1 Introduction

Cancer is a complicated disease that involves many cross-related processes occurring over several spatial scales, ranging from genes to cells to tissues. The abilities of cancer cells to activate invasion and metastasis, to sustain proliferative signalling, to evade growth suppressors, to enable replicative immortality, to induce angiogenesis and to resist cell death have been initially identified as the six main hallmarks of cancer (Hanahan and Weinberg 2000). A growing knowledge about cancer over the last decade has shed more light on the whole picture of the disease, and another four hallmarks were added, namely the ability of cancer to avoid immune destruction, to deregulate cellular energetics, to develop tumour-promoting inflammations, alongside genome instability and mutations (Hanahan and Weinberg 2011).

Highlighted as one of the hallmarks of cancer, cancer cell invasion is a landmark event that transforms a locally growing tumour into a systemic, metastatic and fatal disease. The past four decades have witnessed great scientific efforts focussed on gaining a better understanding of the processes involved during cancer invasion, which is of highly importance in designing early detection strategies and attempting effective therapies.

A malignant tumour includes a complex heterotypic community of cells (such as cancer cells, immuno-inflammatory cells, stromal cell, fibroblasts, endothelial cells and macrophages) that are mixed with ECM. This community is enhanced by vastly complex signalling pathways underpinning intense molecular processes that mediates the crosstalk between the various cell populations composing the tumours, such as the interaction between the cancer cells and the peritumoural stroma cells occurring during cancer invasion (Hanahan and Weinberg 2011; E and Engquist 2003; Qian and Pollard 2010; Joyce and Pollard 2009; Kalluri and Zeisberg 2006). As one of the main factors that affect the way the cancer cell migrate and invade, the peritumoural ECM not only plays the role of a scaffold for the tissues and physical barriers during cell migration but also conveys the signalling pathway processes, enabling the cells to communicate. These give rise to specific conditions within the tumour microenvironment that locally regulate cell migration, proliferation and differentiation. Specifically, the secretion of proteolytic enzymes such as urokinase plasminogen activator (uPA) and matrix metalloproteinases (MMPs) by the tumour cells and interactions of these enzymes with the ECM components lead to proteolytic degradation and remodelling of the ECM and represent a key step in the cancer invasion process.

In order to decipher the mechanisms behind the complicated processes involved in cancer invasion, clinical investigations and experimental observations carried out over the past few decades have started being increasingly accompanied by mathematical modelling (Adam 1986; Anderson et al. 2000; Byrne and Chaplain 1996; Chaplain et al. 2006; Gatenby and Gawlinski 1996; Greenspan 1976; Perumpanani et al. 1996, 1998; Webb et al. 1999). These initial modelling attempts used reaction-diffusion systems to describe the interaction between malignant and normal cells and focused on several important invasion aspects, leading to the first qualitative mathematical modelling approaches and paving the way towards a better understanding of the contribution of proteolytic enzymes in cancer invasion. This was further explored with modelling focused the role of proteolytic activities of some specific hydrolytic enzymes such as urokinase plasminogen activator (uPA) and matrix metalloproteinases (MMPs) in tumour invasive behaviour (Andasari et al. 2011; Anderson 2005; Byrne et al. 2001; Chaplain and Lolas 2005; Deakin and Chaplain 2013; Perumpanani et al. 1996, 1998). For instance, while Chaplain and Lolas (2005) proposed a system of reaction–diffusion–taxis partial differential equations to explore the role of the uPA system (including uPA, uPA-inhibitors, plasmin and host tissue) in cancer invasion, the model introduced in Deakin and Chaplain (2013) evaluates the role of two types of simultaneously expressed MMPs on cancer growth and spread, i.e. membrane-bound matrix MT1-MMP and soluble MMP-2.

Alongside the chemotactic and haptotactic movement assumed in all the models mentioned above, cell–cell and cell-matrix adhesion were also recognised as playing a crucial role in the growth and development of carcinomas (Byrne and Chaplain 1996). Advances in addressing the importance of cell adhesion in the cancer invasion process were obtained via several models based on systems of non-local integro-differential equations (Armstrong et al. 2006; Chaplain et al. 2011; Domschke et al. 2014; Gerisch and Chaplain 2008) that account for dynamic interactions within an appropriately small sensing radius R between potentially mutating cancer cell populations, ECM, and the involved matrix degradation enzymes.

Since the local tissue invasion of a malignant tumour could be regarded also as a free moving boundary problem, several appropriate numerical techniques were considered in the computational modelling of cancer invasion. Particularly, the level-set method was intensely used to study solid tumour growth in homogeneous microenvironments (Frieboes et al. 2006; Macklin and Lowengrub 2005, 2006, 2007; Zheng et al. 2005). A new ghost cell/level- set method (based on a nonlinear nutrient equation coupled with a pressure equation with geometry-dependent jump boundary conditions) was developed in Macklin and Lowengrub (2008) and applied to models of tumour invasive growth in complex, heterogeneous tissues. This model was later extended into an improved model of tumour invasion including the process of tumour-induced angiogenesis (Macklin et al. 2009). Finally, alongside the level-set method, multiphase models based on the theory of mixtures were also developed and used to investigate tumour growth and spread (Byrne and Preziosi 2003; Chaplain et al. 2006; Frieboes et al. 2010; Preziosi and Tosin 2009; Wise et al. 2008, 2011). In these approaches, the tumour was regarded as a system consisting of different phases (e.g. cellular phase, liquid phase) and the development of a solid tumour was modelled by exploring the mass and momentum balances alongside the constitutive laws that distinguish the phases in the system.

However, while recognising the multiscale nature of cancer growth and spread, over the past two decades or so most computational and mathematical modelling has focused mainly on one scale, either at the tissue, cell or molecular scale, with the first attempts towards linking these scales being revised in Deisboeck et al. (2011). Recently, following a series of important developments within the general multiscale framework, and based on strong insights from atomistic-to-continuum methods (Lin 2007), homogenisation techniques (Allaire 1992; Trucu et al. 2012), and heterogeneous multiscale finite element methodology (Abdulle and Schwab 2005; E and Engquist 2003; Ren and E 2005), a genuinely new multiscale moving boundary model for cancer invasion that links the tissue, cellular and subcellular scales was proposed in Trucu et al. (2013). In this new framework, PDE modelling at the tissue scale for the cell population dynamics and PDE modelling at the cell scale for the molecular mechanics of the proteolytic enzymes population are linked together in a two-scale model through top-down and bottom-up permanent links. The top-down link provides the source for the microscale dynamics, which is induced in a non-local manner by the macrodynamics. On the other hand, the bottom-up link enables the microdynamics to provide the macrodynamics with a law for the macroscale boundary movement, whose direction and displacement magnitude is determined at microscale. This is fundamentally different from previous modelling perspectives such as the one-scale modelling presented in Anderson et al. (2000), or the one proposed in Ramis-Conde et al. (2008) where the authors considered continuous modelling at the microscale but an individual-based model at the macroscale and where no top-down links were assumed.

