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Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. During the complex interaction between cancer cell population and the extracellular matrix (ECM), of key importance is the role played by both bulk two-scale dynamics of ECM fibres within collective movement of the tumour cells and...
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... macro-and leading-edge MMP-2 micro-dynamics summarised in (20) take forward and enhance the multiscale moving-boundary framework introduced earlier in Shuttleworth and Trucu (2019) leading to a new extended framework that incorporates two interconnected multiscale systems, which is schematically represented in Fig. 2. These two multiscale systems share the same macro-scale tissue-level dynamics (summarised in (20a)) and at the same time capture the interactions between two micro-scale systems that are different in nature that but which are linked to the two macro-dynamics through two double feedback ...
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... ψ is the standard mollifier detailed in "Appendix A" that acts within a radius γ << Δx 3 from ∂B((2, 2), 0.5 − γ ) to smooth out the characteristic function χ B ((2,2),0.5−γ ) . ...
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... a micro-fibres degradation rate of d f = 1 in previous results, thus implying a slower progression of the tumour. The macroscopic fibre orientations are aligned differently in Fig. 19e, f, where the fibres are directed inwards to the centre of the tumour, confining the tumour to the centre of the domain (Fig. 18a). Moving on to the final stage (Fig. 20), the tumour region is smaller than in previous results in Shuttleworth and Trucu (2020). The bulk of the cancer cells stick closely to the tumour boundary and both the fibrous and non-fibrous ECM phases have undergone a higher level of degradation within the tumour region. Finally, to further our understanding of the effects of ...
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... density, at time 25Δt (Fig. 21), there is very little difference when compared to simulations presented in Shuttleworth and Trucu (2020). Moving on to later stages, it is obvious that the process of boundary micro-fibre degradation causes a slower rate of tumour progression. The tumour region is considerably smaller in Figs. 22 and 23 when com- Fig. 22 Simulations at time 50Δt with a homogeneous distribution of the non-fibrous phase and 15% heterogeneous fibrous phase of the ECM with a micro-fibres degradation rate of d f = 1 pared with previous results (Shuttleworth and Trucu 2020). The bulk of tumour cells display a similar pattern, however much closer to the tumour boundary, ...
Citations
... Finally, we leverage clinical data from T1, T1+C, T2, and DTI scans to account for factors like brain structure, tumour location and extent, and oedema. Shuttleworth and Trucu (32), we denote by Ω(t) the expanding 3dimensional (3D) tumour region that progresses over the time interval [0,T] within a maximal tissue cube Y ⊂R 3 . At any macro-scale spatiotemporal point (x,t) ∈ Y ×[0,T], we consider a cancer cell population, denoted by c(x,t), which interacts with a two-phase heterogeneous ECM (consisting of: a non-fibre l(x,t) and fibre F(x,t) ECM phases (7)), while consuming the available nutrients, denoted by s(x,t), which are present in the environment. ...
Glioblastoma multiforme (GBM), the most aggressive primary brain tumour, exhibits low survival rates due to its rapid growth, infiltrates surrounding brain tissue, and is highly resistant to treatment. One major challenge is oedema infiltration, a fluid build-up that provides a path for cancer cells to invade other areas. MRI resolution is insufficient to detect these infiltrating cells, leading to relapses despite chemotherapy and radiotherapy. In this work, we propose a new multiscale mathematical modelling method, to explore the oedema infiltration and predict tumour relapses. To address tumour relapses, we investigated several possible scenarios for the distribution of remaining GBM cells within the oedema after surgery. Furthermore, in this computational modelling investigation on tumour relapse scenarios were investigated assuming the presence of clinically relevant chemo-radio therapy, numerical results suggest that a higher concentration of GBM cells near the surgical cavity edge led to limited spread and slower progression of tumour relapse. Finally, we explore mathematical and computational avenues for reconstructing relevant shapes for the initial distributions of GBM cells within the oedema from available MRI scans. The results obtained show good overlap between our simulation and the patient’s serial MRI scans taken 881 days into the treatment. While still under analytical investigation, this work paves the way for robust reconstruction of tumour relapses from available clinical data.
... Following the work from Trucu et al. (2013); Shuttleworth and Trucu (2020); Suveges et al. (2021), we denote by Ω(t) the expanding 3-dimensional (3D) tumour region that progresses over the time interval ...
