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Mesoscopic and continuum modelling of angiogenesis

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Angiogenesis is the formation of new blood vessels from pre-existing ones in response to chemical signals secreted by, for example, a wound or a tumour. In this paper, we propose a mesoscopic lattice-based model of angiogenesis, in which processes that include proliferation and cell movement are considered as stochastic events. By studying the dependence of the model on the lattice spacing and the number of cells involved, we are able to derive the deterministic continuum limit of our equations and compare it to similar existing models of angiogenesis. We further identify conditions under which the use of continuum models is justified, and others for which stochastic or discrete effects dominate. We also compare different stochastic models for the movement of endothelial tip cells which have the same macroscopic, deterministic behaviour, but lead to markedly different behaviour in terms of production of new vessel cells.

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  1. We choose the coordinates such that \(x=kh\) corresponds to the midpoint of box \(k\). The precise location of the box midpoint is not essential for the stochastic model, but subtleties arise when combining the stochastic model with PDEs, see Sect. 2.2, and when choosing boundary conditions, see Appendix E.

  2. Note that there are subtleties involved with the space on which the operator acts. In this paper, we are interested in functions depending on the number of cells in the boxes, which is clearly non-negative, so care has to be taken when acting with \(E_{N_k}^{-1}\) on functions with \(N_k=0\).

  3. More precisely, in the model of a corneal assay discussed in Sect. 5 and modelled via PDEs in Byrne and Chaplain (1995), the AF remains close to its steady state configuration and is only slightly influenced by the growing vasculature. Thus, we can assume that the transition rates affecting the endothelial cells remain constant between events associated with the stochastic model. This assumption would cease to hold if the source of the AF were itself dynamic and would change on a similar or faster timescale than the timescale of events affecting tip and vessel cells. For instance, we could include tumour cells as a new species in our model, which would act as a source of AF. The tumour, and hence the resulting distribution of AF, could change faster than the blood vessel dynamics, so we cannot assume any longer that the AF is in a steady state between events affecting tip and vessel cells.

  4. We can interpret this model as a locally averaged version of Case 1: in Case 1, tip cells can also move into a neighbouring box with a higher number of tip cells, but on average, more tip cells would move from the box with the higher tip cell number to the one with lower number. In Case 2, only the net movement is taken into account.

  5. We do not make explicit the dependence on the deterministic AF \(c\), as, similar to the tip cell movement terms, all stochastic events discussed here will occur on a much slower timescale than the timescale governing \(c\).

  6. Fixing \(\delta _R\) means we use different cell scales \(\mu \) when comparing simulations based on the different cases of the transition rate. Again, we emphasise that, as the vessel cells decouple from the dynamics in our test case ignoring sprouting, anastomosis and regression, fixing either \(\mu \) or \(\delta _R\) only leads to a rescaling in the total amount of vessel production. \(\delta _R\) should be an integer, so it is easier to enforce this in the simulation by simply fixing it, rather than calculating it from the lattice constant \(h\) and the cell size \(\mu \).

  7. Not all parameters were listed in Byrne and Chaplain (1995), so we take the remaining one from Mantzaris et al. (2004). Hence, the results we obtain are as in Mantzaris et al. (2004).


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This publication was based on work supported in part by Award No KUK-C1-013-04, made by King Abdullah University of Science and Technology (KAUST). TA and PG gratefully acknowledge the Spanish Ministry for Science and Innovation (MICINN) for funding under grant MTM2011-29342 and Generalitat de Catalunya for funding under grant 2009SGR345. PKM was partially supported by the National Cancer Institute, National Institutes of Health grant U54CA143970.

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Correspondence to F. Spill.


Appendix A: Detailed derivation of mean field equations

The transition rates discussed in Sect. 2 fall into two categories: some describe tip cell movement and the associated production of vessel cells; others represent local reaction terms which change the cell content inside a single box. We will now analyse the general structure of the mean field equations for these two cases.

