A Multiscale Mathematical Model of Tumour Invasive Growth

Abstract

Known as one of the hallmarks of cancer (Hanahan and Weinberg in Cell 100:57–70, 2000) 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 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 Trucu et al. (Multiscale Model Simul 11(1):309–335, 2013), the modelling that we propose here allows us to study the changes in tissue-scale tumour morphology caused by the cell-scale uPA microdynamics 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 Ito et al. (J Gastroenterol 47:1279–1289, 2012).

Appendices

Appendix 1: The Two-Scale Computational Modelling Method

In the following, we will briefly present the technique introduced in Trucu et al. (2013) and adjust this with all the details to our new situation. For completion, we introduce here all the necessary notations and describe the defining principles that are referred to in the paper, as well as the relevant considerations and explanations concerning our new model.

1.1 Preliminary Considerations and Notations

It is assumed that the domain within which the cancer and extracellular matrix exist is a maximal reference spatial cube $Y \subset \mathbb {R}^n (n=2,3)$ with its centre at the origin. Given a fixed $\epsilon$ representing a negative power of 2 (i.e. $0<\epsilon <1$), the initial Y is uniformly decomposed $\epsilon$-size cubes, $\epsilon Y$, whose union will be referred to as an $\epsilon$-resolution of Y. For any $\epsilon Y$ from the decomposition, the “half-way shifted” cubes in the direction $i {\bar{e}}_1 + j {\bar{e}}_2 + k {\bar{e}}_3$ given by any triplet $(i,j,k) \in \{(i,j,k)|i,j,k \in \{-1,0,1\}\}$ are defined as

$$\begin{aligned} \epsilon Y_{\frac{i}{2}, \frac{j}{2}, \frac{k}{2}} = \epsilon Y + \frac{\epsilon (i {\bar{e}}_1 + j {\bar{e}}_2 + k {\bar{e}}_3) }{2} , \end{aligned}$$

(24)

where,

$$\begin{aligned} {\bar{e}}_1:&=e_1,&{\bar{e}}_2:&=e_2,&\text {and}&,&{\bar{e}}_3:&=\left\{ \begin{array}{ll} e_3 &{} \text {for} \;\;\;\;N=3, \\ 0 &{} \text {for} \;\;\;\;N=2 ,\end{array} \right. \end{aligned}$$

(25)

and $\{e_1,e_2,e_3\}$ is the standard Euclidean basis of $\mathbb {R}^3$. The family of all these $\epsilon -$cubes is denoted by $\mathcal {F}$, i.e.

$$\begin{aligned} \mathcal {F}:= \underset{i,j,k \in \{-1,0,1\}}{ \bigcup } \big \{ \epsilon Y_{\frac{i}{2}, \frac{j}{2}, \frac{k}{2}} \big | \epsilon Y \; \text {is in the} \; \epsilon \text {-resolution of} \; Y \big \}. \end{aligned}$$

(26)

In Fig. 11, the notations mentioned so far are illustrated schematically.

Fig. 11

Schematic diagram showing the cubic region Y centred at the origin $\in \mathbb {R}^3$. The dashed blue lines represent the Euclidean directions $\{e_1,e_2,e_3\}$, the pink region illustrates the cancer cluster ${\varOmega }(t_0 )$, and the solid blue line represents the family of microscopic cubic domains $\epsilon Y$ placed at the boundary $\partial {\varOmega }(t_0 )$ (Color figure online)

In order to capture mathematically the microdynamics that occur in a cell-scale neighbourhood of the tumour boundary $\partial {\varOmega }(t_0)$, out of the initial family $\mathcal {F}$, we will focus our attention of the subfamily denoted by $\mathcal {F}_{{\varOmega }(t_0)}$ which consists of only the $\epsilon -$cubes that cross the interface $\partial {\varOmega }(t_0)$ and have exactly one face included in the interior of ${\varOmega }(t_0)$, namely

$$\begin{aligned} \mathcal {F}_{{\varOmega }(t_0)} : = \{&\epsilon Y \in \mathcal {F} | \epsilon Y \cap (Y\backslash {\varOmega }(t_0)) \ne \emptyset , \nonumber \\&\text {and} \; \epsilon Y \; \text {has only one face included in int}({\varOmega }(t_0)) \}, \end{aligned}$$

(27)

where int(${\varOmega }(t_0)$) is the topological interior of ${\varOmega }(t_0)$ with respect to the natural topology on $\mathbb {R}^n$.

In this context, for each $\epsilon Y \in \mathcal {F}_{{\varOmega }(t_0)}$, we have the following face notations:

$$\begin{aligned} \left\{ \begin{array}{l} {\varGamma }_{\epsilon Y}^{\text {int}} \; \text {denotes the face of}\;\epsilon Y \;\text {that is included in int}({\varOmega }(t_0)),\\ \\ {\varGamma }_{\epsilon Y}^{j,\bot }, j=1,...,2^{N-1} \; \text {denote the faces of} \;\epsilon Y \;\text {that are perpendicular to}\; {\varGamma }_{\epsilon Y}^{\text {int}} \\ \\ {\varGamma }_{\epsilon Y}^{\Vert } \; \text {denotes the face of} \;\epsilon Y \;\text {that is parallel to}\; {\varGamma }_{\epsilon Y}^{\text {int}}. \end{array} \right. \end{aligned}$$

(28)

These are illustrated schematically in Fig. 12.

