Collective tumor cell migration in the presence of fibroblasts

Abstract

In this work we investigate fibroblast-enhanced tumor cell migration in an idealized tumor setting through a computational model based on a multiphase approach consisting of three phases, namely tumor cells, fibroblasts and interstitial fluid. The interaction between fibroblasts and tumor cells has previously been investigated through this model [Urdal, et al. (2019) Biomech. Model Mechanobiol. doi: 10.1007/s10237-019-01128-2] to comply with reported in vitro experimental results [Shieh, et al. (2011) Cancer Res. 71:790-800]. Using the information gained from in vitro single-cell behavior, what will the effect of fibroblast-enhanced tumor cell migration be in a tumor setting? In particular, how will tumor cells migrate in a heterogeneous tumor environment compared to controlled in vitro microfluidic-based experiments? From what we know about the behavior of a tumor, is that collective invasion into adjacent tissue is frequently observed. Here, we want to elucidate how fibroblasts may guide tumor cells towards draining lymphatics to which tumor cells may subsequently intravasate and thus spread to other parts of the body. Fibroblasts can act as leader cells, where they create tracks within the extracellular matrix (ECM) by matrix remodeling and contraction. In addition, a heterotypic mechanical adhesion between fibroblasts and tumor cells also assist the fibroblasts to act as leader cells. Our simulation results show how the interaction between the two cell types yields collective migration of tumor cells outwards from the tumor where fibroblasts dictate the direction of migration. The model also describes how this well-orchestrated invasive behavior is the result of a proper combination of different interaction forces between cell-ECM, fibroblast-ECM, fluid-ECM and cell-fibroblast.

1. Introduction

1.1. Background

One of the distinct properties of cancer is the ability of the cancer cells to spread by invading the adjacent tissue, often followed by local or distant metastasis. Cell migration is often referred to as the movement of individual cells. However, most invasive solid tumors frequently exhibit collective invasion, where cohesive cohorts of cells invade the adjacent stroma while maintaining cell-cell contacts [1]. The morphologically organization of cohesive cohorts invading the stroma can vary considerably.

The invading cell groups may range from strands of just a few cells in diameter, to wide masses of cells. The size and shape of a collective invasion structure is probably determined by specific combinations of cell-cell adhesion, cell-matrix adhesion and proteolysis. Therefore, the organization of the front of the collective cells can vary, likely as a combined function of proteolysis, protrusion and expansion, and the type of tissue encountered [2]. Cells located at the front of the invading group are called leader cells. These cells play a special role during migration by sensing the microenvironment and dictating the speed and the direction of the entire cell cluster [3]. Cancer-associated fibroblasts (CAFs) can act similarly as the front of the collectively invading group, where cancer cells retain their epithelial traits. The CAFs would then lead the cells within tracks in the ECM generated by the fibroblasts [4].

While epithelial-to-mesenchymal transition (EMT), a cell-biological program giving cancer cells multiple malignant traits such as loss of epithelial properties and acquisition of certain mesenchymal features in their stead, is widely accepted as an important mode of invasion, its precise roles in primary tumor behavior is not fully understood. As most primary tumor cells are involved in collective migration rather than the dispersal of individual carcinoma cells, this appears to conflict with the behavior of cells that has gone through EMT and lost cell-cell contacts. Therefore, an EMT program might not be necessary for carcinoma cell dissemination. However, EMT and collective tumor cell migration are perhaps not mutually exclusive [5].

1.2. Purpose

This work is based on the computer models introduced and explored in [6] and [7] but used in a more realistic tumor setting. The model uses a multiphase approach describing three phases: tumor cells, interstitial fluid and fibroblasts. We will try to exemplify how tumor cells can invade to adjacent tissue without the need to undergo EMT by using fibroblasts as leader cells. Similar to [7], we allow direct mechanical interaction/coupling between fibroblasts and tumor cells motivated by the results of Labernadie et al [8]. In addition, the moving fibroblasts may remodel the ECM, causing tumor cells to migrate in the tracks of ECM created by the fibroblasts as observed in [9, 4, 10].

Variable	Description
${\alpha }_c,{\alpha }_f,{\alpha }_w$	volume fraction cell, fibroblast, fluid
$S_c,S_f$	cell growth/death
$u_c,u_f,u_w$	cell, fibroblast, fluid velocity
$\rho ,G,C,H$	ECM, protease, chemokine, TGF
$D_G,D_C,D_H$	diffusion coefficients
${\rho }_M,G_M,C_M,H_M$	maximal concentrations
$P_w$	IF pressure
$\text{Δ}{P}_{cw}$	cell-cell stress
${\text{Λ}}_C,{\text{Λ}}_H$	chemokine, TGF chemotaxis stress
${\widehat{\lambda }}_c,{\widehat{\lambda }}_f,{\widehat{\lambda }}_T$	cell, fibroblast and total mobility
$T_v,T_l$	conductivity vascular/lymphatic wall
${\tilde{P}}_v^{\ast },{\tilde{P}}_l^{\ast }$	effective vascular/lymphatic pressure
$Q_v=T_v({\tilde{P}}_v^{\ast }-P_w)$	produced fluid from vascular system
$Q_l=T_l(P_w-{\tilde{P}}_l^{\ast })$	lymphatic drainage
MC, MH	percentage absorption at lymphatics
λ21, λ22, λ23, λ24	production/consumption ECM
λ31, λ32, λ33	production/consumption protease
λ41, λ42, λ43, λ44	production/consumption chemokine
λ51, λ52, λ53, λ54, λ55	production/consumption TGF
vG, vC, vH	exponents of decay rate terms

In [7] the goal was to replicate the experimental results in [9] through a multiphase approach with regards to tumor cell and fibroblast migration. This was performed in a 1D setting where we, similar to the in vitro experimental setting, impose a global pressure gradient to achieve a constant interstitial fluid flow. This has yielded valuable information with regards to how the different parameters in our model need to be set in order to attain realistic fluid flow and cellular behavior. In this work we apply the same model in a 2D domain comprised of a tumor with a vascular system. Outside of the primary tumor we have placed fibroblast cells and draining lymphatics. This allows us to simulate the observed migration mechanisms in vitro, autologous chemotaxis and fibroblast enhanced tumor cell migration, simultaneously in a tumor environment. As opposed to the experimental setting, where both the fluid flow field and cell migration is essentially one-dimensional, a realistic tumor setting is highly heterogeneous. IF now originates from the vascular system and flows through the tissue to the draining lymphatics, creating a heterogeneous flow field which impacts the concentration distribution of chemical components. We want to investigate how fibroblast-enhanced tumor cell migration may occur in an envisioned tumor setting. In particular, what type of invasion does the inclusion and presence of fibroblasts impart on the tumor cells. In addition, will the fibroblasts act as leader cells to guide tumor cells towards lymphatics?

2. Compact three-phase fibroblast-cell-fluid model

Model description

The model takes the following compact form (see Appendix A for details).

$$\begin{gathered} {\alpha }_{ct}+\nabla \cdot ({\alpha }_c{u}_c)=S_c \\ {\alpha }_{ft}+\nabla \cdot ({\alpha }_f{u}_f)=S_f,{\alpha }_c+{\alpha }_f+{\alpha }_w=1 \\ {\rho }_t=-{\lambda }_{21}G\rho +\rho \left({\lambda }_{22}-{\lambda }_{23}{\alpha }_c-{\lambda }_{24}\left(\frac{\rho }{{\rho }_M}\right)\right) \\ {G}_t=\nabla \cdot (D_G\nabla G)-\nabla \cdot (u_{w}G)-{\lambda }_{31}G \\ +({\alpha }_c+{\alpha }_f)\left({\lambda }_{32}-{\lambda }_{33}{\left(\frac{G}{G_M}\right)}^{v_G}\right) \\ {C}_t=\nabla \cdot (D_C\nabla C)-\nabla \cdot (u_{w}C)-C{M}_C{Q}_l \\ +G\rho \left({\lambda }_{41}-{\lambda }_{42}{\left(\frac{C}{C_M}\right)}^2-{\lambda }_{43}{\left(\frac{C}{C_M}\right)}^{v_C}\right)-{\lambda }_{44}{\alpha }_{c}C \\ {H}_t=\nabla \cdot (D_{H}H)-\nabla \cdot (u_{w}H)-H{M}_H{Q}_l-{\lambda }_{51}H \\ +{\alpha }_f\left({\lambda }_{52}-{\lambda }_{53}{\left(\frac{H}{H_M}\right)}^2-{\lambda }_{54}{\left(\frac{H}{H_M}\right)}^{v_H}\right)-{\lambda }_{55}{\alpha }_{f}H. \end{gathered}$$ (1)

The chemical components G, C and H are described by transport-reaction equations and are advected by the IF velocity u_w. The explicit expressions for the interstitial velocities are as follows:

$$\begin{gathered} u_c=\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)+\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H \\ {u}_f=\frac{{\widehat{f}}_f}{{\alpha }_f}{U}_T+\frac{{\widehat{h}}_2}{{\alpha }_f}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)-\frac{{\widehat{h}}_2+{\widehat{h}}_3}{{\alpha }_f}\nabla {\text{Λ}}_H \\ {u}_w=\frac{{\widehat{f}}_w}{{\alpha }_w}{U}_T+\frac{{\widehat{h}}_1}{{\alpha }_w}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)+\frac{{\widehat{h}}_3}{{\alpha }_w}\nabla {\text{Λ}}_H. \end{gathered}$$ (2)

Note that each phase velocity in (2) is governed by four different terms, representing contributions to the overall phase velocity from separate mechanisms. Functional forms of ${\widehat{f}}_c$, ${\widehat{f}}_f$, ${\widehat{f}}_w$ and ${\widehat{h}}_1$, ${\widehat{h}}_2$, ${\widehat{h}}_3$ and ΔP_cw, Λ_C, Λ_H are described by (28), (29), (15), (16), respectively, in Appendix A. In order to find U_T, we first solve an elliptic equation for P_w which takes the form

$$\begin{gathered} \nabla \cdot ({\widehat{\lambda }}_T\nabla P_w)=-T_v({\tilde{P}}_v^{\ast }-P_w)+T_l(P_w-{\tilde{P}}_l^{\ast }) \\ -\nabla \cdot ({\widehat{\lambda }}_c\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C(C)))-\nabla \cdot ({\widehat{\lambda }}_f\nabla {\text{Λ}}_H) \\ {P_w|}_{\partial \text{Ω}}=P_B^{\ast }. \end{gathered}$$ (3)

The explicit expressions for ${\widehat{\lambda }}_c$ and ${\widehat{\lambda }}_T$ are given by (30)_1,4 in Appendix A. We can use the calculated IF pressure P_w to find the total velocity U_T

$$U_T=-{\widehat{\lambda }}_T\nabla P_w-{\widehat{\lambda }}_c\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)-{\widehat{\lambda }}_f\nabla {\text{Λ}}_H,$$ (4)

where ${\widehat{\lambda }}_f$ is given by (30)₂. U_T is required in the calculation of the interstitial velocities u_c, u_f and u_w in (2). The model (1)–(4) is subject to the boundary conditions

$$\frac{\partial }{\partial v}G{|}_{\partial \text{Ω}}=0,\frac{\partial }{\partial v}C{|}_{\partial \text{Ω}}=0,\frac{\partial }{\partial v}H{|}_{\partial \text{Ω}}=0.$$ (5)

Remark 2.1. Looking at the cell velocity u_c given by (2)₁, we see that it consists of four different terms:

1. fluid generated stress, $\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T$;

2. diffusion, $-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Δ}{P}_{cw}({\alpha }_c));$

3. chemotaxis of cells towards increasing concentration gradient of chemokines, $-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_C(C)$;

4. counter-current effect of fibroblasts chemotaxis towards concentration gradient of TGF, $\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H(H)$.