1.1 A.1 Movement of tip cells

The movement terms discussed in Sect. 2.1 were modelled by transition rates of the form \(\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k},N_{l},R_{k},R_{l}}\), \(l=k\pm 1\). We now restrict ourselves to the interior of the domain, \(k=2,\dots ,k_{max}-1\), and refer to Appendix E for the treatment of the boundary. The term in the master equation (3) describing movement was \(\sum _{k,l\in \langle {k}\rangle }(E_{N_k}^{+1}E_{N_l}^{-1}E_{R_k}^{-\delta _R}-1)\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k},N_{l},R_{k},R_{l}}P\). We see that the contribution to the mean field equations (18) from moving tip cells is given by

$$\begin{aligned} \mathcal {E}^{N_k}_M&= \sum _{\{N_{j}\},\{R_{j}\}}\sum _{m,l\in \langle {m}\rangle }N_k(E_{N_m}^{+1}E_{N_l}^{-1}E_{R_m}^{-\delta _R}-1) \mathcal {T}_{N_{m}-1,N_{l}+1,R_{m}+ \delta _R,R_{l}|N_{m},N_{l},R_{m},R_{l}}P\nonumber \\&= \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle }N_k \left( (E_{N_k}^{+1}E_{N_l}^{-1}E_{R_k}^{-\delta _R}-1) \mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k},N_{l},R_{k},R_{l}} \right. \nonumber \\&\quad +\left. (E_{N_l}^{+1}E_{N_k}^{-1}E_{R_l}^{-\delta _R}-1) \mathcal {T}_{N_{l}-1,N_{k}+1,R_{l}+\delta _R,R_{k}|N_{l},N_{k},R_{l},R_{k}} \right) P\nonumber \\&= \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle }N_k \left( \mathcal {T}_{N_{k},N_{l},R_{k},R_{l}|N_{k}\!+\!1,N_{l}\!-\!1,R_{k}\!-\! \delta _R,R_{l}}P(N_k\!+\!1,N_l\!-\!1,R_k\!-\!\delta _R,R_l)\right. \nonumber \\&\quad -\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k}, N_{l},R_{k},R_{l}}P(N_k,N_l,R_k,R_l)\nonumber \\&\quad +\mathcal {T}_{N_{l},N_{k},R_{l},R_{k}|N_{l}+1,N_{k}-1,R_{l}- \delta _R,R_{k}} P(N_k-1,N_l+1,R_k,R_l-\delta _R)\nonumber \\&\quad -\left. \mathcal {T}_{N_{l}-1,N_{k}+1,R_{l}+ \delta _R,R_{k}|N_{l},N_{k},R_{l},R_{k}} P(N_k,N_l,R_k,R_l)\right) \nonumber \\&= \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle } \left( (N_k-1)\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+ \delta _R,R_{l}|N_{k},N_{l},R_{k},R_{l}}P(N_k,N_l,R_k,R_l)\right. \nonumber \\&\quad -N_k\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+ \delta _R,R_{l}|N_{k},N_{l},R_{k},R_{l}}P(N_k,N_l,R_k,R_l)\nonumber \\&\quad +(N_k+1)\mathcal {T}_{N_{l}-1,N_{k}+1,R_{l}+ \delta _R,R_{k}|N_{l},N_{k},R_{l},R_{k}} P(N_k,N_l,R_k,R_l)\nonumber \\&\quad -N_k\left. \mathcal {T}_{N_{l}-1,N_{k}+1,R_{l}+ \delta _R,R_{k}|N_{l},N_{k},R_{l},R_{k}} P(N_k,N_l,R_k,R_l)\right) \nonumber \\&= \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle } (\mathcal {T}_{N_{l}-1,N_{k}+1,R_{l}+\delta _R,R_k |N_{l},N_{k},R_{l},R_{k}}\nonumber \\&\quad -\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l} |N_{k},N_{l}, R_{k},R_{l}})P. \end{aligned}$$

Here, \(\langle {m}\rangle =m\pm 1\) denotes nearest neighbouring boxes of box \(m\). In the first line, we simply substitute in the movement part of the master equation (3). Then we note that only those parts of the sum which are nearest neighbours of \(k\) contribute: we apply the shift operators to \(\mathcal {T}_{N_{m}-1,N_{l}+1,R_{m}+\delta _R,R_{l}|N_{m},N_{l},R_{m},R_{l}}P\) and then shift the summation index of the sum over the state space, and recover the original sum, so the combination \((E_{N_m}^{+1}E_{N_l}^{-1}E_{R_m}^{-\delta _R}-1)\) will yield zero. For the third equality we apply the shift operators, making only the shifted arguments in \(P\) explicit. \(P\) without arguments denotes, for brevity, a dependence on the unshifted cell numbers. Then, the summation indices are shifted in such a way that all probability densities \(P\) will depend on the unshifted arguments.

