## No full-text available

To read the full-text of this research,

you can request a copy directly from the authors.

Local cancer cell invasion is a complex process involving many cellular and tissue interactions and is an important prerequisite for metastatic spread, the main cause of cancer related deaths. As a tumour increases in malignancy, the cancer cells adopt the ability to mutate into secondary cell subpopulations giving rise to a heterogeneous tumour. This new cell subpopulation often carries higher invasive abilities and permits a quicker spread of the tumour.
Building upon the recent multiscale modelling framework for cancer invasion within a fibrous ECM introduced in Shuttleworth and Trucu, (2019), in this paper we consider the process of local invasion by a heterotypic tumour consisting of two cancer cell populations mixed with a two-phase ECM. To that end, we address the double feedback link between the tissue-scale cancer dynamics and the cell-scale molecular processes through the development of a two-part modelling framework that crucially incorporates the multiscale dynamic redistribution of oriented fibres occurring within a two-phase extra-cellular matrix and combines this with the multiscale leading edge dynamics exploring key matrix-degrading enzymes molecular processes along the tumour interface that drive the movement of the cancer boundary. The modelling framework will be accompanied by computational results that explore the effects of the underlying fibre network on the overall pattern of cancer invasion.

To read the full-text of this research,

you can request a copy directly from the authors.

... In this work we present a new formulation of the link that connects the two scales of the tumour dynamics that was considered in the initial multiscale moving boundary modelling approach presented in [73] as well as in the multiscale cancer invasion modelling developments that followed [4,5,6,71,52,62,63,64,65,67,72]. In particular, the movement of the tumor boundary is here defined by a velocity field instead of a displacement of the interface. ...

... Using the scale separation, we explore the micro-dynamics within a cell-scale neighbourhood of each of the boundary points x ∈ ∂Ω(t) (i.e., at each point of the macroscopic interface) that is enabled by acorresponding εY micro-domain centred at x. In brief, adopting here a multiscale modelling perspective similar to the one proposed in [4,5,6,71,52,62,63,64,65,67,72,73], the coupling of two-scale dynamics of cancer invasion is captured as follows: ...

... Schematic summary of our multiscale moving boundary modelling. The new model that we introduced here falls in the class of heterogeneous multiscale models that were developed over the past two decades not only for multiscale biological processes but also for other multiscale processes arising in material science or fluid-structure interactions [1,4,5,6,22,71,27,28,30,52,62,63,64,65,67,72,73]. Schematically, the two-scale dynamics of our cancer invasion model is coupled across the scales as depicted in Figure 2, and its progression can be summarised in the following three steps: ...

The quest for a deeper understanding of the cancer growth and spread process focuses on the naturally multiscale nature of cancer invasion, which requires an appropriate multiscale modeling and analysis approach. The cross-talk between the dynamics of the cancer cell population on the tissue scale (macroscale) and the proteolytic molecular processes along the tumor border on the cell scale (microscale) plays a particularly important role within the invasion processes, leading to dramatic changes in tumor morphology and influencing the overall pattern of cancer spread. Building on the multiscale moving boundary framework proposed in Trucu et al. (Multiscale Model. Simul 11(1): 309-335), in this work we propose a new formulation of this process involving a novel derivation of the macro scale boundary movement law based on micro-dynamics, involving a transport equation combined with the level-set method. This is explored numerically in a novel finite element macro-micro framework based on cut-cells.

... 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.

... 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.

... Then, following a brief description of the numerical 53 methods, we present the computational simulation results in Section 3. Finally, in Section 4 54 we summarise and discuss these results. 55 56 To model the evolution of glioblastomas within a 3-dimensional brain, we employ a 57 multi-scale moving boundary model that was initially introduced in [20] and later expanded 58 in several other works [26][27][28][29][30]49]. To account for the brain's structure, we aim to use 3D T1 59 weighted and DTI scans that ultimately influence the migration of the cancer cells as well 60 as affect both micro-scale dynamics. ...

... First, we denote by Ω(t) the expanding 3-dimensional 66 (3D) tumour region that progresses over the time interval [0, T] within a maximal tissue 67 cube Y ⊂ R N with N = 3, i.e., Ω(t) ⊂ Y, ∀t ∈ [0, T]; see also Figure 1. Then at any macro- 68 scale spatio-temporal point (x, t) ∈ Y × [0, T] we consider a cancer cell population c(x, t) 69 embedded within a two-phase ECM, consisting of the non-fibre l(x, t) and fibre F(x, t) 70 ECM phases [26][27][28][29][30]. On the one hand, the fibre ECM phase accounts for all major fibrous 71 proteins such as collagen and fibronectin, whose micro-scale distribution induces the spatial 72 orientation of the ECM fibres. ...

... Then at any macro- 68 scale spatio-temporal point (x, t) ∈ Y × [0, T] we consider a cancer cell population c(x, t) 69 embedded within a two-phase ECM, consisting of the non-fibre l(x, t) and fibre F(x, t) 70 ECM phases [26][27][28][29][30]. On the one hand, the fibre ECM phase accounts for all major fibrous 71 proteins such as collagen and fibronectin, whose micro-scale distribution induces the spatial 72 orientation of the ECM fibres. Hence, the macro-scale spatio-temporal distribution of the 73 ECM fibres is represented by an oriented vector field θ f (x, t) that describes their spatial 74 bias, as well as by F(x, t) := θ f (x, t) which denote the amount of fibres at a macro-scale point (x, t) [26][27][28][29][30]. On the other hand, in the non-fibre ECM phase we bundle together 76 every other ECM constituent such as non-fibrous proteins (for example amyloid fibrils), 77 enzymes, polysaccharides and extracellular Ca 2+ ions [26][27][28][29][30]. Furthermore, in this new 78 modelling study we incorporate the structure of the brain by extracting data from DTI and 79 T1 weighted brain scans, and then using this data to parametrise the model. ...

Brain-related experiments are limited by nature, and so biological insights are often restricted 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 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 explicitly the role of brain micro-fibres in the 3D invading tumours, in this study we extend 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.

... We employ here a hybrid modeling framework [15] that combines the off-lattice agent based model MultiCell-LF [16][17][18] to represent the cells, and a multi-scale continuous framework [19][20][21][22][23] to represent the microenvironment. To facilitate the description of this multi-scale hybrid model, let us first introduce some useful notations from both frameworks. ...

... Besides the cell-cell interactions described above, of particular importance, are the cell-ECM adhesions [28][29][30][31] that we explore here through a cell-non-fibre ECM adhesion [32][33][34][35] as well as a cell-fibre ECM adhesion [36,37]. In the existing literature [19][20][21][22][23][38][39][40][41][42], this type of interaction is usually modelled by a non-local adhesion integral with a sensing region B(0, R) of radius R. Since in our hybrid model the two phase ECM is modelled in a continuous manner using densities, we can adopt this approach to describe the present cell-ECM interactions. To this end, let us define the cell-ECM adhesion force/velocity for an arbitrary cell C i as ...

... To this end, we start each simulation with a single cancer cell with well defined properties located at a point (x 0 1 , x 0 2 ) (that will be defined for each simulations) of the computational domain Y [−1280mm, 1280mm] × [−1280mm, 1280mm], and any alteration from this will be stated accordingly. Then, this single cancer cell is considered to be embedded within the following (scaled) non-fibre ECM environment [19][20][21][22][23]: ...

The specific structure of the extracellular matrix (ECM), and in particular the density and orientation of collagen fibres, plays an important role in the evolution of solid cancers. While many experimental studies discussed the role of ECM in individual and collective cell migration, there are still unanswered questions about the impact of nonlocal cell sensing of other cells on the overall shape of tumour aggregation and its migration type. There are also unanswered questions about the migration and spread of tumour that arises at the boundary between different tissues with different collagen fibre orientations. To address these questions, in this study we develop a hybrid multi-scale model that considers the cells as individual entities and ECM as a continuous field. The numerical simulations obtained through this model match experimental observations, confirming that tumour aggregations are not moving if the ECM fibres are distributed randomly, and they only move when the ECM fibres are highly aligned. Moreover, the stationary tumour aggregations can have circular shapes or irregular shapes (with finger-like protrusions), while the moving tumour aggregations have elongate shapes (resembling to clusters, strands or files). We also show that the cell sensing radius impacts tumour shape only when there is a low ratio of fibre to non-fibre ECM components. Finally, we investigate the impact of different ECM fibre orientations corresponding to different tissues, on the overall tumour invasion of these neighbouring tissues.

... In this context, at any macro-scale spatiotemporal point (x, t) ∈ Ω(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 [93,94,95], 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 [93,94,95,100]. ...

... This mixture of cancer cells and macrophages exercise their naturally multiscale dynamics within an extracellular matrix, which, as in [93,94,95], 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 [93,94,95,100]. Therefore, the spatiotemporal distribution of the oriented ECM fibres at the macro-scale point 2 representing the amount of fibres at (x, t). Then, on the other hand, besides these fibrous proteins, the ECM also contains many other components such as non-fibrous proteins (for instance amyloid fibrils), enzymes, polysaccharides and extracellular Ca 2+ ions. ...

... Moreover, in (20) 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 [112,113] 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 [29,33,34,47]). Furthrmore, based on the biological evidence [35,42] which suggests that the emergence of strong and stable cancer self adhesion bonds are positively correlated with the high level of extracellular Ca +2 ions (which is one of the constituents of the non-fibre ECM phase), proceeding as in [93,94,95,100] the cancer cells self-adhesion coefficient S cc is taken here as ...