In this paper, we propose a novel multiscale mathematical model of cancer invasion that explores the tissue-scale cancer progression in conjunction with the cell-scale dynamics of the urokinase plasminogen activation system. Building on the multiscale moving boundary method proposed in Trucu et al. (2013), the new modelling that we propose here allows us to study the changes in tissue-scale (macro) tumour morphology caused by the cell-scale uPA microdynamics occurring in a cell-scale (micro) neighbourhood of the invasive edge of the tumour. Assuming the uPA model developed in Andasari et al. (2011) and Chaplain and Lolas (2005) for the tissue-scale dynamics, we derive governing laws for the degrading enzymes cell-scale dynamics arising in the close proximity of the tumour interface. While this results in prescribing an appropriate form for the top-down link between the macro and microdynamics, by exploring the spatial interaction between the uPA microdynamics and the surrounding ECM from the peritumoural region we obtain a bottom-up link between the micro- and macrodynamics. This allows us to describe the evolution of tumour invasive edge morphology and enables computational predictions for the changes occurring in the macroscopic pattern of cancer during the local invasion process.

2 Biological Background: Components of the uPA System and Their Functions

Proteolytic degradation and remodelling of the extracellular matrix are essential for cancer cell invasion. It enables cancer cells to proliferate and migrate through surrounding tissue. In this context, one of the first steps of invasion is the production and secretion of proteolytic enzymes, i.e. urokinase plasminogen activator (uPA) and matrix metalloproteinases (MMPs) by cancer cells. These enzymes interact with the dynamics of the ECM macromolecules and pave the way for cancer invasion. Specifically, the uPA enzymatic system mainly consists of the urokinase receptor (uPAR), urokinase plasminogen activator (uPA), the matrix-like protein vitronectin (VN), plasminogen activator inhibitor type-1 (PAI-1) and the degrading enzyme plasmin. Here we list the key activities of the main components of the uPA system and the role each plays in cancer invasion (Andreasen et al. 2000, 1997; Dass et al. 2008; Tang and Han 2013).

2.1 Urokinase Plasminogen Activator (uPA)

uPA is an extracellular serine protease produced by cells. Two major functional domains of the uPA molecule are the protease domain and the growth factor domain. The protease part activates plasminogen and turns it into plasmin, which is able to digest basement membrane and extracellular matrix proteins. The growth factor domain has no protease activity but can bind a specific high-affinity cell-surface receptor, uPAR. Finally, uPA has a zymogen form, pro-uPA, which can be activated by plasmin and binds to uPAR.

2.2 Urokinase Plasminogen Activator Receptor (uPAR)

uPAR is a high-affinity cell-surface receptor of uPA (and of its zymogen form pro-uPA), which via the binding process localises the uPA and pro-uPA to the cell surface. Importantly, uPAR contains another binding site for the ECM component called vitronectin (VN), and since VN and uPA binding sites are distinct, uPAR can simultaneously bind both ligands, allowing coordinated regulation of proteolysis, cell adhesion and signalling.

uPAR expression during ECM remodelling is well controlled under normal conditions, for example, in gestational tissues during embryo implantation and placental development and in keratinocytes during epidermal would healing. uPAR is also expressed in many human cancers. It indicates poor prognosis and in some cases is predictive of invasion and metastasis. Importantly, uPAR expression in tumours can occur in tumour cells and/or tumour-associated stromal cells, such as fibroblasts and macrophages. Moreover, there is a certain crosstalk between these two binding processes, as the ligand binding of uPA to uPAR enhances the VN binding by uPAR (Smith and Marshall 2010).

2.3 Vitronectin (VN)

VN is an abundant versatile glycoprotein found in serum and the ECM and promotes cell adhesion and spreading. Vitronectin binds strongly to glass surfaces, as the name indicates (vitro = glass), and it has binding sites for several ligands, including heparin, urokinase plasminogen activator receptor (uPAR), plasminogen activator inhibitor type-1(PAI-1) and integrins, such as \(\alpha _v\beta _3\). When vitronectin binds to uPAR, it is thought to bring PAI-1 closer to uPA, thereby promoting inhibition and clearance of uPA from the receptor.

2.4 Urokinase Plasminogen Activator Inhibitor-1(PAI-1)

One of the inhibitors of urokinase plasminogen activator, PAI-1, belongs to the serpin (serine protein inhibitors) family, and it is believed to be the most abundant, fast-acting inhibitor of uPA in vivo. It can specifically bind to soluble and membrane-bound uPA to inhibit plasminogen activation. When PAI-1 binds to the uPA/uPAR complex, it triggers the internalisation of the uPA/uPAR/PAI-1 complex by receptor-mediated endocytosis, meaning that the complex will be dissociated and PAI-1 and uPA will be digested, but the receptor will be recycled to the cell surface. This process helps with the clearance of PAI-1 from the vicinity of the cell surface. Additionally, as a major binding protein of VN, PAI-1 competes with uPAR for binding to VN.

2.5 Plasmin

Plasmin is a widespread enzyme that cleaves many extracellular matrix proteins, such as fibronectin, laminin, vitronectin and thrombospondin. In addition, plasmin can also activate many matrix metalloproteinases (MMPs), enhancing even more the degradation of extracellular matrix. It can also influence the composition of the extracellular environment by affecting the activity of cytokines and growth factors, for example, decreasing the activation of TGF-\(\beta 1\) (Venkatraman et al. 2012).

3 Mathematical Modelling

In this section, we will detail the three main components of the uPA multiscale model that we propose for cancer invasion, namely the macroscopic dynamics, the microscopic dynamics and regulation of the tumour boundary relocation. Thus, the description will cover the model at both the macrolevel (tissue scale) and the microlevel (cell scale) and will explore the link between these two biological scales.

3.1 The Macroscopic Dynamics

For the tumour macroscopic dynamics, we adopt here the modelling hypotheses formulated in Andasari et al. (2011) and Chaplain and Lolas (2005) which are based on the activities of the key molecules of the uPA system and their invasive roles (Andreasen et al. 2000, 1997; Dass et al. 2008; Tang and Han 2013).

