Research

19 pages, 3883 KiB

Open AccessArticle

Toward Optimal Fitting Parameters for Multi-Exponential DWI Image Analysis of the Human Kidney: A Simulation Study Comparing Different Fitting Algorithms

by Jonas Jasse, Hans-Joerg Wittsack, Thomas Andreas Thiel, Romans Zukovs, Birte Valentin, Gerald Antoch and Alexandra Ljimani

Mathematics 2024, 12(4), 609; https://doi.org/10.3390/math12040609 - 18 Feb 2024

Viewed by 614

Abstract

In DWI, multi-exponential signal analysis can be used to determine signal underlying diffusion components. However, the approach is very complex due to the inherent low SNR, the limited number of signal decay data points, and the absence of appropriate acquisition parameters and standardized [...] Read more.

In DWI, multi-exponential signal analysis can be used to determine signal underlying diffusion components. However, the approach is very complex due to the inherent low SNR, the limited number of signal decay data points, and the absence of appropriate acquisition parameters and standardized analysis methods. Within the scope of this work, different methods for multi-exponential analysis of the diffusion signal in the kidney were compared. To assess the impact of fitting parameters, a simulation was conducted comparing the free non-negative (NNLS) and rigid non-linear least square (NLLS) fitting methods. The simulation demonstrated improved accuracy for NNLS in combination with area-under-curve estimation. Furthermore, the accuracy and stability of the results were further enhanced utilizing optimized parameters, namely 350 logarithmically spaced diffusion coefficients within [0.7, 300] × 10⁻³ mm²/s and a minimal SNR of 100. The NNLS approach shows an improvement over the rigid NLLS method. This becomes apparent not only in terms of accuracy and omitting prior knowledge, but also in better representation of renal tissue physiology. By employing the determined fitting parameters, it is expected that more stable and reliable results for diffusion imaging in the kidney can be achieved. This might enable more accurate DWI results for clinical utilization. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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39 pages, 1344 KiB

Open AccessArticle

Controlling COVID-19 Spreading: A Three-Level Algorithm

by Giovanni Dieguez, Cristiane Batistela and José R. C. Piqueira

Mathematics 2023, 11(17), 3766; https://doi.org/10.3390/math11173766 - 01 Sep 2023

Viewed by 941

Abstract

As the main methods of the coronavirus disease (COVID-19) transmission are air and physical contact, actions to mitigate and suppress its spread must be developed in order to change population dynamics and provide efficient control strategies. Here, these actions are described as a [...] Read more.

As the main methods of the coronavirus disease (COVID-19) transmission are air and physical contact, actions to mitigate and suppress its spread must be developed in order to change population dynamics and provide efficient control strategies. Here, these actions are described as a simple heuristic framework to establish public policies. Two control systems were studied: the first organized in the form of an algorithm stratified into three levels and the second as a minimization problem similar to optimal control strategies, applied to both social distancing and vaccination. The possible effects of these actions are modeled and applied to an extension of the Susceptible - Infected - Removed (SIR) compartmental model. The control system is developed, which is organized in the form of an algorithm stratified into three levels. These levels intend to represent social distancing strategies implemented by sanitary authorities around the globe, representing stronger or weaker grades of isolation intensity according to the ability of the healthcare system to cope with symptomatic individuals. The algorithm control is applied in a simulation, and the results give evidence of the effectiveness of the procedures adopted against the coronavirus. The model dynamics are analyzed and validated with simulations considering parameters obtained from epidemiological data from Brazil and Uruguay and in a more detailed way for three Brazilian states: São Paulo, Minas Gerais and Rio de Janeiro. The model was validated using cumulative data on cases and deaths. For cases of death, the results were satisfactory, while for case data, the response was reasonable, considering the possibility of adding delays or variations in parameters in the model. In addition, the effective reproduction number was proposed for the cities studied in Brazil, the result being relevant because it has a qualitative behavior similar to that published by official centers. This paper also discusses the implementation and optimization of social distancing and vaccination control strategies, considering different parameters and their effects on reducing the number of cases and deaths. Model simulations present promising results for developing strategies to attack COVID-19 dissemination. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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17 pages, 2506 KiB