Cancer invasion of the surrounding tissue is a multiscale process 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 cell 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.

... Hence, at macro-scale, we explore the cancer invasion process occurring within a maximal tissue cube Y ∈ R d for d = 2, 3, where the expanding tumour region denoted by Ω(t) progresses over the time interval [0, T ] (i.e., Ω(t) ⊂ Y , ∀ t ∈ [0, T ]). We adopt the same simplified context as in [78,62,68,69,70] where aside from the tumour cells population c(x, t), the rest of the tumour micro-environment and surrounding tissue is represented here simply by a generic ECM. To that end, while we acknowledge that, besides the tumour cells, some of the tumour microenvironment components are not ECM constituents and are rather only supported by the usual ECM, in this framework we still regard all those constituents (such as VEGF, FGF, TGF-beta, and ions such as Ca 2+ ) as being part of and represented by this extended concept of ECM. ...

... To that end, while we acknowledge that, besides the tumour cells, some of the tumour microenvironment components are not ECM constituents and are rather only supported by the usual ECM, in this framework we still regard all those constituents (such as VEGF, FGF, TGF-beta, and ions such as Ca 2+ ) as being part of and represented by this extended concept of ECM. Furthermore, due to the biologically established importance played within cell migration by the major ECM fibres, namely collagen and fibronectin, as considered also in [68,69,70], we regard this ECM as two-phase matter, consisting of an ECM fibre phase and an ECM non-fibre phase. Specifically, on one hand the ECM fibres phase accounts exclusively for all major fibres components such as collagen and fibronectin (notably characterised by their insolubility properties [40]), and its amount distributed at (x, t) is denoted here by F (x, t). ...

... Specifically, in addition to the situation considered in [68,69,70] (exploring the adhesive interactions of the cells distributed at x ∈ Ω(t) with the other cancer cells as well as with the distribution of non-fibres ECM phase [24,27,28,41] and the oriented ECM fibres phase [86,87] within a sensing region B(x, R) of radius R > 0), here the flux term A(x, t, u, θ f ) explores the key biological evidence underlining the contribution of the macrophages to the directional movement of the tumour cells. Indeed, this explores not only the fact that cancer cells bind themselves to TAMs [11], but also accounts for the experimental evidence detailed in [13,29,82,92] that underscores the existence of a cross-talk between tumour cells and macrophages which is mediated through various chemokines. ...

Invasion of the surrounding tissue is one of the recognised hallmarks of cancer [32], which is accomplished through a complex heterotypic multiscale dynamics involving tissue-scale random and directed movement of the population of both cancer cells and other accompanying cells (including here, the family of tumour-associated macrophages) as well as the emerging cell-scale activity of both the matrix-degrading enzymes and the rearrangement of the cell-scale constituents of the ECM fibres. The involved processes include not only the presence of cell proliferation and cell adhesion (to other cells and to the extracellular matrix), but also the secretion of matrix-degrading enzymes. This is as a result of cancer cells as well as macrophages, which are one of the most abundant types of immune cells in the tumour microenvironment. In large tumours, these tumour-associated macrophages (TAMs) have a tumour promoting phenotype, contributing to tumour proliferation and spread.
In this paper, we extend a previous multi-scale moving-boundary mathematical model for cancer invasion, by considering also the multi-scale effects of TAMs, with special focus on the influence that their directional movement exerts on the overall tumour progression. Numerical investigation of this new model shows the importance of the interactions between pro-tumour TAMs and the fibrous extracellular matrix (ECM), highlighting the impact of the fibres on the spatial structure of solid tumour.

... Finally, as the invasion process is naturally multiscale, with its dynamics ranging from subcellular, cellular to tissue scale, major advances have been witnessed in the multiscale modelling of cancer invasion (Anderson et al. 2007;Peng et al. 2017;Ramis-Conde et al. 2008;Shuttleworth and Trucu 2018;Trucu et al. 2013). In particular, advancements have been made in towards two-scale approaches, modelling and appropriately linking the spatiotemporal dynamics occurring at different scales, as first proposed in Trucu et al. (2013) and extended upon in Peng et al. (2017), Shuttleworth and Trucu (2020) and Shuttleworth and Trucu (2019). ...

... At time 25Δt (Fig. 12), the non- Fig. 9 Simulations at time 25Δt with a homogeneous distribution of the non-fibrous phase and 15% heterogeneous fibres phase of the ECM with a micro-fibres degradation rate of d f = 1 fibrous ECM phase has been degraded by the cancer cells (Fig. 12b), with the highest degradation occurring in the regions of highest cell distribution. The tumour boundary at this stage is larger than in previous simulations, this correlating with results in Shuttleworth and Trucu (2020Trucu ( , 2019 where an initially higher density fibrous ECM phase Fig. 10 Simulations at time 50Δt with a homogeneous distribution of the non-fibrous phase and 15% heterogeneous fibres phase of the ECM with a micro-fibres degradation rate of d f = 1 resulted in an accelerated spread of the tumour. Additionally, the macroscopic fibre density (Fig. 12c) also exhibits different behaviour than in previous simulations, where a similar pattern of fibres was noted in Trucu (2020, 2019). ...

... There is significant non-fibres ECM degradation stretching the entire area of the tumour (Fig. 13b), with this only becom-ing more pronounced at later stages (Fig. 14b). The macroscopic fibre orientations (Fig. 13d, f) display high levels of reorientation with a similar "frenzied" appearance to figures in Shuttleworth and Trucu (2020). At the final time 75Δt (Fig. 14), the cancer cells are dispersed further into the matrix (Fig. 14a) and noticeably in the pattern of degradation of the non-fibres ECM phase (Fig. 14b). ...

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 the multiscale leading edge dynamics driven by proteolytic activity of the matrix-degrading enzymes (MDEs) that are secreted by the cancer cells. As these two multiscale subsystems share and contribute to the same tumour macro-dynamics, in this work we develop further the model introduced in Shuttleworth and Trucu (Bull Math Biol 81:2176–2219, 2019. https://doi.org/10.1007/s11538-019-00598-w) by exploring a new aspect of their interaction that occurs at the cell scale. Specifically, here we will focus on understanding the cell-scale cross talk between the micro-scale parts of these two multiscale subsystems which get to interact directly in the peritumoural region, with immediate consequences both for MDE micro-dynamics occurring at the leading edge of the tumour and for the cell-scale rearrangement of the naturally oriented ECM fibres in the peritumoural region, ultimately influencing the way tumour progresses in the surrounding tissue. To that end, we will propose a new modelling that captures the ECM fibres degradation not only at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour.

... Finally, as the invasion process is naturally multiscale, with its dynamics ranging from sub-cellular-, cellular-to tissue-scale, major advances have been witnessed in the multiscale modelling of cancer invasion [4,24,28,31,37]. In particular, advancements have been made in towards two-scale approaches, modelling and appropriately linking the spatio-temporal dynamics occurring at different scales, as first proposed in [37] and extended upon in [24,32,33]. ...

... At stage 25∆t, Figure 11, the non-fibres ECM phase has been degraded by the cancer cells, subfigure 11b, with the highest degradation occurring in the regions of highest cell distribution. The tumour boundary at this stage is larger than in previous simulations, this correlating with results in [32,33] where an initially higher density fibre ECM phase resulted in an accelerated spread of the tu- mour. Additionally, the macroscopic fibre density, subfigure 11c, also exhibits different behaviour than in previous simulations, where a similar pattern of fibres was noted in [32,33]. ...

... The tumour boundary at this stage is larger than in previous simulations, this correlating with results in [32,33] where an initially higher density fibre ECM phase resulted in an accelerated spread of the tu- mour. Additionally, the macroscopic fibre density, subfigure 11c, also exhibits different behaviour than in previous simulations, where a similar pattern of fibres was noted in [32,33]. This pattern of fibres is witnessed because the tumour boundary is expanding faster than the micro-fibres are being rearranged, thus the distributions are not found to build on the proliferating edge. ...

Local cancer invasion of tissue is a complex, multiscale process which plays an essential role in tumour progression. Occurring over many different temporal and spatial scales, the first stage of invasion is the secretion of matrix degrading enzymes (MDEs) by the cancer cells that consequently degrade the surrounding extracellular matrix (ECM). This process is vital for creating space in which the cancer cells can progress and it is driven by the activities of specific matrix metalloproteinases (MMPs). In this paper, we consider the key role of two MMPs by developing further the novel two-part multiscale model introduced in [33] to better relate at micro-scale the two micro-scale activities that were considered there, namely, the micro-dynamics concerning the continuous rearrangement of the naturally oriented ECM fibres within the bulk of the tumour and MDEs proteolytic micro-dynamics that take place in an appropriate cell-scale neighbourhood of the tumour boundary. Focussing primarily on the activities of the membrane-tethered MT1-MMP and the soluble MMP-2 with the fibrous ECM phase, in this work we investigate the MT1-MMP/MMP-2 cascade and its overall effect on tumour progression. To that end, we will propose a new multiscale modelling framework by considering the degradation of the ECM fibres not only to take place at macro-scale in the bulk of the tumour but also explicitly in the micro-scale neighbourhood of the tumour interface as a consequence of the interactions with molecular fluxes of MDEs that exercise their spatial dynamics at the invasive edge of the tumour.