Fig. 12

Schematic diagram illustrating the notations introduced in (28), (32). For the arbitrary microdomain $\epsilon Y \in \mathcal {P}_{\epsilon }$, we indicate with a black arrow the features: ${\varGamma }_{\epsilon Y}^{int}$ , ${\varGamma }_{\epsilon Y}^{j_1,\bot }$, and ${\varGamma }_{\epsilon Y}^{j_2,\bot },\; j_1,j_2\in \{1,...,2^{N-1} \}$, ${\varGamma }_{\epsilon Y}^{\Vert }$ , $x_{\epsilon Y}^{c}$, $\mu _{\epsilon Y}$ , and $x_{\epsilon Y}^*$ . The arbitrary cube $\epsilon Y \in \mathcal {P}_{\epsilon }^*$ is shown in green, while the corresponding half-way shifted $\epsilon Y_{\frac{i}{2}}^{sign} \in \mathcal {P}_{\epsilon }$that are not chosen in $\mathcal {P}_{\epsilon }^*$ are shown in the blue dashed line (Color figure online)

Furthermore, for each $\epsilon Y \in \mathcal {F}_{{\varOmega }(t_0)}$, the topological closure of the only connected component of ${\varOmega }(t_0) \cap \epsilon Y$ that is confined between $[\partial {\varOmega }(t_0)]_{\epsilon Y}$ and ${\varGamma }_{\epsilon Y}^{\text {int}}$ is denoted by $[{\varOmega }(t_0)]_{\epsilon Y}$. Moreover, denoting by $[\partial {\varOmega }(t_0)]_{\epsilon Y}$ the connected component part of $\partial {\varOmega }(t_0) \cap \epsilon Y$ with the property that

$$\begin{aligned}{}[\partial {\varOmega }(t_0)]_{\epsilon Y} \cap {\varGamma }_{\epsilon Y}^{j,\bot } \ne \emptyset \qquad \text {for any} j=1,2,...,2^{n-1}, \end{aligned}$$

(29)

we can observe that $[\partial {\varOmega }(t_0)]_{\epsilon Y}$ represents the part of $\partial {\varOmega }(t_0) \cap \epsilon Y$ that corresponds to $[{\varOmega }(t_0)]_{\epsilon Y}$, and is actually the only connected component of this intersection that has property (29). Finally, using this observation, for the currently fixed $\epsilon$, the subfamily denoted by $\mathcal {P}_{\epsilon }$ consisting of all those $\epsilon -$cubes that have $[{\varOmega }(t_0)]_{\epsilon Y}$ not touching ${\varGamma }_{\epsilon Y}^{\parallel }$ is selected as follows:

$$\begin{aligned} \mathcal {P}_{\epsilon } := \{ \epsilon Y \in \mathcal {F}_{{\varOmega }(t_0)} |\; [{\varOmega }(t_0)]_{\epsilon Y} \subset \epsilon Y \;\text {and}\; [\partial {\varOmega }(t_0)]_{\epsilon Y} \cap {\varGamma }_{\epsilon Y}^{\parallel } = \emptyset \}. \end{aligned}$$

(30)

Leaving now $\epsilon$ to take all the negative powers of 2, the union

$$\begin{aligned} \underset{\epsilon \in \{2^{-k}\,|\,k\in \mathbb {N}\}}{\bigcup } \mathcal {P}_{\epsilon } \end{aligned}$$

provides an infinite covering of $\partial {\varOmega }(t_0)$. Since $\partial {\varOmega }(t_0)$ is compact, using standard compactness arguments, a finite complete sub-covering of $\partial {\varOmega }(t_0)$ that consist only of small cubes an equal size $\epsilon ^*$ is denoted by $\mathcal {P}_{\epsilon }^*$, i.e.

$$\begin{aligned} \partial {\varOmega }(t_0) \subset \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y. \end{aligned}$$

(31)

Together with this finite complete covering $\mathcal {P}_{\epsilon }^*$ of the tumour interface $\partial {\varOmega }(t_0)$, at each time of the tumour evolution we obtain also the size of the microscale $\epsilon ^*$ (Trucu et al. 2013). For simplicity, in this paper, the size of the cell-scale $\epsilon ^*$ will still be denoted by $\epsilon$. Finally, for each $\epsilon Y \in \mathcal {P}_{\epsilon }^*$, we distinguish the following topological details:

$$\begin{aligned} \left\{ \begin{array}{l} x_{\epsilon Y}^{c} \; \text {denotes the centre of the face}\;{\varGamma }_{\epsilon Y}^{\text {int}} ,\\ \\ \mu _{\epsilon Y} \; \text {is the line that passes through} \; x_{\epsilon Y}^{c} \;\text {and is perpendicular on}\; {\varGamma }_{\epsilon Y}^{\text {int}} \\ \\ x_{\epsilon Y}^{*} \in [\partial {\varOmega }(t_0)]_{\epsilon Y} \; \text {which will be referred to as the ``midpoint'' of} \; [\partial {\varOmega }(t_0)]_{\epsilon Y} \\ \text {represents the point from the intersection}\; \mu _{\epsilon Y} \cap [\partial {\varOmega }(t_0)]_{\epsilon Y} \; \text {that is located} \\ \text {at the smallest distance with respect to} \; x_{\epsilon Y}^{c}. \end{array} \right. \end{aligned}$$