Open AccessArticle

Invadopodia Formation in Cancer Cell: The Mathematical and Computational Modelling Based on Free Boundary Problem

by Muhammad Akmal Ramlee, Nuha Loling Othman and Takashi Suzuki

Mathematics 2023, 11(14), 3044; https://doi.org/10.3390/math11143044 - 09 Jul 2023

Viewed by 1041

Abstract

We present a mathematical model of an individual cell to expand the simulation of invadopodia formation to a three-dimensional (3D) domain for a more realistic complexity. Simulating invadopodia replication in order for it to be biologically relevant is important since it helps us [...] Read more.

We present a mathematical model of an individual cell to expand the simulation of invadopodia formation to a three-dimensional (3D) domain for a more realistic complexity. Simulating invadopodia replication in order for it to be biologically relevant is important since it helps us to understand cancer invasion and metastasis better as well as giving some insight into investigating ways to stop the spread of this fatal disease. Invadopodia formation is formulated using the Stefan problem approach, where the free boundary is characterised by the Stefan free boundary condition, in which the boundary membrane is not known in advance. Level set method is proposed to indicate the behaviour of the cell interface and the motion of the plasma membrane. An enthalpy method (phase-transition problem) is used to describe the cell membrane diffusion. In addition to this, we were able to improve the simulation outcome, giving it a more realistic complexity by using a different simulation technique and domain as well as a different data set. Singularities and instabilities were eliminated. The results that were achieved have the potential to be helpful for novel approaches or to be extended to other methods in the development of a more accurate numerical simulation. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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23 pages, 3845 KiB

Open AccessArticle

Epidemiological Investigation: Important Measures for the Prevention and Control of COVID-19 Epidemic in China

by Cheng-Cheng Zhu, Jiang Zhu and Jie Shao

Mathematics 2023, 11(13), 3027; https://doi.org/10.3390/math11133027 - 07 Jul 2023

Cited by 1 | Viewed by 930

Abstract

Based on China’s summary of three years of experience and measures in the prevention and control of the COVID-19 epidemic, we have built a COVID-19 prevention and control model integrating health and medical detection, big data information technology to track the trend of [...] Read more.

Based on China’s summary of three years of experience and measures in the prevention and control of the COVID-19 epidemic, we have built a COVID-19 prevention and control model integrating health and medical detection, big data information technology to track the trend of the epidemic throughout the whole process, isolation of key epidemic areas, and dynamic prevention and control management throughout the whole process. This model provides a simple, feasible, and theoretically reliable prevention and control model for future large-scale infectious disease prevention and control. The Lyapnov functional method is replaced by the global exponential attractor theory, which provides a new mathematical method for studying the global stability of the multi parameter, multi variable infectious disease prevention and control system. We extracted mathematical methods and models suitable for non-mathematical infectious disease researchers from profound and difficult to understand mathematical theories. Using the results of the global exponential Attractor theory obtained in this paper, we studied the global dynamics of the COVID-19 model with an epidemiological investigation. The results demonstrated that the non-constant disease-free equilibrium is globally asymptotically stable when

λ^{*} < 0

, and the COVID-19 epidemic is persisting uniformly when

λ^{*} > 0

. In order to understand the impact of the epidemiological investigation under different prevention and control stages in China, we compare the control effects of COVID-19 under different levels of epidemiological investigation policies. We visually demonstrate the global stability and global exponential attractiveness of the COVID-19 model with transferors between regions and epidemiological investigation in a temporal-spatial heterogeneous environment with the help of numerical simulations. We find that the epidemiological investigation really has a significant effect on the prevention and control of the epidemic situation, and we can also intuitively observe the relationship between the flow of people (including tourism, shopping, work and so on) and epidemiological investigation policies. Our model is adapted to different stages of prevention and control; the emergency “circuit breaker” mechanism of the model is also consistent with actual prevention and control. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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25 pages, 2202 KiB

Open AccessArticle

On Incidence-Dependent Management Strategies against an SEIRS Epidemic: Extinction of the Epidemic Using Allee Effect

by Tri Nguyen-Huu, Pierre Auger and Ali Moussaoui

Mathematics 2023, 11(13), 2822; https://doi.org/10.3390/math11132822 - 23 Jun 2023

Cited by 2 | Viewed by 843

Abstract

We developed a mathematical model to study the effects of non-pharmaceutical interventions (NPIs) on the dynamics of an epidemic. The level of intervention was assessed as a fraction of the population being isolated and depended on the level of incidence of the epidemic [...] Read more.