... In approach 1, reaction-diffusion PDEs are defined for concentrations of one or more matrix degrading enzyme(s) such as MMP, populations of tumor cells, and density of ECM [41][42][43] (see references cited in Ref. [42] for historical overview). Approach 2 involves continuous modeling at the macroscale and microscale with a moving boundary front representing where ECM remodeling occurs at the microscale at the edge of the macroscopic tumor mass [44][45][46][47][48]. Approach 3 employs hybrid modeling using ABMs at the cellular level and PDEs for ECM and chemical factors [49][50][51]. ...

... A cell-scale cross-talk interaction between cell migration and ECM remodeling along the moving boundary was refined in Ref. [46]. Heterogeneous tumor cell populations were simulated in Ref. [47]. The model was extended to consider the influence of tumor-associated macrophages on ECM remodeling in Ref. [48]. ...

Multiscale computational modeling aims to connect the complex networks of effects at different length and/or time scales. For example, these networks often include intracellular molecular signaling, crosstalk and other interactions between neighboring cell populations, and higher levels of emergent phenomena across different regions of tissues and among collections of tissues or organs interacting with each other in the whole body. Recent applications of multiscale modeling across intracellular, cellular, and/or tissue levels are highlighted here. These models incorporated the roles of biochemical and biomechanical modulation in processes that are implicated in the mechanisms of several diseases including fibrosis, joint and bone diseases, respiratory infectious diseases, and cancers.

... with δ 0 > 0 here being considered known. A usual choice for Q 2 (v) is of the form [26,27]: ...

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.

... with δ 0 > 0 here being considered known. A usual choice for Q 2 (v) is of the form [26,27]: ...

... 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.

In this study we investigate computationally tumour-oncolytic virus (OV) interactions that take place within a heterogeneous extracellular matrix (ECM). The ECM is viewed as a mixture of two constitutive phases, namely a fibre phase and a non-fibre phase. The multiscale mathematical model presented here focuses on the nonlocal cell-cell and cell-ECM interactions, and how these interactions might be impacted by the infection of cancer cells with the OV. At macroscale we track the kinetics of cancer cells, virus particles and the ECM. At microscale we track (i) the degradation of ECM by matrix degrading enzymes (MDEs) produced by cancer cells, which further influences the movement of tumour boundary; (ii) the re-arrangement of the microfibres that influences the re-arrangement of macrofibres (i.e., fibres at macroscale). With the help of this new multiscale model, we investigate two questions: (i) whether the infected cancer cell fluxes are the result of local or non-local advection in response to ECM density; and (ii) what is the effect of ECM fibres on the the spatial spread of oncolytic viruses and the outcome of oncolytic virotherapy.

In this study we investigate computationally tumour-oncolytic virus(OV) interactions that take place within a heterogeneous ExtraCellular Matrix (ECM). The ECM is viewed as a mixture of two constitutive phases, namely a fibre phase and a non-fibre phase. The multiscale mathematical model presented here focuses on the nonlocal cell-cell and cell-ECM interactions, and how these interactions might be impacted by the infection of cancer cells with the OV. At macroscale we track the kinetics of cancer cells, virus particles and the ECM. At microscale we track (i) the degradation of ECM by matrix degrading enzymes (MDEs) produced by cancer cells, which further influences the movement of tumour boundary; (ii) the re-arrangement of the microfibres that influences the re-arrangement of macrofibres (i.e., fibres at macroscale). With the help of this new multiscale model, we investigate two questions: (i) whether the infected cancer cell fluxes are the result of local or non-local advection in response to ECM density; and (ii) what is the effect of ECM fibres on the the spatial spread of oncolytic viruses and the outcome of oncolytic virotherapy.

This book presents original papers reflecting topics featured at the international symposium entitled “Fusion of Mathematics and Biology” and organized by the editor of the book. The symposium, held in October 2020 at Osaka University in Japan, was the core event for the final year of the research project entitled “Establishing International Research Networks of Mathematical Oncology.” The project had been carried out since April 2015 as part of the Core-to-Core Program of Japan Society for the Promotion of Science (JSPS). In this book, the editor presents collaborative research from prestigious organizations in France, the UK, and the USA. By utilizing their individual strengths and realizing the fusion of life science and mathematical science, the project achieved a combination of mathematical analysis, verification by biomedical experiments, and statistical analysis of chemical databases.
Mathematics is sometimes regarded as a universal language. It is a valuable property that everyone can understand beyond the boundaries of culture, religion, and language. This unifying force of mathematics also applies to the various fields of science. Mathematical oncology has two aspects, i.e., data science and mathematical modeling, and definitely helps in the prediction and control of biological phenomena observed in cancer evolution.
The topics addressed in this book represent several methods of applying mathematical modeling to scientific problems in the natural sciences. Furthermore, novel reviews are included that may motivate many mathematicians to become interested in biological research.

A defining feature of cancer is the capability to spread locally into the surrounding tissue, with cancer cells spreading beyond any normal boundaries. Cancer invasion is a complex phenomenon involving many inter-connected processes at different spatial and temporal scales. A key component of invasion is the ability of cancer cells to alter and degrade the extracellular matrix through the secretion of matrix-degrading enzymes. Combined with excessive cell proliferation and cell migration (individual and collective), this facilitates the spread of cancer cells into the local tissue. Along with tumour-induced angiogenesis, invasion is a critical component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we present an overview of the various mathematical models and different modelling techniques and approaches that have been developed over the past 25 years or so and which focus on various aspects of the invasive process.

We propose and study computationally a novel non-local multiscale moving boundary mathematical model for tumour and oncolytic virus (OV) interactions when we consider the go or grow hypothesis for cancer dynamics. This spatio-temporal model focuses on two cancer cell phenotypes that can be infected with the OV or remain uninfected, and which can either move in response to the extracellular-matrix (ECM) density or proliferate. The interactions between cancer cells, those among cancer cells and ECM, and those among cells and OV occur at the macroscale. At the micro-scale, we focus on the interactions between cells and matrix degrading enzymes (MDEs) that impact the movement of tumour boundary. With the help of this multiscale model we explore the impact on tumour invasion patterns of two different assumptions that we consider in regard to cell-cell and cell-matrix interactions. In particular we investigate model dynamics when we assume that cancer cell fluxes are the result of local advection in response to the density of extracellular matrix (ECM), or of non-local advection in response to cell-ECM adhesion. We also investigate the role of the transition rates between mainly-moving and mainly-growing cancer cell sub-populations, as well as the role of virus infection rate and virus replication rate on the overall tumour dynamics.

Oncolytic virus (OV) therapy is a promising treatment for cancer due to the OVs selective ability to infect and replicate inside cancer cells, thus killing them, without harming healthy cells. In this work, we present a new non-local multiscale moving boundary model for the spatio-temporal cancer-OV interactions. This model explores an important double feedback loop that links the macro-scale dynamics of cancer-virus interactions and the micro-scale dynamics of proteolytic activity taking place at the tumour interface. The cancer cell-cell and cell-matrix interactions are assumed to be nonlocal, while the cell-virus interactions are assumed local. With the help of this model we investigate computationally various cancer treatment scenarios involving oncolytic viruses (i.e., the effect of injecting the OV inside the tumour, or outside it). Moreover, we investigate the effect of different cell-cell and cell-matrix interaction strengths on the success of OV spreading throughout the tumour, and the effect of constant or density-dependent virus diffusion coefficients on viral spread.

Recognised as one of the hallmarks of cancer, local cancer cell invasion is a complex multiscale process that combines the secretion of matrix-degrading enzymes with a series of altered key cell processes (such as abnormal cell proliferation and changes in cell–cell and cell–matrix adhesion leading to enhanced migration) to degrade important components of the surrounding extracellular matrix (ECM) and this way spread further in the human tissue. In order to gain a deeper understanding of the invasion process, we pay special attention to the interacting dynamics between the cancer cell population and various constituents of the surrounding tumour microenvironment. To that end, we consider the key role that ECM plays within the human body tissue, and in particular we focus on the special contribution of its fibrous proteins components, such as collagen and fibronectin, which play an important part in cell proliferation and migration. In this work, we consider the two-scale dynamic cross-talk between cancer cells and a two-component ECM (consisting of both a fibre and a non-fibre phase). To that end, we incorporate the interlinked two-scale dynamics of cell–ECM interactions within the tumour support that contributes simultaneously both to cell adhesion and to the dynamic rearrangement and restructuring of the ECM fibres. Furthermore, this is embedded within a multiscale moving boundary approach for the invading cancer cell population, in the presence of cell adhesion at the tissue scale and cell-scale fibre redistribution activity and leading edge matrix-degrading enzyme molecular proteolytic processes. The overall modelling framework will be accompanied by computational results that will explore the impact on cancer invasion patterns of different levels of cell adhesion in conjunction with the continuous ECM fibres rearrangement.

Cancer-associated fibroblasts (CAFs) are major components of the carcinoma microenvironment that promote tumor progression. However, the mechanisms by which CAFs regulate cancer cell migration are poorly understood. In this study, we show that fibronectin (Fn) assembled by CAFs mediates CAF–cancer cell association and directional migration. Compared with normal fibroblasts, CAFs produce an Fn-rich extracellular matrix with anisotropic fiber orientation, which guides the cancer cells to migrate directionally. CAFs align the Fn matrix by increasing nonmuscle myosin II- and platelet-derived growth factor receptor α–mediated contractility and traction forces, which are transduced to Fn through α5β1 integrin. We further show that prostate cancer cells use αv integrin to migrate efficiently and directionally on CAF-derived matrices. We demonstrate that aligned Fn is a prominent feature of invasion sites in human prostatic and pancreatic carcinoma samples. Collectively, we present a new mechanism by which CAFs organize the Fn matrix and promote directional cancer cell migration.