We denote the cancer cell density by c, the extracellular matrix density by v (without making the distinction between ECM and its component VN), the urokinase plasminogen activator (uPA) concentration by u, the plasminogen activator inhibitor (PAI-1) concentration by p and the plasmin concentration by m. Further, since we assume a fixed average number of receptors uPAR located on each cancer cell surface, there is no explicit modelling of uPAR. Therefore, the concentration of uPAR is considered to be proportional to the cancer cell density. Another important assumption is that the supply of plasminogen is unlimited in this model. Finally, the macroscopic model is obtained by accounting for the biological considerations described in previous section in conjunction with the following presumptions:

3.1.1 The Cancer Cell Dynamics

It is assumed that cancer cell migration is mainly governed by diffusion, chemotaxis due to uPA, and PAI-1 (Degryse et al. 2001; Resnati et al. 2002; Roussos et al. 2011) and haptotaxis due to VN and other ECM components. Additionally, a logistic growth law is used to model cancer cell proliferation. Thus, the mathematical equation for cancer cell density that is considered here is as follows:

$$\begin{aligned} \frac{\partial c}{\partial t} = \underbrace{D_c \Delta c}_{\text {diffusion}} - \nabla \cdot [ \underbrace{\chi _u c \nabla u}_{\text {uPA-chemo}} + \underbrace{\chi _p c \nabla p}_{\text {PAI-1-chemo}} + \underbrace{\chi _v c \nabla v}_{\text {VN-hapo}} ] + \underbrace{\mu _1 c(1-\frac{c}{c_0})}_{\text {profieration}}, \end{aligned}$$
(1)

where \(D_c\) is the diffusion coefficient of cancer cells, \(\chi _u\) and \(\chi _p\) are the chemotaxis coefficients relevant to uPA and PAI-1, respectively, \(\chi _v\) is the VN-mediated haptotaxis rate, \(\mu _1\) is the cancer cell proliferation rate and \(c_0\) is the maximum carrying capacity for cancer cells.

3.1.2 The ECM/VN Dynamics

As ECM is not assumed to move, we rule out any migration terms in the governing law. While VN (which is an important ECM component) is degraded in contact with enzymes m, the binding of PAI-1 to uPA inhibits the activation of plasminogen, leading to the protection of VN and other ECM molecular constituents and indirectly contributing to their production. Simultaneously, the binding of PAI-1 to VN results in less binding to cell-surface receptors such as uPAR, and so, through the regulation of cell-matrix-associated signal transduction pathways, this inhibits the production of VN. Therefore, assuming a logistic ECM remodelling, the governing equation for ECM is given by

$$\begin{aligned} \frac{\partial v}{\partial t}= -\underbrace{\delta v m}_{\text {degradation}} +\underbrace{\phi _{21} u p}_{\text {uPA/PAI-1}} -\underbrace{\phi _{22} v p}_{\text {PAI-1/VN} }+ \underbrace{\mu _2 v(1-\frac{v}{v_0})}_{\text {remodelling}}. \end{aligned}$$
(2)

where \(\delta \) is the rate of ECM degradation by plasmin, \(\phi _{21}\) is the binding rate of PAI-1 to uPA, \(\phi _{22}\) is the binding rate of PAI-1 to VN and \(\mu _2\) is the matrix remodelling rate.

3.1.3 The uPA Dynamics

The mathematical modelling of the uPA concentration dynamics accounts for the following aspects. While being produced by the cancer cells and removed from the system due to its binding with PAI-1 and uPAR, per unit time the uPA exercises a local diffusion. Therefore, this can be formalised mathematically as follows:

$$\begin{aligned} \frac{\partial u}{\partial t} = \underbrace{D_u \Delta u}_{\text {diffusion} }- \underbrace{\phi _{31} p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{33} c u}_{\text {uPA/uPAR}} +\underbrace{\alpha _{31} c}_{\text {production}}, \end{aligned}$$
(3)

where \(D_u\) is the diffusion coefficient, \(\phi _{31}\) and \(\phi _{33}\) are binding rates of uPA/PAI-1 and uPA/uPAR accordingly, and \(\alpha _{31}\) is the production rate of uPA by the cancer cells.

3.1.4 The PAI-1 Dynamics

Similarly, the equation for PAI-1 simply includes a diffusion term with coefficient \(D_p\), removal caused by binding to uPA and VN with binding rates \(\phi _{41}\) and \(\phi _{42}\), respectively, and production as a result of plasmin formation at a rate \(\alpha _{41}\). Thus, these considerations lead us to the following governing equation:

$$\begin{aligned} \frac{\partial p}{\partial t} = \underbrace{D_p \Delta p}_{\text {diffusion}} - \underbrace{\phi _{41}p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{42} p v}_{\text {PAI-1/VN}} +\underbrace{\alpha _{41} m}_{\text {production}}. \end{aligned}$$
(4)

3.1.5 The Plasmin Dynamics

The evolution of plasmin concentration is modelled as follows. While assuming that per unit time this exercises a local diffusion, we consider that the binding of uPA to uPAR provides an opportunity for pericellular proteolytic activity through plasminogen activation leading to plasmin formation. Moreover, the binding of PAI-1 to VN indirectly enhances the binding of uPA to uPAR, therefore bringing additional contribution to plasmin formation. Thus, these assumptions give us the following evolution law:

$$\begin{aligned} \frac{\partial m}{\partial t} = \underbrace{D_m \Delta m}_{\text {diffusion}} + \underbrace{\phi _{52} p v}_{\text {PAI-1/VN}}+\underbrace{\phi _{53} c u}_{\text {uPA/uPAR}}-\underbrace{\phi _{54} m}_{\text {degradation}}, \end{aligned}$$
(5)

where \(D_m\) is the diffusion coefficient, \(\phi _{52}\) and \(\phi _{53}\) are the binding rates of PAI-1/VN and uPAR/uPA accordingly and \(\phi _{54}\) is the rate of decay of plasmin.

To summarise the macrodynamics, the dimensionless mathematical model of the uPA system adopted here is the one that was initially proposed in Chaplain and Lolas (2005), namely,