The first term represents a stress caused by the flowing IF on the cancer cells. This is a co-current transport effect. Cancer cells will to a large extent resist the direct pushing of fluid flow, as reported in [11, 9]. The next three terms represent counter-current transport effects. The second term represents migration due to diffusion, i.e., a more or less weak non-directional migration, causing fibroblasts and fluid to be squeezed in the opposite direction to give room for the tumor cells. This effect is also created in the third term as tumor cells chemotax towards positive chemokine gradients. The fourth term represents tumor cells that are pushed in the opposite direction of fibroblasts as fibroblasts chemotax towards positive TGF gradients. Note that Λ_C(C), Λ_H(H) are decreasing functions.

Remark 2.2. Regarding the mechanical coupling between cells and fibroblasts, this is expressed through ${\widehat{\zeta }}_{cf}$ in the general momentum balance equations (14)_4,5. Still focusing on u_c described by (2)₁, this term will have an impact both on ${\widehat{f}}_c$ and ${\widehat{h}}_2$, as is seen from the expressions (28)₁ and (29)₂. Having that ${\widehat{\zeta }}_{cf}>0$ we may look at the transport effect through ${\widehat{h}}_2$ which is given by

$${\widehat{h}}_2({\alpha }_c,{\alpha }_f)=\frac{{\widehat{\lambda }}_c{\widehat{\lambda }}_f}{{\widehat{\lambda }}_T}-\frac{{\alpha }_c{\alpha }_f{\widehat{\zeta }}_{cf}}{{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}({\widehat{\zeta }}_c+{\widehat{\zeta }}_f)}.$$ (6)

If we increase the cell-fibroblast interaction ${\widehat{\zeta }}_{\mathit{cf}}$, ${\widehat{h}}_2$ will eventually become negative [7]. This means that there is acceleration of tumor cells through $\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H$ of u_c, which essentially is fibroblast migration toward a positive gradient of TGF and felt by the nearby tumor cells that connect to fibroblasts mechanically. In addition, there is a corresponding deceleration of fibroblasts through $-\frac{{\widehat{h}}_2+{\widehat{h}}_3}{{\alpha }_f}\nabla {\text{Λ}}_H$ of u_f in (2)₂.

Remark 2.3. We include both TGF and tumor cells in our model, yet we have not implemented any direct effect TGF may have on tumor cells. In reality, TGF-β1 regulates a variety of tumor promoting and suppressive effect depending on which stage of development the tumor is in. During the early stages of development TGF acts as a tumor suppressor by inducing cell death and growth arrest. At later stages, TGF switches roles and enhances migration, invasion and survival of tumor cells [12]. However, the mathematical model is developed to comply with the experimental observations of [9] where TGF had no direct effect on the cancer cells.

3. Results

3.1. Choice of parameters

Most of the parameters used in this work are the same as in [7, 6]. We refer to Table 1 for a precise description. We have performed thorough investigations in a 1D setting with regard to the choice of different parameters where these are set such that the model can capture the IF pressure and velocity behavior as well as the cellular behavior of in vitro 1D experimental results found in [9, 11]. In this work we perform 2D simulations in a more realistic tumor setting where the interstitial fluid flow field is created due to leaky blood vessels at the tumor margin and functional lymphatic vessels somewhere within the peritumoral region. This will potentially create a heterogenous IF velocity field. The main objective is to visualize to what extent the enhanced tumor cell behavior reported from in vitro experiments can give rise to more aggressive tumor cell behavior in this envisioned heterogenoues tumor setting. Hence, this summarizes our method:

1. Train our model using data from experimental results performed in a controlled in vitro setting, as reported in [9].

2. Expand the model, using the experience and parameters from step 1, to mimic a real-world tumor in a two-dimensional setting.

Table 1:

Model parameters (dimensional)

Parameter	Description	Value	Unit
Reference variables
T*	Time	104	s
L*	Length	10−2	m
u*	Velocity	10−6	m/s
D*	Diffusion	10−8	m2/s
ρ*	ECM density	1	kg/m3
G*	Protease	10−4	kg/m3
C*	Chemokine	10−4	kg/m3
H*	TGF	10−4	kg/m3
P*	Pressure	104	Pa
ρM	Maximum ECM density	ρ*	kg/m3
GM	Maximum protease density	0.5G*	kg/m3
CM	Maximum chemokine density	0.3G*	kg/m3
HM	Maximum TGF density	0.5H*	kg/m3
Material constants
DG	diffusion coefficient of protease	8 · 10−12	m2/s
DC	diffusion coefficient of chemokine	7 · 10−14	m2/s
DH	diffusion coefficient of TGF	8 · 10−12	m2/s
Production/decay rates
λ 21	degradation of ECM	10	m3/kgs
λ 22	reconstruction of ECM	1.25 · 10−3	1/s
λ 23	release of ECM	0	1/s
λ 24	release of ECM	1.25 · 10−3	1/s
λ 31	decay of protease	2.5 · 10−3	1/s
λ 32	cell production of protease	2 · 10−6	kg/m3s
λ 33	logistic term constant (protease)	2 · 10−6	kg/m3s
νG	exponent in logistic function of protease	1	-
λ 41	proteolytically freed chemokine	3.2 · 10−3	m3/kgs
λ 42	logistic term constant (chemokine)	1.44 · 10−4	m3/kgs
λ 43	logistic term constant (chemokine)	3.2 · 10−3	m3/kgs
λ 44	cell consumption of chemokine	1 · 10−9	1/s
νC	exponent in logistic function of chemokine	0.2	-
λ 51	decay of TGF	1 · 10−5	1/s
λ 52	production of TGF	8.75 · 10−7	kg/m3s
λ 53	logistic term constant (TGF)	5.5 · 10−7	kg/m3s
λ 54	logistic term constant (TGF)	0	kg/m3s
λ 55	fibroblast consumption of TGF	2 · 10−6	1/s
νH	exponent in logistic function of TGF	0.2	-
Potential function, chemokine
ξC	parameter characterizing ΛC	8 · 104	m3/kg
ΛC0	parameter characterizing ΛC	0	Pa
ΛC1	parameter characterizing ΛC	2.5 · 104	Pa
Potential function, TGF
ξH	parameter characterizing ΛH	1.6 · 105	m3/kg
ΛH0	parameter characterizing ΛH	0	Pa
ΛH1	parameter characterizing ΛH	2.5 · 104	Pa
Capillary pressure function
γ	parameter characterizing ΔP	103	Pa
δ	parameter characterizing ΔP	0.01	-

3.1.1. Interaction parameters

We have three distinct phases: cancer cells, fibroblasts and interstitial fluid. These three phases interact with ECM (the matrix structure) and possibly also with one another. The interaction terms are expressed by ζ_c, ζ_w, ζ_f and ζ_cf, as reflected by the general momentum balance equations (14)_4,5,6. The explicit correlations are specified in (17–20). The interaction terms between the cellular phases and the fluid phase have been neglected in this work, but is investigated in depth in [13, 14], which in turn is motivated by the experimental findings in [15]. As far as the correlations in (17–20) is concerned, the purpose of the coefficients ${\widehat{k}}_w$, ${\widehat{k}}_c$ and ${\widehat{k}}_f$ is to represent dynamic properties of ECM, and is initially set as ${\widehat{k}}_w$ = ${\widehat{k}}_c$ = ${\widehat{k}}_f$ = 1. I_w, I_c and I_f are static parameters that reflect the conductivity of the tumor microenvironment.

The process of setting parameters starts with obtaining a reasonable pathological IF fluid flow velocity u_w and IFP P_w throughout the domain. The interstitial fluid flow is mainly constructed by the hydraulic conductivity of the interstitial space, i.e. the resistance to the fluid flow in porous and fibrous media. A high hydraulic conductivity yields a faster flowing fluid through the interstitial space [17]. We have set the conductivity associated with the tumor microenvironment as $I_w^{-1}=\frac{K}{{\mu }_w}$ = 5 · 10⁻¹³ m²/Pa s to achieve interstitial fluid velocity of u_w = [0.1 – 0.7]μm/s. Then, we can apply the resistance forces ζ_c and ζ_f, seen in (18) and (19), between the cellular phases and ECM to achieve valid behavior of the cells based on expected behavior seen in experiments. The parameters involved in ζ_c and ζ_f in combination with the parameters used to express the chemotactic strength of the cells, see (16), can be considered unique for one type of cell line. Changes to these parameters will allow the tumor cells in the model to be more or less aggressive, more single cell invasion, or collective invasion and so on. Thus by tuning these parameters, we can simulate the effects of fibroblasts on metastatic progression for cancers of different origin, for example to determine whether fibroblasts play a more or less significant role in guiding single cell migration, as observed in fibrosarcoma or glioblastoma, or in guiding collective cell migration as seen in cancers of epithelial origin [18]. Furthermore, we also have the ability to vary essential aspects of the tumor microenvironment through the parameter ζ_w given by (17) in combination with the placement and production/absorption rate of the vascular and lymphatic system, i.e., the parameters involved in the expression for Q_v and Q_l given by (21) and (22). These interaction forces are imperative for the behavior of the model, as they largely impact the functions ${\widehat{f}}_c$, ${\widehat{f}}_f$, ${\widehat{f}}_w$ and ${\widehat{h}}_1$, ${\widehat{h}}_2$, ${\widehat{h}}_3$ as given by (28) and (29), which determine the phase velocities u_c, u_f, u_w in (2). The following values are used as default:

$$\begin{gathered} I_w=\frac{{\mu }_w}{K}=2\cdot {10}^{12}\text{Pa s}/{\text{m}}^2{\widehat{k}}_w=1,r_w=0.0, \\ {I}_c=2000I_w\text{Pa s}/{\text{m}}^2{\widehat{k}}_c=1,r_c=0.6, \\ {I}_f=100I_w\text{Pa s}/{\text{m}}^2{\widehat{k}}_f=1,r_f=0.6, \\ {I}_{cf}=1000I_w,\text{Pa s}/{\text{m}}^2{r}_{cf}=r_{fc}=0.5. \end{gathered}$$ (7)