(32)

The well-posedness of these topological features is discussed in Trucu et al. (2013), and these are illustrated in Fig. 12.

1.2 The Multiscale Moving Boundary Approach for the Proposed Cancer Invasion Model

In the following, we will explain how the set of midpoints $\{x_{\epsilon Y}^* \}_{\epsilon Y \in \mathcal {P}_{\epsilon }^*}$ defined on the boundary of tumour at the current time moves to a set of new spatial positions $\{\widetilde{x_{\epsilon Y}^* }\}_{\epsilon Y \in \mathcal {P}_{\epsilon }^*}$ to form the new boundary at the very next time, by describing the movement of one such midpoint $x_{\epsilon Y}^* \in [\partial {\varOmega }(t_0)]_{\epsilon Y}$ for any $\epsilon Y \in \mathcal {P}_{\epsilon }^*$.

Based on biological observations that, on any microdomain $\epsilon Y$, provided that a sufficient amount of plasmin has been produced across the invading edge and it is the pattern of the front of the advancing spatial distribution of plasmin that characterised ECM degradation, therefore it is assumed that each boundary midpoint $x_{\epsilon Y}^* \in [\partial {\varOmega }(t_0)]_{\epsilon Y}$ will be potentially relocated in a movement direction and by a certain displacement magnitude dictated by the spatial distribution of plasmin obtained via the microprocess on $\epsilon Y$ at the final microtime $\tau _f:=\Delta t$, namely $m(\cdot , \tau _f)$. In the following, we explain how the movement direction and displacement magnitude are defined for each $x_{\epsilon Y}^* \in [\partial {\varOmega }(t_0)]_{\epsilon Y}$.

For any given threshold $\delta > 0$ and any fixed $\epsilon Y \in \mathcal {P}_{\epsilon }^*$, the regularity property of Lebesgue measure (Halmos 1974) is used to select the first dyadic decomposition $\{ D_j\}_{j \in \mathcal {J}_{\delta }}$ of $\epsilon Y$ such that

$$\begin{aligned} \lambda \bigg ( [\epsilon Y \backslash {\varOmega }(t_0) ] \;\backslash \; \underset{ \{j \in \mathcal {J}_{\delta }\,| \,D_{j} \subset \epsilon Y \backslash {\varOmega }(t_0) \} }{\bigcup } \mathcal {D}_j\bigg ) \le \delta . \end{aligned}$$

(33)

which simply means that $\epsilon Y \backslash {\varOmega }(t_0)$ is approximated with accuracy $\delta$ by the union of all the dyadic cubes that this includes. Once this dyadic decomposition is selected, we denote by $y_{j}$ the barycenters of $D_{j}$, for all $j\in \mathcal {J}_{\delta }$. As discussed in Trucu et al. (2013) for all $\epsilon Y \in \mathcal {P}_{\epsilon }^*$, this provides a resolution at which we read the further away part of the level set $\frac{1}{\lambda (\epsilon Y \backslash {\varOmega }(t_0))} \int _{\epsilon Y \backslash {\varOmega }(t_0)} m(y, \cdot ) \text {d}y$ in the distribution of the advancing degrading enzymes $m(\cdot ,\cdot )$ outside ${\varOmega }(t_{0})$ in radial direction with respect to the midpoint $x_{\epsilon Y}^*$. Therefore, this enables us to locate dyadic pixels $D_{l}$ that support the peaks at the tip of the plasmin front with significant contribution in degrading the ECM. Hence, at the final microscopic time $\tau _{f}$, the pixels supporting these peaks are therefore selected as

$$\begin{aligned} \mathcal {\mathcal {I}_{\delta }}:= \left\{ l\in \mathcal {J}_{\delta } \left| \begin{array}{l} \exists r\in S^{1} \text { such that, if the index }i\in \mathcal {J}_{\delta } \text { has the properties:}\\ 1)\mathcal {D}_i\cap \{x\in \mathbb {R}^{n}\,| x=x_{\epsilon Y}^*+\alpha r, \alpha \in \mathbb {R}\} \ne \emptyset , \\ 2)\mathcal {D}_i\subset \epsilon Y \backslash {\varOmega }(t_0),\\ 3) \frac{1}{\lambda (\mathcal {D}_i)} \int _{\mathcal {D}_i} m(y,\tau _f) \text {d}y\ge \frac{1}{\lambda (\epsilon Y \backslash {\varOmega }(t_0))} \int _{\epsilon Y \backslash {\varOmega }(t_0)} m(y, \tau _f) \text {d}y , \\ \text {then} \\ l=\text { argmax }\{d(x_{\epsilon Y}^*, y_{i})\,|\,i\in \mathcal {J}_{\delta } \text { satisfies: } 1), 2), \text { and }\,3)\} \end{array} \right. \right\} , \end{aligned}$$