Likewise, we can derive a general form for the mean field equations for the vessels:

$$\begin{aligned} \mathcal {E}^{R_k}_M&=\! \sum _{\{N_{j}\},\{R_{j}\}}\sum _{m,l\in \langle {m}\rangle }R_k(E_{N_m}^{+1}E_{N_l}^{-1}E_{R_m}^{-\delta _R}-1) \mathcal {T}_{N_{m}-1,N_{l}+1,R_{m}+\delta _R,R_{l}|N_{m},N_{l},R_{m},R_{l}}P\nonumber \\&=\! \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle }R_k\left( (E_{N_k}^{+1}E_{N_l}^{-1} E_{R_k}^{-\delta _R}-1)\mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k}, N_{l},R_{k},R_{l}}\right) P\nonumber \\&=\! \sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle }\!R_k\!\left( \mathcal {T}_{N_{k}, N_{l},R_{k},R_{l}|N_{k}+1,N_{l}-1,R_{k}-\delta _R,R_{l}}P(N_k\!+\!1,N_l\!-\!1,R_k\!-\! \delta _R,R_l)\right. \nonumber \\&\ \quad -\left. \mathcal {T}_{N_{k}-1,N_{l}+1,R_{k}+\delta _R,R_{l}|N_{k},N_{l}, R_{k},R_{l}}P(N_k,N_l,R_k,R_l)\right) \nonumber \\&= \!\sum _{\{N_{j}\},\{R_{j}\}}\sum _{l\in \langle {k}\rangle }\delta _R\mathcal {T}_{N_{k} -1,N_{l}+1,R_{k}+\delta _R,R_{l} |N_{k},N_{l},R_{k},R_{l}}P. \end{aligned}$$

The derivation is similar to the derivation of (48), but the shift of summation index here is by \(\delta _R\) rather than by \(\pm \)1. We see that the qualitative difference between (49) and the mean field equation for the tips, (48), is that to the rate of change of mean of vessels in box \(k\) only those transition rates contribute which account for tips jumping out of box \(k\). In contrast, the mean number of tips in box \(k\) changes with the difference of transition rates describing incoming and outgoing tips from box \(k\). This difference is clearly understood by the fact that we model vessel cells to be static, so there is no loss of vessel cells in any box due to movement. There will only be a loss of vessel cells by other mechanisms such as regression.

1.2 A.2 Local source and sink terms

We now discuss the general structure of the master equation which takes into account sprouting, i.e. the production of a new tip cell. This involves a transition rate of the form

$$\begin{aligned} \mathcal {T}_{N_{k}+1|N_{k}}. \end{aligned}$$

The term in the master equation (3) describing only tip birth takes the form \(\sum _{k}(E_{N_k}^{-1}-1)\mathcal {T}_{N_{k}+1|N_{k}}P\). From this we are led to the contribution to the mean field equation (18)

$$\begin{aligned} \mathcal {E}^{N_k}_S&= \sum _{\{N_{j}\},\{R_{j}\}} N_k(E_{N_k}^{-1}-1)\mathcal {T}_{N_{k}+1|N_{k}}P\nonumber \\&=\sum _{\{N_{j}\},\{R_{j}\}} N_k\left( \mathcal {T}_{N_{k}|N_{k}-1}P(N_k-1) -\mathcal {T}_{N_{k}+1|N_{k}}P(N_k)\right) \\&=\sum _{\{N_{j}\},\{R_{j}\}} \left( (N_k+1)\mathcal {T}_{N_{k}|N_{k}-1}P(N_k) -N_k\mathcal {T}_{N_{k}+1|N_{k}}P(N_k)\right) \nonumber \\&=\sum _{\{N_{j}\},\{R_{j}\}} \mathcal {T}_{N_{k}+1|N_{k}}P\nonumber \\ \mathcal {E}^{R_k}_S&= 0.\nonumber \end{aligned}$$

The derivation again makes use of the application of the shift operators and subsequent shift of summation indices, as for the derivation of Eqs. (48),(49). Similar expressions can be derived for contributions of anastomosis and vessel regression to the mean field equations.