We have set I_c, how strong cells are anchored to the ECM structure, to be a 2000-fold larger than the fluid-ECM resistance force (I_c = 2000I_w). This is greater than we have operated with before. However, it serves better to illustrate how fibroblasts may enhance tumor cell migration despite the fact that the initial cell-ECM resistance force keeps cancer cells mostly stationary. As fibroblasts are much more mobile than cancer cells, the fibroblast-ECM resistance force is set 100 times larger than fluid-ECM resistance force, i.e. I_f = 100I_w. We have assumed that not all fibroblast directly interact with cancer cells. Thus, the strength of this interaction is set to I_cf = 1000I_w to determine the strength of ${\widehat{\zeta }}_{\mathit{cf}}$ in (20). If we were to set I_cf → ∞, the tumor cells and fibroblasts would act as one phase in the regions where both phases are present.

3.1.2. Vascular and lymphatic flow parameters

The following values are used for parameters related to the vascular flow, Q_v, given by (21) and involved in the continuity equation for the IF (14)₃:

$$T_v=5\cdot {10}^{-7}1/\text{Pa s},{\tilde{P}}_v^{\ast }=4000\text{Pa}$$ (8)

and for lymphatic absorption Q_l in (22) we use

$$T_l=3.5\cdot {10}^{-7}1/\text{Pa s},{\tilde{P}}_l^{\ast }=1000\text{Pa}.$$ (9)

Equipped with the above values yield an IF velocity around 0.1-0.7 μm/s and IF pressure at the tumor margin around 3000-4000 Pa (i.e., 20-30 mmHg). We have assumed non-functional lymphatic vessels inside the tumor, and therefore no absorption of fluid within the tumor. Information about the placement of lymphatics is found in Figure 2.

Figure 2: Pressure and fluid flow velocity:

(A) Interstitial fluid pressure P_w is elevated at the center of the tumor and decreases significantly towards its periphery. (B). Same interstitial fluid pressure as in (A) but seen from another angle. (C) Fluid velocity field corresponding to the interstitial pressure in (B), where we have the highest fluid velocity at the tumor periphery and leads to the draining lymphatics outside of the primary tumor. (D) Fluid velocity field zoomed in at x ∈ [0.5, 1] and y ∈ [0.3, 0.8].

3.2. Initial and Boundary data

The initial primary tumor cell distribution is given by

$${\alpha }_c(x,y,t=0)=0.4\exp \left(-100{(x-0.5)}^2-100{(y-0.5)}^2\right)$$ (10)

where (x, y) ∈ Ω = [0, 1] × [0, 1] (dimensionless). The initial fibroblast volume fraction α_f is restricted to surround one side of the primary tumor periphery. This will better illustrate the impact the fibroblasts have on the cancer cells. The initial condition can be seen in Figure 1. The position of fibroblasts is motivated by [16, 19], where CAFs are situated around the primary tumor close to its margin.

Figure 1: Initial cell volume fractions:

Fibroblasts are placed on one side of the tumor to make it easier to compare simulated cell migration with and without fibroblasts in the same figure.

In particular, using the values assigned in (7–10), we get the IFP and the corresponding IF flow field seen in Figure 2. Figure 2 (A) shows where the vascular and lymphatic system are placed as well; vascular system in the center of the domain (inside the tumor), whereas four draining lymphatics are placed near the edge of the domain. In some of the simulation cases the lymphatics have changed positions and/or the conductivity of the tissue is decreased, causing a different flow pattern. This will be described within the respective cases. Figure 3 shows IFP and IF flow field when the conductivity of the tissue is reduced by a factor 10 by increasing the parameter ${\widehat{k}}_w$ in (17) from 1 to 10. In order to maintain the same level of the IF velocity we increase the transcapillary coefficient T_v by a factor of 10 and T_l by a factor of 2 as well as raise the intratumoral vascular pressure ${\tilde{P}}_v^{\ast }$ and effective lymphatic pressure ${\tilde{P}}_l^{*}$ by a factor of 1.5. Clearly, this leads to a higher level of the intratumoral IFP as well as a sharper drop of the IF pressure at the tumor margin (panel A and B), but also a slightly lower maximal IF velocity at the margin (panel C and D).

Figure 3: Pressure and fluid flow velocity with decreased tissue conductivity:

(A) Interstitial fluid pressure P_w is elevated at the center of the tumor and decreases significantly towards its periphery. The hydrostatic pressure in the vascular and lymphatic system is now ${\tilde{P}}_v^{*}$ = 6000 Pa and ${\tilde{P}}_l^{*}$ = 1500 Pa respectively. (B) Same interstitial fluid pressure as in (A) but seen from another angle. (C) Fluid velocity field is heavily dependent on the interstitial fluid pressure, thus we have outgoing fluid flow from the tumor periphery to the four lymphatics and nearly stagnant at the tumor core. (D) Fluid velocity field zoomed in at x ∈ [0.5, 1] and y ∈ [0.3, 0.8]

3.3. Collective behavior due to the presence of fibroblasts

In [7] we developed a model that incorporates fibroblasts as a separate phase, in addition to tumor cells and interstitial fluid. We used the experimental results from [9] as a means to describe the behavior of fibroblasts. Furthermore, [4] has shown that tumor cells may be led by fibroblast while migrating. We wanted to examine how this could play out in a more realistic tumor setting, using the mathematical model developed in [7]. In this work, we include the following two fibroblast-enhanced cancer cell migration mechanisms.

1. Fibroblasts remodel the ECM, priming the matrix to enhance the tumor cell invasion.

2. Tumor cells migrate in the same direction of fibroblasts due to a direct mechanical interaction between them.

We represent ECM remodeling by fibroblast (i.e. point 1. above) through the dynamic cell-ECM interaction parameter ${\widehat{k}}_c$. We use the same correlation as in [7], which is given by

$${\widehat{k}}_c({\alpha }_f)=1-A\left(1-\exp (-B{\alpha }_f)\right),$$ (11)

with prescribed constants A and B. We use values as found for the in vitro study [7] with A = 0.7 and B = 50. Hence, ${\widehat{k}}_c$ acts as a function of fibroblasts, α_f, where the presence of fibroblasts decreases the cell-ECM resistance force ${\widehat{\zeta }}_c$ (see (18)). This accounts for the effect that fibroblasts prime the matrix in the direction they migrate and therefore pave the way for cells to follow their path.

Regarding point 2 above, we assign the direct fibroblast-cell mechanical interaction strength I_cf from (7) in such a way that it can be interpreted as not all fibroblasts link themselves to the tumor cells. This still allows, however, that fibroblasts create tracks within the ECM so that tumor cells may follow them without the necessity of direct interaction.

3.3.1. Partial mechanical coupling and ECM remodeling

As prescribed in (7), the strength of the mechanical coupling between cancer cells and fibroblasts is set to I_cf = 1000I_w. This implies that not all fibroblasts create a direct coupling with the cancer cells, yet some will maintain this ability. The simulated results are displayed in Figure 4 and 5 after a time period of T ≈ 5.8 days. Main observations are:

• Tumor cell migration is manifested as a strong finger like migration pattern outwards from the tumor, see Figure 4, panel (A). Fibroblasts shown in panel (B) effectively migrate in the IF flow direction and pull cancer cells out of the primary tumor region so that the two cell types migrate in a coupled fashion where tumor cells follow fibroblasts through a primed matrix.

• The chemical components, chemokine and TGF-β, create pericellular concentration gradients in the direction of the lymphatics (Figure 4, panel (C) and (D)). This causes both tumor cells and fibroblasts to migrate in that direction due to autologous chemotaxis, as reflected by panel (A) and (B).

• As fibroblasts migrate at a higher velocity through the ECM and are not fully anchored to the cancer cells, they also prime the matrix further from their initial position. This results in cells following the fibroblasts farther through the ECM, both by the now primed ECM and the direct mechanical coupling between cells and fibroblasts, as observed in Figure 4 (A).

• Figure 5 shows the total cell velocity u_c (A) and its different velocity components (B-E), as mentioned in Remark 2.1. Panel (A) illustrates that the migration is highest at the invasive strand-like front. Apparently, the more motile fibroblasts, enable the cancer cells that are coupled with the fibroblasts to migrate even farther from their initial position.

• Figure 5 (F), which visualizes the different components of (2)₁, tells us that tumor cell migration due to chemotaxis towards chemokine (yellow region) is dominant close to the tumor periphery behind the more invasive front whereas migration owing to the mechanical coupling between fibroblasts and tumor cells is dominant outside of the periphery (orange region) of the primary tumor. The simulation suggests that the more aggressive tumor cell behavior is a result of the cell-fibroblast interaction.

• Comparing the left and right side of Figure 4 (A) shows that fibroblasts are necessary to activate the invasive behavior of tumor cells.

Figure 4: Combined partial mechanical coupling and ECM remodeling:

All variables are dimensionless. (A) Cancer cell volume fraction α_c shows migration towards the lymphatic vessel placed on the edge of the domain. The tumor cells follows the tracks made by the fibroblasts. On the right side of the tumor there clearly is a very small degree of invasion. (B) Fibroblast cell volume fraction α_f migrate at a higher velocity than tumor cells towards the edges of the domain. Fibroblasts chemotact towards the lymphatics as TGF-β accumulates and creates higher concentrations (positive gradients) in that direction. (C) Chemokine concentration C is proteolytically released from the ECM and creates a concentration gradient towards the lymphatics due to advection. (D) TGF-β concentration H is produced by fibroblasts and is advected towards the lymphatics by the interstitial fluid.