Glioblastoma multiforme (GBM), the most aggressive primary brain tumour, exhibits low survival rates due to its rapid growth, infiltrates surrounding brain tissue, and is highly resistant to treatment. One major challenge is oedema infiltration, a fluid buildup that provides a path for cancer cells to invade other areas. MRI resolution is insufficient to detect these infiltrating cells, leading to relapses despite chemotherapy and radiotherapy. In this work, we propose a new multiscale mathematical modelling method, to explore the oedema infiltration and predict tumour relapses. To address tumour relapses, we investigated several possible scenarios for the distribution of remaining GBM cells within the oedema after surgery. Furthermore, in this computational modelling investigation on tumour relapse scenarios were investigated assuming the presence of clinically relevant chemoradio therapy numerical results suggest that a higher concentration of GBM cells near the surgical cavity edge led to limited spread and slower progression of tumour relapse. Finally, we explore mathematical and computational avenues for reconstructing relevant shapes for the initial distributions of GBM cells within the oedema from available MRI scans. The results obtained show good overlap between our simulation and the patient serial MRI scans taken 881 days into the treatment. While still under analytical investigation, this work paves the way for robust reconstruction of tumour relapses from available clinical data.
... Simply put, the ECM works as a wall that regulates the flow of nutrients around the tumor. Several authors (Chaplain et al. 2011;Engwer et al. 2017;Stinner et al. 2014;Sfakianakis et al. 2020;Shuttleworth and Trucu 2020;Sciumè et al. 2014b;Ng and Frieboes 2017;Preziosi and Tosin 2009) have examined the ECM in reaction-diffusion-type tumor models. We investigated the ECM in a Cahn-Hilliard-type model (Fritz et al. 2019a), and it was also included in our successive research (Fritz et al. 2021a, b). ...
In this survey article, a variety of systems modeling tumor growth are discussed. In accordance with the hallmarks of cancer, the described models incorporate the primary characteristics of cancer evolution. Specifically, we focus on diffusive interface models and follow the phase-field approach that describes the tumor as a collection of cells. Such systems are based on a multiphase approach that employs constitutive laws and balance laws for individual constituents. In mathematical oncology, numerous biological phenomena are involved, including temporal and spatial nonlocal effects, complex nonlinearities, stochasticity, and mixed-dimensional couplings. Using the models, for instance, we can express angiogenesis and cell-to-matrix adhesion effects. Finally, we offer some methods for numerically approximating the models and show simulations of the tumor's evolution in response to various biological effects.
... Trucu et al. model invasion as a moving value problem with ECM represented as a scalar field [26]. Building off of that work, Shuttleworth et al. 2019 and 2020 add non-local effects as well as orientation to the underlying moving boundary value of tumor invasion and adds matrix degradation respectively [27,28]. ...
Extracellular matrix (ECM) is a key part of the cellular microenvironment and critical in multiple disease and developmental processes. Representing ECM and cell–ECM interactions is a challenging multi–scale problem that acts across the tissue and cell scales. While several computational frameworks exist for ECM modeling, they typically focus on very detailed modeling of individual ECM fibers or represent only a single aspect of the ECM. Using the PhysiCell agent–based modeling platform, we combine aspects of previous modeling efforts and develop a framework of intermediate detail that addresses direct cell–ECM interactions. We represent a small region of ECM as an ECM element containing 3 variables: anisotropy, density, and orientation. We then place many ECM elements through a space to form an ECM. Cells have a mechanical response to the local ECM variables and remodel ECM based on their velocity. We demonstrate aspects of this framework with a model of cell invasion where the cell's motile phenotype is driven by the ECM microstructure patterned by prior cells' movements. Investigating the limit of high–speed communication and with stepwise introduction of the framework features, we generate a range of cellular dynamics and ECM patterns — from recapitulating a homeostatic tissue, to indirect communication of paths (stigmergy), to collective migration. When we relax the high–speed communication assumption, we find that the behaviors persist but can be lost as rate of signal generation declines. This result suggests that cell–cell communication mitigated via the ECM can constitute an important mechanism for pattern formation in dynamic cellular patterning while other processes likely also contribute to leader-follower behavior.
... Simply put, the ECM functions as a wall around the tumor which regulates the flow of nutrients. The ECM has been considered in [19,41,123,125,126,129] in tumor models of reaction-diffusion type. Our group was the first to analyze the ECM in a Cahn-Hilliard type model, see [61], and it was also included in our subsequent works [56,57]. ...
Different systems for modeling tumor growth are presented. We follow the path of diffusive interface models and investigate these tumor models with respect to their well-posedness. Many biological phenomena, such as temporal and spatial nonlocal effects, complex nonlinearities, and mixed-dimensional couplings, are involved in mathematical oncology. As a result, detailed analysis of these complex systems is required, and we provide rigorous proofs for this.
... with δ 0 > 0 here being considered known. A usual choice for Q 2 (v) is of the form [26,27]: ...
... 26) where, for any n ∈ {0, . . . , L}, we havec m,n 1 :=c m 1 (t n ),c m,n 2 :=c m 2 (t n ), andṽ m,n :=ṽ m (t n ), whileH 1 (c m,n 1 ,c m,n 2 ,ṽ m,n , m) := H 1 (c m 1 (t n ), c m 2 (t n ),ṽ m (t n ), m), H 2 (c m,n 1 ,c m,n 2 ,ṽ m,n , m) := H 2 (c m 1 (t n ), c m 2 (t n ),ṽ m (t n ), m), H 3 (c m,n 1 ,c m,n 2 ,ṽ m,n ) := H 3 (c m 1 (t n ),c m 2 (t n ),ṽ m (t n )). ...
