Tumor Growth with Nutrients: Regularity and Stability

Jacobs, Matt; Kim, In-Won; Tong, Jianliang

doi:10.48550/arxiv.2204.07572

Cited by 4 publications

(4 citation statements)

References 13 publications

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“…Many studies build models for cancer cells living in a free environment [28]. Some other studies consider the effect of nutrients [25] and drugs [3].…”

Section: Population Dynamicsmentioning

confidence: 99%

Three facets of mathematical cancer biology research

Wang¹

2023

Preprint

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Cancer, as the uncontrollable cell growth, is related to many branches of biology. In this review, we will discuss three mathematical approaches for studying cancer biology: population dynamics, gene regulation, and developmental biology. If we understand all biochemical mechanisms of cancer cells, we can directly calculate how the cancer cell population behaves. Inversely, just from the cell count data, we can use population dynamics to infer the mechanisms. Cancer cells emerge from certain genetic mutations, which affect the expression of other genes through gene regulation. Therefore, knowledge of gene regulation can help with cancer prevention and treatment. Developmental biology studies acquisition and maintenance of normal cellular function, which is inspiring to cancer biology in the opposite direction. Besides, cancer cells implanted into an embryo can differentiate into normal tissues, which provides a possible approach of curing cancer. This review illustrates the role of mathematics in these three fields: what mathematical models are used, what data analysis tools are applied, and what mathematical theorems need to be proved. We hope that applied mathematicians and even pure mathematicians can find meaningful mathematical problems related to cancer biology.

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“…Many studies build models for cancer cells living in a free environment [28]. Some other studies consider the effect of nutrients [25] and drugs [3].…”

Section: Population Dynamicsmentioning

confidence: 99%

Three facets of mathematical cancer biology research

Wang¹

2023

Preprint

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“…However, the advantage might take years to become significant, and it is common to find cells with different numbers of mutations (0, 1, or 2). This type of gene-related growth patterns might lead to complicated phenomena [10,15,61]. For patients having cells with both JAK2 and TET2 mutations, an important question is to determine which mutation appears first.…”

Section: Introductionmentioning

confidence: 99%

Determine transposable genes when the orders of genes are different

Wang

2023

Preprint

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We can find JAK2 V617F and TET2 mutations in some patients of myeloproliferative neoplasm. JAK2 V617F mutation has different effects on regulating gene expressions, depending on whether TET2 mutation is present or not. In addition, when JAK2 V617F and TET2 mutations are both present, which mutation appears first also affects certain properties of patients. We use nonlinear ordinary differential equation models and Markov chain models to explain such phenomena.

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“…As tumor growth is a rather complex biological process, it develops in distinguishable phases and is affected by various factors. Many mathematicians are devoted to incorparate these elements in modeling and analyze their individual and synergistic effects, such as nutrient concentration [14,24], degree of vascularization [5,41], cell reproduction and apoptosis [16,17], chemotaxis [36,37]. However, the development of the model library also raises an alarming issue, the model identification and the parameter calibrations in the equations are becoming significantly more challenging as well.…”

Section: Introductionmentioning

confidence: 99%

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

Liu

Zhou

2023

Preprint

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In this paper, we use the Bayesian inversion approach to study the data assimilation problem for a family of tumor growth models described by porous-medium type equations. The models contain uncertain parameters and are indexed by a physical parameter $m$, which characterizes the constitutive relation between density and pressure. Based on these models, we employ the Bayesian inversion framework to infer parametric and nonparametric unknowns that affect tumor growth from noisy observations of tumor cell density. We establish the well-posedness and the stability theories for the Bayesian inversion problem and further prove the convergence of the posterior distribution in the so-called incompressible limit, $m \rightarrow \infty$. Since the posterior distribution across the index regime $m\in[2,\infty)$ can thus be treated in a unified manner, such theoretical results also guide the design of the numerical inference for the unknown. We propose a generic computational framework for such inverse problems, which consists of a typical sampling algorithm and an asymptotic preserving solver for the forward problem. With extensive numerical tests, we demonstrate that the proposed method achieves satisfactory accuracy in the Bayesian inference of the tumor growth models, which is uniform with respect to the constitutive relation.

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Tumor Growth with Nutrients: Regularity and Stability

Cited by 4 publications

References 13 publications

Three facets of mathematical cancer biology research

Three facets of mathematical cancer biology research

Determine transposable genes when the orders of genes are different

A unified Bayesian inversion approach for a class of tumor growth models with different pressure laws

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