Figure 5: Combined partial mechanical coupling and ECM remodeling: Tumor cell velocity.

The highest tumor cell migration velocity is in the range 10-20 μm/hr. (A) Tumor cell velocity u_c. The invading tumor cells furthest away from the primary tumor are migrating with the highest velocity. (B) Tumor cell velocity due to fluid generated stress, $u_{c,\mathit{fluid}–\mathit{stress}}=\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T$. Fluid generated stress imposed on the tumor cell does only slightly contribute to the total tumor cell velocity. (C) Tumor cell velocity by cell-cell interaction, $u_{c,\mathit{cell}-\mathit{cell}}=-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Λ}{P}_{cw})$. Through our choice of parameters, random migration of tumor cells by diffusion is very low. (D) Tumor cell velocity due to chemotaxis of fibroblasts towards concentration gradient of TGF, $u_{c,\mathit{chemotaxis},H}=\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H$. Through the mechanical interaction between the two cell types, we have momentum transfer between them. (E) Tumor cell velocity due to chemotaxis of tumor cells towards concentration gradient of chemokine, $u_{c,\mathit{chemotaxis},C}=-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_C$. Chemotaxis towards chemokine contributes the most to the overall tumor cell migration behind the invasive front. (F) The components of u_c, shown in subfigures (B)-(E), which is largest in magnitude in a given area. Each component is represented by a color, referenced on the right of the figure.

3.3.2. Partial mechanical coupling and ECM remodeling with reduced tissue conductivity

The mechanical properties of the tumor microenvironment affect the tumor cell invasiveness. We change the tissue conductivity, as mentioned above, by increasing ${\widehat{k}}_w$ with a factor 10. The corresponding fluid flow field is similar to Figure 2 but the magnitude is decreased, as seen in Figure 3. With a decreased conductivity there is also a stronger connection between the vascular and the lymphatic system, causing the fluid flow to concentrate towards the lymphatics more directly than the previous case. The simulated results are shown in Figure 6 and 7. Main observations are:

• Tumor cells do not seem to be more aggressive in terms of penetration distance or tumor cell velocity, as seen in Figure 6 (A) and Figure 7 (A). Rather than small fraction of tumor cells migrating from the primary tumor as in Figure 4 (A), the cells now migrate in a more collective sheet-like manner away from the primary tumor, Figure 6 (A).

• By decreasing tissue conductivity the tumor cells have increased directionality towards the lymphatics, as the direction of flow through the tissue plays a much larger part in this case. The fluid generated stress imposed on tumor cells, seen in Figure 7 (B) and (F), is upregulated. In addition, the fluid flow creates both chemokine and fibroblast gradients towards lymphatics in a more direct manner, Figure 6 (C and D).

Figure 6: Combined partial mechanical coupling and ECM remodeling with reduced tissue conductivity:

All variables are dimensionless. (A) Cancer cell volume fraction α_c. Tumor cells migrate slowly towards the lymphatic, but in a more direct manner. (B) Fibroblast cell volume fraction α_f. Fibroblasts migrate outwards from their initial position. They also guide the tumor cells to migrate in their path, such that the tumor cells experience less resistance towards the lymphatic. (C) Chemokine concentration. Due to the decreased fluid flow velocity there is a greater concentration gradient of chemokine, as it will not be rapidly advected by the fluid. (D) TGF-β concentration. TGF-β concentration similarly to chemokine, but is instead produced by fibroblasts.

Figure 7: Combined partial mechanical coupling and ECM remodeling with reduced tissue conductivity: Tumor cell velocity.

The highest tumor cell migration velocity is in the range 10-20 μm/hr. (A) Tumor cell velocity u_c. (B) Tumor cell velocity due to fluid generated stress, $u_{c,\mathit{fluid}–\mathit{stress}}=\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T$. With decreased conductivity, fluid-generated stress has a larger impact on the total tumor cell velocity. (C) Tumor cell velocity by cell-cell interaction, $u_{c,cell–cell}=–\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Λ}{P}_{cw})$. (D) Tumor cell velocity due to chemotaxis of fibroblasts towards concentration gradient of TGF, $u_{c,\mathit{chemotaxis},H}=\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H$. Tumor cells follow fibroblasts through a direct mechanical coupling. (E) Tumor cell velocity due to chemotaxis of tumor cells towards concentration gradient of chemokine, $u_{c,\mathit{chemotaxis},C}=-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_C$. Tumor cells chemotact towards the lymphatics. Tumor cell velocity due to chemotaxis is increased on the left side due to ECM remodeling by fibroblasts. (F) The components of u_c, shown in subfigures (B)-(E), which contributes the most to the total tumor cell velocity in a given area. Each component is represented by a color, referenced on the right of the figure.

3.3.3. Diagonally placed lymphatics

In the next example we have placed the lymphatics at the corners of the domain rather than in the center of the sides. We continue to use the same parameters as in the first case. In this instance we exemplify how fibroblasts guide cancer cells towards the lymphatics. Having the same initial conditions for tumor cells and fibroblasts, shown in Figure 1, we expect that fibroblasts may change the direction of migration and thereby lead tumor cells to the lymphatics through ECM remodeling and direct mechanical adhesion.

The results can be seen in Figure 8 and 9.

• The migration of fibroblasts are clearly diverted towards the lymphatics. This also causes tumor cells to follow the fibroblasts towards the lymphatics, as seen in Figure 8 (A), (B) and Figure 9 (F).

• The direction of the concentration gradients of both TGF and chemokine guides both cell phases towards the lymphatics, shown in Figure 8 (C) and (D). The chemical components still accumulate near the lymphatics.

Figure 8: Diagonally placed lymphatics:

All variables are dimensionless. (A) Cancer cell volume fraction α_c. Tumor cells that are invading tend to veer towards the lymphatics which are now placed on the diagonals of the domain. (B) Fibroblast cell volume fraction α_f. Fibroblasts can clearly be seen to migrate in the direction of lymphatics. (C) Chemokine concentration. Chemokine continue to accumulate at the lymphatics.(D) TGF-β concentration. Similarly to chemokine, TGF-β accumulates at the lymphatics.

Figure 9: Diagonally placed lymphatics: Tumor cell velocity.

The highest tumor cell migration velocity is in the range 10-20 μm/hr. (A) Tumor cell velocity u_c. Tumor cells migrate towards the lymphatics. (B) Tumor cell velocity due to fluid generated stress, $u_{c,\mathit{fluid}–\mathit{stress}}=\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T$. Fluid generated stress imposed on the tumor cells does not have a strong contribution to the total velocity. (C) Tumor cell velocity by cell-cell interaction, $u_{c,cell–cell}=-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Λ}{P}_{cw})$. (D) Tumor cell velocity due to chemotaxis of fibroblasts towards concentration gradient of TGF, $u_{c,\mathit{chemotaxis},H}=\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H$. Through the direct mechanical coupling between fibroblasts and tumor cells, fibroblasts acts as leader cells and guide tumor cells toward the lymphatics. (E) Tumor cell velocity due to chemotaxis of tumor cells towards concentration gradient of chemokine, $u_{c,\mathit{chemotaxis},C}=-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_C$. The chemokine concentration gradient acts as a guide due to the accumulation of chemokine near the lymphatics. (F) The components of u_c, shown in subfigures (B)-(E), which has the highest relative tumor cell migration velocity in a given area. Each component is represented by a color, referenced on the right of the figure.

4. Discussion

4.1. Main conclusions

The multiphase model (1)–(5) has demonstrated how tumor cells can collectively invade the adjacent tissue by using fibroblasts as their leader cells to guide them towards the lymphatic system. There are two mechanisms accounted for in the computer model through which the fibroblasts enhance tumor cell migration: (i) Fibroblasts generate tracks in the tissue through remodeling and tumor cells are more inclined to invade in that direction. We express this by reducing cell-ECM drag force in ${\widehat{\zeta }}_c$ given by (18) as expressed by the correlation (11), letting tumor cells invade more aggressively where fibroblasts are present. (ii) Tumor cells directly attach themselves to fibroblasts and since fibroblasts are more mobile, they increase the tumor cell velocity and thus they increase tumor cell aggressiveness and ability to invade. This effect is expressed in the ${\widehat{h}}_2$-function in (29)₂ since this becomes negative when the cell-fibroblast interaction term ${\widehat{\zeta }}_{\mathit{cf}}$ given by (20) becomes sufficiently large. This allows fibroblasts chemotaxis towards a positive TGF gradient to increase tumor cell velocity in the same direction, as expressed through the last term of u_c in (2)₁. It is evident that under these circumstances, fibroblasts are necessary to initiate aggressive tumor cell behavior.

One important aspect of the second effect, (ii), is that fibroblasts increase the aggressiveness of the tumor cells through mechanical coupling, but decreases the mobility of fibroblasts. Tumor cells, being less mobile than fibroblasts, slow the fibroblasts down through this coupling. However, fibroblasts also remodel the ECM, increasing tumor cell velocity. Therefore, this can be considered a positive feedback loop. Fibroblasts will migrate more aggressively if tumor cells do the same, and since fibroblasts increases tumor cell aggressiveness, fibroblasts themselves will in turn increase their aggressiveness.

In our simulation cases we study the effect of fibroblasts on tumor cell migration in a tumor setting, under different conditions with regards to placement of the lymphatic system and the tissue conductivity. The results indicate that

• Placement of lymphatics greatly impacts the direction in which both fibroblasts and tumor cells migrate. This is mainly a result of the pericellular chemical concentration gradient of TGF and chemokine skewed in the direction of IF flow, which originates from the vascular system and is drained by the lymphatics. The fluid flow field is dependent on the location of the lymphatics.

• By decreasing the conductivity of the tissue, the fluid-ECM interaction term ${\widehat{\zeta }}_w$ in (17) increases, i.e., the fluid feels a stronger resistance force. This results in a higher contribution by the fluid generated stress imposed on the cells, letting the direction of fluid flow also guide tumor cells. Tumor cells thus displayed an enhanced collective migration towards the lymphatics when the conductivity was reduced. This is evident in Figure 7 (F).

• Two different types of aggressive behavior have been observed. When hydraulic conductivity is relatively high, the cancer cells are inclined to develop elongated strands of connected tumor cells at the invasive front that move farther away from the primary tumor, as illustrated in Figure 4. When hydraulic conductivity is reduced, the cancer cells tend to move collectively with a lower speed but more direction-specific towards a nearby draining lymphatic vessel, see Figure 6. This is a result of the fact that changes in the tissue microenvironment will change the relative strength of the different tumor cell migration components, as expressed by (2)₁.