We developed a mathematical model to study the effects of non-pharmaceutical interventions (NPIs) on the dynamics of an epidemic. The level of intervention was assessed as a fraction of the population being isolated and depended on the level of incidence of the epidemic in the population. We performed a mathematical analysis of the model and showed that, depending on the choice of the prevalence-dependent isolation function, it is possible to create new endemic equilibria and to change the stability of the disease-free equilibrium for which the epidemic vanishes. The model was then applied to the case of the COVID-19 pandemic. Several NPI management strategies were considered. In the case of an NPI intensity increasing with the level of infection, it is possible to avoid the initial epidemic peak of great amplitude that would have occurred without intervention and to stabilize the epidemic at a chosen and sufficiently low endemic level. In the case of an NPI intensity decreasing with the level of infection, the epidemic can be driven to extinction by generating an “Allee” effect: when the incidence is below a given level, the epidemic goes extinct whereas, above it, the epidemic will still be able take hold at a lower endemic level. Simulations illustrate that appropriate NPIs could make the COVID-19 vanish relatively fast. We show that, in the context of the COVID-19 pandemic, most countries have not chosen to use the most efficient strategies. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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22 pages, 6055 KiB

Open AccessArticle

A Mathematical Model to Optimize the Neoadjuvant Chemotherapy Treatment Sequence for Triple-Negative Locally Advanced Breast Cancer

by Juan C. López-Alvarenga, Antonmaria Minzoni-Alessio, Arturo Olvera-Chávez, Gustavo Cruz-Pacheco, Juan C. Chimal-Eguia, Joselín Hernández-Ruíz, Mario A. Álvarez-Blanco, María Y. Bautista-Hernández and Rosa M. Quispe-Siccha

Mathematics 2023, 11(11), 2410; https://doi.org/10.3390/math11112410 - 23 May 2023

Cited by 2 | Viewed by 1320

Abstract

Background: Triple-negative locally advanced breast cancer is an aggressive tumor type. Currently, the standard sequence treatment is applied, administering anthracyclines first and then a taxane plus platinum. Clinical studies for all possible treatment combinations are not practical or affordable, but mathematical modeling of [...] Read more.

Background: Triple-negative locally advanced breast cancer is an aggressive tumor type. Currently, the standard sequence treatment is applied, administering anthracyclines first and then a taxane plus platinum. Clinical studies for all possible treatment combinations are not practical or affordable, but mathematical modeling of the active mitotic cell population is possible. Our study aims to show the regions with the tumor’s most substantial cellular population variation by utilizing all possible values of the parameters

(α_{s}^{i})

that define the annihilatory drug capacity according to the proposed treatment. Method: A piecewise linear mathematical model was used to analyze the cell population growth by applying four treatments: standard sequences of 21 days (SS21) and 14 days (SS14), administering anthracyclines first, followed by a taxane plus platinum, and inverted sequences of 21 days (IS21) and 14 days (IS14), administering a taxane plus platinum first then anthracyclines. Results: The simulation showed a higher effect of IS14 over SS14 when the rate of drug resistance was larger in the cell population during DNA synthesis (G1 and S) compared to cells in mitosis (G2 and M). However, if the proportion of resistant cells in both populations was equivalent, then treatments did not differ. Conclusions: When resistance is considerable, IS14 is more efficient than SS14, reducing the tumor population to a minimum. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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20 pages, 716 KiB

Open AccessArticle

Modeling and Analyzing Homogeneous Tumor Growth under Virotherapy

by Chayu Yang and Jin Wang

Mathematics 2023, 11(2), 360; https://doi.org/10.3390/math11020360 - 10 Jan 2023

Viewed by 1038

Abstract

We present a mathematical model based on ordinary differential equations to investigate the spatially homogeneous state of tumor growth under virotherapy. The model emphasizes the interaction among the tumor cells, the oncolytic viruses, and the host immune system that generates both innate and [...] Read more.