In this paper, we develop a non-local mathematical model describing cancer cell invasion and movement as a result of integrin-controlled cell–cell adhesion and cell–matrix adhesion, and transforming growth factor-beta (TGF-β) effect on cell proliferation and adhesion, for two cancer cell populations with different levels of mutation. The model consists of partial integro-differential equations describing the dynamics of two cancer cell populations, coupled with ordinary differential equations describing the extracellular matrix (ECM) degradation and the production and decay of integrins, and with a parabolic PDE governing the evolution of TGF-β concentration. We prove the global existence of weak solutions to the model. We then use our model to explore numerically the role of TGF-β in cell aggregation and movement.

Tumour cell-extracellular matrix (ECM) interactions are fundamental for discrete steps in breast cancer progression. In particular, cancer cell adhesion to ECM proteins present in the microenvironment is critical for accelerating tumour growth and facilitating metastatic spread. To assess the utility of tumour cell-ECM adhesion as a means for discovering prognostic factors in breast cancer survival, here we perform a systematic phenotypic screen and characterise the adhesion properties of a panel of human HER2 amplified breast cancer cell lines across six ECM proteins commonly deregulated in breast cancer. We determine a gene expression signature that defines a subset of cell lines displaying impaired adhesion to laminin. Cells with impaired laminin adhesion showed an enrichment in genes associated with cell motility and molecular pathways linked to cytokine signalling and inflammation. Evaluation of this gene set in the Molecular Taxonomy of Breast Cancer International Consortium (METABRIC) cohort of 1,964 patients identifies the F12 and STC2 genes as independent prognostic factors for overall survival in breast cancer. Our study demonstrates the potential of in vitro cell adhesion screens as a novel approach for identifying prognostic factors for disease outcome.

Known as one of the hallmarks of cancer [30], cancer cell invasion of human body tissue is a complicated spatio-temporal multiscale process which enables a localised solid tumour to transform into a systemic, metastatic and fatal disease. This process explores and takes advantage of the reciprocal relation that solid tumours establish with the extracellular matrix (ECM) components and other multiple distinct cell types from the surrounding microenvironment. Through the secretion of various proteolytic enzymes such as matrix metalloproteinases (MMP) or the urokinase plasminogen activator (uPA), the cancer cell population alters the configuration of the surrounding ECM composition and overcomes the physical barriers to ultimately achieve local cancer spread into the surrounding tissue. The active interplay between the tissue-scale tumour dynamics and the molecular mechanics of the involved proteolytic enzymes at the cell-scale underlines the biologically multiscale character of invasion, and raises the challenge of modelling this process with an appropriate multiscale approach. In this paper, we present a new two-scale moving boundary model of cancer invasion that explores the tissue scale tumour dynamics in conjunction with the molecular dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in [58], the modelling that we propose here allows us to study the changes in tissue scale tumour morphology caused by the cell-scale uPA micro-dynamics occurring along the invasive edge of the tumour. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as the tumour infiltrative growth patterns discussed in [33].

Cells use a combination of changes in adhesion, proteolysis and motility (directed and random) during the process of migration. Proteolysis of the extracellular matrix (ECM) results in the creation of haptotactic gradients, which cells use to move in a directed fashion. The proteolytic creation of these gradients also results in the production of digested fragments of ECM. In this study we show that in the human fibrosarcoma cell line HT1080, matrix metalloproteinase-2 (MMP-2)-digested fragments of fibronectin exert a chemotactic pull stronger than that of undigested fibronectin. During invasion, this gradient of ECM fragments is established in the wake of an invading cell, running counter to the direction of invasion. The resultant chemotactic pull is anti-invasive, contrary to the traditional view of the role of chemotaxis in invasion. Uncontrolled ECM degradation by high concentrations of MMP can thus result in steep gradients of ECM fragments, which run against the direction of invasion. Consequently, the invasive potential of a cell depends on MMP production in a biphasic manner, implying that MMP inhibitors will upregulate invasion in high-MMP-expressing cells. Hence the therapeutic use of protease inhibitors against tumours expressing high levels of MMP could produce an augmentation of invasion.

The invasive capability is fundamental in determining the malignancy of a solid tumor. Revealing biomedical strategies that are able to partially decrease cancer invasiveness is therefore an important approach in the treatment of the disease and has given rise to multiple in vitro and in silico models. We here develop a hybrid computational framework, whose aim is to characterize the effects of the different cellular and subcellular mechanisms involved in the invasion of a malignant mass. In particular, a discrete Cellular Potts Model is used to represent the population of cancer cells at the mesoscopic scale, while a continuous approach of reaction-diffusion equations is employed to describe the evolution of microscopic variables, as the nutrients and the proteins present in the microenvironment and the matrix degrading enzymes secreted by the tumor. The behavior of each cell is then determined by a balance of forces, such as homotypic (cell-cell) and heterotypic (cell-matrix) adhesions and haptotaxis, and is mediated by the internal state of the individual , i.e. its motility. The resulting composite model quantifies the influence of changes in the mechanisms involved in tumor invasion and, more interestingly, puts in evidence possible therapeutic approaches, that are potentially effective in decreasing the malignancy of the disease, such as the alteration in the adhesive properties of the cells, the inhibition in their ability to remodel and the disruption of the haptotactic movement. We also extend the simulation framework by including cell proliferation which, following experimental evidence, is regulated by the intracellular level of growth factors. Interestingly, in spite of the increment in cellular density, the depth of invasion is not significantly increased, as one could have expected.

Directional mesenchymal cell invasion in vivo is understood to be a stimulated event and to be regulated by cytokines, chemokines and types of extracellular matrix (ECM). Instead, by focusing on the cellular response to ECM stiffness, we found that soft ECM (low stiffness) itself is sufficient to prevent stable cell-to-cell adherens junction (AJ) formation, up-regulate MMP secretion, promote MMP activity and induce invadosome-like protrusion (ILP) formation. Consistently, similar ILP formation was also detected in a 3D directional invasion assay in soft matrix. Primary human fibroblasts spontaneously form ILPs in a very narrow range of ECM stiffness (0.1 ∼ 0.4 kPa) and such ILP formation is Src family kinase (SFK) dependent. In contrast, spontaneous ILP formation in malignant cancer cells and fibrosarcoma cells occurs across a much wider range of ECM stiffness, and these tumor cell ILPs are also more prominent at lower stiffness. These findings suggest that ECM softness is a natural stimulator for cellular invasiveness.

Cell polarization (cued or uncued) is a fundamental mechanism in cell biology. As an alternative to the classical Turing bifurcation, it has been proposed that the onset of cell polarity might arise by means of the well-known phenomenon of wave-pinning [Gamba et al., Proc. Natl. Acad. Sci. USA 102, 16927 (2005)]. A particularly simple and elegant deterministic model of cell polarization based on the wave-pinning mechanism has been proposed by Edelstein-Keshet and coworkers [Biophys. J. 94, 3684 (2008)]. This model consists of a small biomolecular network where an active membrane-bound factor interconverts into its inactive form that freely diffuses in the cell cytosol. However, biomolecular networks do communicate with other networks as well as with the external world. Thus, their dynamics must be considered as perturbed by extrinsic noises. These noises may have both a spatial and a temporal correlation, and in any case they must be bounded to preserve the biological meaningfulness of the perturbed parameters. Here we numerically show that the inclusion of external spatiotemporal bounded parametric perturbations in the above wave-pinning-based model of cellular polarization may sometimes destroy the polarized state. The polarization loss depends on both the extent of temporal and spatial correlations and on the kind of noise employed. For example, an increase of the spatial correlation of the noise induces an increase of the probability of cell polarization. However, if the noise is spatially homogeneous then the polarization is lost in the majority of cases. These phenomena are independent of the type of noise. Conversely, an increase of the temporal autocorrelation of the noise induces an effect that depends on the model of noise.

This article discusses some paradoxical results that arise when modelling uncertainties in models of anti-tumor chemotherapies
using Gaussian noise. The effects of intrinsic and environmental perturbations and uncertainties on the dynamics of tumor
growth and anti-tumor chemotherapy delivered via continuous infusion are considered.

Cell migration is vitally important in a wide variety of biological contexts ranging from embryonic development and wound healing to malignant diseases such as cancer. It is a very complex process that is controlled by intracellular signaling pathways as well as the cell's microenvironment. Due to its importance and complexity, it has been studied for many years in the biomedical sciences, and in the last 30 years it also received an increasing amount of interest from theoretical scientists and mathematical modelers. Here we propose a force-based, individual-based modeling framework that links single-cell migration with matrix fibers and cell-matrix interactions through contact guidance and matrix remodelling. With this approach, we can highlight the effect of the cell's environment on its migration. We investigate the influence of matrix stiffness, matrix architecture, and cell speed on migration using quantitative measures that allow us to compare the results to experiments.