Cancer cell mutations occur when cells undergo multiple cell divisions, and these mutations can be spontaneous or environmentally-induced. The mechanisms that promote and sustain these mutations are still not fully understood.
This study deals with the identification (or reconstruction) of the usually unknown cancer cell mutation law, which lead to the transformation of a primary tumour cell population into a secondary, more aggressive cell population. We focus on local and nonlocal mathematical models for cell dynamics and movement, and identify these mutation laws from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. In a local cancer invasion model, we first reconstruct the mutation law when we assume that the mutations depend only on the surrounding cancer cells (i.e., the ECM plays no role in mutations). Second, we assume that the mutations depend on the ECM only, and we reconstruct the mutation law in this case. Third, we reconstruct the mutation when we assume that there is no prior knowledge about the mutations. Finally, for the nonlocal cancer invasion model, we reconstruct the mutation law that depends on the cancer cells and on the ECM. For these numerical reconstructions, our approximations are based on the finite difference method combined with the finite elements method. As the inverse problem is ill-posed, we use the Tikhonov regularisation technique in order to regularise the solution. Stability of the solution is examined by adding additive noise into the measurements.
... In this context, at any macroscale spatio-temporal point (x, t) ∈ (t) × [0, T], we consider a mixed cell population consisting of distributions of: (a) cancer cells c(x, t); (b) M1-like macrophages, M 1 (x, t), briefly addressed here as M1 TAM; and (c) M2-like macrophages, M 2 (x, t), which are briefly referred to as M2 TAM. This mixture of cancer cells and macrophages exercise their naturally multiscale dynamics within an extracellular matrix, which, as in Shuttleworth and Trucu [29,41,42], is regarded as consisting of two major phases, namely a fibrous and a non-fibrous one. Specifically, on the one hand, we have the fibre ECM phase, accounting for all major fibrous proteins (such as collagen and fibronectin), whose micro-scale structure enables a spatial bias for withstanding incoming spatial cell fluxes, inducing this way an intrinsic ECM fibres spatial orientation [29,30,41,42]. ...
... This mixture of cancer cells and macrophages exercise their naturally multiscale dynamics within an extracellular matrix, which, as in Shuttleworth and Trucu [29,41,42], is regarded as consisting of two major phases, namely a fibrous and a non-fibrous one. Specifically, on the one hand, we have the fibre ECM phase, accounting for all major fibrous proteins (such as collagen and fibronectin), whose micro-scale structure enables a spatial bias for withstanding incoming spatial cell fluxes, inducing this way an intrinsic ECM fibres spatial orientation [29,30,41,42]. Therefore, the spatiotemporal distribution of the oriented ECM fibres at the macroscale point (x, t) is described by a vector field θ f (x, t), where ...
... Moreover, in Equation (19) S cM > 0 represents the strength of the adhesion relationship between the cancer cells and M1 and M2 TAMs, S cF > 0 is the strength of the cell-fibre ECM adhesion [91] and S cl > 0 corresponds to strength of adhesion between the cancer cells and the non-fibre ECM phase (that includes for instance amyloid fibrils, which can support cell-adhesion processes [92]). Furthrmore,as high level of extracellular Ca +2 ions (which form one of the constituents of the non-fibre ECM phase) are necessary for cell-cell adhesion [93,94], proceeding as in Shuttleworth and Trucu [29,41,42], and Suveges et al. [30] the cancer cells self-adhesion coefficient S cc is taken here as ...
Cancer invasion of the surrounding tissue is a multiscale process of collective cell movement that involves not only tumour cells but also other immune cells in the environment, such as the tumour-associated macrophages (TAMs). The heterogeneity of these immune cells, with the two extremes being the pro-inflammatory and anti-tumour M1 cells, and the anti-inflammatory and pro-tumour M2 cells, has a significant impact on cancer invasion as these cells interact in different ways with the tumour cells and with the ExtraCellular Matrix (ECM). Experimental studies have shown that cancer cells co-migrate with TAMs, but the impact of these different TAM sub-populations (which can change their phenotype and re-polarise depending on the microenvironment) on this co-migration is not fully understood. In this study, we extend a previous multi-scale moving boundary mathematical model, by introducing the M1-like macrophages alongside with their exerted multi-scale effects on the tumour invasion process. With the help of this model we investigate numerically the impact of re-polarising the M2 TAMs into the anti-tumoral M1 phenotype and how such a strategy affects the overall tumour progression. In particular, we investigate numerically whether the M2→M1 re-polarisation could depend on time and/or space, and what would be the macroscopic effects of this spatial- and temporal-dependent re-polarisation on tumour invasion.
