathematics 2020, 8(8), 1204; https://0-doi-org.brum.beds.ac.uk/10.3390/math8081204 - 22 Jul 2020

Cited by 12 | Viewed by 2226

Abstract

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A [...] Read more.

A spatially-distributed continuous mathematical model of solid tumor growth and treatment by fractionated radiotherapy is presented. The model explicitly accounts for three time and space-dependent factors that influence the efficiency of radiotherapy fractionation schemes—tumor cell repopulation, reoxygenation and redistribution of proliferative states. A special algorithm is developed, aimed at finding the fractionation schemes that provide increased tumor cure probability under the constraints of maximum normal tissue damage and maximum fractional dose. The optimization procedure is performed for varied radiosensitivity of tumor cells under the values of model parameters, corresponding to different degrees of tumor malignancy. The resulting optimized schemes consist of two stages. The first stages are aimed to increase the radiosensitivity of the tumor cells, remaining after their end, sparing the caused normal tissue damage. This allows to increase the doses during the second stages and thus take advantage of the obtained increased radiosensitivity. Such method leads to significant expansions in the curative ranges of the values of tumor radiosensitivity parameters. Overall, the results of this study represent the theoretical proof of concept that non-uniform radiotherapy fractionation schemes may be considerably more effective that uniform ones, due to the time and space-dependent effects. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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19 pages, 2617 KiB

Open AccessArticle

Hemodynamic Effects of Alpha-Tropomyosin Mutations Associated with Inherited Cardiomyopathies: Multiscale Simulation

by Fyodor Syomin, Albina Khabibullina, Anna Osepyan and Andrey Tsaturyan

Mathematics 2020, 8(7), 1169; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071169 - 16 Jul 2020

Cited by 2 | Viewed by 1754

Abstract

The effects of two cardiomyopathy-associated mutations in regulatory sarcomere protein tropomyosin (Tpm) on heart function were studied with a new multiscale model of the cardiovascular system (CVS). They were a Tpm mutation, Ile284Val, associated with hypertrophic cardiomyopathy (HCM), and an Asp230Asn one associated [...] Read more.

The effects of two cardiomyopathy-associated mutations in regulatory sarcomere protein tropomyosin (Tpm) on heart function were studied with a new multiscale model of the cardiovascular system (CVS). They were a Tpm mutation, Ile284Val, associated with hypertrophic cardiomyopathy (HCM), and an Asp230Asn one associated with dilated cardiomyopathy (DCM). When the molecular and cell-level changes in the Ca²⁺ regulation of cardiac muscle caused by these mutations were introduced into the myocardial model of the left ventricle (LV) while the LV shape remained the same as in the model of the normal heart, the cardiac output and arterial blood pressure reduced. Simulations of LV hypertrophy in the case of the Ile284Val mutation and LV dilatation in the case of the Asp230Asn mutation demonstrated that the LV remodeling partially recovered the stroke volume and arterial blood pressure, confirming that both hypertrophy and dilatation help to preserve the LV function. The possible effects of changes in passive myocardial stiffness in the model according to data reported for HCM and DCM hearts were also simulated. The results of the simulations showed that the end-systolic pressure–volume relation that is often used to characterize heart contractility strongly depends on heart geometry and cannot be used as a characteristic of myocardial contractility. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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17 pages, 557 KiB

Open AccessArticle

Dynamics of Periodic Waves in a Neural Field Model

by Nikolai Bessonov, Anne Beuter, Sergei Trofimchuk and Vitaly Volpert

Mathematics 2020, 8(7), 1076; https://0-doi-org.brum.beds.ac.uk/10.3390/math8071076 - 02 Jul 2020

Cited by 1 | Viewed by 1891

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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13 pages, 2386 KiB

Open AccessArticle

Drift of Scroll Waves in a Mathematical Model of a Heterogeneous Human Heart Left Ventricle

by Sergey Pravdin, Pavel Konovalov, Hans Dierckx, Olga Solovyova and Alexander V. Panfilov

Mathematics 2020, 8(5), 776; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050776 - 12 May 2020

Cited by 4 | Viewed by 2447

Abstract

Rotating spiral waves of electrical excitation underlie many dangerous cardiac arrhythmias. The heterogeneity of myocardium is one of the factors that affects the dynamics of such waves. In this paper, we present results of our simulations for scroll wave dynamics in a heterogeneous [...] Read more.

Rotating spiral waves of electrical excitation underlie many dangerous cardiac arrhythmias. The heterogeneity of myocardium is one of the factors that affects the dynamics of such waves. In this paper, we present results of our simulations for scroll wave dynamics in a heterogeneous model of the human left ventricle with analytical anatomically based representation of the geometry and anisotropy. We used a set of 18 coupled differential equations developed by ten Tusscher and Panfilov (TP06 model) which describes human ventricular cells based on their measured biophysical properties. We found that apicobasal heterogeneity dramatically changes the scroll wave dynamics. In the homogeneous model, the scroll wave annihilates at the base, but the moderate heterogeneity causes the wave to move to the apex and then continuously rotates around it. The rotation speed increased with the degree of the heterogeneity. However, for large heterogeneity, we observed formation of additional wavebreaks and the onset of complex spatio-temporal patterns. Transmural heterogeneity did not change the dynamics and decreased the lifetime of the scroll wave with an increase in heterogeneity. Results of our numerical experiments show that the apex may be a preferable location of the scroll wave, which may be important for development of clinical interventions. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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19 pages, 702 KiB

Open AccessArticle

Mathematical Modeling Shows That the Response of a Solid Tumor to Antiangiogenic Therapy Depends on the Type of Growth

by Maxim Kuznetsov

Mathematics 2020, 8(5), 760; https://0-doi-org.brum.beds.ac.uk/10.3390/math8050760 - 11 May 2020

Cited by 9 | Viewed by 2163

Abstract

It has been hypothesized that solid tumors with invasive type of growth should possess intrinsic resistance to antiangiogenic therapy, which is aimed at cessation of the formation of new blood vessels and subsequent shortage of nutrient inflow to the tumor. In order to [...] Read more.

It has been hypothesized that solid tumors with invasive type of growth should possess intrinsic resistance to antiangiogenic therapy, which is aimed at cessation of the formation of new blood vessels and subsequent shortage of nutrient inflow to the tumor. In order to investigate this effect, a continuous mathematical model of tumor growth is developed, which considers variables of tumor cells, necrotic tissue, capillaries, and glucose as the crucial nutrient. The model accounts for the intrinsic motility of tumor cells and for the convective motion, arising due to their proliferation, thus allowing considering two types of tumor growth—invasive and compact—as well as their combination. Analytical estimations of tumor growth speed are obtained for compact and invasive tumors. They suggest that antiangiogenic therapy may provide a several times decrease of compact tumor growth speed, but the decrease of growth speed for invasive tumors should be only modest. These estimations are confirmed by numerical simulations, which further allow evaluating the effect of antiangiogenic therapy on tumors with mixed growth type and highlight the non-additive character of the two types of growth. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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20 pages, 2323 KiB

Open AccessArticle

Nonlocal Reaction–Diffusion Model of Viral Evolution: Emergence of Virus Strains

by Nikolai Bessonov, Gennady Bocharov, Andreas Meyerhans, Vladimir Popov and Vitaly Volpert

Mathematics 2020, 8(1), 117; https://0-doi-org.brum.beds.ac.uk/10.3390/math8010117 - 12 Jan 2020

Cited by 8 | Viewed by 4726

Abstract

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 [...] Read more.

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. Full article

(This article belongs to the Special Issue Mathematical Modelling in Biomedicine)

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