$$\begin{aligned} \frac{\partial c}{\partial t}&= \underbrace{D_c \Delta c}_{\text {diffusion}} - \nabla \cdot [ \underbrace{\chi _u c \nabla u}_{\text {uPA-chemo}} + \underbrace{\chi _p c \nabla p}_{\text {PAI-1-chemo}} + \underbrace{\chi _v c \nabla v}_{\text {VN-hapo}} ] + \underbrace{\mu _1 c(1-c)}_{\text {profieration} } \; , \end{aligned}$$
(6)
$$\begin{aligned} \frac{\partial v}{\partial t}&= -\underbrace{\delta v m}_{\text {degradation}} +\underbrace{\phi _{21} u p}_{\text {uPA/PAI-1}} -\underbrace{\phi _{22} v p}_{\text {PAI-1/VN} }+ \underbrace{\mu _2 v(1-v)}_{\text {remodelling}}, \end{aligned}$$
(7)
$$\begin{aligned} \frac{\partial u}{\partial t}&= \underbrace{D_u \Delta u}_{\text {diffusion} }- \underbrace{\phi _{31} p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{33} c u}_{\text {uPA/uPAR}} +\underbrace{\alpha _{31} c}_{\text {production}}, \end{aligned}$$
(8)
$$\begin{aligned} \frac{\partial p}{\partial t}&= \underbrace{D_p \Delta p}_{\text {diffusion}} - \underbrace{\phi _{41}p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{42} p v}_{\text {PAI-1/VN}} +\underbrace{\alpha _{41} m}_{\text {production}}, \end{aligned}$$
(9)
$$\begin{aligned} \frac{\partial m}{\partial t}&= \underbrace{D_m \Delta m}_{\text {diffusion}} + \underbrace{\phi _{52} p v}_{\text {PAI-1/VN}}+\underbrace{\phi _{53} c u}_{\text {uPA/uPAR}}-\underbrace{\phi _{54} m}_{\text {degradation}}. \end{aligned}$$
(10)

We note that the above model was investigated numerically in some depth in Chaplain and Lolas (2005), where wide-ranging computational simulations revealed rich spatio-temporal dynamics. Further numerical investigations on variations of the original model (including a reduced model involving only the three variables c, v, u) were undertaken in Chaplain and Lolas (2006) where once again solutions displaying a persistent “dynamic heterogeneity” were observed. An analysis (linear stability) of the model was later carried out by Andasari et al. (2011) who demonstrated that the dynamic heterogeneous spatio-temporal solutions were caused by a taxis-driven instability of an underlying spatially homogeneous steady state. In particular, three key parameters controlling the bifurcations were noted to be \(D_c\) (the cancer cell diffusion coefficient), \(\mu _1\) (the cancer cell proliferation rate) and \(\phi _{53}\) (the rate of plasmin production due to uPA/uPAR binding). We also note that a rigorous mathematical analysis of the simplified three-variable system involving only cancer cells, ECM and uPA (i.e. no PAI-1 and no plasmin), was carried out in Hillen et al. (2013) [following a preliminary numerical study in Painter and Hillen (2011)]. This rigorous analysis has shown that the complex dynamics of the model are driven largely by cancer cell proliferation and cancer cell chemotaxis. Finally, we note that a study from a numerical analysis point of view concerning mesh adaptivity has been undertaken in Kolbe et al. (2014).

3.2 The Microscopic Dynamics

Turning now our attention to the microscale setting, in the following we will derive and propose a system of three coupled PDEs to describe the microdynamics of the plasminogen activation system taking place within a cell-scale \(\epsilon \)-neighbourhood \(\mathcal {P}_{\epsilon }\) of the tumour invasive edge \(\partial {\varOmega }(t_{0})\) (which is introduced in “Appendix 1”).

Assuming that PAI-1 and uPAR are uniformly expressed on the cell surface of various cell types in the tumour, this system will capture the leading edge microdynamics of the uPA, PAI-1 and plasmin by accounting for the following biological considerations. On one hand, the urokinase plasminogen activator (uPA) is assumed to bind to the cancer surface receptor uPAR to activate plasminogen, leading to degradation of pericellular ECM through a series of proteolytic activities. On the other hand, the membrane-bound MMPs (such as MT1-MMP) are secreted from within the tumour cell population distributed on the outer proliferating rim along the entire tumour periphery. Their region of proteolytic activities is therefore restricted around the tumour interface (Deakin and Chaplain 2013; Sabeh et al. 2009). Thus, based on these considerations, we propose a coupled governing law for the leading edge microdynamics, which is detailed as follows:

3.2.1 The uPA Microdynamics

In each microregion \(\epsilon Y\), the dynamics of the uPA molecular population is governed by a diffusion process whose source is induced from the tumour cell macrodynamics. At each point \(y\in \epsilon Y\), a source of uPA arises as a collective contribution of the tumour cells distributed within a certain neighbouring area within the tumour’s outer proliferating rim. Therefore, this source is denoted by \(f_{\epsilon Y}(\cdot ,\cdot )\times [0,\Delta t]:\epsilon Y\rightarrow \mathbb {R}_+\) and is defined by:

$$\begin{aligned} f_1^{\epsilon Y} (y, \tau ) = \left\{ \begin{array}{ll} \frac{1}{\lambda (B(y,\gamma )\cap {\varOmega }(t_0))} \int \limits _{B(y,\gamma )\cap {\varOmega }(t_0)} c\;(x,t_0+\tau ) \;\text {d}x, &{} \quad y \in \epsilon Y \cap {\varOmega }(t_{0}),\\ 0, &{} \text {outside cancer}, \end{array} \right. \end{aligned}$$
(11)

where \(\lambda (\cdot )\) is the standard Lebesgue measure and \(\gamma \) represents the maximal thickness of the outer proliferating rim. Thus, per unit time, under the presence of source (11) the uPA is locally diffusing and is binding to both PAI-1 and uPAR, and so its microdynamics can be formally written as:

$$\begin{aligned} \frac{\partial u}{\partial \tau } = \underbrace{D_u \Delta u}_{\text {diffusion} }- \underbrace{\phi _{31} p u}_{\text {uPA/PAI-1}} + \;( \underbrace{\alpha _{31} }_{\text {production}} -\underbrace{\phi _{33} u}_{\text {uPA/uPAR}}) f_1^{\epsilon Y} (y, \tau ) \end{aligned}$$
(12)

3.2.2 The PAI-1 Microdynamics

The equation for PAI-1 accounts for diffusive motion, production due to plasmin activation, and loss due to binding with uPA and VN. Specifically, the binding between PAI-1 and VN is as a collective effect of the ECM distribution within \(\epsilon Y\). Therefore, proceeding similarly to the case of the source term in (11), we define

$$\begin{aligned} f_2^{\epsilon Y} (y, \tau ) = \frac{1}{\lambda (B(y,2\epsilon ))} \int _{B(y,2\epsilon )} v\;(x,t_0+\tau ) \;\text {d}x,\;\;\;y \in \epsilon Y, \end{aligned}$$
(13)

which finally enable us to write the following governing law for the PAI-1 microdynamics, namely,

$$\begin{aligned} \frac{\partial p}{\partial \tau } = \underbrace{D_p \Delta p}_{\text {diffusion}} - \underbrace{\phi _{41}p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{42} p\; f_2^{\epsilon Y} (y, \tau )}_{\text {PAI-1/VN}} +\underbrace{\alpha _{41} m}_{\text {production}}. \end{aligned}$$
(14)