In this work we introduce a spatio-temporal bounded noise derived by the sine-Wiener noise and by the spatially colored unbounded noise proposed by García-Ojalvo, Sancho, and Ramírez-Piscina (GSR noise). We characterize the behavior of the equilibrium distribution of this novel noise by showing its dependence on both the temporal and the spatial autocorrelation lengths. In particular, we show that the distribution experiences a stochastic transition from bimodality to trimodality.
Then, we employ the noise here defined to study the emergence of phase transitions in the real Ginzburg–Landau model. Various phenomena are evidenced by means of numerical simulations, among which reentrant transitions, as well as differences in the response of the system to “equivalent” GSR additive noise perturbations.

Cell motion and interaction with the extracellular matrix is studied deriving a kinetic model and considering its diffusive limit. The model takes into account the chemotactic and haptotactic effects, and obtains friction as a result of the interactions between cells and between cells and the fibrous environment. The evolution depends on the fibre distribution, as cells preferentially move along the fibre direction and tend to cleave and remodel the extracellular matrix when their direction of motion is not aligned with the fibre direction. Simulations are performed to describe the behavior of an ensemble of cells under the action of a chemotactic field and in the presence of heterogeneous and anisotropic fibre networks.

In this work, we introduce two spatio-temporal colored bounded noises, based
on the zero-dimensional Cai-Lin and Tsallis-Borland noises. We then study and
characterize the dependence of the defined bounded noises on both a temporal
correlation parameter $\tau$ and on a spatial coupling parameter $\lambda$. The
boundedness of these noises has some consequences on their equilibrium
distributions. Indeed in some cases varying $\lambda$ may induce a transition
of the distribution of the noise from bimodality to unimodality. With the aim
to study the role played by bounded noises on nonlinear dynamical systems, we
investigate the behavior of the real Ginzburg-Landau time-varying model
additively perturbed by such noises. The observed phase transitions
phenomenology is quite different from the one observed when the perturbations
are unbounded. In particular, we observed an inverse "order-to-disorder"
transition, and a re-entrant transition, with dependence on the specific type
of bounded noise.

Dynamic remodeling of the extracellular matrix (ECM) is essential for development, wound healing and normal organ homeostasis. Life-threatening pathological conditions arise when ECM remodeling becomes excessive or uncontrolled. In this Perspective, we focus on how ECM remodeling contributes to fibrotic diseases and cancer, which both present challenging obstacles with respect to clinical treatment, to illustrate the importance and complexity of cell-ECM interactions in the pathogenesis of these conditions. Fibrotic diseases, which include pulmonary fibrosis, systemic sclerosis, liver cirrhosis and cardiovascular disease, account for over 45% of deaths in the developed world. ECM remodeling is also crucial for tumor malignancy and metastatic progression, which ultimately cause over 90% of deaths from cancer. Here, we discuss current methodologies and models for understanding and quantifying the impact of environmental cues provided by the ECM on disease progression, and how improving our understanding of ECM remodeling in these pathological conditions is crucial for uncovering novel therapeutic targets and treatment strategies. This can only be achieved through the use of appropriate in vitro and in vivo models to mimic disease, and with technologies that enable accurate monitoring, imaging and quantification of the ECM.

In this paper we present two types of mathematical model which describe the invasion of host tissue by tumour cells. In the models, we focus on three key variables implicated in the invasion process, namely, tumour cells, host tissue (extracellular matrix) and matrix-degradative enzymes associated with the tumour cells. The first model focusses on the macro-scale structure (cell population level) and considers the tumour as a single mass. The mathematical model consists of a system of partial differential equations describing the production and/or activation of degradative enzymes by the tumour cells, the degradation of the matrix and the migratory response of the tumour cells. Numerical simulations are presented in one and two space dimensions and compared qualitatively with experimental and clinical observations. The second type of model focusses on the micro-scale (individual cell) level and uses a discrete technique developed in previous models of angiogenesis. This technique enables one to model migration and invasion at the level of individual cells and hence it is possible to examine the implications of metastatic spread. Finally, the results of the models are compared with actual clinical observations and the implications of the model for improved surgical treatment of patients are considered.

The ability of cancer cells to break out of tissue compartments and invade locally gives solid tumours a defining deadly characteristic. One of the first steps of invasion is the remodelling of the surrounding tissue or extracellular matrix (ECM) and a major part of this process is the over-expression of proteolytic enzymes, such as the urokinase-type plasminogen activator (uPA) and matrix metalloproteinases (MMPs), by the cancer cells to break down ECM proteins. Degradation of the matrix enables the cancer cells to migrate through the tissue and subsequently to spread to secondary sites in the body, a process known as metastasis. In this paper we undertake an analysis of a mathematical model of cancer cell invasion of tissue, or ECM, which focuses on the role of the urokinase plasminogen activation system. The model consists of a system of five reaction-diffusion-taxis partial differential equations describing the interactions between cancer cells, uPA, uPA inhibitors, plasmin and the host tissue. Cancer cells react chemotactically and haptotactically to the spatio-temporal effects of the uPA system. The results obtained from computational simulations carried out on the model equations produce dynamic heterogeneous spatio-temporal solutions and using linear stability analysis we show that this is caused by a taxis-driven instability of a spatially homogeneous steady-state. Finally we consider the biological implications of the model results, draw parallels with clinical samples and laboratory based models of cancer cell invasion using three-dimensional invasion assay, and go on to discuss future development of the model.

In this paper we consider a mathematical model of cancer cell invasion of tissue (extracellular matrix). Two crucial components of tissue invasion are (i) cancer cell proliferation, and (ii) over-expression and secretion of proteolytic enzymes by the cancer cells. The proteolytic enzymes are responsible for the degradation of the tissue, enabling the proliferating cancer cells to actively invade and migrate into the degraded tissue. Our model focuses on the role of nonlocal kinetic terms modelling competition for space and degradation. The model consists of a system of reaction-diffusion-taxis partial differential equations, with nonlocal (integral) terms describing the interactions between cancer cells and the host tissue. First of all we prove results concerning the local existence, uniqueness and regularity of solutions. We then prove global existence. Using Green's functions, we transform our original nonlocal equations into a coupled system of parabolic and elliptic equations and we undertake a numerical analysis of this equivalent system, presenting computational simulation results from our model showing the effect of the nonlocal terms (travelling waves we observed have the shape closely linked to the nonlocal terms). Finally, in the discussion section, concluding remarks are made and open problems are indicated.

The generation of invasiveness in transformed cells represents an essential step of tumor progression. We show here, first, that nontransformed Madin-Darby canine kidney (MDCK) epithelial cells acquire invasive properties when intercellular adhesion is specifically inhibited by the addition of antibodies against the cell adhesion molecule uvomorulin; the separated cells then invade collagen gels and embryonal heart tissue. Second, MDCK cells transformed with Harvey and Moloney sarcoma viruses are constitutively invasive, and they were found not to express uvomorulin at their cell surface. These data suggest that the loss of adhesive function of uvomorulin (which is identical to E-cadherin and homologous to L-CAM) is a critical step in the promotion of epithelial cells to a more malignant, i.e., invasive, phenotype. Similar modulation of intercellular adhesion might also occur during invasion of carcinoma cells in vivo.

The E-cadherin-catenin complex plays a crucial role in epithelial cell-cell adhesion and in the maintenance of tissue architecture. Perturbation in the expression or function of this complex results in loss of intercellular adhesion, with possible consequent cell transformation and tumour progression. Recently, much progress has been made in understanding the interaction between the different components of this protein complex and how this cell-cell adhesion complex is modulated in cancer cells.
This is an update of the role of the E-cadherin-catenin complex in human cancers. It emphasizes new features and the possible role of the complex in clinical practice, discussed in the light of 165 references obtained from the Medline database from 1995 to 1999.
More evidence is now appearing to suggest that disturbance in protein-protein interaction in the E-cadherin-catenin adhesion complex is one of the main events in the early and late steps of cancer development. An inverse correlation is found between expression of the E-cadherin-catenin complex and the invasive behaviour of tumour cells. Therefore, E-cadherin-catenin may become a significant prognostic marker for tumour behaviour. Besides its role in establishing tight cell-cell adhesion, beta- catenin plays a major role in cell signalling and promotion of neoplastic growth. This suggests its dual role as a tumour suppressor and as an oncogene in human cancers.
Recent developments show that the E-cadherin-catenin complex is more than a 'sticky molecular complex'. Further studies may yield greater insight into the early molecular interactions critical to the initiation and progression of tumours. This should aid the development of novel strategies for both prevention and treatment of cancer.

Tumor cells often show a decrease in cell-cell and/or cell-matrix adhesion. An increasing body of evidence indicates that this reduction in cell adhesion correlates with tumor invasion and metastasis. Two main groups of adhesion molecules, cadherins and CAMs, have been implicated in tumor malignancy. However, the specific role that these proteins play in the context of tumor progression remains to be elucidated. In this review, we discuss recent data pointing to a causal relationship between the loss of cell adhesion molecules and tumor progression. In addition, the direct involvement of these molecules in specific signal transduction pathways will be considered, with particular emphasis on the alterations of such pathways in transformed cells. Finally, we review recent observations on the molecular mechanisms underlying metastatic dissemination. In many cases, spreading of tumor cells from the primary site to distant organs has been characterized as an active process involving the loss of cell-cell adhesion and gain of invasive properties. On the other hand, various examples of metastases exhibiting a relatively benign (i.e. not invasive) phenotype have been reported. Together with our recent results on a mouse tumor model, these findings indicate that 'passive' metastatic dissemination can occur, in particular as a consequence of impaired cell-matrix adhesion and of tumor tissue disaggregation.

