Research

15 pages, 730 KiB

Open AccessArticle

Viral Infection Spreading and Mutation in Cell Culture

by Latifa Ait Mahiout, Bogdan Kazmierczak and Vitaly Volpert

Mathematics 2022, 10(2), 256; https://0-doi-org.brum.beds.ac.uk/10.3390/math10020256 - 14 Jan 2022

Cited by 8 | Viewed by 1573

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized [...] Read more.

A new model of viral infection spreading in cell cultures is proposed taking into account virus mutation. This model represents a reaction-diffusion system of equations with time delay for the concentrations of uninfected cells, infected cells and viral load. Infection progression is characterized by the virus replication number

R_{v}

, which determines the total viral load. Analytical formulas for the speed of propagation and for the viral load are obtained and confirmed by numerical simulations. It is shown that virus mutation leads to the emergence of a new virus variant. Conditions of the coexistence of the two variants or competitive exclusion of one of them are found, and different stages of infection progression are identified. Full article

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

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15 pages, 4960 KiB

Open AccessArticle

Rotational Activity around an Obstacle in 2D Cardiac Tissue in Presence of Cellular Heterogeneity

by Pavel Konovalov, Daria Mangileva, Arsenii Dokuchaev, Olga Solovyova and Alexander V. Panfilov

Mathematics 2021, 9(23), 3090; https://0-doi-org.brum.beds.ac.uk/10.3390/math9233090 - 30 Nov 2021

Cited by 4 | Viewed by 1647

Abstract

Waves of electrical excitation rotating around an obstacle is one of the important mechanisms of dangerous cardiac arrhythmias occurring in the heart damaged by a post-infarction scar. Such a scar is also surrounded by the region of heterogeneity called a gray zone. In [...] Read more.

Waves of electrical excitation rotating around an obstacle is one of the important mechanisms of dangerous cardiac arrhythmias occurring in the heart damaged by a post-infarction scar. Such a scar is also surrounded by the region of heterogeneity called a gray zone. In this paper, we perform the first comprehensive numerical study of various regimes of wave rotation around an obstacle surrounded by a gray zone. We use the TP06 cellular ionic model for human cardiomyocytes and study how the period and the pattern of wave rotation depend on the radius of a circular obstacle and the width of a circular gray zone. Our main conclusions are the following. The wave rotation regimes can be subdivided into three main classes: (1) functional rotation, (2) scar rotation and the newly found (3) gray zone rotation regimes. In the scar rotation regime, the wave rotates around the obstacle, while in the gray zone regime, the wave rotates around the gray zone. As a result, the period of rotation is determined by the perimeter of the scar, or gray zone perimeter correspondingly. The transition from the scar to the gray rotation regimes can be determined from the minimal period principle, formulated in this paper. We have also observed additional regimes associated with two types of dynamical instabilities which may affect or not affect the period of rotation. The results of this study can help to identify the factors determining the period of arrhythmias in post-infarction patients. Full article

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

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15 pages, 4743 KiB

Open AccessArticle

Period of Arrhythmia Anchored around an Infarction Scar in an Anatomical Model of the Human Ventricles

by Daria Mangileva, Pavel Konovalov, Arsenii Dokuchaev, Olga Solovyova and Alexander V. Panfilov

Mathematics 2021, 9(22), 2911; https://0-doi-org.brum.beds.ac.uk/10.3390/math9222911 - 15 Nov 2021

Cited by 1 | Viewed by 1405

Abstract

Rotating nonlinear waves of excitation in the heart cause dangerous cardiac arrhythmias. Frequently, ventricular arrhythmias occur as a result of myocardial infarction and are associated with rotation of the waves around a post-infarction scar. In this paper, we perform a detailed in silico [...] Read more.