3.2.3 The Plasmin Microdynamics

For the spatio-temporal evolution of plasmin, it is assumed that, per unit time, this exercises a local diffusion in the presence of the following source and decay circumstances. Considering that the binding of uPA to uPAR is required to provide the cell surface with a potential proteolytic activity, the plasmin source accounts on one hand on the contribution of the binding uPA/uPAR. On the other hand, as PAI-1 collectively competes with uPAR for binding to VN, the binding of PAI-1 to VN gives more opportunities to uPAR to bind with uPA and indirectly results in more plasmin formation. Finally, plasmin can be deactivated either by degradation or by the action of the plasmin inhibitor \(\alpha _2\)-antiplasmin. Thus, the equation that we obtain to describe these biological interactions is:

$$\begin{aligned} \frac{\partial m}{\partial \tau } = \underbrace{D_m \Delta m}_{\text {diffusion}} + \underbrace{\phi _{52} p \; f_2^{\epsilon Y} (y, \tau )}_{\text {PAI-1/VN}}+\underbrace{\phi _{53} u \; f_1^{\epsilon Y} (y, \tau ) }_{\text {uPA/uPAR}}-\underbrace{\phi _{53} m}_{\text {degradation}} \end{aligned}$$
(15)

In summary, the leading edge microdynamics are therefore given by the following system:

$$\begin{aligned} \frac{\partial u}{\partial \tau }&= \underbrace{D_u \Delta u}_{\text {diffusion} }- \underbrace{\phi _{31} p u}_{\text {uPA/PAI-1}} + \;( \underbrace{\alpha _{31} }_{\text {production}} -\underbrace{\phi _{33} u}_{\text {uPA/uPAR}}) f_1^{\epsilon Y} (y, \tau ), \end{aligned}$$
(16)
$$\begin{aligned} \frac{\partial p}{\partial \tau }&= \underbrace{D_p \Delta p}_{\text {diffusion}} - \underbrace{\phi _{41}p u}_{\text {uPA/PAI-1}} -\underbrace{\phi _{42} p\; f_2^{\epsilon Y} (y, \tau )}_{\text {PAI-1/VN}} +\underbrace{\alpha _{41} m}_{\text {production}}, \end{aligned}$$
(17)
$$\begin{aligned} \frac{\partial m}{\partial \tau }&= \underbrace{D_m \Delta m}_{\text {diffusion}} + \underbrace{\phi _{52} p \; f_2^{\epsilon Y} (y, \tau )}_{\text {PAI-1/VN}}+\underbrace{\phi _{53} u \; f_1^{\epsilon Y} (y, \tau ) }_{\text {uPA/uPAR}}-\underbrace{\phi _{54} m}_{\text {degradation}}. \end{aligned}$$
(18)

3.3 The Macroscopic Tumour Boundary Relocation Induced by the Leading Edge Microdynamics

Following the multiscale approach described in “Appendix 1”, the set of points \(\{x_{\epsilon Y}^* \}\) on the boundary of tumour at the current time moves towards a set of new spatial positions \(\{\widetilde{x_{\epsilon Y}^* }\}\) to form the new boundary at the next multiscale stage, provided that the local transitional probability \(q^{*}\) is in agreement with the circumstances in the surrounding peritumoural microenvironment. When the invading strength is above a tissue threshold, the point \(x_{\epsilon Y}^* \) will relocate to a new position \(\widetilde{x_{\epsilon Y}^*}\) following a direction and displacement magnitude that represents the choreographic movement of all the points from the part of the invasive edge captured by the microdomain \(\epsilon Y\).

As described in “The multiscale moving boundary approach for the proposed cancer invasion model” section of “Appendix 1”, on any microdomain \(\epsilon Y\), provided that a sufficient amount of plasmin has been produced across the invading edge, it is the pattern of the front of the advancing spatial distribution of plasmin that characterises ECM degradation. Therefore, the movement direction and displacement magnitude of the part of the invading edge of the tumour caputred by the current microdomain \(\epsilon Y\) will be determined by the spatial distribution pattern of the advancing front of plasmin \(m(\cdot , \tau _f)\) in the peritumoural region. As detailed in “The multiscale moving boundary approach for the proposed cancer invasion model” section of “Appendix 1”, these movement characteristics are obtained by accounting the contribution of all peaks (baricentred at the spatial points \(y_{l}\)) at the front of advancing plasmin that are above the mean value of the entire mass of plasmin produced on \(\epsilon Y \backslash {\varOmega }(t_0)\) and are located at the furthest away Euclidean distance from \(\{x_{\epsilon Y}^* \}\). Thus, under these conditions, the moving direction \(\eta _{\epsilon Y}\) and displacement magnitude \(\xi _{\epsilon Y} \) derived in “The multiscale moving boundary approach for the proposed cancer invasion model” section of “Appendix 1” in (35)–(36) have the following expressions:

$$\begin{aligned} \eta _{\epsilon Y}= & {} x_{\epsilon Y}^* + \nu \underset{l \in \mathcal {I}_{\delta }}{\sum } \bigg ( \int _{\mathcal {D}_l} m(y,\tau _f) \text {d}y \bigg ) (y_l - x_{\epsilon Y}^*), \nu \in [0,\infty ),\\ \xi _{\epsilon Y}:= & {} \underset{l \in \mathcal {I}_{\delta }}{\sum } \frac{\int _{\mathcal {D}_l} m(y,\tau _f) \text {d}y }{\underset{l \in \mathcal {I}_{\delta }}{\sum } \int _{\mathcal {D}_l} m(y,\tau _f) \text {d}y } \big | \overrightarrow{x_{\epsilon Y}^*y_l} \big |. \end{aligned}$$

Finally, the transitional probability \(q^*\) defined in (37) is a quantification of the amount of plasmin in \(\epsilon Y \backslash {\varOmega }(t_0)\) relative to the total amount of plasmin concentration in \(\epsilon Y\) and characterises the invading strength. Therefore, the point \(x^*_{\epsilon Y}\) will exercise the movement into the new spatial position \(\widetilde{x^*_{\epsilon Y}}\) if and only if \(q(x^*_{\epsilon Y}):=q^*(\epsilon Y)\) exceeds a certain tissue local threshold \(\omega (\beta , \epsilon Y) \in (0,1)\) associated with the microdomain \(\epsilon Y\) under a given state of favourable conditions \(\beta \).