(34)

where $S^{1}\subset \mathbb {R}^{n}$ represents the unit sphere and $d(\cdot ,\cdot )$ is the Euclidean distance on $\mathbb {R}^{n}$. Thus, cumulating the driving ECM degradation forces spanned by each front peak of plasmin given by the dyadic pixels $\mathcal {D}_{l}$ with $l \in {\mathcal {I}_{\delta }}$ in the direction of the position vectors $\overrightarrow{x_{\epsilon Y}^*, y_l}$ and appropriately representing the amount of plasmin that each $D_l$ supports, the revolving direction of movement $\eta _{\epsilon Y}$ for the potential displacement of $x_{\epsilon Y}^*$ is given by:

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

(35)

Further, the displacement magnitude of the point $x_{\epsilon Y}^*$ is defined as:

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

(36)

Finally, as debated in Trucu et al. (2013), although a displacement magnitude and a moving direction are derived for each $x_{\epsilon Y}^*$, this will only exercise the movement if and only if the ECM degradation were of a certain local strength. The strength of ECM degradation within $\epsilon Y$ is explored by the transitional probability

$$\begin{aligned} q^* : \sum \left( \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y\right) \rightarrow \mathbb {R}_{+} \end{aligned}$$

defined as

$$\begin{aligned} q^*(G):= \frac{1}{\int _{G} m(y,\tau _f)\text {d}y} \int _{G\backslash {\varOmega }(t_0)} m(y,\tau _f) \text {d}y, \qquad \text { for all } G\in \sum \left( \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y\right) \end{aligned}$$

(37)

where $\sum \left( \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y\right)$ represents the Borel $\sigma -$algebra of $\underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y$. Locally, in each $\epsilon Y$, equation (37) is in fact a quantification of the amount of plasmin in $\epsilon Y \backslash {\varOmega }(t_0)$ relative to the total amount of plasmin concentration in $\epsilon Y$. In conjunction with the local tissue conditions, this characterises whether the point $x_{\epsilon Y}^*$ is likely to relocate to the new spatial position ${\widetilde{x_{\epsilon Y}^*}}$ or not.

Now, by assuming that the point $x^*_{\epsilon Y}$ is moved to the position ${\widetilde{x^*_{\epsilon Y}}}$ if and only if $q^{*}(x^*_{\epsilon Y}):=q^{*}(\epsilon Y)$ exceeds a certain threshold $\omega _{ \epsilon Y} \in (0,1)$, we find that the new boundary $\partial {\varOmega }(t_0+\Delta t)$ will be the interpolation of the following set of points:

$$\begin{aligned} \{ x^*_{\epsilon Y} | \epsilon Y \in \mathcal {P}_{\epsilon Y}^* \;\text {and}\; q(x^*_{\epsilon Y}) < \omega _{\epsilon Y} \} \;\cup \; \{\widetilde{x^*_{\epsilon Y}} | \epsilon Y \in \mathcal {P}_{\epsilon Y}^* \;\text {and}\; q(x^*_{\epsilon Y}) \ge \omega _{\epsilon Y} \} \end{aligned}$$

(38)

Finally before moving to the next time step of the whole macro-micro two-scale system, we replace the initial conditions of the macroscopic dynamics with the solution at the final time of the previous invasion step as follows:

$$\begin{aligned} c_{{\varOmega }(t_0+\Delta t)}(x,t_0)&:= c(x,t_0+\Delta t)( \chi _{_{{\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }), \nonumber \\ v_{{\varOmega }(t_0+\Delta t)}(x,t_0)&:= v(x,t_0+\Delta t)(\chi _{_{Y \backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y}} *\psi _{\gamma }), \nonumber \\ u_{{\varOmega }(t_0+\Delta t)}(x,t_0)&:= u(x,t_0+\Delta t)( \chi _{_{{\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }), \nonumber \\ p_{{\varOmega }(t_0+\Delta t)}(x,t_0)&:= p(x,t_0+\Delta t)( \chi _{_{{\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }), \nonumber \\ m_{{\varOmega }(t_0+\Delta t)}(x,t_0)&:= m(x,t_0+\Delta t)( \chi _{_{{\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }). \end{aligned}$$

(39)