In this paper we present a hybrid mathematical model of the invasion of healthy tissue by a solid tumour. In particular we consider early vascular growth, just after angiogenesis has occurred. We examine how the geometry of the growing tumour is affected by tumour cell heterogeneity caused by genetic mutations. As the tumour grows, mutations occur leading to a heterogeneous tumour cell population with some cells having a greater ability to migrate, proliferate or degrade the surrounding tissue. All of these cell properties are closely controlled by cell-cell and cell-matrix interactions and as such the physical geometry of the whole tumour will be dependent on these individual cell interactions. The hybrid model we develop focuses on four key variables implicated in the invasion process: tumour cells, host tissue (extracellular matrix), matrix-degradative enzymes and oxygen. The model is considered to be hybrid since the latter three variables are continuous (i.e. concentrations) and the tumour cells are discrete (i.e. individuals). With this hybrid model we examine how individual-based cell interactions (with one another and the matrix) can affect the tumour shape and discuss which of these interactions is perhaps most crucial in influencing the tumour's final structure.

In this paper mesoscopic (individual based) and macroscopic (population based) models for mesenchymal motion of cells in fibre networks are developed. Mesenchymal motion is a form of cellular movement that occurs in three-dimensions through tissues formed from fibre networks, for example the invasion of tumor metastases through collagen networks. The movement of cells is guided by the directionality of the network and in addition, the network is degraded by proteases. The main results of this paper are derivations of mesoscopic and macroscopic models for mesenchymal motion in a timely varying network tissue. The mesoscopic model is based on a transport equation for correlated random walk and the macroscopic model has the form of a drift-diffusion equation where the mean drift velocity is given by the mean orientation of the tissue and the diffusion tensor is given by the variance-covariance matrix of the tissue orientations. The transport equation as well as the drift-diffusion limit are coupled to a differential equation that describes the tissue changes explicitly, where we distinguish the cases of directed and undirected tissues. As a result the drift velocity and the diffusion tensor are timely varying. We discuss relations to existing models and possible applications.

Cells adhere to each other through the binding of cell adhesion molecules at the cell surface. This process, known as cell-cell adhesion, is fundamental in many areas of biology, including early embryo development, tissue homeostasis and tumour growth. In this paper we develop a new continuous mathematical model of this phenomenon by considering the movement of cells in response to the adhesive forces generated through binding. We demonstrate that our model predicts the aggregation behaviour of a disassociated adhesive cell population. Further, when the model is extended to represent the interactions between multiple populations, we demonstrate that it is capable of replicating the different types of cell sorting behaviour observed experimentally. The resulting pattern formation is a direct consequence of the relative strengths of self-population and cross-population adhesive bonds in the model. While cell sorting behaviour has been captured previously with discrete approaches, it has not, until now, been observed with a fully continuous model.

Stromal-epithelial interactions are of particular significance in breast tissue as misregulation of these interactions can promote tumorigenesis and invasion. Moreover, collagen-dense breast tissue increases the risk of breast carcinoma, although the relationship between collagen density and tumorigenesis is not well understood. As little is known about epithelial-stromal interactions in vivo, it is necessary to visualize the stroma surrounding normal epithelium and mammary tumors in intact tissues to better understand how matrix organization, density, and composition affect tumor formation and progression.
Epithelial-stromal interactions in normal mammary glands, mammary tumors, and tumor explants in three-dimensional culture were studied with histology, electron microscopy, and nonlinear optical imaging methodologies. Imaging of the tumor-stromal interface in live tumor tissue ex vivo was performed with multiphoton laser-scanning microscopy (MPLSM) to generate multiphoton excitation (MPE) of endogenous fluorophores and second harmonic generation (SHG) to image stromal collagen.
We used both laser-scanning multiphoton and second harmonic generation microscopy to determine the organization of specific collagen structures around ducts and tumors in intact, unfixed and unsectioned mammary glands. Local alterations in collagen density were clearly seen, allowing us to obtain three-dimensional information regarding the organization of the mammary stroma, such as radiating collagen fibers that could not have been obtained using classical histological techniques. Moreover, we observed and defined three tumor-associated collagen signatures (TACS) that provide novel markers to locate and characterize tumors. In particular, local cell invasion was found predominantly to be oriented along certain aligned collagen fibers, suggesting that radial alignment of collagen fibers relative to tumors facilitates invasion. Consistent with this observation, primary tumor explants cultured in a randomly organized collagen matrix realigned the collagen fibers, allowing individual tumor cells to migrate out along radially aligned fibers.
The presentation of these tumor-associated collagen signatures allowed us to identify pre-palpable tumors and see cells at the tumor-stromal boundary invading into the stroma along radially aligned collagen fibers. As such, TACS should provide indications that a tumor is, or could become, invasive, and may serve as part of a strategy to help identify and characterize breast tumors in animal and human tissues.

Mammographically dense breast tissue is one of the greatest risk factors for developing breast carcinoma. Despite the strong clinical correlation, breast density has not been causally linked to tumorigenesis, largely because no animal model has existed for studying breast tissue density. Importantly, regions of high breast density are associated with increased stromal collagen. Thus, the influence of the extracellular matrix on breast carcinoma development and the underlying molecular mechanisms are not understood.
To study the effects of collagen density on mammary tumor formation and progression, we utilized a bi-transgenic tumor model with increased stromal collagen in mouse mammary tissue. Imaging of the tumors and tumor-stromal interface in live tumor tissue was performed with multiphoton laser-scanning microscopy to generate multiphoton excitation and spectrally resolved fluorescent lifetimes of endogenous fluorophores. Second harmonic generation was utilized to image stromal collagen.
Herein we demonstrate that increased stromal collagen in mouse mammary tissue significantly increases tumor formation approximately three-fold (p < 0.00001) and results in a significantly more invasive phenotype with approximately three times more lung metastasis (p < 0.05). Furthermore, the increased invasive phenotype of tumor cells that arose within collagen-dense mammary tissues remains after tumor explants are cultured within reconstituted three-dimensional collagen gels. To better understand this behavior we imaged live tumors using nonlinear optical imaging approaches to demonstrate that local invasion is facilitated by stromal collagen re-organization and that this behavior is significantly increased in collagen-dense tissues. In addition, using multiphoton fluorescence and spectral lifetime imaging we identify a metabolic signature for flavin adenine dinucleotide, with increased fluorescent intensity and lifetime, in invading metastatic cells.
This study provides the first data causally linking increased stromal collagen to mammary tumor formation and metastasis, and demonstrates that fundamental differences arise and persist in epithelial tumor cells that progressed within collagen-dense microenvironments. Furthermore, the imaging techniques and signature identified in this work may provide useful diagnostic tools to rapidly assess fresh tissue biopsies.

Cancer invasion of tissue is a key aspect of the growth and spread of cancer and is crucial in the process of metastatic spread, i.e., the growth of secondary cancers. Invasion consists in cancer cells secreting various matrix degrading enzymes (MDEs) which destroy the surrounding tissue or extracellular matrix (ECM). Through a combination of proliferation and migration, the cancer cells then actively spread locally into the surrounding tissue. Thus processes occurring at the level of individual cells eventually give rise to processes occurring at the tissue level. In this paper we introduce a new type of multiscale model describing the process of cancer invasion of tissue. Our multiscale model is a two-scale model which focuses on the macroscopic dynamics of the distributions of cancer cells and of the surrounding extracellular matrix, and on the microscale dynamics of the MDEs, produced at the level of the individual cancer cells. These microscale dynamics take place at the interface of the cancer cells and the ECM and give rise to a moving boundary at the macroscale. On the computational side, in order to approximate the newly proposed model, we have developed a novel computational scheme based on a combination of finite elements at the microscale with a new finite difference technique at the macroscale, linking together in a moving boundary formulation of the problem. This two-scale numerical scheme is organized in such a way that it enables us to accurately model all the key processes of cancer invasion at both the macroscale and microscale.

Gliomas are a class of rarely curable tumors arising from abnormal glia cells in the human brain. The understanding of glioma spreading patterns is essential for both radiological therapy as well as surgical treatment. Diffusion tensor imaging (DTI) allows to infer the white matter fibre structure of the brain in a noninvasive way. Painter and Hillen (J Theor Biol 323:25-39, 2013) used a kinetic partial differential equation to include DTI data into a class of anisotropic diffusion models for glioma spread. Here we extend this model to explicitly include adhesion mechanisms between glioma cells and the extracellular matrix components which are associated to white matter tracts. The mathematical modelling follows the multiscale approach proposed by Kelkel and Surulescu (Math Models Methods Appl Sci 23(3), 2012). We use scaling arguments to deduce a macroscopic advection-diffusion model for this process. The tumor diffusion tensor and the tumor drift velocity depend on both, the directions of the white matter tracts as well as the binding dynamics of the adhesion molecules. The advanced computational platform DUNE enables us to accurately solve our macroscopic model. It turns out that the inclusion of cell binding dynamics on the microlevel is an important factor to explain finger-like spread of glioma.