... with δ 0 > 0 here being considered known. A usual choice for Q 2 (v) is of the form [26,27]: ...
... 26) where, for any n ∈ {0, . . . , L}, we havec m,n 1 :=c m 1 (t n ),c m,n 2 :=c m 2 (t n ), andṽ m,n :=ṽ m (t n ), whileH 1 (c m,n 1 ,c m,n 2 ,ṽ m,n , m) := H 1 (c m 1 (t n ), c m 2 (t n ),ṽ m (t n ), m), H 2 (c m,n 1 ,c m,n 2 ,ṽ m,n , m) := H 2 (c m 1 (t n ), c m 2 (t n ),ṽ m (t n ), m), H 3 (c m,n 1 ,c m,n 2 ,ṽ m,n ) := H 3 (c m 1 (t n ),c m 2 (t n ),ṽ m (t n )). ...
... with δ 0 > 0 here being considered known. A usual choice for Q 2 (v) is of the form [33,34]: ...
Cancer cell mutations occur when cells undergo multiple cell divisions, and these mutations can be spontaneous or environmentally-induced. The mechanisms that promote and sustain these mutations are still not fully understood. This study deals with the identification (or reconstruction) of the usually unknown cancer cell mutation law, which lead to the transformation of a primary tumour cell population into a secondary, more aggressive cell population. We focus on local and nonlocal mathematical models for cell dynamics and movement, and identify these mutation laws from macroscopic tumour snapshot data collected at some later stage in the tumour evolution. In a local cancer invasion model, we first reconstruct the mutation law when we assume that the mutations depend only on the surrounding cancer cells (i.e., the ECM plays no role in mutations). Second, we assume that the mutations depend on the ECM only, and we reconstruct the mutation law in this case. Third, we reconstruct the mutation when we assume that there is no prior knowledge about the mutations. Finally, for the nonlocal cancer invasion model, we reconstruct the mutation law that depends on the cancer cells and on the ECM. For these numerical reconstructions, our approximations are based on the finite difference method combined with the finite elements method. As the inverse problem is ill-posed, we use the Tikhonov regularisation technique in order to regularise the solution. Stability of the solution is examined by adding additive noise into the measurements.
... However, tumour progression is characterised by various biological processes occurring on different scales, and thus their effects on the overall tumour dynamics cannot be neglected. Hence, recent efforts have been made to establish new multi-scale frameworks for tumour progression [26][27][28][29][30][35][36][37]49], which were able to capture some of these underlying multiscale biological processes usually involving the extracellular matrix (ECM). ...
... To model the evolution of glioblastomas within a three-dimensional brain, we employ a multi-scale moving boundary model that was initially introduced in [20] and later expanded in several other works [26][27][28][29][30]49]. To account for the brain's structure, we aim to use 3D T1 weighted and DTI scans that ultimately influence the migration of the cancer cells as well as affect both micro-scale dynamics. ...
... First, we denote by Ω(t) the expanding 3-dimensional (3D) tumour region that progresses over the time interval [0, T] within a maximal tissue cube Y ⊂ R N with N = 3, i.e., Ω(t) ⊂ Y, ∀t ∈ [0, T]; as can also be seen in Figure 1. Then, at any macro-scale spatio-temporal point (x, t) ∈ Y × [0, T], we consider a cancer cell population c(x, t) that is placed within and interacts with a two-phase ECM: the non-fibre l(x, t) and fibre F(x, t) ECM phases [26][27][28][29][30]. On the one hand, the fibre ECM phase accounts for all major fibrous proteins such as collagen and fibronectin, whose micro-scale distribution induces the spatial orientation of ECM fibres. ...
Brain-related experiments are limited by nature, and so biological insights are often limited or absent. This is particularly problematic in the context of brain cancers, which have very poor survival rates. To generate and test new biological hypotheses, researchers have started using mathematical models that can simulate tumour evolution. However, most of these models focus on single-scale 2D cell dynamics, and cannot capture the complex multi-scale tumour invasion patterns in 3D brains. A particular role in these invasion patterns is likely played by the distribution of micro-fibres. To investigate the explicit role of brain micro-fibres in 3D invading tumours, in this study, we extended a previously introduced 2D multi-scale moving-boundary framework to take into account 3D multi-scale tumour dynamics. T1 weighted and DTI scans are used as initial conditions for our model, and to parametrise the diffusion tensor. Numerical results show that including an anisotropic diffusion term may lead in some cases (for specific micro-fibre distributions) to significant changes in tumour morphology, while in other cases, it has no effect. This may be caused by the underlying brain structure and its microscopic fibre representation, which seems to influence cancer-invasion patterns through the underlying cell-adhesion process that overshadows the diffusion process.