Here $\chi _{_{{\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }$ and $\chi _{_{Y\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y} }*\psi _{\gamma }$ are the characteristic functions corresponding to the sets ${\varOmega }(t_0)\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y$ and $Y\backslash \underset{\epsilon Y \in \mathcal {P}_{\epsilon }^*}{\bigcup } \epsilon Y$, and choosing $\gamma \ll \frac{\epsilon }{3}$, $\psi _{\gamma }: \mathbb {R}^n \rightarrow \mathbb {R}_+$ is constructed as a smooth compact support function with $\text {sup}(\psi _{\gamma }) = \{z \in \mathbb {R}^n | || z ||_2 \le \gamma \}$. This is defined by the standard mollifier $\psi : \mathbb {R}^n \rightarrow \mathbb {R}_+$, namely,

$$\begin{aligned} \psi _{\gamma }(x) := \frac{1}{\gamma ^n} \psi (\frac{x}{\gamma }), \end{aligned}$$

(40)

and,

$$\begin{aligned} \psi (x):= \left\{ \begin{array}{ll} \frac{ \text {exp}(\frac{1}{||{x}||_2^2-1}) }{ \underset{\{z \in \mathbb {R}^n | || z ||_2 \le \gamma \}}{\int } \text {exp}(\frac{1}{||z||_2^2-1}) dz } &{}\;\; \text {if}\;\; ||x||_2<1, \\ 0&{}\;\; \text {if}\;\; ||x||_2 \ge 1, \end{array} \right. \end{aligned}$$

(41)

Then, the invasion process will continue on the new expanded domain ${\varOmega }(t_0)$ with the macroscopic system and the new initial conditions in (39) at macrolevel followed by proteolytic microprocesses around its boundary, which again governs the movement of the boundary of the next time multiscale stage.

Appendix 2: Description of the Multiscale Numerical Approach

We compute and solve the multiscale model in a two-dimensional setting by using computational approach based on a finite difference scheme for macrodynamics and finite element approximation for the microdynamics occurring on each of the boundary $\epsilon Y$ microdomains. In the following subsections, we detail the computational approach and present the steps of the overall multiscale algorithm.

1.1 The Macroscopic Stage of the Numerical Scheme

Since the macroscopic dynamics are taking place in the cube Y, we discretise the entire Y by considering a uniform spatial mesh of size $h:=\frac{\epsilon }{2}$, i.e. $\Delta x =\Delta y = h$. And, the time interval $[t_0,t_0+\Delta t]$ is discretised in k uniformly distributed time steps, i.e. using the uniform time step $\delta \tau :=\frac{\Delta t}{k}$. The temporal discretisation of the reaction-diffusion system (6)–(10) that we used here is a second-order trapezoidal scheme, while the diffusion term and haptotactic terms are approximated with a second-order midpoint rule. For instance, for the diffusion and haptotactic terms involved in (6), we approximate $\nabla \cdot (\nabla c)_{i,j}^{n}$ and $\nabla \cdot (c \nabla v)_{i,j}^{n}$ as follows:

$$\begin{aligned} \nabla \cdot (\nabla c)_{i,j}^{n}&= \text {div}(\nabla c)_{i,j}^{n} \\&\simeq \frac{(c_x)_{i+\frac{1}{2},j}^{n} - (c_x)_{i-\frac{1}{2},j}^{n}}{\Delta x} +\frac{ (c_y)_{i,j+\frac{1}{2}}^{n} - (c_y)_{i,j-\frac{1}{2}}^{n} }{\Delta y}, \end{aligned}$$

and

$$\begin{aligned} \nabla \cdot (c \nabla v)_{i,j}^{n}&=\text {div} (c \nabla v)_{i,j}^{n} \\&\simeq \frac{c_{i+\frac{1}{2},j}^{n} (v_x)_{i+\frac{1}{2},j}^{n} - c_{i-\frac{1}{2},j}^{n} (v_x)_{i-\frac{1}{2},j}^{n} }{\Delta x} +\frac{ c_{i,j+\frac{1}{2}}^{n} (v_y)_{i,j+\frac{1}{2}}^{n} - c_{i,j-\frac{1}{2}}^{n} (v_y)_{i,j-\frac{1}{2}}^{n} }{\Delta y}, \end{aligned}$$

where

$$\begin{aligned} \left\{ \begin{array}{l l} c_{i,j+\frac{1}{2}}^{n} &{}:= \frac{c_{i,j}^{n} + c_{i,j+1}^{n}}{2}, \\ c_{i,j-\frac{1}{2}}^{n} &{}:= \frac{c_{i,j}^{n} + c_{i,j-1}^{n}}{2}, \\ c_{i+\frac{1}{2},j}^{n} &{}:= \frac{c_{i,j}^{n} + c_{i+1,j}^{n}}{2}, \\ c_{i-\frac{1}{2},j}^{n} &{}:= \frac{c_{i,j}^{n} + c_{i-1,j}^{n}}{2}, \end{array} \right. \end{aligned}$$