Cancer invasion, recognised as one of the hallmarks of cancer, is a complex, multiscale phenomenon involving many inter-related genetic, biochemical, cellular and tissue processes at different spatial and temporal scales. Central to invasion is the ability of cancer cells to alter and degrade extracellular matrix. Combined with abnormal excessive proliferation and migration which is enabled and enhanced by altered cell-cell and cell-matrix adhesion, the cancerous mass can invade the neighbouring tissue. Along with tumour-induced angiogenesis, invasion is a key component of metastatic spread, ultimately leading to the formation of secondary tumours in other parts of the host body. In this paper we explore the spatio-temporal dynamics of a model of cancer invasion, where cell-cell and cell-matrix adhesion are accounted for through non-local interaction terms in a system of partial integro-differential equations. The change of adhesion properties during cancer growth and development is investigated here through time-dependent adhesion characteristics within the cell population as well as those between the cells and the components of the extracellular matrix. Our computational simulation results demonstrate a range of heterogeneous dynamics which are qualitatively similar to the invasive growth patterns observed in a number of different types of cancer, such as tumour infiltrative growth patterns (INF).

We consider an anisotropic diffusion equation of the form ut
= ∇∇(D(x)u) in two dimensions, which arises in various applications, including the modelling of wolf movement along seismic lines and the invasive spread of certain brain tumours along white matter neural fibre tracts. We consider a degenerate case, where the diffusion tensor D(x) has a zero-eigenvalue for certain values of x. Based on a regularisation procedure and various pointwise and integral a priori estimates, we establish the global existence of very weak solutions to the degenerate limit problem. Moreover, we show that in the large time limit these solutions approach profiles that exhibit a Dirac-type mass concentration phenomenon on the boundary of the region in which diffusion is degenerate, which is quite surprising for a linear diffusion equation. The results are illustrated by numerical examples.

The process of invasion of tissue by cancer cells is crucial for metastasis - the formation of secondary tumours - which is the main cause of mortality in patients with cancer. In the invasion process itself, adhesion, both cell-cell and cell-matrix, plays an extremely important role. In this paper, a mathematical model of cancer cell invasion of the extracellular matrix is developed by incorporating cell-cell adhesion as well as cell-matrix adhesion into the model. Considering the interactions between cancer cells, extracellular matrix and matrix degrading enzymes, the model consists of a system of reaction-diffusion partial integro-differential
equations, with nonlocal (integral) terms describing the adhesive interactions between cancer cells and the host tissue, i.e. cell-cell adhesion and cell-matrix adhesion. Having formulated the model, we prove the existence and uniqueness of global in time classical solutions which are uniformly bounded. Then, using computational simulations, we investigate the effects of the relative importance of cell-cell adhesion and cell-matrix adhesion on the invasion process. In particular, we examine the roles of cell-cell adhesion and cell-matrix adhesion in generating heterogeneous spatio-temporal solutions. Finally, in the discussion section, concluding remarks are made and open problems are indicated.

Evidence for the potent influence of stromal organization and function on invasion and metastasis of breast tumors is ever growing. Here we have performed a rigorous examination of the relationship of a tumor-associated collagen signature (TACS-3), to the long term survival rate of human patients. TACS-3 is characterized by bundles of straightened and aligned collagen fibers that are oriented perpendicular to the tumor boundary. An evaluation of TACS-3 was performed in biopsied tissue sections from 196 patients by second harmonic generation (SHG) imaging of the backscattered signal generated by collagen. Univariate analysis of a Cox proportional hazard model demonstrated that the presence of TACS-3 was associated with poor disease-specific and disease free survival, resulting in hazard ratios between 3.0-3.9. Furthermore, TACS-3 was confirmed to be an independent prognostic indicator regardless of tumor grade and size, ER or PR status, HER-2 status, node status, and tumor subtype. Interestingly, TACS-3 was positively correlated to expression of stromal syndecan-1, a receptor for several extracellular matrix proteins including collagens. Because of the strong statistical evidence for poor survival in patients with TACS, and since the assessment can be performed in routine histopathologic samples imaged via SHG or using picrosirius, we propose that quantifying collagen alignment is a viable, novel paradigm for the prediction of human breast cancer survival.
Citation Format: {Authors}. {Abstract title} [abstract]. In: Proceedings of the 102nd Annual Meeting of the American Association for Cancer Research; 2011 Apr 2-6; Orlando, FL. Philadelphia (PA): AACR; Cancer Res 2011;71(8 Suppl):Abstract nr 4749. doi:10.1158/1538-7445.AM2011-4749

The ability to invade tissue is one of the hallmarks of cancer. Cancer cells achieve this through the secretion of matrix degrading enzymes, cell proliferation, loss of cell–cell adhesion, enhanced cell–matrix adhesion and active migration. Invasion of tissue by the cancer cells is one of the key components in the metastatic cascade, whereby cancer cells spread to distant parts of the host and initiate the growth of secondary tumours (metastases). A better understanding of the complex processes involved in cancer invasion may ultimately lead to treatments being developed which can localise cancer and prevent metastasis. In this paper we formulate a novel continuum model of cancer cell invasion of tissue which explicitly incorporates the important biological processes of cell–cell and cell–matrix adhesion. This is achieved using non-local (integral) terms in a system of partial differential equations where the cells use a so-called “sensing radius” R to detect their environment. We show that in the limit as R→0 the non-local model converges to a related system of reaction–diffusion–taxis equations. A numerical exploration of this model using computational simulations shows that it can form the basis for future models incorporating more details of the invasion process.

In this paper we consider a simple continuous model to describe cell invasion, incorporating the effects of both cell-cell adhesion and cell-matrix adhesion, along with cell growth and proteolysis by cells of the surrounding extracellular matrix (ECM). We demonstrate that the model is capable of supporting both noninvasive and invasive tumour growth according to the relative strength of cell-cell to cell-matrix adhesion. Specifically, for sufficiently strong cell-matrix adhesion and/or sufficiently weak cell-cell adhesion, degradation of the surrounding ECM accompanied by cell-matrix adhesion pulls the cells into the surrounding ECM. We investigate the criticality of matrix heterogeneity on shaping invasion, demonstrating that a highly heterogeneous ECM can result in a "fingering" of the invasive front, echoing observations in real-life invasion processes ranging from malignant tumour growth to neural crest migration during embryonic development.

The extracellular matrix (ECM) is a highly organised structure with the capacity to direct cell migration through their tendency to follow matrix fibres, a process known as contact guidance. Amoeboid cell populations migrate in the ECM by making frequent shape changes and have minimal impact on its structure. Mesenchymal cells actively remodel the matrix to generate the space in which they can move. In this paper, these different types of movement are studied through simulation of a continuous transport model. It is shown that the process of contact guidance in a structured ECM can spatially organise cell populations. Furthermore, when combined with ECM remodelling, it can give rise to cellular pattern formation in the form of "cell-chains" or networks without additional environmental cues such as chemoattractants. These results are applied to a simple model for tumour invasion where it is shown that the interactions between invading cells and the ECM structure surrounding the tumour can have a profound impact on the pattern and rate of cell infiltration, including the formation of characteristic "fingering" patterns. The results are further discussed in the context of a variety of relevant processes during embryonic and adult stages.

Detachment of cell-cell adhesion is indispensable for the first step of invasion and metastasis of cancer. This mechanism is frequently associated with the impairment of either E-cadherin expression or function. However, mechanisms of such abnormalities have not been fully elucidated. In this study, we demonstrated that the function of E-cadherin was completely abolished in the human gastric cancer cell line HSC-39, despite the high expression of E-cadherin, because of mutations in one of the E-cadherin-associated cytoplasmic proteins, beta-catenin. Although immunofluorescence staining of HSC-39 cells by using an anti-E-cadherin antibody (HECD-1) revealed the strong and uniform expression of E-cadherin on the cell surface, cell compaction and cell aggregation were not observed in this cell. Western blotting (immunoblotting) using HECD-1 exhibited a 120-kDa band which is equivalent to normal E-cadherin. Northern (RNA) blotting demonstrated a 4.7-kb band, the same as mature E-cadherin mRNA. Immunoprecipitation of metabolically labeled proteins with HECD-1 revealed three bands corresponding to E-cadherin, alpha-catenin, and gamma-catenin and a 79-kDa band which was apparently smaller than that of normal beta-catenin, indicating truncated beta-catenin. The 79-kDa band was immunologically identified as beta-catenin by using immunoblotting with anti-beta-catenin antibodies. Examination of beta-catenin mRNA by the reverse transcriptase-PCR method revealed a transcript which was shorter than that of normal beta-catenin. The sequencing of PCR product for beta-catenin confirmed deletion in 321 bases from nucleotides +82 to +402. Southern blotting of beta-catenin DNA disclosed mutation at the genomic level. Expression vectors of Beta-catenin were introduced into HSC-39 cells by transfection. In the obtained transfectants, E-cadherin-dependent cell-cell adhesiveness was recovered, as revealed by cell compaction, cell aggregation, and immunoflourescence staining. From these results, it was concluded that in HSC-39 cells, impaired cell-cell adhesion is due to mutations in beta-catenin which results in the dysfunction of E-cadherin.

25 years ago, then President Nixon "declared" War on Cancer. In this personal commentary, the war is reviewed. There have been obvious triumphs, for instance in cure of acute lymphocytic leukaemia and other childhood cancers, Hodgkin's disease, and testicular cancer. However, substantial advances in molecular oncology have yet to impinge on mortality statistics. Too many adults still die from common epithelial cancers. Failure to appreciate that local invasion and distant metastasis rather then cell proliferation itself are lethal, obsession with cure of advanced disease rather than prevention of early disease, and neglect of the need to arrest preneoplastic lesions may all have served to make victory elusive.