Appendix B: Angiogenesis model in higher dimensions

We will now comment on the generalisation of our model to higher spatial dimensions \(d\). Consider the transition rate describing tip movement from box \(k\) to box \(l\), with the structure

$$\begin{aligned} \mathcal {T}_{N_k-1,N_l+1,R_k+\delta _R,R_l|N_k,N_l,R_k,R_l}. \end{aligned}$$

In one dimension, we had \(l=k\pm 1\). In higher dimensions, we can think of \(k,l\) as multiindices, i.e.

$$\begin{aligned} k = \{k_1,k_2,\dots ,k_d\}. \end{aligned}$$

Here, \(k_j = 1,\dots ,k_{max,j}\).

In this way, the above transition rate requires no change at all. We need only define the nearest neighbours of \(k\). On a regular grid, we define the nearest neighbours \(l\) to be any of

$$\begin{aligned} l&=\{k_1\pm 1,k_2,\dots ,k_d\}\\ l&=\{k_1,k_2 \pm 1,\dots ,k_d\}\\ \ldots \\ l&=\{k_1,k_2,\dots ,k_d\pm 1\}. \end{aligned}$$

With this definition, we can calculate the mean field equations, take the continuum limit, and obtain the continuum equations for the different cases of movement given in Eqs. (23), (27) and (30), with the Laplace and Nabla operators now defined in \(d\) dimensions. The norms are \(L_1\) norms, due to the choice of our grid. The relation between the discrete and continuous variables is now given by

$$\begin{aligned} n(t,x_1,\dots ,x_d)&= \frac{N_k}{h^d}, \end{aligned}$$

with \(x_j=k_j h\).

Concerning the generalisation of the local interaction terms, which were used in the model to describe sprouting, anastomosis and regression, we remark that one needs to scale the parameters such that these rates scale like \(\mathcal {T} = h^d f(\frac{N_k}{h^d},\frac{R_k}{h^d},\frac{C_k}{h^d})\), where \(f\) is a function of the intensive variables only, which means these variables do not scale with system size.

Appendix C: Stochastic treatment of the angiogenic factor

It is straightforward in our model to treat the AF stochastically. For this purpose, we introduce a new variable \(C_k\), \(k=1,\ldots ,k_{max}\), which is related to the continuous, deterministic \(c\) used in this paper by

$$\begin{aligned} \left\langle C_k\right\rangle = h c(x). \end{aligned}$$

As before, \(x=kh\) in one spatial dimension. Then, a state in the stochastic model is specified by \(N_k, R_k\) and \(C_k\). We can now introduce additional transition rates in the stochastic model describing the movement of molecules:

$$\begin{aligned} {\mathcal {T}}_{C_k-1,C_l+1|C_k,C_l} = D_c^hC_k. \end{aligned}$$

As for the cell motility terms, we have \(D_c^h = \frac{D_c}{h^2}\). Furthermore, we introduce a transition rate describing the consumption and degradation of the AF:

$$\begin{aligned} {\mathcal {T}}_{C_k-1|C_k} = \lambda C_k + \frac{a_1}{h}H(C_k-\hat{C})N_kC_k. \end{aligned}$$