are the midpoint approximations for c and

$$\begin{aligned} \left\{ \begin{array}{l l} (c_y)_{i,j+\frac{1}{2}}^{n} &{}:= \frac{c_{i,j+1}^{n} - c_{i,j}^{n}}{\Delta y}, \\ (c_y)_{i,j-\frac{1}{2}}^{n} &{}:= \frac{c_{i,j}^{n} - c_{i,j-1}^{n}}{\Delta y}, \\ (c_x)_{i+\frac{1}{2},j}^{n} &{}:= \frac{c_{i+1,j}^{n} - c_{i,j}^{n}}{\Delta x}, \\ (c_x)_{i-\frac{1}{2},j}^{n} &{}:= \frac{c_{i,j}^{n} - c_{i-1,j}^{n}}{\Delta x}, \end{array} \right. , \;\;\;\;&\text {and} \;\;\;\;&\left\{ \begin{array}{ll} (v_y)_{i,j+\frac{1}{2}}^{n} &{}:= \frac{v_{i,j+1}^{n} - v_{i,j}^{n}}{\Delta y}, \\ (v_y)_{i,j-\frac{1}{2}}^{n} &{}:= \frac{v_{i,j}^{n} - v_{i,j-1}^{n}}{\Delta y}, \\ (v_x)_{i+\frac{1}{2},j}^{n} &{}:= \frac{v_{i+1,j}^{n} - v_{i,j}^{n}}{\Delta x}, \qquad \\ (v_x)_{i-\frac{1}{2},j}^{n} &{}:= \frac{v_{i,j}^{n} - v_{i-1,j}^{n}}{\Delta x}, \end{array} \right. \end{aligned}$$

represent the central differences for spatial derivatives of c and v. Note that $n = 0,1,...,k$ are index of time step and (i, j) are spatial nodes where $i = 1,...q$ are the indices for the x-direction and $j = 1,...q$ are the indices for the y-direction. The diffusion terms in Eqs. (8)–(10) are approximated in the same way as it is in Eqs. (6) and (7).

1.2 The Computational Microscopic Scheme and Its Relation to the Macroscopic Level

In this section, we describe our computational scheme for the micro scale dynamics occurring on each microdomains $\epsilon Y \in \mathcal {P}_{\epsilon }^*$, which are cubes of size $\epsilon$ located at the boundary $\partial {\varOmega }(t_0)$. We have each microdomain $\epsilon Y$ centred at a boundary point form the macroscopic mesh, with the neighbouring $\epsilon$-cubes staring from the centre of the current one (i.e. they are appropriately “half-way shifted” copies of $\epsilon Y \in \mathcal {P}_{\epsilon }^*$), due to the purposely chosen macroscopic mesh size $h=\frac{\epsilon }{2}$ and the properties of the family $\mathcal {P}_{\epsilon Y}^*$. Moreover, the centre point of the microdomains is coincidentally the midpoint induced by $\epsilon Y$ on $[\partial {\varOmega }(t_0)]_{\epsilon Y}$, i.e. $x_{\epsilon Y}^*$.

In order to compute the integrals in the source terms (i.e. $f_1^{\epsilon Y}$ and $f_2^{\epsilon Y}$) in the microscopic system (12)–(15), a midpoint rule is proposed and the constitutive details are given below. Assuming that K denotes a generic element domain in a finite element subdivision with either triangular or square elements of a given region $A \subset \mathbb {R}^2$, this “midpoint rule” consists of approximating the integral of a function f over K as the product between the value of f at the centre of mass of K, $K_{\text {centre}}$, and the Lebesgue measure of K, namely,

$$\begin{aligned} \int \limits _{K} f = f(K_{\text {centre}}) \lambda (K). \end{aligned}$$

(42)

For an arbitrarily chosen $\epsilon Y \in \mathcal {P}_{\epsilon }^*$, we consider a finite element approach involving triangular elements on a uniform micromesh, which is maintained with identical structure for all the microdomains. Further, we consider a time-constant approximation ${\tilde{f}}_1^{\epsilon Y}$ of $f_1^{\epsilon Y}$ on the time interval $[0, \Delta t]$. In this context, using the computed final-time values of $c(\cdot , t_0+\Delta t)$ at the macromesh points that are included on the current microdomain, $x_1,x_2,...,x_{P_{\epsilon Y}} \in \epsilon Y \cap {\varOmega }(t_0)$, we take:

$$\begin{aligned} {\tilde{f}}_1^{\epsilon Y} (x_s) = \frac{1}{\lambda (B(x_s,2\epsilon )\cap {\varOmega }(t_0))} \int \limits _{B(x_s,2\epsilon )\cap {\varOmega }(t_0)} c\;(x_s,t_0+\Delta t) \;\text {d}x, \end{aligned}$$

(43)

where $s = 1,..., P_{\epsilon Y}$, and the integrals are computed via the midpoint rule. For the rest of the points y on the micromesh, the value of ${\tilde{f}}_1^{\epsilon Y}$ is obtained in terms of the set of finite element basis functions considered at the contact points, i.e. $\{\phi _{x_s} | s = 1,..., P_{\epsilon Y} \}$. Finally, we observe that for any micromesh point $y \in \epsilon Y$ we have two possibilities:

Case 1 If there exists three overlapping points $x_{i_{1}},x_{i_{2}},x_{i_{3}} \in \{x_1,x_2,...,x_{P_{\epsilon Y}}\}$ which belongs to the same connected component of $\epsilon Y \cap {\varOmega }(t_0)$ and y belongs to the convex closure of the set, i.e. $y \in \text {Conv} \{ x_{i_{1}},x_{i_{2}},x_{i_{3}} \}$, then we have:

$$\begin{aligned} {\tilde{f}}_1^{\epsilon Y}(y) = {\tilde{f}}_1^{\epsilon Y}(x_{i_1}) \phi _{x_{i_1}}(y) +{\tilde{f}}_1^{\epsilon Y}(x_{i_{2}}) \phi _{x_{i_2}}(y) +{\tilde{f}}_1^{\epsilon Y}(x_{i_3}) \phi _{x_{i_3}}(y). \end{aligned}$$

(44)

Case 2 If y does not satisfy the conditions in Case 1, then we have

$$\begin{aligned} {\tilde{f}}_1^{\epsilon Y}(y) = 0. \end{aligned}$$

(45)

For the source term $f_2^{\epsilon Y}$, we use the same approximation method as above, except that there is only one case taken into consideration which is similar in Eq. (44) according to the definition of $f_2^{\epsilon Y}$. Now we could obtain the source terms ${\tilde{f}}_1^{\epsilon Y}$ and ${\tilde{f}}_2^{\epsilon Y}$ on each microdomain $\epsilon Y$ with zero initial condition and Neumann boundary conditions and furthermore use the finite element method to solve the reaction-diffusion equations (12)–(15) on $\epsilon Y$ over the time interval $[0,t_0 + \Delta t]$. Then, we use bilinear elements on a square mesh, the numerical scheme for the microprocesses occurring on each $\epsilon Y$ is finally obtained by involving a trapezoidal predictor-corrector method for the time integration.

Then, for each microdomain we use the midpoint rule to compute the transitional probability described in (37). For simplicity, now the numerical implementation of the multiscale model for cancer invasion proposed above is slightly simplified in the following way: provided that the transitional probability exceeds an associated threshold $\omega _{\epsilon Y} \in (0, 1)$, the boundary mesh-point $x_{\epsilon Y}^*$ will move in direction $\eta _{\epsilon Y}$ to the macromesh point from $\partial \epsilon Y \backslash [{\varOmega }(t_0)]_{\epsilon Y}$ that is closest (in Euclidean distance) to $x_{\epsilon Y}^*$ . If the threshold is not satisfied, then $x_{\epsilon Y}^*$ remains at the same spatial location. Therefore, the new boundary $\partial {\varOmega }(t_0+\Delta t)$ is now obtained by the interpolation of the set of points given in (38) , and the computational process is continued on the new domain ${\varOmega }(t_0+\Delta t)$ by using as a discretised version of (39) as a new initial condition at the macroscopic stage, i.e.

$$\begin{aligned} c(x_{i,j}, t_0+\Delta t)&= \left\{ \begin{array} {ll} c_{i,j}^k, &{} x_{i,j} \in \overline{{\varOmega }(t_0)},\\ \frac{1}{4}(c_{i-1,j}^k+c_{i+1,j}^k+c_{i,j-1}^k+c_{i,j+1}^k), &{} x_{i,j} \in \overline{\mathbf {B}(\overline{{\varOmega }(t_0)},h)} \backslash \overline{{\varOmega }(t_0)},\\ 0, &{} x_{i,j} \notin \overline{\mathbf {B}(\overline{{\varOmega }(t_0)},h)}, \end{array} \right. \end{aligned}$$

(46)

and,

$$\begin{aligned} v(x_{i,j}, t_0+\Delta t)&= v_{i,j}^k,&u(x_{i,j}, t_0+\Delta t)&= u_{i,j}^k, \nonumber \\ p(x_{i,j}, t_0+\Delta t)&= p_{i,j}^k,&m(x_{i,j}, t_0+\Delta t)&= m_{i,j}^k. \end{aligned}$$

(47)

where $\{x_{i,j} \ i,j = 1,...,q \}$ is the macroscopic mesh in Y, $\overline{{\varOmega }(t_0)}$ is the topological closure of ${\varOmega }(t_0)$, and $\overline{\mathbf {B}(\overline{{\varOmega }(t_0)},h)}$ represents the topological closure of the h-bundle of $\overline{{\varOmega }(t_0)}$., i.e. $\overline{\mathbf {B}(\overline{{\varOmega }(t_0)},h)} := \{ x \in Y | \exists z_x \in \overline{{\varOmega }(t_0)} \; \text {such that} \; || x-z_x ||_2 \le h \}$.

1.3 Overall Algorithm Steps

To sum up, the overall algorithm package of the macromicroscopic method consists of the following steps:

Step 1 At the very begining time $t_0$, first of all, we discretise the macrodomain $[a,b] \times [c,d]$ by

$$\begin{aligned} a&= x_0,\ldots ,x_i = a+ i\Delta x ,\ldots , x_m = a+m\Delta x =b, \\ c&= y_0,\ldots , y_j = c+ j\Delta y ,\ldots , y_n = c+n\Delta y =d. \\ \end{aligned}$$

where $\Delta x = \Delta y = h$, $h:=\frac{\epsilon }{2}$ and let $a=c=0, c=d=4$. Also, we number each point on the macrodomain and record their coordinates all sorts of data of the domain that might be used later.