Rotating nonlinear waves of excitation in the heart cause dangerous cardiac arrhythmias. Frequently, ventricular arrhythmias occur as a result of myocardial infarction and are associated with rotation of the waves around a post-infarction scar. In this paper, we perform a detailed in silico analysis of scroll waves in an anatomical model of the human ventricles with a generic model of the infarction scar surrounded by the gray zone with modified properties of the myocardial tissue. Our model includes a realistic description of the heart shape, anisotropy of cardiac tissue and a detailed description of the electrical activity in human ventricular cells by a TP06 ionic model. We vary the size of the scar and gray zone and analyze the dependence of the rotation period on the injury dimensions. Two main regimes of wave scrolling are observed: the scar rotation, when the wave rotates around the scar, and the gray zone rotation, when the wave rotates around the boundary of the gray zone and normal tissue. The transition from the gray zone to the scar rotation occurs for the width of gray zone above 10–20 mm, depending on the perimeter of the scar. We compare our results with simulations in 2D and show that 3D anisotropy reduces the period of rotation. We finally use a model with a realistic shape of the scar and show that our approach predicts correctly the period of the arrhythmia. Full article

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

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27 pages, 4701 KiB

Open AccessArticle

Anatomical Model of Rat Ventricles to Study Cardiac Arrhythmias under Infarction Injury

by Roman Rokeakh, Tatiana Nesterova, Konstantin Ushenin, Ekaterina Polyakova, Dmitry Sonin, Michael Galagudza, Tim De Coster, Alexander Panfilov and Olga Solovyova

Mathematics 2021, 9(20), 2604; https://0-doi-org.brum.beds.ac.uk/10.3390/math9202604 - 15 Oct 2021

Cited by 2 | Viewed by 1999

Abstract

Species-specific computer models of the heart are a novel powerful tool in studies of life-threatening cardiac arrhythmias. Here, we develop such a model aimed at studying infarction injury in a rat heart, the most common experimental system to investigate the effects of myocardial [...] Read more.

Species-specific computer models of the heart are a novel powerful tool in studies of life-threatening cardiac arrhythmias. Here, we develop such a model aimed at studying infarction injury in a rat heart, the most common experimental system to investigate the effects of myocardial damage. We updated the Gattoni2016 cellular ionic model by fitting its parameters to experimental data using a population modeling approach. Using four selected cellular models, we studied 2D spiral wave dynamics and found that they include meandering and break-up. Then, using an anatomically realistic ventricular geometry and fiber orientation in the rat heart, we built a model with a post-infarction scar to study the electrophysiological effects of myocardial damage. A post-infarction scar was simulated as an inexcitable obstacle surrounded by a border zone with modified cardiomyocyte properties. For cellular models, we studied the rotation of scroll waves and found that, depending on the model, we can observe different types of dynamics: anchoring, self-termination or stable rotation of the scroll wave. The observed arrhythmia characteristics coincide with those measured in the experiment. The developed model can be used to study arrhythmia in rat hearts with myocardial damage from ischemia reperfusion and to examine the possible arrhythmogenic effects of various experimental interventions. Full article

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

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19 pages, 1545 KiB

Open AccessArticle

Blood Clotting Decreases Pulmonary Circulation during the Coronavirus Disease

by Anastasia Mozokhina, Anass Bouchnita and Vitaly Volpert

Mathematics 2021, 9(19), 2401; https://0-doi-org.brum.beds.ac.uk/10.3390/math9192401 - 27 Sep 2021

Cited by 3 | Viewed by 1625

Abstract

Spontaneous blood clotting in pulmonary circulation caused by thrombo-inflammation is one of the main mortality causes during the COVID-19 disease. Blood clotting leads to reduced pulmonary circulation and blood oxygenation. Lung inflammation can be evaluated with noninvasive diagnostic techniques. However, the correlation of [...] Read more.

Spontaneous blood clotting in pulmonary circulation caused by thrombo-inflammation is one of the main mortality causes during the COVID-19 disease. Blood clotting leads to reduced pulmonary circulation and blood oxygenation. Lung inflammation can be evaluated with noninvasive diagnostic techniques. However, the correlation of the severity of the inflammation with the pulmonary blood flow has not been established. To address this question, in this work, we develop a multiscale model taking into account the interaction of a local model of thrombus growth with 1D hemodynamics in a vessel network. Flux reduction depending on the level of lung obstruction is evaluated. In particular, the model obtains that an obstruction level of 5% leads to a 12% reduction of blood flux. The suggested approach can be used to investigate the interaction of blood clotting and flow not only in the pulmonary network but also in other complex vessel networks. Full article

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

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12 pages, 1342 KiB

Open AccessArticle

Numerical Modelling of Multicellular Spheroid Compression: Viscoelastic Fluid vs. Viscoelastic Solid

by Ruslan Yanbarisov, Yuri Efremov, Nastasia Kosheleva, Peter Timashev and Yuri Vassilevski

Mathematics 2021, 9(18), 2333; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182333 - 20 Sep 2021

Cited by 3 | Viewed by 1893

Abstract

Parallel-plate compression of multicellular spheroids (MCSs) is a promising and popular technique to quantify the viscoelastic properties of living tissues. This work presents two different approaches to the simulation of the MCS compression based on viscoelastic solid and viscoelastic fluid models. The first [...] Read more.