In the continuum limit, we obtain

$$\begin{aligned} \frac{\partial c}{\partial t}&= D_c\Delta c -\lambda c - a_1H(c-\hat{c})n c, \end{aligned}$$

reproducing (8) and (9). Other reaction terms can be implemented in a similar way. Let us now estimate the size of the stochastic effects associated with the AF. As stochastic effects are expected to be stronger when smaller numbers of molecules are involved, we are conservative in underestimating the number of molecules. In the corneal assay modelled in Sect. 5, the concentration of the AF (here, acidic fibroblast growth factor) was taken to be \(10^{-10}\)M (Byrne and Chaplain 1995), and the distance between tumour and initial vessel, i.e. the size of our modelling domain, was \(3\) mm. There are various estimates of the thickness of the cornea, and a conservative choice is given by 100 \(\upmu \)m (Henriksson et al. 2009). Then a box in a \(2D\) model with a discretisation of 50 sites in both dimensions would represent a volume of 100 \(\upmu \)m \((\frac{3~\hbox {mm}}{50})^2\), which is again a conservative choice for the \(1D\) model where we integrate over one dimension. Thus, the total number of molecules of the AF in one box is of the order

$$\begin{aligned} C_k\!\approx \! 10^{-10}M 100\,\upmu m \left( \frac{3~\hbox {mm}}{50}\right) ^2 \!\approx \! 6\times 10^{23}10^{-10}\frac{1}{10^{-3}\,\hbox {m}^3} \frac{9}{2{,}500}10^{2-6-6}\,\hbox {m}^3\!\approx \! 10^{4}. \end{aligned}$$

If we solve (56) and the stochastic model defined by (54) and (55) for the case considered in Sect. 5, i.e. Dirichlet boundary conditions, with a source on the left boundary, \(C_1=10{,}000\), and a sink with \(C_{50}=0\) on the left, and a lattice of \(50\) sites, we obtain the steady state profile presented in Fig. 13. With \(C_1=10{,}000\) and \(C_{50}=0\) we typically have \(C_k\le 10{,}000\), giving again a conservative estimate of the real stochastic fluctuations. We see in Fig. 13 that the stochastic fluctuations are relatively small throughout the domain, justifying the use of the PDE for the AF throughout the domain. We note also that it is justified to use the same spatial discretisation to solve the PDE as was used to simulate the stochastic model, as variations between neighbouring lattice sites are small.

Fig. 13
figure 13

The steady state profile of the angiogenic factor simulated from the PDE (56) and the stochastic model defined by (54) and (55), but no vessel cells present. Dirichlet boundary conditions where imposed such that \(C_1=10{,}000\) and \(C_{50}=0\)

Appendix D: Numerical solution of the model

1.1 D.1 Stochastic model

To simulate a realisation of the stochastic model, we use the Gillespie algorithm (Gillespie 1976, 1977), see also the review (Erban et al. 2007). This means that if we let \(\alpha _m\), \(m=1,\ldots ,m_{tot}\) denote all non-zero transition rates in the model, enumerated by integers \(m\), then \(\alpha _0=\sum _m\alpha _m\) denotes the total rate. The time to the next event is exponentially distributed, so we draw a uniformly distributed random number \(r_1\in [0,1]\) such that the time to the next event will be

$$\begin{aligned} \tau =\frac{1}{\alpha _0}\log {\frac{1}{r_1}}. \end{aligned}$$

We then draw a second uniformly distributed random number \(r_2\in [0,1]\) which determines which event will take place. In particular, if

$$\begin{aligned} \frac{\sum _m^{s-1}\alpha _m}{\alpha _0}\le r_2 <\frac{\sum _m^{s}\alpha _m}{\alpha _0}, \end{aligned}$$

with \(0<s\le m_{tot}\), then the event corresponding to rate \(\alpha _s\) will occur. A principal assumption concerning the AF is that it diffuses and reacts on a timescale much faster than that of events affecting the cells, which are movement, sprouting, anastomosis or regression. Hence, between consecutive Gillespie events we solve the PDE describing the evolution of the concentration of the AF \(c\) using a finite difference approximation, with a forward Euler method.

1.2 D.2 Solution of the PDEs

The PDEs appearing in this paper were solved with Mathematica 9 using the method of lines, and in most cases we could rely on the built-in solver to choose the correct time integration method. We have compared our results with those obtained using several, alternative ODE integration methods such as explicit and implicit 4th order Runge-Kutta, and also to a Matlab implementation using the pdepe function, and a C++ implementation of a finite difference scheme with explicit Euler integration. In the reduced models of tip cell movement and vessel production shown in Sect. 4, the tips evolve according to pure diffusion or pure chemotaxis with linear gradient. In these cases, the PDEs admit explicit solutions by Fourier expansion or direct integration, respectively. For the case of pure chemotaxis as well as the simulations of the full model in Sect. 5, the explicit Euler method is unstable and cannot be used.