Step 2 Define initial conditions for cancer and ECM distribution on macrodomain:

$$\begin{aligned} c(x,t_0)&=: c_0(x),&x\in {\varOmega }(t_0)\\ v(x,t_0)&=: v_0(x),&x\in {\varOmega }(t_0)\\ u(x,t_0)&=: u_0(x),&x\in {\varOmega }(t_0)\\ p(x,t_0)&=: p_0(x),&x\in {\varOmega }(t_0)\\ m(x,t_0)&=: m_0(x),&x\in {\varOmega }(t_0) \end{aligned}$$

where $c(x,t_0)$ is set as zero at the mesh points located outside the closure of the macroscopic domain ${\varOmega }(t_0)$.

Fig. 13

Plot shows the relocation of one point on the boundary moves to a new spatial position in the microdomain $\epsilon Y$

Step 3 Start the main time loop (from time stage 1 to certain time stage N), and at the current time stage,

• a) Run the macrosolver, which applies the finite difference scheme mentioned above, to obtain the distribution of components in the system $c_{i,j}^{n+1}$ $v_{i,j}^{n+1}$, $u_{i,j}^{n+1}$, $p_{i,j}^{n+1}$, and $m_{i,j}^{n+1}$, where $i,j = 1,...,q$.

• b) Run the microsolver, in which we loop over each points that was on the boundary of tumour at previous time, and at an arbitrary boundary points,

  • i. Define the microdomain $\epsilon Y$ centring at the current point on the boundary, which consists of nine points on macrodomain. For simplicity, we first construct the domain on $[0,\epsilon ] \times [0,\epsilon ]$, and on this domain, compute the source terms $f_1^{\epsilon Y}$ and $f_2^{\epsilon Y}$, and by interpolation, we uniformly decompose the domain into sixty-four square elements which consists of eighty-one points in total, with the source term values and concentration values for uPA, PAI-1 and plasmin on a finer mesh (see Fig. 13).

  • ii. On the microdomain $\epsilon Y$, apply the finite element method to solve the microscopic dynamics system (12)–(15), to obtain the spatial distribution of plasmin at the final microtime $m(\cdot , \tau _f)$ (involving a proposed midpoint rule formula for the integral source terms $f_1^{\epsilon Y}$ and $f_2^{\epsilon Y}$, and for time integration a trapezoidal predictor-corrector), which will be used in the regulation functions of cancer cells’ movement.

  • iii. Translated the coordinates on this microdomain back to where the microspatial position was before.

  • iv. Using the transitional probability $q^{*}$ defined in (37), compute the invasion strength as $q^{*}(x_{\epsilon Y}^*):=q^{*}(\epsilon Y)$.

  • v. If and only if the microenvironment induced probability $q^*(x_{\epsilon Y}^*)$ is greater than some tissue threshold value $\omega _{\epsilon Y} \in (0,1)$, we further compute the direction $\eta _{\epsilon Y}$ and magnitude $\xi _{\epsilon Y}$ of the movement as described by (35) and (36).

• c) Once finishing both macrosolver and microsolver at the current time stage, we obtained new macroscopic distribution for each components in the system $c_{i,j}^{n+1}$, $v_{i,j}^{n+1}$, $u_{i,j}^{n+1}$, $p_{i,j}^{n+1}$ and $m_{i,j}^{n+1}$; also for each midpoint $x_{\epsilon Y}^*$ on the tumour boundary, we have the possibility $q^*(x_{\epsilon Y}^*)$, direction $\eta _{\epsilon Y}$ and magnitude $\xi _{\epsilon Y}$ of their movement; therefore, we could use all these information to determine the new position $\widetilde{x^*_{\epsilon Y}}$ and the points remain where they were on the cancer interface $\partial {\varOmega }(t_0+\Delta t)$. This is schematically shown in Fig. 13

  where the red dots represent the discrete macromesh location for the where microscale source induced by the macroscale was calculated via the integral formula (43).

• d) Finally, by using approximations shown in (46) and (47), replace the initial values of cancer and ECM distribution in macroscopic dynamics with the solution at the final time of the previous macrostep.

Step 4 Using the new initial conditions for macroscopic dynamics, continue the invasion process by coupling the next-step macroprocess given by the systems (6) and (10) on the expand domain ${\varOmega }(t_0+\Delta t)$ with the corresponding microprocesses (12)–(15) occurring on its boundary, which means repeating the Step 3 above with new initial conditions for macroscopic dynamics and new boundary of cancer.

Appendix 3: Table for the Parameter Set $\mathscr {P}$

In Table 1, we present a description of the parameters included in $\mathscr {P}$.

Table 1 The parameters in $\mathscr {P}$