Parallel-plate compression of multicellular spheroids (MCSs) is a promising and popular technique to quantify the viscoelastic properties of living tissues. This work presents two different approaches to the simulation of the MCS compression based on viscoelastic solid and viscoelastic fluid models. The first one is the standard linear solid model implemented in ABAQUS/CAE. The second one is the new model for 3D viscoelastic free surface fluid flow, which combines the Oldroyd-B incompressible fluid model and the incompressible neo-Hookean solid model via incorporation of an additional elastic tensor and a dynamic equation for it. The simulation results indicate that either approach can be applied to model the MCS compression with reasonable accuracy. Future application of the viscoelastic free surface fluid model is the MCSs fusion highly-demanded in bioprinting. Full article

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

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19 pages, 755 KiB

Open AccessArticle

Computational Analysis of Haemodynamic Indices in Synthetic Atherosclerotic Coronary Netwroks

by Sergey Simakov, Timur Gamilov, Fuyou Liang and Philipp Kopylov

Mathematics 2021, 9(18), 2221; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182221 - 10 Sep 2021

Cited by 8 | Viewed by 1747

Abstract

Haemodynamic indices are widely used in clinical practice when deciding on a particular type of treatment. Low quality of the computed tomography data and tachycardia complicate interpretation of the measured or simulated values. In this work, we present a novel approach for evaluating [...] Read more.

Haemodynamic indices are widely used in clinical practice when deciding on a particular type of treatment. Low quality of the computed tomography data and tachycardia complicate interpretation of the measured or simulated values. In this work, we present a novel approach for evaluating resistances in terminal coronary arteries. Using 14 measurements from 10 patients, we show that this algorithm retains the accuracy of 1D haemodynamic simulations in less detailed (truncated) geometric models of coronary networks. We also apply the variable systole fraction model to study the effect of elevated heart rate on the values of fractional flow reserve (FFR), coronary flow reserve (CFR) and instantaneous wave-free ratio (iFR). We conclude that tachycardia may produce both overestimation or underestimation of coronary stenosis significance. Full article

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

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27 pages, 885 KiB

Open AccessArticle

Combined Influence of Nutrient Supply Level and Tissue Mechanical Properties on Benign Tumor Growth as Revealed by Mathematical Modeling

by Maxim Kuznetsov

Mathematics 2021, 9(18), 2213; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182213 - 09 Sep 2021

Cited by 5 | Viewed by 1864

Abstract

A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which [...] Read more.

A continuous mathematical model of non-invasive avascular tumor growth in tissue is presented. The model considers tissue as a biphasic material, comprised of a solid matrix and interstitial fluid. The convective motion of tissue elements happens due to the gradients of stress, which change as a result of tumor cells proliferation and death. The model accounts for glucose as the crucial nutrient, supplied from the normal tissue, and can reproduce both diffusion-limited and stress-limited tumor growth. Approximate tumor growth curves are obtained semi-analytically in the limit of infinite tissue hydraulic conductivity, which implies instantaneous equalization of arising stress gradients. These growth curves correspond well to the numerical solutions and represent classical sigmoidal curves with a short initial exponential phase, subsequent almost linear growth phase and a phase with growth deceleration, in which tumor tends to reach its maximum volume. The influence of two model parameters on tumor growth curves is investigated: tissue hydraulic conductivity, which links the values of stress gradient and convective velocity of tissue phases, and tumor nutrient supply level, which corresponds to different permeability and surface area density of capillaries in the normal tissue that surrounds the tumor. In particular, it is demonstrated, that sufficiently low tissue hydraulic conductivity (intrinsic, e.g., to tumors arising from connective tissue) and sufficiently high nutrient supply can lead to formation of giant benign tumors, reaching tens of centimeters in diameter, which are indeed observed clinically. Full article