Appendix E: Boundary conditions in the stochastic model

We will now briefly discuss different choices of boundary conditions in the stochastic model. We only need to focus on the part of the model involving the transition rate describing tip cell migration, \(\mathcal {T}_{N_k-1,N_l+1,R_k+\delta _R,R_l|N_k,N_l,R_k,R_l}\), as the other transition rates are local, depending on a single box.

To facilitate comparison of the stochastic and PDE models, it is useful to choose slightly different conventions from the main part of this paper and center the boxes at \(x=(k-1)h\), so that box \(k\) extends from \(x\in [k-\frac{3h}{2},k-\frac{h}{2}]\). Then the domain extends from \(x\in [-\frac{h}{2},L+\frac{h}{2}]\), if we let \(k=1,\dots ,k_{max}\), and \((k_{max}+1)h = L\). This has the advantage that, when choosing Dirichlet boundary conditions, we can directly relate the contents of boxes \(k=1, k_{max}\) to corresponding values of the PDE model at \(x=0,L\). As we assume \(h\ll L\) this redefinition of the domain does not affect the solution of the PDE.

Dirichlet boundary conditions We implement the boundary conditions by specifying the tip numbers in boxes \(k=1\) and \(k=k_{max}\), so

$$\begin{aligned} \begin{aligned} N_1(t)&=f_L(t),\\ N_{k_{max}}(t)&=f_R(t). \end{aligned} \end{aligned}$$

Here, we assume that \(f_L\) and \(f_R\) are slowly varying functions of time, compared to the timescale of events occurring in the stochastic model. This means we can hold \(N_1\) and \(N_{k_{max}}\) fixed between successive stochastic events. Another option is to implement Dirichlet boundary conditions in terms of stochastic reactions. Rather than fixing \(N_1\) and \(N_{k_{max}}\) deterministically, they could undergo birth and death processes so that only on average would we get \(N_1=f_L(t)\), \(N_{k_{max}}=f_R(t)\). For simplicity, in this paper we view \(N_1\) and \(N_{k_{max}}\) as deterministic.

Neumann boundary conditions For simplicity we restrict our study to zero flux boundary conditions. A simple implementation is such that for box \(k=1\), there is only one outgoing transition rate,

$$\begin{aligned} \mathcal {T}_{N_1-1,N_2+1,R_1+\delta _R,R_2|N_1,N_2,R_1,R_2}, \end{aligned}$$

and similarly for box \(k=k_{max}\). In this way, there is automatically no flux through the boundary. However, the number of vessel cells produced in the boundary boxes will be considerably smaller than the number produced in boxes \(k=2,\dots ,k_{max}-1\), as there will be less net movement. One way to overcome this is to add a reflection transition rate

$$\begin{aligned} \mathcal {T}_{N_1,R_1+\delta _R|N_1,R_1}\!, \end{aligned}$$

and similarly for \(k=k_{max}\). This has the interpretation that when a cell attempts to move to the left, it is reflected at the impenetrable boundary, and, as a result, remains in the same box from where it starts. It will still leave some vessel cells behind in this process. Such a term would only make sense for random movement Case 1, as outlined in Sect. 2.1, since Case 2, as well as chemotaxis, depend on the difference in cell numbers or concentrations in neighbouring boxes. Here, the incoming and outgoing boxes are identical.

An alternative implementation of zero-flux boundary conditions is similar to that commonly used for boundary conditions in finite difference schemes of PDEs. One fixes the number of tips in boxes \(1\) and \(k_{max}\), respectively, as

$$\begin{aligned} N_1&= N_2,\\ N_{k_{max}}&= N_{k_{max}-1}. \end{aligned}$$

For simplicity, we implemented Neumann no-flux boundary conditions using the first method (61).

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Spill, F., Guerrero, P., Alarcon, T. et al. Mesoscopic and continuum modelling of angiogenesis. J. Math. Biol. 70, 485–532 (2015).

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