Studies have shown that tumors that had developed lymph node metastases expressed higher interstitial fluid pressure and elevated interstitial fluid velocity compared to tumors that had not metastasized [20, 21]. In our simulations we reproduce the measured IFP in metastasizing tumors, which lies in the range of 30-40 mmHg (4-5.5 kPa). The source of elevated IFP in tumors may be influenced by many features of the local and adjacent stroma. Through our model, we can see that by decreasing the conductivity of the tissue we need to set a higher effective vascular pressure, ${\tilde{P}}_v$, and conductivity of the vascular wall, T_v, to maintain realistic interstitial fluid velocity. It is known that tumors develop elevated IFP because they show high resistance to blood flow (i.e., $P_v^{\ast }$ in (21) is high), low resistance to transcapillary fluid flow (i.e., T_v in (21) is high), and impaired lymphatic drainage (i.e., Q_l in (22) is located to the peritumoral region) [21].

As the conductivity is decreased, and thus the IFP is elevated, we observe in our results increased collective directionality towards the lymphatics. This may account for development of lymph node metastasis of in vivo tumors seen in tumors with elevated IFP [20, 21]. In addition, a study performed in vitro showed that cells cultured on stiffer substrates displayed sheet-like migration pattern, which is a slow collective migration but is exhibiting increased directionality, whereas cells cultured on softer tissue displayed more mesenchymal behavior [22]. Collectively migrating tumor cells are most frequently observed in solid tumors [1], which may explain the observations with regards to the increased number of lymph node metastasis found in [21] in tumors where IFP is increased.

Based on the work by Gaggioli et al. [4], where fibroblasts create tracks within the tissue for the tumor cells to follow, we included fibroblasts with the ability to act as leader cells through ECM remodeling and direct mechanical interaction. Like tumor cells, fibroblasts also sense the decreased conductivity and have a reduced migration velocity, but increased directionality [23]. This reduces the effect of the fibroblast-enhanced tumor cell migration as tumor cells are dependent on how the fibroblasts migrate. In summary, a multiphase approach has been used to illustrate the migration of tumor cells and fibroblasts in a realistic tumor setting, a model trained by experimental data observed in vitro. Fibroblast-enhanced tumor cell migration is present through two distinct mechanisms, i.e. through ECM remodeling and direct mechanical coupling between the phases. Moreover, we observe a collective migration of tumor cells outwards from the tumor and reducing the conductivity of the adjacent tissue increases the collective directionality. This yields results which can elucidate how fibroblasts guides tumor cells towards the lymphatics through collective migration. Furthermore, our model suggests that tumor cells are able to reach functioning lymphatic vessels through solely using fibroblasts as their guide as leader cells and without changes in the intrinsic migratory behavior of the cancer cells. These results support the strategy of targeting fibroblast-cancer cell interactions as a method to decrease metastasis in patients.

4.2. Future directions

Dense interstitial matrix and elevated interstitial fluid pressure have long been known to be barriers for drug delivery to solid tumors [24], and an emerging strategy for improving drug delivery has been to enzymatically degrade the matrix to increase hydraulic conductivity [25, 26, 27]. However, there is some indication that degrading matrix leads to an increased rate of metastasis [28], and expression of matrix-targeting proteases by tumors has long been established as a marker for metastatic potential [29]. The results in the present study demonstrate a dominating collective directional migration toward the lymphatics with decreased hydraulic conductivity. However, with higher conductivity the tumor cell behavior becomes more aggressive with a deeper penetration (higher velocity) and formation of a strand-like invasive front. This suggests a possible mechanism for the increased metastasis. More broadly, it demonstrates that enzymatically targeting the interstitium and thereby increasing the hydraulic conductivity, could have detrimental side-effects that promote metastasis. Rather, since fibroblasts contribute greatly to the metastatic and aggressive behavior of cancer cells, as observed on the left versus right side of the computational domain in this study, the model suggests that targeting fibroblasts for therapeutic treatment would decrease the invasion of cancer cells. This is consistent with experimental work that has identified CAFs as among the predominant cell types present within the tumor microenvironment [30]. A high concentration of stromal CAFs are often associated with poor prognosis in humans, as fibroblasts have abilities that can support and promote tumors during growth and metastasis [31, 32]. Natural future work and development of the computer model could be to tune it towards preclinical model data and use it in a search for associations between metastatic propensity and conditions pertaining to the TMEs, as reported, for instance in the recent work [21]. Another natural application would be to add a therapautic agent to the model with a prescribed impact on aspects of the TME or the migratory features of the cells, and then explore systematically for possibly barriers for efficient delivery and effect of this drug.

Appendix A: Three-phase fibroblast-cell-fluid model

In this work we use the multiphase model developed in [7] which in turn is motivated by the formulation in [35]. The approach is closely related to modelling of creeping fluid flow in porous media by mixture theory [36, 37, 38]. However, in this Appendix we also include porosity in the derivation of the model, where choice of parameters and assuming a constant porosity in space and time will reduce the model to its original form found in [7].

The tumor microenvironment contains the extracellular matrix (ECM) that occupies a volume fraction ϕ_m where the rest of the volume is represented by pore space ϕ_p. Fibroblasts, tumor cells and interstitial fluid reside within the pore space. We have that

$${\varphi }_m+{\varphi }_p=1.$$ (12)

In the derivation we will use $\varphi ={\varphi }_p=\frac{V_p}{V_T}$ to represent the pore space and ${\varphi }_m=\frac{V_m}{V_T}=1-\varphi$ to represent the matrix, where V_T is total tissue volume (V_T = V_p + V_m). Thus, the tumor environment is considered a mixture of four interacting continua [33, 34]: a stagnant matrix occupying a volume 1 − ϕ, a tumor cell phase represented by a volume fraction α_c moving within the pore space ϕ with a velocity $u_c^p$, and similarly for the fibroblasts and interstitial fluid (volume fractions α_f and α_w with velocities $u_f^p$ and $u_w^p$). The pore space is filled by the three phases, giving the closure relation

$${\alpha }_c+{\alpha }_f+{\alpha }_w=1(\text{i.e.},\varphi {\alpha }_c+\varphi {\alpha }_f+\varphi {\alpha }_w=\varphi )$$ (13)

In addition to mass and momentum balance equations for the three phases we have included the following components:

• ECM component $\rho =\frac{m_{\rho }}{V_T}$ associated with matrix (mass per total tissue volume V_T).

• Protease $G=\frac{m_G}{\varphi {\alpha }_w{V}_T}$ (mass per volume of solution) secreted by tumor cells.

• Chemokine $C=\frac{m_C}{\varphi {\alpha }_w{V}_T}$ (mass per volume of solution) released by proteolytic activity.

• Transforming growth factor $H=\frac{m_H}{\varphi {\alpha }_w{V}_T}$ (mass per volume of solution) released by fibroblasts.

Note that we can multiply the concentration of G, C and H by ϕα_w in order to express the concentration as mass per total volume tissue V_T. The resulting model, with inclusion of porosity and concentrations expressed in terms of mass per total volume tissue, becomes:

$$\begin{gathered} {(\varphi {\alpha }_c)}_t+\nabla \cdot (\varphi {\alpha }_c{u}_c^p)=S_c,S_c={\alpha }_c\left({\lambda }_{11}-{\lambda }_{12}{\alpha }_c-{\lambda }_{13}\frac{\rho }{{\rho }_M}\right) \\ {(\varphi {\alpha }_f)}_t+\nabla \cdot (\varphi {\alpha }_f{u}_f^p)=S_f, \\ {(\varphi {\alpha }_w)}_t+\nabla \cdot (\varphi {\alpha }_w{u}_w^p)=-S_c-S_f+Q,Q=Q_v-Q_l \\ {\alpha }_c\nabla (P_w+\text{Δ}{P}_{cw}+{\text{Λ}}_C)=-{\widehat{\zeta }}_c{u}_c^p+{\widehat{\zeta }}_{cf}(u_f^p-u_c^p) \\ {\alpha }_f\nabla (P_w+{\text{Λ}}_H)=-{\widehat{\zeta }}_f{u}_f^p-{\widehat{\zeta }}_{cf}(u_f^p-u_c^p) \\ {\alpha }_w\nabla P_w=-{\widehat{\zeta }}_w{u}_w^p \\ {\rho }_t=-\frac{{\lambda }_{21}}{\varphi }G\rho +\frac{\rho }{\varphi }\left({\lambda }_{22}-{\lambda }_{23}{\alpha }_c-{\lambda }_{24}(\frac{\rho }{{\rho }_M})\right) \\ {(\varphi {\alpha }_{w}G)}_t=\nabla \cdot (D_G\nabla G)-\nabla \cdot (\varphi {\alpha }_w{u}_w^{p}G)-{\lambda }_{31}G \\ +({\alpha }_c+{\alpha }_f)\left({\lambda }_{32}-{\lambda }_{33}{(\frac{G}{G_M})}^{v_G}\right) \\ {(\varphi {\alpha }_{w}C)}_t=\nabla \cdot (D_C\nabla C)-\nabla \cdot (\varphi {\alpha }_w{u}_w^{p}C)-C{M}_C{Q}_l \\ +G\rho \left({\lambda }_{41}-{\lambda }_{42}{(\frac{C}{C_M})}^2-{\lambda }_{43}{(\frac{C}{C_M})}^{v_C}\right)-{\lambda }_{44}{\alpha }_{c}C, \\ {(\varphi {\alpha }_{w}H)}_t=\nabla \cdot (D_H\nabla H)-\nabla \cdot (\varphi {\alpha }_w{u}_w^{p}H)-H{M}_H{Q}_l-{\lambda }_{51}H \\ +{\alpha }_f\left({\lambda }_{52}-{\lambda }_{54}{(\frac{H}{H_M})}^2-{\lambda }_{54}{(\frac{H}{H_M})}^{v_H}\right)-{\lambda }_{55}{\alpha }_{f}H \end{gathered}$$ (14)

where $u_i^p=(u_i^x,u_i^y)$ for i = c, w, f are interstitial velocities. The first six equations are mass and momentum balance equations for each of the phases, whereas the remaining equations account for ECM, protease, chemokine and TGF. The chemical components move by diffusion and advection. S _c and S _f are proliferation/apoptosis terms whereas the source term Q in (14)₃ describes the produced IF flow Q_v from the leaky vasculature and Q_l is the collected fluid by functional lymphatics in the peritumoral region. Similar to [35, 7] we use the following function for cell-cell interaction stress ΔP_cw(α_c)

$$\text{Δ}{P}_{cw}({\alpha }_c)=\gamma J({\alpha }_c)=-\gamma \ln [\delta +(1-{\alpha }_c)]$$ (15)

where γ > 0 is a coefficient (unit Pa) that depends linearly on the surface tension (unit Pa m) whereas J(α_c) is a monotonic increasing dimensionless function with respect to the cell volume fraction α_c. This accounts for the effect that tumor cells will try to reduce the cell-cell stress by moving towards a region with fewer tumor cells. The ability of the cancer cells and fibroblasts to generate a force and move is expressed through the potential function Λ_A(A) with A = C, H given by

$${\text{Λ}}_A(A)={\text{Λ}}_{A0}-\frac{{\text{Λ}}_{A1}}{1+\exp [-{\xi }_A(A-A_M)]}$$ (16)

where Λ_A0, Λ_A1 and ξ_A are constant parameters with units, respectively, as $[{\mathrm{\Lambda }}_{A0},{\mathrm{\Lambda }}_{A1}]=Pa$ and [ξ_A] = m³/kg. Note that Λ_A(A) for A = C, H is a decreasing function reflecting that cells/fibroblasts will try to reduce the additional stress associated with it by moving towards a higher concentration of A.

