Breast Cancer: Modelling and Detection

Gavaghan, David; Brady, J. Michael; Behrenbruch, Christian; Highnam, Ralph; Maini, Philip K.

doi:10.1080/10273660290015233

Cited by 8 publications

(6 citation statements)

References 42 publications

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“…Calcium deposits form within the remains of these cells, which can be identified through imaging. Since mammography images may be difficult to interpret, Gavaghan et al [232] used breast carcinoma to illustrate how mathematical modelling can aid in detection, diagnosis and treatment. The authors discussed models underpinning mammography image analysis, which complement tumour growth models.…”

Section: Continuum Modellingmentioning

confidence: 99%

Nonlinear modelling of cancer: bridging the gap between cells and tumours

et al. 2009

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Despite major scientific, medical and technological advances over the last few decades, a cure for cancer remains elusive. The disease initiation is complex, and including initiation and avascular growth, onset of hypoxia and acidosis due to accumulation of cells beyond normal physiological conditions, inducement of angiogenesis from the surrounding vasculature, tumour vascularization and further growth, and invasion of surrounding tissue and metastasis. Although the focus historically has been to study these events through experimental and clinical observations, mathematical modelling and simulation that enable analysis at multiple time and spatial scales have also complemented these efforts. Here, we provide an overview of this multiscale modelling focusing on the growth phase of tumours and bypassing the initial stage of tumourigenesis. While we briefly review discrete modelling, our focus is on the continuum approach. We limit the scope further by considering models of tumour progression that do not distinguish tumour cells by their age. We also do not consider immune system interactions nor do we describe models of therapy. We do discuss hybrid-modelling frameworks, where the tumour tissue is modelled using both discrete (cell-scale) and continuum (tumour-scale) elements, thus connecting the micrometre to the centimetre tumour scale. We review recent examples that incorporate experimental data into model parameters. We show that recent mathematical modelling predicts that transport limitations of cell nutrients, oxygen and growth factors may result in cell death that leads to morphological instability, providing a mechanism for invasion via tumour fingering and fragmentation. These conditions induce selection pressure for cell survivability, and may lead to additional genetic mutations. Mathematical modelling further shows that parameters that control the tumour mass shape also control its ability to invade. Thus, tumour morphology may serve as a predictor of invasiveness and treatment prognosis.

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Section: Continuum Modellingmentioning

confidence: 99%

Nonlinear modelling of cancer: bridging the gap between cells and tumours

et al. 2009

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“…Theoretical and computational cancer modeling that integrates different mechanisms for neoplasm progression, when appropriately linked with experimental and clinical data, offers a promising avenue for a better understanding of tumor evolution [21][22][23][24][25]. Although a variety of mathematical models have been devised for proliferative tumors [26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41], models for invasive tumors incorporating the aforementioned tumor-host [13]. (b) Emergence of the rough surface on a solid tumor (ductal carcinoma in situ), which might lead to further invasive behavior.…”

Section: Introductionmentioning

confidence: 99%

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

Jiao

Torquato

2013

Phys. Rev. E

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The emergence of invasive and metastatic behavior in malignant tumors can often lead to fatal outcomes for patients. The collective malignant tumor behavior resulting from the complex tumor-host interactions and the interactions between the tumor cells is currently poorly understood. In this paper, we employ a cellular automaton (CA) model to investigate microenvironment-enhanced malignant behaviors and morphologies of in vitro avascular invasive solid tumors in three dimensions. Our CA model incorporates a variety of microscopic-scale tumor-host interactions, including the degradation of the extracellular matrix by the malignant cells, nutrient-driven cell migration, pressure buildup due to the deformation of the microenvironment by the growing tumor, and its effect on the local tumor-host interface stability. Moreover, the effects of cell-cell adhesion on tumor growth are explicitly taken into account. Specifically, we find that while strong cell-cell adhesion can suppress the invasive behavior of the tumors growing in soft microenvironments, cancer malignancy can be significantly enhanced by harsh microenvironmental conditions, such as exposure to high pressure levels. We infer from the simulation results a qualitative phase diagram that characterizes the expected malignant behavior of invasive solid tumors in terms of two competing malignancy effects: the rigidity of the microenvironment and cell-cell adhesion. This diagram exhibits phase transitions between noninvasive and invasive behaviors. We also discuss the implications of our results for the diagnosis, prognosis, and treatment of malignant tumors.

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“…Among finite difference method, finite volume method, finite element method (FEM), and methods of lines are few to mention here; see the list of examples. [15][16][17][18][19][20][21][22][23] Finite element methods have become popular for numerical simulations of reaction-diffusion equations due to their good approximation and feasibility to work with any domains. Therefore, we proposed a finite element scheme for the considered cancer invasion model in this work.…”

Section: Introductionmentioning

confidence: 99%

“…Apart from the theoretical methods, some numerical methods have been proposed in the literature for cancer invasion mathematical models also. Among finite difference method, finite volume method, finite element method (FEM), and methods of lines are few to mention here; see the list of examples . Finite element methods have become popular for numerical simulations of reaction‐diffusion equations due to their good approximation and feasibility to work with any domains.…”

Section: Introductionmentioning

confidence: 99%

Solvability and numerical simulations for tumor invasion model with nonlinear diffusion

Manimaran

Shangerganesh

2019

Comp and Math Methods

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This paper is devoted to study the nonlinear diffusion parabolic partial differential equations, which describes the invasion of cancer cells into the normal tissues. We prove the existence of a weak solution of the considered model using the Faedo‐Galerkin approximation method, a priori estimates, and compactness arguments. Then, the considered model is discretized in space with the standard Galerkin finite elements and in time with the Crank‐Nicolson method. Furthermore, nonlinear terms in the proposed model is treated semi‐implicitly in the finite element scheme. Numerical results are validated by convergence study and existing numerical results of similar problems. Finally, the influence of nonlinear diffusion is studied to understand the effect in the cancer invasion model.

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Breast Cancer: Modelling and Detection

Cited by 8 publications

References 42 publications

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Nonlinear modelling of cancer: bridging the gap between cells and tumours

Evolution and morphology of microenvironment-enhanced malignancy of three-dimensional invasive solid tumors

Solvability and numerical simulations for tumor invasion model with nonlinear diffusion

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