Background:
The matrix metalloproteinases (MMPs) have a role in gastrointestinal malignancy. This role is reviewed, with particular reference to the gelatinase subgroup of enzymes.
Methods:
All relevant papers derived from the Medline and Enbase databases between 1984 and early 1996 were reviewed.
Result and conclusion:
There is now strong evidence that MMPs play a major role in tumour invasion and metastasis. The development of MMP inhibitors may lead to important new treatment for the control of malignant disease.

We wish to thank Terry Schoop of Biomed Arts Associates, San Francisco, for preparation of the figures, Cori Bargmann and Zena Werb for insightful comments on the manuscript, and Normita Santore for editorial assistance. In addition, we are indebted to Joe Harford and Richard Klausner, who allowed us to adapt and expand their depiction of the cell signaling network, and we appreciate suggestions on signaling pathways from Randy Watnick, Brian Elenbas, Bill Lundberg, Dave Morgan, and Henry Bourne. R. A. W. is a Ludwig Foundation and American Cancer Society Professor of Biology. His work has been supported by the Department of the Army and the National Institutes of Health. D. H. acknowledges the support and encouragement of the National Cancer Institute. Editorial policy has rendered the citations illustrative but not comprehensive.

Agonist-evoked, intracellular Ca2+-signalling events are associated with active extrusion of Ca2+ across the plasma membrane, implying a local increase in Ca2+ concentration ([Ca2+]) at the extracellular face of the cell. The possibility that these external [Ca2+] changes may have specific physiological functions has received little consideration in the past. Here we show that, at physiological ambient [Ca2+], Ca2+ mobilization in one cell produces an extracellular signal that can be detected in nearby cells expressing the extracellular Ca2+-sensing receptor (CaR), a cell-surface receptor for divalent cations with a widespread tissue distribution. The CaR may therefore mediate a universal form of intercellular communication that allows cells to be informed of the Ca2+-signalling status of their neighbours.

E-cadherin-mediated cell-cell adhesion is reduced in epithelial tumors, which is thought to be a prerequisite to acquire invasive properties. We observed that several pancreatic carcinoma cell lines with high metastatic potential expressed normal levels of E-cadherin and possessed functional E-cadherin/catenin adhesion complexes. When the cell lines PANC-1, BxPC-3, and PaTu8988s were cultured either on type I or type III collagen, E-cadherin gene expression was repressed, and E-cadherin and catenin protein concentrations were reduced. In contrast, growth on fibronectin and collagen type IV had no influence. Collagen type I- or type III-dependent reduction of E-cadherin expression led to decreased cell-cell adhesion, increased proliferation, and migratory activity as well as morphological transformation. Overexpression of activated c-Src in PANC-1 cells mimicked collagen-induced E-cadherin down-regulation and changed the elevated cell proliferation and migration. Conversely, treatment of cells with the Src-inhibitors PP1 or herbimycin A resulted in complete suppression of collagen type I-induced E-cadherin decrease. Our data demonstrate that specific collagens are able to promote metastatic behavior by down-regulation of E-cadherin gene expression in a Src-kinase-dependent manner. This points toward a novel mechanism for substrate-dependent signaling and underlines the significance of extracellular matrix environment for tumor growth and invasiveness.

It is 40 years since the first member of what came to be known as the matrix metalloproteinase (MMP) family was described. Structural, molecular and biochemical approaches have subsequently contributed to piecing together the puzzle of how MMPs work, and how they contribute to various disease processes.

We develop a discrete model of malignant invasion using a thermodynamic argument. An extension of the Potts model is used to simulate a population of malignant cells experiencing interactions due to both homotypic and heterotypic adhesion while also secreting proteolytic enzymes and experiencing a haptotactic gradient. In this way we investigate the influence of changes in cell-cell adhesion on the invasion process. We demonstrate that the morphology of the invading front is influenced by changes in the adhesiveness parameters, and detail how the invasiveness of the tumour is related to adhesion. We show that cell-cell adhesion has less of an influence on invasion compared with cell-medium adhesion, and that increases in both proteolytic enzyme secretion rate and the coefficient of haptotaxis act in synergy to promote invasion. We extend the simulation by including proliferation, and, following experimental evidence, develop an algorithm for cell division in which the mitotic rate is explicitly related to changes in the relative magnitudes of homotypic and heterotypic adhesiveness. We show that although an increased proliferation rate usually results in an increased depth of invasion into the extracellular matrix, it does not invariably do so, and may, indeed, cause invasiveness to be reduced.

Angiogenesis, the growth of a network of blood vessels, is a crucial component of solid tumor growth, linking the relatively harmless avascular and the potentially fatal vascular growth phases of the tumor. As a process, angiogenesis is a well-orchestrated sequence of events involving endothelial cell migration and proliferation; degradation of tissue; new capillary vessel formation; loop formation (anastomosis) and, crucially, blood flow through the network. Once there is flow associated with the nascent network, subsequent growth evolves both temporally and spatially in response to the combined effects of angiogenic factors, migratory cues via the extracellular matrix, and perfusion-related hemodynamic forces in a manner that may be described as both adaptive and dynamic. In this article, we first present a review of previous theoretical and computational models of angiogenesis and then indicate how recent developments in flow models are providing insight into antiangiogenic and chemotherapeutic drug treatment of solid tumors.

Migrating cells tend to continue moving in the same direction, a property called persistence. During migration, cells, by definition, form new adhesions at their front and break old adhesions at the rear. We hypothesize that the distinction between new adhesions at the front and older adhesions at the rear plays a major role in directional persistence. We propose specific mechanisms of persistence on the basis of known properties of integrin signals, in hope of stimulating investigation of these ideas.

The ability to connect to the actin cytoskeleton is a key part of the adhesive function of integrins. This linkage between integrins and the cytoskeleton involves a large complex of integrin-associated proteins that function in both the assembly and disassembly of the link. Genetic evidence has helped to clarify the relative contributions of different components of this link. In different contexts integrins can either stimulate or suppress actin based structures, indicating the variety of pathways leading from integrins to the cytoskeleton. The cytoskeleton also contributes to the extent of the integrin junction, allowing an adhesive contact to attain sufficient strength to resist contractile forces involved in cellular movement and function.

An siRNA directed against the extracellular calcium-sensing receptor (CaSR) was used to down-regulate this protein in CBS colon carcinoma cells. In additional studies, we utilized a variant of the parental CBS line that demonstrates CaSR expression but does not upregulate this protein in response to extracellular Ca(2+). In neither the siRNA-transfected cells nor the Ca(2+)-nonresponsive variant cells did inclusion of Ca(2+) in the culture medium inhibit proliferation or induce morphological alterations. Extracellular Ca(2+) also failed to induce E-cadherin production or a shift in beta-catenin from the cytoplasm to the cell membrane. In mock-transfected cells and in a Ca(2+)-responsive variant line derived from the same parental CBS cells, Ca(2+) treatment resulted in growth-reduction. This was accompanied by increased E-cadherin production and a shift in beta-catenin distribution from the cytoplasm to the cell membrane. Additionally, down-regulation of c-myc and cyclin D1 expression was observed in mock-transfected cells and in the Ca(2+)-responsive variant line (along with reduced T cell factor transcriptional activation). Neither c-myc nor cyclin D1 was significantly down-regulated in the siRNA-transfected cells or in the Ca(2+)-nonresponsive variant cells upon Ca(2+) stimulation. In histological sections of human colon carcinoma CaSR was significantly reduced as compared to the level in normal colonic crypt epithelial cells. Where CaSR expression was high, strong surface staining for E-cadherin and beta-catenin was observed. Where CaSR expression was reduced, beta-catenin surface expression was likewise reduced.

The complex interactions of cells with extracellular matrix (ECM) play crucial roles in mediating and regulating many processes, including cell adhesion, migration, and signaling during morphogenesis, tissue homeostasis, wound healing, and tumorigenesis. Many of these interactions involve transmembrane integrin receptors. Integrins cluster in specific cell-matrix adhesions to provide dynamic links between extracellular and intracellular environments by bi-directional signaling and by organizing the ECM and intracellular cytoskeletal and signaling molecules. This mini review discusses these interconnections, including the roles of matrix properties such as composition, three-dimensionality, and porosity, the bi-directional functions of cellular contractility and matrix rigidity, and cell signaling. The review concludes by speculating on the application of this knowledge of cell-matrix interactions in the formation of cell adhesions, assembly of matrix, migration, and tumorigenesis to potential future therapeutic approaches.

In this article, we show, using a mathematical multiscale model, how cell adhesion may be regulated by interactions between E-cadherin and beta-catenin and how the control of cell adhesion may be related to cell migration, to the epithelial-mesenchymal transition and to invasion in populations of eukaryotic cells. E-cadherin mediates cell-cell adhesion and plays a critical role in the formation and maintenance of junctional contacts between cells. Loss of E-cadherin-mediated adhesion is a key feature of the epithelial-mesenchymal transition. beta-catenin is an intracellular protein associated with the actin cytoskeleton of a cell. E-cadherins bind to beta-catenin to form a complex which can interact both with neighboring cells to form bonds, and with the cytoskeleton of the cell. When cells detach from one another, beta-catenin is released into the cytoplasm, targeted for degradation, and downregulated. In this process there are multiple protein-complexes involved which interact with beta-catenin and E-cadherin. Within a mathematical individual-based multiscale model, we are able to explain experimentally observed patterns solely by a variation of cell-cell adhesive interactions. Implications for cell migration and cancer invasion are also discussed.