There is a drag force between the extracellular fluid, represented by u_w, and the ECM fibers. We use the following expression for this force

$${\widehat{\zeta }}_w=I_w{\widehat{k}}_w\varphi {\alpha }_w^{r_w},{\widehat{k}}_w>0,r_w<2,$$ (17)

with $I_w=\frac{{\mu }_w}{K}$ and K is the permeability of the porous media and μ_w the fluid viscosity. The coefficient r_w plays a similar role to the use of relative permeability functions in standard Darcy’s equation approach extended to several phases. Similarly, there is a drag force between the cells and the ECM

$${\widehat{\zeta }}_c=I_c{\widehat{k}}_c\varphi {\alpha }_c^{r_c},{\widehat{k}}_c>0,r_c<2,$$ (18)

where I_c (Pa s/m²), ${\widehat{k}}_c$ and r_c must be specified (the last two are dimensionless). In addition, there is a similar drag force between the fibroblasts and the ECM

$${\widehat{\zeta }}_f=I_f{\widehat{k}}_f\varphi {\alpha }_f^{r_f},{\widehat{k}}_f>0,r_f<2,$$ (19)

where I_f (Pa s/m²), ${\widehat{k}}_f$ and r_f must be specified. Finally, there is a drag force between the cell phase and the fibroblast phase which accounts for the mechanical coupling between the two cell types. This drag force represent momentum transfer from the faster moving fluid (fibroblasts) to the slower moving fluid (cancer cells)

$${\widehat{\zeta }}_{cf}=I_{cf}\varphi {\alpha }_c^{r_{cf}}{\alpha }_f^{r_{fc}},$$ (20)

I_cf is a positive constant determining the order of magnitude of the cell-fibroblast interaction and r_cf, r_fc are related exponents determining further details of this interaction. The form of the different interaction terms ζ_w, ζ_c, ζ_f and ζ_cf is consistent with traditional modeling of multiphase flow in porous media based on Darcy’s extended law [36, 37, 38, 39, 40].

Interstitial flow is a relatively slow fluid movement through the interstitium driven by hydrostatic and osmotic pressure differences between the arterial and lymphatic vessels [41]. The transcapillary exchange and formation of interstitial fluid is determined by a modified Starling’s Law, where Q_v in (14)₃ is given by

$$\begin{gathered} Q_v=T_v\left(P_v^{\ast }-P_w-{\sigma }_T({\pi }_v^{\ast }-{\pi }_w)\right)=T_v\left({\tilde{P}}_v^{\ast }-P_w\right), \\ {T}_v=L_v\frac{S_v}{V} \end{gathered}$$ (21)

with ${\tilde{P}}_v^{\ast }=P_v^{\ast }-{\sigma }_T({\pi }_v^{\ast }-{\pi }_w)$. Here, L_v is the hydraulic conductivity of the capillaries (m²s/kg = m/Pa s); S _v/V is the exchange area available for filtration per unit volume of tissues V; $P_v^{\ast }$ and P_w are the hydrostatic pressures in the blood capillary and the interstitial compartments, respectively; ${\pi }_v^{*}$ and π_w are the osmotic pressure in the capillary and interstitial compartments, respectively. σ_T is the capillary reflection coefficient.

The lymphatic vessels drain excessive fluid from the interstitial space, expressed by Q_l in (14)₃. The lymphatic system therefore regulates the fluid balance in tissues and prevents formation of edema. In a tumor microenvironment, the increased hydrostatic pressure causes the lymphatics within the tumor to be compressed and non-functional. Similar to (21), Q_l is expressed by

$$Q_l=T_l\left(P_w-{\tilde{P}}_l^{\ast }\right),T_l=L_l\frac{S_l}{V}$$ (22)

Here L_l is the hydraulic conductivity of the lymphatics; S_l/V is the surface area of the lymphatics per volume unit of tissues V and ${\tilde{P}}_l^{*}$ is the effective lymphatic pressure.

Remark 4.1. The model (14) is essentially the same as the one discussed in [7]. One difference is the appearance of the porosity ϕ which is set to a constant and will not have a direct impact on the simulation results. In addition, since we now use the model in an envisioned tumor setting, which is different from the experimental setup explored in [7], some other changes can be found. First, we need the source term Q in (14)₃ to account for the characteristic fluid flow from the the intratumoral vascular system to the draining peritumoral lymphatics. We have also added some source terms to the equation of chemical components (G, C and H) in (14)₈,₉,₁₀. This is done primarily to be able to better control the production and decay of the chemical components, and add some more realistic features to the equations. In particular, we have added decay terms –CM_CQ_l and –HM_HQ_l to ensure that the accumulation of chemokine and TGF at the lymphactis do not reach unreasonable high values.

4.3. Rewritten version of the model

We have implicit expressions for the velocities, represented by the momentum balance equations (14)_4,5,6. We can replace these equations with explicit expressions for the phase velocities (see [7] for more details). First, we introduce velocities u_c, u_f and u_w that are phase velocities relatively pore space, i.e.

$$u_i=\varphi u_i^p$$ (23)

In addition, we assume that α_wG ≈ G, α_wC ≈ C and α_wH ≈ H in (14)_8,9,10, meaning that in the region outside the tummor α_w is close to 1 (i.e. the pore space outside the tumor is dominated by fluid) and we are mainly interested in the role played by the chemical components in generating migration of tumor cells and fibroblasts in this area. From (14), after we have made it dimensionless (see Appendix B), we then have

$$\begin{gathered} {({\alpha }_c)}_t+\nabla \cdot ({\alpha }_c{u}_c)=S_c,S_c={\alpha }_c\left({\lambda }_{11}-{\lambda }_{12}{\alpha }_c-{\lambda }_{13}\frac{\rho }{{\rho }_M}\right) \\ {({\alpha }_f)}_t+\nabla \cdot ({\alpha }_f{u}_f)=S_f, \\ {({\alpha }_w)}_t+\nabla \cdot ({\alpha }_w{u}_w)=-S_c-S_f+Q,Q=Q_v-Q_l \\ {\alpha }_c\nabla (P_w+\text{Δ}{P}_{cw}+{\text{Λ}}_C)=-\frac{\widehat{\zeta }c}{\varphi }{u}_c+\frac{{\widehat{\zeta }}_{cf}}{\varphi }(u_f-u_c) \\ {\alpha }_f\nabla (P_w+{\text{Λ}}_H)=-\frac{{\widehat{\zeta }}_f}{\varphi }{u}_f-\frac{{\widehat{\zeta }}_{\mathit{cf}}}{\varphi }(u_f-u_c) \\ {\alpha }_w\nabla P_w=-\frac{{\widehat{\zeta }}_w}{\varphi }{u}_w \\ {\rho }_t=-{\lambda }_{21}G\rho +\rho \left({\lambda }_{22}-{\lambda }_{23}{\alpha }_c-{\lambda }_{24}(\frac{\rho }{{\rho }_M})\right) \\ {G}_t=\nabla \cdot (D_G\nabla G)-\nabla \cdot (u_{w}G)-{\lambda }_{31}G \\ +({\alpha }_c+{\alpha }_f)\left({\lambda }_{32}-{\lambda }_{33}{(\frac{G}{G_M})}^{v_G}\right) \\ {C}_t=\nabla \cdot (D_C\nabla C)-\nabla \cdot (u_{w}C)-C{M}_c{Q}_l \\ +G\rho \left({\lambda }_{41}-{\lambda }_{42}{(\frac{C}{C_M})}^2-{\lambda }_{43}{(\frac{C}{C_M})}^{v_C}\right)-{\lambda }_{44}{\alpha }_{c}C, \\ {H}_t=\nabla \cdot (D_H\nabla H)-\nabla \cdot (u_{w}H)-H{M}_H{Q}_l-{\lambda }_{51}H \\ +{\alpha }_f\left({\lambda }_{52}-{\lambda }_{53}{(\frac{H}{H_M})}^2-{\lambda }_{54}{(\frac{H}{H_M})}^{v_H}\right)-{\lambda }_{55}{\alpha }_{f}H \end{gathered}$$ (24)

with $u_i=(u_i^x,u_i^y,u_i^z)$ for i = c, w, f. The model is combined with the boundary condition

$$\begin{gathered} {P_w|}_{\partial \text{Ω}}=P_B^{\ast },\frac{\partial }{\partial v}G{|}_{\partial \text{Ω}}=0, \\ \frac{\partial }{\partial v}C{|}_{\partial \text{Ω}}=0,\frac{\partial }{\partial v}H{|}_{\partial \text{Ω}}=0,t>0 \end{gathered}$$ (25)

where v is the outward normal on ∂Ω. The corresponding initial data are

$$\begin{gathered} {\alpha }_c(x,t=0)={\alpha }_{c0}(x),{\alpha }_f(x,t=0)={\alpha }_{f0}(x),\rho (x,t=0)={\rho }_0(x), \\ G(x,t=0)=G_0(x),C(x,t=0)=C_0(x),H(x,t=0)=H_0(x) \end{gathered}$$ (26)

We can find the explicit expressions for the cell velocity u_c, fibroblast velocity u_f and the IF velocity u_w (refer [7] for details).

