Mathematical modelling of tumour growth and treatment

Fasano, Antonio; Bertuzzi, Alessandro; Gandolfi, Alberto

doi:10.1007/88-470-0396-2_3

Cited by 43 publications

(35 citation statements)

References 60 publications

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“…Theoretical nonlinear analyses of various tumour models mentioned above have been performed (e.g. [53, 78, 122, 131–134, 136–142, 155, 182, 199, 202–204, 207, 208, 389, 404, 480, 515, 517, 539, 558, 562, 564, 576, 577]). Recently, mixture models have been developed that are capable of describing the detailed interactions among multiple solid (cell) species and extra-/intra- cellular liquids (see section 2.5).…”

Section: Continuum Modellingmentioning

confidence: 99%

“…oxygen, glucose, matrix-degrading enzymes, etc) are described using continuum fields while the cell-based components are discrete. See the reviews by Alber et al [12], Moreira and Deutsch [381], Drasdo [159], Araujo and McElwain [34], Quaranta et al [439], Hatzikirou et al [260], Nagy [389], Abbott et al [1] Byrne et al [88], Fasano et al [182], Galle et al [213], Drasdo and Höhme [163], Thorne et al [519], Anderson et al [27], Deisboeck et al [150], Anderson and Quaranta [30], Quaranta et al [438] and Zhang et al [573]. Here, we briefly present several approaches to discrete modelling.…”

Section: Discrete Modellingmentioning

confidence: 99%

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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%

Section: Discrete Modellingmentioning

confidence: 99%

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

et al. 2009

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“…A second example of a multiphysics problem is avascular tumour growth. There is extensive literature devoted to modelling tumour growth (see Fasano et al (2006) for a recent review). The growth of a tumour displaces surrounding tissue.…”

Section: Formulationmentioning

confidence: 99%

Multi-scale computational modelling in biology and physiology

Southern

Pitt-Francis

Whiteley

et al. 2008

Progress in Biophysics and Molecular Biology

163

125

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Recent advances in biotechnology and the availability of ever more powerful computers have led to the formulation of increasingly complex models at all levels of biology. One of the main aims of systems biology is to couple these together to produce integrated models across multiple spatial scales and physical processes. In this review, we formulate a definition of multi-scale in terms of levels of biological organisation and describe the types of model that are found at each level. Key issues that arise in trying to formulate and solve multi-scale and multi-physics models are considered and examples of how these issues have been addressed are given for two of the more mature fields in computational biology: the molecular dynamics of ion channels and cardiac modelling. As even more complex models are developed over the coming few years, it will be necessary to develop new methods to model them (in particular in coupling across the interface between stochastic and deterministic processes) and new techniques will be required to compute their solutions efficiently on massively parallel computers. We outline how we envisage these developments occurring.

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“…Continuum models such as single-phase and multiphase mixture models treat tumors as a collection of cells at larger scales and principles from continuum mechanics such as mass and momentum conservation are used to construct partial differential equations and integro-differential equations governing the motion of cell densities, or volume fractions, stresses and cell velocities. See, for example, the recent reviews (Ribba et al, 2004; Quaranta et al, 2005; Hatzikirou et al, 2005; Nagy, 2005; Wodarz et al, 2005; Byrne et al, 2006; Fasano et al, 2006; van Leeuwen et al, 2007; Roose et al, 2007; Graziano et al, 2007; Harpold et al, 2007; Drasdo and H ö hme, 2007; Friedman et al, 2007; Sanga et al, 2007; Anderson and Quaranta, 2008; Bellomo et al, 2008; Cristini et al, 2008; Deisboeck et al, 2009; Byrne, 2010; Rejniak and McCawley, 2010; Lowengrub et al, 2010; Deisboeck et al, 2011; Frieboes et al, 2011; Kim et al, 2011; Kam et al, 2012; Hatzikirou et al, 2012; Szab ó et al, 2013; Baldock et al, 2013; Katira et al, 2013) for a collection of recent results.…”

Section: Introductionmentioning

confidence: 99%

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

Chen

Lowengrub

2014

Journal of Theoretical Biology

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We develop a mathematical model of tumor growth in complex, dynamic microenvironments with active, deformable membranes. Using a diffuse domain approach, the complex domain is captured implicitly using an auxiliary function and the governing equations are appropriately modified, extended and solved in a larger, regular domain. The diffuse domain method enables us to develop an efficient numerical implementation that does not depend on the space dimension or the microenvironmental geometry. We model homotypic cell-cell adhesion and heterotypic cell-basement membrane (BM) adhesion with the latter being implemented via a membrane energy that models cell-BM interactions. We incorporate simple models of elastic forces and the degradation of the BM and ECM by tumor-secreted matrix degrading enzymes. We investigate tumor progression and BM response as a function of cell-BM adhesion and the stiffness of the BM. We find tumor sizes tend to be positively correlated with cell-BM adhesion since increasing cell-BM adhesion results in thinner, more elongated tumors. Prior to invasion of the tumor into the stroma, we find a negative correlation between tumor size and BM stiffness as the elastic restoring forces tend to inhibit tumor growth. In order to model tumor invasion of the stroma, we find it necessary to downregulate cell-BM adhesiveness, which is consistent with experimental observations. A stiff BM promotes invasiveness because at early stages the opening in the BM created by MDE degradation from tumor cells tends to be narrower when the BM is stiffer. This requires invading cells to squeeze through the narrow opening and thus promotes fragmentation that then leads to enhanced growth and invasion. In three dimensions, the opening in the BM was found to increase in size even when the BM is stiff because of pressure induced by growing tumor clusters. A larger opening in the BM can increase the potential for further invasiveness by increasing the possibility that additional tumor cells could invade the stroma.

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Mathematical modelling of tumour growth and treatment

Cited by 43 publications

References 60 publications

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

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

Multi-scale computational modelling in biology and physiology

Tumor growth in complex, evolving microenvironmental geometries: A diffuse domain approach

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