4 Multiscale Computational Simulation Results

The multiscale model of cancer invasion that we proposed here was numerically solved in a rectangular region \(Y:= [0,4 \times [0,4]\). For all our simulations, we discretise the entire cube Y uniformly, using the spatial mesh size: \(\Delta x=\Delta y=\frac{\epsilon }{2} = 0.03125\). In this context, the initial conditions for the macrodynamics are specified as follows. The cancer cell population on the initially considered tumour region \({\varOmega }(0):=\mathbf {B}((2,2),0.5)\) is assumed to be given by a translated Gaussian centred at (2, 2) and mollified to 0 after a radius of 0.5, namely,

$$\begin{aligned} c(x,0) = \frac{\bigg (\exp \big (-\frac{||x-(2,2)||_2^2}{\sqrt{\Delta x \Delta y}}\big ) - \exp (-28.125)\bigg )\big (\chi _{_{\mathbf {B}((2,2),0.5-\gamma )}}*\psi _{_{\gamma }}\big ) }{2},\quad x \in Y,\nonumber \\ \end{aligned}$$
(19)

where \(\psi _{\gamma }\) is the mollifier defined in (40)–(41) with \(\gamma \ll \frac{\Delta x}{3}\). Further, the initial macroscopic conditions for the enzymatic components entering the uPA system are considered as follows:

$$\begin{aligned} u(x,0)= & {} 1-\frac{1}{2}c(x,0), \qquad \quad x\in Y\nonumber \\ p(x,0)= & {} \frac{1}{2}c(x,0), \qquad \qquad x\in Y \nonumber \\ m(x,0)= & {} \frac{1}{20}c(x,0), \qquad \qquad x\in Y \end{aligned}$$
(20)

At the same time, in all the subsequent simulations, we consider the following homogeneous and heterogeneous initial condition for ECM, namely,

$$\begin{aligned} \begin{array}{lll} \text {homogeneous case:}\quad &{} v(x,0) = 1 - c(x,0), &{}\quad x\in Y, \\ \text {heterogeneous case:}\quad &{} v(x,0) = \frac{1\,+\,0.3\sin {(4\pi ||x ||_2)}\,+\,\sin {(4\pi || (4,0)\,-\,x ||_2)}}{2}, &{}\quad x\in Y. \end{array}\qquad \end{aligned}$$
(21)

Figure 1 shows the initial conditions (19) and (21) of ECM and cancer cell distributions for both homogeneous and heterogeneous cases that are used in all the simulations presented in this paper.

Fig. 1
figure 1

Initial conditions of the distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) for the homogeneous case (top row) and the heterogeneous case (bottom row) (Color figure online)

Full size image

Finally, in the absence of medical data, for the tissue threshold \(\omega (\beta , \epsilon Y)\) controlling whether or not a point on the boundary will exercised the movement according to the direction and displacement magnitude defined in “The multiscale moving boundary approach for the proposed cancer invasion model” section of “Appendix 1”, we adopt the following functional form:

$$\begin{aligned} \omega (\beta , \epsilon Y) := \left\{ \begin{array}{l l} \text {sin}\bigg (\frac{\pi }{2} \bigg (1-\frac{1}{\beta } \frac{v_{\omega (t_0)}(x_{\epsilon Y}^*, t_0+\Delta t)}{\underset{\xi \in \partial {\varOmega }(t_0)}{\mathrm {sup}} v_{{\varOmega }(t_0)}(\xi ,t_0 +\Delta t)}\bigg )\bigg ) &{} \quad \text {if } \frac{v_{\omega (t_0)}(x_{\epsilon Y}^*, t_0+\Delta t)}{\underset{\xi \in \partial {\varOmega }(t_0)}{\mathrm {sup}} v_{{\varOmega }(t_0)}(\xi ,t_0 +\Delta t)} \le \beta \\ \text {sin}\bigg (\frac{\pi }{2(1-\beta )} \bigg (\frac{v_{\omega (t_0)}(x_{\epsilon Y}^*, t_0+\Delta t)}{\underset{\xi \in \partial {\varOmega }(t_0)}{\mathrm {sup}} v_{{\varOmega }(t_0)}(\xi ,t_0 +\Delta t)} - \beta \bigg )\bigg ) &{} \quad \text {if } \frac{v_{\omega (t_0)}(x_{\epsilon Y}^*, t_0+\Delta t)}{\underset{\xi \in \partial {\varOmega }(t_0)}{\mathrm {sup}} v_{{\varOmega }(t_0)}(\xi ,t_0 +\Delta t)} > \beta \end{array} \right. \end{aligned}$$
(22)

where \(\beta \in (0,1)\) is a parameter that controls a certain “optimal level” of ECM degradation and consider this as being the indicator of the most favourable invasion conditions at the level of tumour and tissue microenvironment. This functional form of \(\omega \) rules out any invasion if either a complete destruction or a very superficial degradation of the surrounding ECM is performed by the MDEs. This is due to the fact that while for invasion the cancer cells need considerable level of the ECM degradation, a total destruction of the surrounding ECM structure prevents them to advance further into the tissue, as they need some of the ECM components to adhere to in order to migrate. Finally, while this functional form of the tissue threshold aimed to showcase the proposed multiscale modelling framework, future work seeks to infer \(\omega (\cdot , \epsilon Y)\) from medical imaging data of the peritumoural tissue.

The following figures show the simulation results of the evolving cancer cell and ECM spatial distributions and of the invasive tumour boundary at macrotime stage 20, 40, 60. The images are presented in two columns, with the left columns representing the cancer cell distribution and right columns showing the corresponding ECM concentrations. Furthermore, all these images include the tumour boundary.

In the subsequent simulations, for the macroscopic part of the model, except otherwise stated, we will generally be using the following basic set of parameter values \(\mathscr {P}:\)

$$\begin{aligned} D_n&= 4.3 \times 10^{-3},&\chi _u&= 3.05 \times 10^{-2},&\chi _p&= 3.75 \times 10^{-2},&\chi _v&= 2.85 \times 10^{-2}, \nonumber \\ \mu _1&= 0.25,&\delta&=1.5,&\phi _{21}&=0.75,&\phi _{22}&=0.55, \nonumber \\ \mu _2&=0.15,&D_u&= 2.5 \times 10^{-3} ,&\phi _{31}&=0.75,&\phi _{33}&=0.3, \nonumber \\ \alpha _{31}&=0.215,&D_p&= 3.5 \times 10^{-3},&\phi _{41}&=0.75,&\phi _{42}&=0.55, \nonumber \\ \alpha _{41}&=0.5 ,&D_m&= 4.91 \times 10^{-3} ,&\phi _{52}&=0.11,&\phi _{53}&= 0.75, \nonumber \\ \phi _{54}&=0.5, \end{aligned}$$
(23)

which are detailed in Table 1 in “Table for the parameter Set \(\mathscr {P}\)” section of “Appendix 3”. However, in order to investigate different cancer growth patterns, as described in the following simulation, we will perform numerical tests also for slightly changed values for the diffusion coefficient of cancer cells \(D_c\), the ECM proliferation rate \(\mu _2\) or the ECM degradation rate \(\delta \). Also, we will explore a range of values for parameter \(\beta \), to highlight the correlation between changes in tissue microenvironment conditions and the resulting cancer invasion patterns.