$$\begin{gathered} u_c=\frac{{\widehat{f}}_c}{{\alpha }_c}{U}_T-\frac{{\widehat{h}}_1+{\widehat{h}}_2}{{\alpha }_c}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)+\frac{{\widehat{h}}_2}{{\alpha }_c}\nabla {\text{Λ}}_H \\ {u}_f=\frac{{\widehat{f}}_f}{{\alpha }_f}{U}_T+\frac{{\widehat{h}}_2}{{\alpha }_f}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)-\frac{{\widehat{h}}_2+{\widehat{h}}_3}{{\alpha }_f}\nabla {\text{Λ}}_H \\ {u}_w=\frac{{\widehat{f}}_w}{{\alpha }_w}{U}_T+\frac{{\widehat{h}}_1}{{\alpha }_w}\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)+\frac{{\widehat{h}}_3}{{\alpha }_w}\nabla {\text{Λ}}_H \end{gathered}$$ (27)

with fractional flow functions which describes co-current flow ${\widehat{f}}_c$, ${\widehat{\zeta }}_{\mathit{fw}}$ and ${\widehat{f}}_f$, respectively, for the cell, fluid and fibroblast phase given by

$$\begin{gathered} {\widehat{f}}_c({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_c}{{\widehat{\lambda }}_T} \\ =\frac{{\alpha }_c[{\alpha }_f{\widehat{\zeta }}_{cf}+{\alpha }_c({\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_f)]}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)} \\ {\widehat{f}}_f({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_f}{{\widehat{\lambda }}_T} \\ =\frac{{\alpha }_f[{\alpha }_c{\widehat{\zeta }}_{cf}+{\alpha }_f({\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c)]}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)} \\ {\widehat{f}}_w({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_w}{{\widehat{\lambda }}_T} \\ =\frac{\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}, \end{gathered}$$ (28)

and the $\widehat{h}$ functions describing counter-current flow are given by

$$\begin{gathered} {\widehat{h}}_1({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_c{\widehat{\lambda }}_w}{{\widehat{\lambda }}_T} \\ =\frac{{\alpha }_c\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}[{\alpha }_c({\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf})+{\alpha }_f{\widehat{\zeta }}_{cf}]}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}\varphi \\ {\widehat{h}}_2({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_c{\widehat{\lambda }}_f}{{\widehat{\lambda }}_T}-\frac{{\alpha }_c{\alpha }_f{\widehat{\zeta }}_{cf}}{{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}({\widehat{\zeta }}_c+{\widehat{\zeta }}_f)}\varphi \\ =\frac{{\alpha }_c{\alpha }_f({\alpha }_c{\alpha }_f-\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}{\widehat{\zeta }}_{cf})}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}\varphi \\ {\widehat{h}}_3({\alpha }_c,{\alpha }_f)≔\frac{{\widehat{\lambda }}_f{\widehat{\lambda }}_w}{{\widehat{\lambda }}_T} \\ =\frac{{\alpha }_f\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}[{\alpha }_c{\widehat{\zeta }}_{cf}+{\alpha }_f({\widehat{\zeta }}_c+{\widehat{\zeta }}_{cf})]}{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}\varphi . \end{gathered}$$ (29)

where the coefficients ${\widehat{\lambda }}_c$, ${\widehat{\lambda }}_w$, ${\widehat{\lambda }}_f$ and ${\widehat{\lambda }}_T$ are generalized mobility functions given by

$$\begin{gathered} {\widehat{\lambda }}_c=\frac{{\alpha }_c[{\alpha }_c({\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_f)+{\alpha }_f{\widehat{\zeta }}_{cf}]}{{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}({\widehat{\zeta }}_c+{\widehat{\zeta }}_f)}\varphi \\ {\widehat{\lambda }}_f=\frac{{\alpha }_f[{\alpha }_c{\widehat{\zeta }}_{cf}+{\alpha }_f({\widehat{\zeta }}_c+{\widehat{\zeta }}_{cf})]}{{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}({\widehat{\zeta }}_c+{\widehat{\zeta }}_f)}\varphi \\ {\widehat{\lambda }}_w=\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}\varphi \\ {\widehat{\lambda }}_T=\frac{{({\alpha }_c+{\alpha }_f)}^2{\widehat{\zeta }}_{cf}+{\alpha }_c^2{\widehat{\zeta }}_f+{\alpha }_f^2{\widehat{\zeta }}_c+\frac{{\alpha }_w^2}{{\widehat{\zeta }}_w}({\widehat{\zeta }}_c{\widehat{\zeta }}_{cf}+{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}{\widehat{\zeta }}_f)}{{\widehat{\zeta }}_c{\widehat{\zeta }}_f+{\widehat{\zeta }}_{cf}({\widehat{\zeta }}_c+{\widehat{\zeta }}_f)}\varphi , \end{gathered}$$ (30)

Using the chosen correlations for the biomechanical interaction forces ζ_w, ζ_c, ζ_f and ζ_cf in (17)–(20) and by assuming constant porosity the model (24) then takes the simpler form

$$\begin{gathered} {\alpha }_{ct}+\nabla \cdot ({\alpha }_c{u}_c)=S_c \\ {\alpha }_{ft}+\nabla \cdot ({\alpha }_f{u}_f)=S_f \\ {\rho }_t=-{\lambda }_{21}G\rho +\rho \left({\lambda }_{22}-{\lambda }_{23}{\alpha }_c-{\lambda }_{24}\left(\frac{\rho }{{\rho }_M}\right)\right) \\ {G}_t=\nabla \cdot (D_G\nabla G)-\nabla (u_{w}G)-{\lambda }_{31}G \\ +({\alpha }_c+{\alpha }_f)\left({\lambda }_{32}-{\lambda }_{33}{\left(\frac{G}{G_M}\right)}^{v_G}\right) \\ {C}_t=\nabla \cdot (D_C\nabla C)-\nabla \cdot (u_{w}C)-C{r}_C{Q}_l \\ +G\rho \left({\lambda }_{41}-{\lambda }_{42}{\left(\frac{C}{C_M}\right)}^2-{\lambda }_{43}{\left(\frac{C}{C_M}\right)}^{v_C}\right)-{\lambda }_{44}{\alpha }_{c}C \\ {H}_t=\nabla \cdot (D_{H}H)-\nabla \cdot (u_{w}H)-{\lambda }_{51}H \\ +{\alpha }_f\left({\lambda }_{52}-{\lambda }_{53}{\left(\frac{H}{H_M}\right)}^2-{\lambda }_{54}{\left(\frac{H}{H_M}\right)}^{v_H}\right)-{\lambda }_{55}{\alpha }_{f}H, \end{gathered}$$ (31)

The cell velocity u_c is given by (27)₁, fibroblast velocity u_f is given by (27)₂ and finally the interstital fluid velocity u_w is given by (27)₃. u_w appears in the advective terms in the transport-reaction equations for G, C and H which means that the chemical components flows with the interstitial fluid. In order to compute U_T, which is used to calculate the interstitial velocities of the phases, we first solve the elliptic problem for P_w

$$\begin{gathered} \nabla \cdot ({\widehat{\lambda }}_T\nabla P_w),=-T_v({\tilde{P}}_v^{\ast }-P_w)+T_l(P_w-{\tilde{P}}_l^{\ast }) \\ -\nabla \cdot ({\widehat{\lambda }}_c\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C(C)))-\nabla \cdot ({\widehat{\lambda }}_f\nabla {\text{Λ}}_H) \\ {P}_w\mid \partial \text{Ω}=P_B^{\ast } \end{gathered}$$ (32)

Knowing IF pressure P_w we find the total velocity U_T by

$$\begin{gathered} U_T=U_c+U_f+U_w \\ =-{\widehat{\lambda }}_T\nabla P_w-{\widehat{\lambda }}_c\nabla (\text{Δ}{P}_{cw}+{\text{Λ}}_C)-{\widehat{\lambda }}_f\nabla {\text{Λ}}_H, \end{gathered}$$ (33)

where U_i, i = w, c, f, is the superficial velocity and are given as the product of the interstitial phase velocity and its respective volume fraction U_i = α_iu_i.

Appendix B: Non-Dimensionalization

In this section we want to obtain the dimensionless version of the model (14). We introduce characteristic length L* and time T* in addition to characteristic concentration and pressure: G*, C*, H*, ρ*, P* with corresponding characteristic velocity u* and diffusion D*

$$u^{\ast }=\frac{L^{\ast }}{T^{\ast }},D^{\ast }=\frac{{(L^{\ast })}^2}{T^{\ast }}$$

with dimensionless space and time variables

$$\tilde{x}=\frac{x}{L^{\ast }},\tilde{t}=\frac{t}{\varphi T^{\ast }},$$

where the tilde emphasizes that it is a dimensionless variable. In addition, we choose dimensionless variables related to the concentrations of the chemical components, the phase pressures and velocities, in the following way

$$\begin{gathered} \tilde{\rho }=\frac{\rho }{{\rho }^{\ast }},\tilde{G}=\frac{G}{G^{\ast }},\tilde{C}=\frac{C}{C^{\ast }},\tilde{H}=\frac{H}{H^{\ast }}, \\ {\tilde{D}}_G=\frac{D_G}{D^{\ast }},{\tilde{D}}_C=\frac{D_C}{D^{\ast }},{\tilde{D}}_H=\frac{D_H}{D^{\ast }}, \\ {\tilde{P}}_l=\frac{P_l}{P^{\ast }},{\tilde{u}}_l=\frac{u_l}{u^{\ast }},(l=c,f,w)\tilde{Q}=Q{T}^{\ast }. \end{gathered}$$