We present a mathematical model based on ordinary differential equations to investigate the spatially homogeneous state of tumor growth under virotherapy. The model emphasizes the interaction among the tumor cells, the oncolytic viruses, and the host immune system that generates both innate and adaptive immune responses. We conduct a rigorous equilibrium analysis and derive threshold conditions that determine the growth or decay of the tumor under various scenarios. Numerical simulation results verify our analytical predictions and provide additional insight into the tumor growth dynamics. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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14 pages, 501 KiB

Open AccessArticle

On the Effects of Boundary Conditions in One-Dimensional Models of Hemodynamics

by Gerasim V. Krivovichev

Mathematics 2022, 10(21), 4058; https://doi.org/10.3390/math10214058 - 01 Nov 2022

Cited by 1 | Viewed by 1017

Abstract

The paper is devoted to the theoretical analysis of the effects of boundary conditions on the solutions of the system of one-dimensional (1D) hemodynamics. The integral inequalities, which realize the energy inequalities for the solutions of initial-boundary-value problems, are obtained. It is demonstrated [...] Read more.

The paper is devoted to the theoretical analysis of the effects of boundary conditions on the solutions of the system of one-dimensional (1D) hemodynamics. The integral inequalities, which realize the energy inequalities for the solutions of initial-boundary-value problems, are obtained. It is demonstrated that the unphysical unbounded solutions can take place for the case of bounded functions from boundary conditions. For the periodic boundary conditions, the integral estimation illustrates the correct behavior of the solution. For this case of boundary conditions, the effective Fourier method for the analytical solution is proposed. The analytical solutions, obtained by this approach, can be used for the comparison of different 1D blood-flow models. The results obtained in the paper allow for an the alternatively view of the stated boundary conditions and can explain some problems, which can arise in numerical simulations. They expand the possibilities of the application of analytical methods in the field of blood-flow simulation. The results can be useful for the specialists on blood-flow modeling. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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15 pages, 5058 KiB

Open AccessFeature PaperArticle

Graph-Based Integration of Histone Modification Profiles

by Federica Baccini, Monica Bianchini and Filippo Geraci

Mathematics 2022, 10(11), 1842; https://doi.org/10.3390/math10111842 - 27 May 2022

Cited by 2 | Viewed by 1607

Abstract

In this work, we introduce a similarity-network-based approach to explore the role of interacting single-cell histone modification signals in haematopoiesis—the process of differentiation of blood cells. Histones are proteins that provide structural support to chromosomes. They are subject to chemical modifications—acetylation or methylation—that [...] Read more.

In this work, we introduce a similarity-network-based approach to explore the role of interacting single-cell histone modification signals in haematopoiesis—the process of differentiation of blood cells. Histones are proteins that provide structural support to chromosomes. They are subject to chemical modifications—acetylation or methylation—that affect the degree of accessibility of genes and, in turn, the formation of different phenotypes. The concentration of histone modifications can be modelled as a continuous signal, which can be used to build single-cell profiles. In the present work, the profiles of cell types involved in haematopoiesis are built based on all the major histone modifications (i.e., H3K27ac, H3K27me3, H3K36me3, H3K4me1, H3K4me3, H3K9me3) by counting the number of peaks in the modification signals; then, the profiles are used to compute modification-specific similarity networks among the considered phenotypes. As histone modifications come as interacting signals, we applied a similarity network fusion technique to integrate these networks in a unique graph, with the aim of studying the simultaneous effect of all the modifications for the determination of different phenotypes. The networks permit defining of a graph-cut-based separation score for evaluating the homogeneity of subgroups of cell types corresponding to the myeloid and lymphoid phenotypes in the classical representation of the haematopoietic tree. Resulting scores show that separation into myeloid and lymphoid phenotypes reflects the actual process of haematopoiesis. Full article

(This article belongs to the Special Issue Mathematical Modeling and Data Science for Biology and Medicine)

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