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

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23 pages, 5547 KiB

Open AccessArticle

Impact of Material Stiffness and Anisotropy on Coaptation Characteristics for Aortic Valve Cusps Reconstructed from Pericardium

by Alexey Liogky, Pavel Karavaikin and Victoria Salamatova

Mathematics 2021, 9(18), 2193; https://0-doi-org.brum.beds.ac.uk/10.3390/math9182193 - 08 Sep 2021

Cited by 8 | Viewed by 1833

Abstract

The numerical assessment of reconstructed aortic valves competence and leaflet design optimization rely on both coaptation characteristics and the diastolic valve configuration. These characteristics can be evaluated by the shell or membrane formulations. The membrane formulation is preferable for surgical aortic valve neocuspidization [...] Read more.

The numerical assessment of reconstructed aortic valves competence and leaflet design optimization rely on both coaptation characteristics and the diastolic valve configuration. These characteristics can be evaluated by the shell or membrane formulations. The membrane formulation is preferable for surgical aortic valve neocuspidization planning since it is easy to solve. The results on coaptation zone sensitivity to the anisotropy of aortic leaflet material are contradictive, and there are no comparisons of coaptation characteristics based on shell and membrane models for anisotropic materials. In our study, we explore for the first time how the reduced model and anisotropy of the leaflet material affect the coaptation zone and the diastolic configuration of the aortic valve. To this end, we propose the method to mimic the real, sutured neo-leaflet, and apply our numerical shell and membrane formulations to model the aortic valve under the quasi-static diastolic pressure varying material stiffness and anisotropy directions. The shell formulation usually provides a lesser coaptation zone than the membrane formulation, especially in the central zone. The material stiffness does influence the coaptation zone: it is smaller for stiffer material. Anisotropy of the leaflet material does not affect significantly the coaptation characteristics, but can impact the deformed leaflet configuration and produce a smaller displacement. Full article

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

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16 pages, 4993 KiB

Open AccessArticle

Impact of Arrhythmia on Myocardial Perfusion: A Computational Model-Based Study

by Xinyang Ge, Sergey Simakov, Youjun Liu and Fuyou Liang

Mathematics 2021, 9(17), 2128; https://0-doi-org.brum.beds.ac.uk/10.3390/math9172128 - 02 Sep 2021

Cited by 1 | Viewed by 2027

Abstract

(1) Background: Arrhythmia, which is an umbrella term for various types of abnormal rhythms of heartbeat, has a high prevalence in both the general population and patients with coronary artery disease. So far, it remains unclear how different types of arrhythmia would affect [...] Read more.

(1) Background: Arrhythmia, which is an umbrella term for various types of abnormal rhythms of heartbeat, has a high prevalence in both the general population and patients with coronary artery disease. So far, it remains unclear how different types of arrhythmia would affect myocardial perfusion and the risk/severity of myocardial ischemia. (2) Methods: A computational model of the coronary circulation coupled to the global cardiovascular system was employed to quantify the impacts of arrhythmia and its combination with coronary artery disease on myocardial perfusion. Furthermore, a myocardial supply–demand balance index (MSDBx) was proposed to quantitatively evaluate the severity of myocardial ischemia under various arrhythmic conditions. (3) Results: Tachycardia and severe irregularity of heart rates (HRs) depressed myocardial perfusion and increased the risk of subendocardial ischemia (evaluated by MSDBx), whereas lowering HR improved myocardial perfusion. The presence of a moderate to severe coronary artery stenosis considerably augmented the sensitivity of MSDBx to arrhythmia. Further data analyses revealed that arrhythmia induced myocardial ischemia mainly via reducing the amount of coronary artery blood flow in each individual cardiac cycle rather than increasing the metabolic demand of the myocardium (measured by the left ventricular pressure-volume area). (4) Conclusions: Both tachycardia and irregular heartbeat tend to increase the risk of myocardial ischemia, especially in the subendocardium, and the effects can be further enhanced by concomitant existence of coronary artery disease. In contrast, properly lowering HR using drugs like β-blockers may improve myocardial perfusion, thereby preventing or relieving myocardial ischemia in patients with coronary artery disease. Full article

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

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