For production, decay and consumption of the chemical agents, we are going to use the following set of dimensionless expressions

$$\begin{gathered} \rho :{\tilde{\lambda }}_{21}={\lambda }_{21}{T}^{\ast }{G}^{\ast },{\tilde{\lambda }}_{22}={\lambda }_{22}{T}^{\ast },{\tilde{\lambda }}_{23}={\lambda }_{23}{T}^{\ast },{\tilde{\lambda }}_{24}={\lambda }_{24}{T}^{\ast }, \\ G:{\tilde{\lambda }}_{31}={\lambda }_{31}{T}^{\ast },{\tilde{\lambda }}_{32}=\frac{{\lambda }_{32}{T}^{\ast }}{G^{\ast }},{\tilde{\lambda }}_{33}=\frac{{\lambda }_{33}{T}^{\ast }}{G^{\ast }}, \\ C:{\tilde{\lambda }}_{41}=\frac{{\lambda }_{41}{T}^{\ast }{G}^{\ast }{\rho }^{\ast }}{C^{\ast }},{\tilde{\lambda }}_{42}=\frac{{\lambda }_{42}{T}^{\ast }{G}^{\ast }{\rho }^{\ast }}{C^{\ast }},{\tilde{\lambda }}_{43}=\frac{{\lambda }_{43}{T}^{\ast }{G}^{\ast }{\rho }^{\ast }}{C^{\ast }}, \\ {\tilde{\lambda }}_{44}={\lambda }_{44}{T}^{\ast }, \\ H:{\tilde{\lambda }}_{51}={\lambda }_{51}{T}^{\ast },{\tilde{\lambda }}_{52}=\frac{{\lambda }_{52}{T}^{\ast }}{H^{\ast }},{\tilde{\lambda }}_{53}=\frac{{\lambda }_{53}{T}^{\ast }}{H^{\ast }},{\tilde{\lambda }}_{54}=\frac{{\lambda }_{54}{T}^{\ast }}{H^{\ast }}, \\ {\tilde{\lambda }}_{55}={\lambda }_{55}{T}^{\ast }. \end{gathered}$$

The potential and capillary pressure functions are having units of pressure, and therefore they can be made dimensionless by dividing by the reference pressure, P*:

$$\begin{gathered} {\tilde{\mathrm{\Lambda }}}_C=\frac{{\mathrm{\Lambda }}_C}{P^{\ast }},{\tilde{\mathrm{\Lambda }}}_{C0}=\frac{{\mathrm{\Lambda }}_{C0}}{P^{\ast }},{\tilde{\mathrm{\Lambda }}}_{C1}=\frac{{\mathrm{\Lambda }}_{C1}}{P^{\ast }},{\tilde{\xi }}_C={\xi }_C{C}^{\ast }, \\ {\tilde{\mathrm{\Lambda }}}_C=\frac{{\mathrm{\Lambda }}_C}{P^{\ast }},{\tilde{\mathrm{\Lambda }}}_{H0}=\frac{{\mathrm{\Lambda }}_{H0}}{P^{\ast }},{\tilde{\mathrm{\Lambda }}}_{H1}=\frac{{\mathrm{\Lambda }}_{H1}}{P^{\ast }},{\tilde{\xi }}_H={\xi }_H{H}^{\ast }, \\ \text{Δ}{\tilde{P}}_{cw}=\frac{\text{Δ}{P}_{cw}}{P^{\ast }},\tilde{\gamma }=\frac{\gamma }{P^{\ast }}. \end{gathered}$$

Note that using (16) the chemokine potential function can now be written (similar for TGF)

$$\begin{gathered} {\tilde{\mathrm{\Lambda }}}_C=\frac{{\text{Λ}}_C}{P^{\ast }}={\tilde{\mathrm{\Lambda }}}_{C0}-\frac{{\tilde{\mathrm{\Lambda }}}_{C1}}{1+\exp [-{\xi }_C(C-C_M)]} \\ ={\tilde{\mathrm{\Lambda }}}_{C0}-\frac{{\tilde{\mathrm{\Lambda }}}_{C1}}{1+\exp [-{\tilde{\xi }}_C(\tilde{C}-{\tilde{C}}_M)]}. \end{gathered}$$

Interaction coefficients

$${\tilde{\widehat{\zeta }}}_c={\widehat{\zeta }}_c\frac{D^{\ast }}{P^{\ast }},{\tilde{\widehat{\zeta }}}_f={\widehat{\zeta }}_f\frac{D^{\ast }}{P^{\ast }},{\tilde{\widehat{\zeta }}}_w={\widehat{\zeta }}_w\frac{D^{\ast }}{P^{\ast }},{\tilde{\widehat{\zeta }}}_{cf}={\widehat{\zeta }}_{cf}\frac{D^{\ast }}{P^{\ast }}.$$

Let’s proceed with the details of rewriting the model (14) subject to the condition that phases are incompresible and using (23), which gives us

$$\begin{gathered} {(\varphi {\alpha }_c)}_t+\nabla \cdot ({\alpha }_c{u}_c)=S_c \\ {(\varphi {\alpha }_f)}_t+\nabla \cdot ({\alpha }_f{u}_f)=S_f, \\ {(\varphi {\alpha }_w)}_t+\nabla \cdot ({\alpha }_w{u}_w)=-S_c-S_f+Q \\ {\alpha }_c\nabla (P_w+\text{Δ}{P}_{cw}+{\text{Λ}}_C)=-\frac{{\widehat{\zeta }}_c}{\varphi }{u}_c+\frac{{\widehat{\zeta }}_{cf}}{\varphi }(u_f-u_c) \\ {\alpha }_f\nabla (P_w+{\text{Λ}}_H)=-\frac{{\widehat{\zeta }}_f}{\varphi }{u}_f-\frac{{\widehat{\zeta }}_{cf}}{\varphi }(u_f-u_c) \\ {\alpha }_w\nabla P_w=-\frac{{\widehat{\zeta }}_w}{\varphi }{u}_w \\ {(\rho \varphi )}_t=-{\lambda }_{21}G\rho +\rho \left({\lambda }_{22}-{\lambda }_{23}{\alpha }_c-{\lambda }_{24}(\frac{\rho }{{\rho }_M})\right) \\ {(\varphi {\alpha }_{w}G)}_t=\nabla \cdot (D_G\nabla G)-\nabla \cdot (u_{w}G)-{\lambda }_{31}G \\ +({\alpha }_c+{\alpha }_f)\left({\lambda }_{32}-{\lambda }_{33}{(\frac{G}{G_M})}^{v_G}\right) \\ {(\varphi {\alpha }_{w}C)}_t=\nabla \cdot (D_C\nabla C)-\nabla \cdot (u_{w}C)-C{M}_c{Q}_l \\ +G\rho \left({\lambda }_{41}-{\lambda }_{42}{(\frac{C}{C_M})}^2-{\lambda }_{43}{(\frac{C}{C_M})}^{v_C}\right)-{\lambda }_{44}{\alpha }_{c}C, \\ {(\varphi {\alpha }_{w}H)}_t=\nabla \cdot (D_H\nabla H)-\nabla \cdot (u_{w}H)-H{M}_H{Q}_l-{\lambda }_{51}H \\ +{\alpha }_f\left({\lambda }_{52}-{\lambda }_{53}{(\frac{H}{H_M})}^2-{\lambda }_{54}{(\frac{H}{H_M})}^{v_H}\right)-{\lambda }_{55}{\alpha }_{f}H \end{gathered}$$ (34)

We use the transformation (x, t) → ($\tilde{x}$, $\tilde{t}$) in combination with the dimensionless variables defined above. This yields the following dimensionless version of the model

$$\begin{gathered} {({\alpha }_c)}_{\tilde{t}}+\nabla \cdot ({\alpha }_c{\tilde{u}}_c)={\tilde{S}}_c \\ {({\alpha }_f)}_{\tilde{t}}+\nabla \cdot ({\alpha }_f{\tilde{u}}_f)={\tilde{S}}_f, \\ {({\alpha }_w)}_{\tilde{t}}+\nabla \cdot ({\alpha }_w{\tilde{u}}_w)=-{\tilde{S}}_c-{\tilde{S}}_f+\tilde{Q} \\ {\alpha }_c\nabla ({\tilde{P}}_w+\text{Δ}{\tilde{P}}_{cw}+{\tilde{\mathrm{\Lambda }}}_C)=-\frac{{\tilde{\widehat{\zeta }}}_c}{\varphi }{\tilde{u}}_c+\frac{{\widehat{\tilde{\zeta }}}_{cf}}{\varphi }({\tilde{u}}_f-{\tilde{u}}_c) \\ {\alpha }_f\nabla ({\tilde{P}}_w+{\tilde{\mathrm{\Lambda }}}_H)=-\frac{{\tilde{\widehat{\zeta }}}_f}{\varphi }{\tilde{u}}_f-\frac{{\tilde{\widehat{\zeta }}}_{cf}}{\varphi }({\tilde{u}}_f-{\tilde{u}}_c) \\ {\alpha }_w\nabla {\tilde{P}}_w=-\frac{{\tilde{\widehat{\zeta }}}_w}{\varphi }{\tilde{u}}_w \\ {\tilde{\rho }}_t=-{\tilde{\lambda }}_{21}\tilde{G}\tilde{\rho }+\tilde{\rho }\left({\tilde{\lambda }}_{22}-{\tilde{\lambda }}_{23}{\alpha }_c-{\tilde{\lambda }}_{24}(\frac{\tilde{\rho }}{{\tilde{\rho }}_M})\right) \\ {({\alpha }_w\tilde{G})}_t=\nabla \cdot ({\tilde{D}}_G\nabla \tilde{G})-\nabla \cdot (\widetilde{u_w}\tilde{G})-{\tilde{\lambda }}_{31}\tilde{G} \\ +({\alpha }_c+{\alpha }_f)\left({\tilde{\lambda }}_{32}-{\tilde{\lambda }}_{33}{(\frac{\tilde{G}}{{\tilde{G}}_M})}^{v_G}\right) \\ {({\alpha }_w\tilde{C})}_t=\nabla \cdot ({\tilde{D}}_C\nabla \tilde{C})-\nabla \cdot ({\tilde{u}}_w\tilde{C})-\tilde{C}{M}_c{\tilde{Q}}_l \\ +\tilde{G}\tilde{\rho }\left({\tilde{\lambda }}_{41}-{\tilde{\lambda }}_{42}{(\frac{\tilde{C}}{{\tilde{C}}_M})}^2-{\tilde{\lambda }}_{43}{(\frac{\tilde{C}}{{\tilde{C}}_M})}^{v_C}\right)-{\tilde{\lambda }}_{44}{\alpha }_c\tilde{C}, \\ {({\alpha }_w\tilde{H})}_t=\nabla \cdot ({\tilde{D}}_H\nabla \tilde{H})-\nabla \cdot ({\tilde{u}}_w\tilde{H})-\tilde{H}{M}_H{\tilde{Q}}_l-{\tilde{\lambda }}_{51}\tilde{H} \\ +{\alpha }_f\left({\tilde{\lambda }}_{52}-{\tilde{\lambda }}_{53}{(\frac{\tilde{H}}{{\tilde{H}}_M})}^2-{\tilde{\lambda }}_{54}{(\frac{\tilde{H}}{{\tilde{H}}_M})}^{v_H}\right)-{\tilde{\lambda }}_{55}{\alpha }_f\tilde{H} \end{gathered}$$ (35)