In order to analyse the effect of each variables (namely ECM initial condition, cancer cell diffusion coefficient, ECM proliferation and degradation rates, and threshold coefficient), we will split the results into four groups as follows:

Fig. 2
figure 2

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the homogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c~=~4.3 \times ~10^{-3}\), \(\beta = 0.775\), \(\mu _2 = 0.01\) and \(\delta = 1.5\) (Color figure online)

Full size image
Fig. 3
figure 3

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\beta = 0.775\), \(\mu _2 = 0.01\) and \(\delta = 1.5\) (Color figure online)

Full size image

4.1 ECM Initial Condition

To investigate what effect do different ECM initial conditions have on the whole dynamics of the model, we apply the same threshold function for both homogeneous and heterogeneous ECM scenario.

The ECM heterogeneity gives rise to a tumour–tissue interaction that is naturally more complex than in the case of homogeneous ECM. A direct consequence of this is that the ECM heterogeneity triggers a corresponding intrinsic variability in the tissue thresholds, which explore the peritumoural tissue conditions that the cancer interacts with during invasion. This results in a higher degree of complexity in the spatial structure of the regions with most favourable conditions for the tumour to progress further in the surrounding region, within the directions and displacement magnitudes specified by the microscale dynamics. Ultimately, this gives rise to a higher level of fingering and infiltrative patterns in the heterogeneous case as opposed to the homogeneous case. This is confirmed by our results presented in Fig. 3 that are obtained for the heterogeneous ECM, which exhibit more “fingered” and infiltrative spreading of the tumour compared with those obtained in the homogeneous ECM case shown in Fig. 2. This type of fingering patterns is often observed in medical imaging data, such as the one reported in the case of oesophageal and lung cancer by the authors in Ito et al. (2012) and Masuda et al. (2012), respectively.

4.2 Cancer Cell Diffusion Coefficient \(D_c\)

As was demonstrated in Hillen et al. (2013) and Painter and Hillen (2011), the chemotaxis terms in the cancer cell Eq. (6) are the main causes of the occurrence of heterogeneous patterns inside the tumour domain. Therefore, if the cancer cell diffusion coefficient (\(D_c\)) is increased to be one order magnitude larger than the chemotaxis coefficients (\(\chi _u\) and \(\chi _p\)), which becomes the dominant mechanism of cell movement, then no heterogeneous dynamics will occur inside the tumour as shown in Fig. 4. In Fig. 5, the chemotaxis coefficients are one order magnitude larger than the diffusion coefficient, and as a consequence, we obtain heterogeneous pattern formation of cancer cells, which leads to a more dynamic tumour boundary deformation.

Fig. 4
figure 4

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 1.4\times 10^{-2}\), \(\beta = 0.775\), \(\mu _2 = 0\) and \(\delta = 0.75\) (Color figure online)

Full size image
Fig. 5
figure 5

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\beta = 0.775\), \(\mu _2 = 0\) and \(\delta = 0.75\) (Color figure online)

Full size image
Fig. 6
figure 6

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\beta = 0.7625\), \(\mu _2 = 0.005\) and \(\delta = 1.5\) (Color figure online)

Full size image
Fig. 7
figure 7

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at various macro-micro stages: stage 20, 40, 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\beta = 0.7625\), \(\mu _2 = 0\) and \(\delta = 0.75\) (Color figure online)

Full size image
Fig. 8
figure 8

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at macro-micro stage 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\beta = 0.7625\), \(\delta = 1.5\), and for rows 1 to 3 of images we consider \(\mu _2=0.0005\), \(\mu _2=0.001\), and \(\mu _2=0.005\), respectively (Color figure online)

Full size image
Fig. 9
figure 9

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at macro-micro stage 60. Starting from the heterogeneous initial conditions shown in Fig. 1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\mu _2=0\), \(\beta = 0.7625\), and for rows 1 to images we consider \(\delta = 0.5\), \(\delta = 0.75\), and \(\delta = 1\), respectively (Color figure online)

Full size image
Fig. 10
figure 10

Simulation results showing distributions of cancer cells (left column) and ECM (right column) and the invasive boundary of the tumour (white line) at macro-micro stage 60. Starting from the homogeneous initial conditions shown in Fig.  1, these results were obtained for \(D_c = 4.3 \times 10^{-3}\), \(\delta = 1.5\), \(\mu _2 = 0.01\), and for rows 1 to 3 of images we consider \(\beta =0.7625\), \(\beta =0.775\), and \(\beta =0.7875\), respectively (Color figure online)

Full size image

4.3 ECM Proliferation Rate \(\mu _2\) and Degradation Rate \(\delta \)

From all the simulation results presented so far, we conclude that the degradation of ECM facilitates cancer invasion. However, as shown also in several interdisciplinary biological investigations (Gatenby and Gawlinski 1996; Gatenby et al. 2006; Stetler-Stevenson et al. 1993), the invasion process will stop where ECM is degraded a lot. This captures the biological scenario in 2D that when cell-matrix adhesion is too low, no focal adhesions or stress fibres are formed, and the cells do not move. In order to investigate the effect of ECM proliferation and degradation on the invasion process, we compare two groups of parameters: (1) \(\mu _2 = 0.01\), \(\delta = 1.5\) (nonzero proliferation rate with relatively large degradation rate, Fig. 6); (2) \(\mu _2 = 0\), \(\delta = 0.75\) (no proliferation with relatively small degradation rate, Fig. 7). From these two groups of images, we observe that when proliferation is present and the degradation rate is relatively large, deformations of the boundary are not as dynamic as that when the proliferation term is absent with a relatively small degradation rate. The reason could be that the proliferation term will reduce the degree of heterogeneity of the distribution of ECM, which leads to a less fingered spreading of the cancer cell population.

Figure 8 shows the results of simulations at the macro-micro stage 60 where the ECM proliferation rate parameter \(\mu _2\) was increased over a range of values in the interval [0.0005, 0.005]. The figures show that, over the range chosen, there is little difference in either the overall extent of invasion or the morphology of the invading cancer. Figure 9 shows the results of simulations at the same macro-micro stage 60 where the degradation parameter \(\delta \) was increased over the interval [0.5, 1]. The figures show that, over the range chosen, the extent of the invasion is similar, but the morphology of the invading cancer changes slightly from a more fingered boundary to a less fingered boundary. Finally, as the two parameters \(\mu _2\) and \(\delta \) are varied, both Figs.  8 and 9 exhibit spatial consistency aspects in the invasion pattern.

4.4 Threshold Coefficient \(\beta \)

Finally, Fig. 10 shows us comparative results at macro-micro stage 60 for several values of the threshold coefficients \(\beta \) in the interval [0.7625, 0.7875]. Since \(\beta \) controls the “optimal level” of ECM density for cancer cells to migrate, its variation gives us different invasion morphologies, as expected. Again, as we noticed also in the cases of Figs. 8 and 9, a quick comparison between consecutive rows of images (from top to bottom) in Fig. 10 seems to indicate a certain degree of consistency in the changes occurring in the tumour morphology and “fingered” boundary deformations with respect to increasing \(\beta \) parameter. This consistency aspect is currently under investigation and will form the topic of a separate research work.

5 Conclusion

In this paper, we presented and developed a mathematical model of cancer invasion based on the previous work in Andasari et al. (2011), Chaplain and Lolas (2005) and Trucu et al. (2013). We adapted and extended the two-scale technique in Trucu et al. (2013) to cope with the different settings that enable the coupling of the finite difference uPA macrosolver and a new finite element microsolver developed for the newly proposed leading edge microdynamics. This enabled us to simulate the multiscale process of cancer invasion by exploring the link between the macroscopic dynamics of the spatio-temporal distribution of cancer cells and ECM taking place on a macroscopic domain, and the matrix degrading enzymes microdynamics developed on the microscopic domains that are in close proximity to tumour boundary.

We derived a new governing law for the microdynamics based exclusively on the molecular mechanics of the uPA system occurring in close proximity to the tumour boundary. This is based on the dynamics of the uPA system including uPA, the inhibitor uPA-1 and plasmin and considers the source terms related to macroscopic components, i.e. cancer cells and ECM molecules, in a collective fashion, and we solve this new microscopic uPA system by finite element method. While this is sourced from within the macrodynamics via a top-down link, in our multiscale method (described in “The multiscale moving boundary approach for the proposed cancer invasion model” section of “Appendix 1”) the microdynamics occurring at the cell-scale neighbourhood of the tumour is represented back at the macroscale through a bottom-up feedback by defining the movement direction and displacement magnitude of the tissue-scale tumour boundary. By coupling this new microscopic governing law for the leading edge microdynamics with the macroscopic model for the uPA system and cancer invasion proposed in Andasari et al. (2011) and Chaplain and Lolas (2005), we are able to capture an important class of multiscale dynamic interactions between cancer cells, ECM molecules, cancer associated matrix degrading enzymes and the peritumoural tissue conditions, leading to significant changes in tumour morphology during invasion.

From the computational simulation results of our model, we can see that the extended two-scale technique coupled with the uPA system and more specific modelling of the pericellular proteolytic activities gives more adverse dynamics in the invading cancer. Values of the ECM initial condition, the cancer cell diffusion coefficient, the threshold coefficient, and the ECM proliferation and degradation rates all have an impact on the deformations of the tumour boundary. While for simulating the proposed non-dimensional model we considered functional formulations for the initial conditions and tissue thresholds, future work will attempt to assimilate the heterogeneity of ECM and as well as the peritumoural tissue conditions from imaging data. Finally, concerning the transitional probability that arises naturally at the microscale and intervenes in the bottom-up link between micro- and macroscales, while the simulations presented in this paper are deterministic, future work will assess the stochastic character of the overall model.

The conclusions that can be drawn from the qualitative results presented in this paper are: (1) a heterogeneous ECM initial condition leads to more fingered spreading of the tumour compared with that in the homogeneous ECM; (2) in order to obtain heterogeneous patterns of cancer cells inside the tumour region, chemotaxis must be dominant to drive the cells migration; (3) the changes of threshold coefficient will definitely affect the boundary deformations, and there is a tendency that the increase in \(\beta \) reduces the number of ‘fingers’ of the interface; and (4) without a proliferation term of ECM coupled with a relatively small degradation rate, deformations of the boundary show more fingering. However, further investigations are required to analyse the observed fingering. For instance, the dependence of the width of the fingers on the size of the microscale and potentially on other regulatory parameters remains an open question and is an important objective of a future work on the propose modelling framework.

It is useful to compare these results with other models of cancer growth and invasion, particularly those that adopt an alternative modelling approach such as hybrid continuum-discrete, cellular-Potts or cellular automaton, as well as partial differential equations (PDE) models. Many PDE models of solid tumour growth and development (including invasion) have modelled cell–cell adhesion at the outer boundary (or invading edge) of the tumour as a surface-tension-like force (Byrne and Chaplain 1996, 1997). A reduction in surface tension, i.e. interpreted as a loss of cell–cell adhesion, then leads to an instability at the invading edge which manifests itself in a subsequent growth consisting of finger-like protrusions. This effect was shown computationally by Cristini et al. (2003) and Macklin and Lowengrub (2007) who demonstrated a range of fingering patterns as a cell adhesion parameter was varied. Similar spectra of invasive patterns have been observed when adopting either a cellular-Potts approach (Popławski et al. 2009) or a hybrid continuum-discrete approach (Anderson 2005), i.e. by varying a key cell–cell adhesion parameter of the model, invasive fingering patterns can either be enhanced or suppressed. In the specific case of glioma invasion, four different approaches—PDE, cellular-Potts, lattice-gas automaton, cellular automaton—have each investigated the role of cell–cell adhesion and compared computational simulation results with experimental data (Aubert et al. 2006; Frieboes et al. 2007; Rubenstein and Kaufman 2008; Tektonidis et al. 2011. Given these results, we are currently extending our current model to include cell–cell adhesion.

As the main purpose of this work was to formulate a novel non-dimensional two-scale modelling platform that simultaneously explores spatio-temporal dynamics at both macroscale (cell population level) and microscale alongside the links in between the two scales, the simulations presented here have a qualitative character. Future work will explore the possibilities of dimensionalising and calibrating the proposed multiscale model with measured data at both macroscopic (tissue-scale) and microscopic (cell-scale) levels to obtain quantitative simulation that we could compare with clinical observations.

The two-scale modelling described provides a useful mathematical platform to capture and investigate processes at different levels during cancer invasion. However, further details remain to be explained on the dynamics and interactions of the tumour cell community at the macrolevel and the microlevel as well as regarding the links in between these two scales. Various avenues could be pursued to extend our modelling by exploring for instance other ways to determine the cancer cell population macrodynamics (for example change the definition of the threshold function \(\omega \) characterising the interaction with the peritumoural tissue) or by accounting for more complex signalling mechanisms in the establishment of more detailed microdynamics as well as the appropriate top-down and bottom-up links between these different levels (scales) of the invasion process.