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How Tumor Cells Can Make Use of Interstitial Fluid Flow in a Strategy for Metastasis

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Abstract

Introduction

The phenomenon of lymph node metastasis has been known for a long time. However, the underlying mechanism by which malignant tumor cells are able to break loose from the primary tumor site remains unclear. In particular, two competing fluid sensitive migration mechanisms have been reported in the experimental literature: (i) autologous chemotaxis (Shields et al. in Cancer Cell 11:526–538, 2007) which gives rise to downstream migration; (ii) an integrin-mediated and strain-induced upstream mechanism (Polacheck et al. in PNAS 108:11115–11120, 2011). How can these two competing mechanisms be used as a means for metastatic behavior in a realistic tumor setting? Excessive fluid flow is typically produced from leaky intratumoral blood vessels and collected by lymphatics in the peritumoral region giving rise to a heterogeneous fluid velocity field and a corresponding heterogeneous cell migration behavior, quite different from the experimental setup.

Method

In order to shed light on this issue there is a need for tools which allow one to extrapolate the observed single cell behavior in a homogeneous microfluidic environment to a more realistic, higher-dimensional tumor setting. Here we explore this issue by using a computational multiphase model. The model has been trained with data from the experimental results mentioned above which essentially reflect one-dimensional behavior. We extend the model to an envisioned idealized two-dimensional tumor setting.

Result

A main observation from the simulation is that the autologous chemotaxis migration mechanism, which triggers tumor cells to go with the flow in the direction of lymphatics, becomes much more aggressive and effective as a means for metastasis in the presence of realistic IF flow. This is because the outwardly directed IF flow generates upstream cell migration that possibly empowers small clusters of tumor cells to break loose from the primary tumor periphery. Without this upstream stress-mediated migration, autologous chemotaxis is inclined to move cells at the rim of the tumor in a homogeneous and collective, but space-demanding style. In contrast, inclusion of realistic IF flow generates upstream migration that allows two different aspects to be synthesized: maintain the coherency and solidity of the the primary tumor and at the same time cleave the outgoing waves of tumor cells into small clusters at the front that can move collectively in a more specific direction.

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Acknowlegments

We are grateful for highly instructive comments from the anonymous reviewers that helped improving certain parts of a first version of this manuscript. We also thank Tia R. Tidwell for useful input.

Conflict of interest

Steinar Evje and Jahn Otto Waldeland declare that they have no conflicts of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.

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Corresponding author

Correspondence to Steinar Evje.

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Associate Editor Partha Roy oversaw the review of this article.

Appendices

Appendix A: A Swartz–Kamm Model

The cell–fluid two-phase model we are interested in takes the following form (we refer to Refs. 3, 4, 6, 15, 28, 34, 35, and 36 for details):

$$\begin{aligned}&(\alpha _{\text{c}}\rho _{\text{c}})_{\text{t}} + \nabla \cdot (\alpha _{\text{c}}\rho _{\text{c}} {\mathbf {u}}_{\text{c}})= \rho _{\text{c}} S_{\text{c}}, \nonumber \\&(\alpha _{\text{w}}\rho _{\text{w}})_{\text{t}} + \nabla \cdot (\alpha _{\text{w}}\rho _{\text{w}} {\mathbf {u}}_{\text{w}})=- \rho _{\text{w}} S_{\text{c}} + \rho _{\text{w}} Q, \nonumber \\&\alpha _{\text{c}} \nabla P_{\text{c}}= -{\hat{\zeta }}_{\text{c}}{\mathbf {u}}_{\text{c}}+ {\hat{\zeta }}({\mathbf {u}}_{\text{w}}- {\mathbf {u}}_{\text{c}}) \nonumber \\&\alpha _{\text{w}}\nabla P_{\text{w}}= -{\hat{\zeta }}_{\text{w}}{\mathbf {u}}_{\text{w}}- {\hat{\zeta }}({\mathbf {u}}_{\text{w}}- {\mathbf {u}}_{\text{c}}) \nonumber \\&\rho _{\text{t}} ={-\lambda _{21}G \rho } + \rho \left (\lambda _{22} - \lambda _{23}\alpha _{\text{c}} - \lambda _{24}\left (\frac{\rho }{\rho _{\text{M}}}\right )\right )\nonumber \\&G_{\text{t}} = \nabla \cdot (D_{\text{G}} \nabla G) - \nabla \cdot (G {\mathbf {u}}_{\text{w}})- \lambda _{31} G \nonumber \\ {}&\qquad +\,\,\alpha _{\text{c}}\left ( \lambda _{32}-\lambda _{33}\left (\frac{G}{G_{\text{M}}}\right )^{\nu _1} \right ) \nonumber \\&C_{\text{t}} = \nabla \cdot (D_{\text{C}} \nabla C) - \nabla \cdot (C {\mathbf {u}}_{\text{w}})- \lambda _{44}\alpha _{\text{c}}\nonumber \\ {}&\qquad + \,\,G \rho \left ({\lambda }_{41} - {\lambda }_{42}\left (\frac{C}{C_{\text{M}}}\right )^2\nonumber -\lambda _{43}\left (\frac{C}{C_{\text{M}}}\right )^{\nu _2}\right ) ,\nonumber \\&\text {with } S_{\text{c}}=\alpha _{\text{c}}\left (\lambda _{11} - \lambda _{12}\alpha _{\text{c}} - \lambda _{13}\frac{\rho }{\rho _{\text{M}}}\right ) \nonumber \\&\qquad \,\,\, Q=Q_{\text{v}}-Q_{\text{l}}, \end{aligned}$$
(19)

where \({\mathbf {u}}_i=(u_i^x,u_i^y)\) for \(i={\text{c}}, {\text{w}}\). The above model must be combined with the closure relation

$$\begin{aligned} \alpha _{\text{c}}+\alpha _{\text{w}}=1 \end{aligned}$$
(20)

and appropriate pressure-density closure relations \(\rho _i=\rho _i(P_i)\), \(i={\text{c}}, {\text{w}}\). We implicitly treat the cell phase as a fluid like phase but where we add cell-specific features to the momentum Eq. (19)3 by letting the cell phase pressure \(P_{\text{c}}\) feel additional stress due to cell–cell interaction and migration-related stress due to chemotaxis through the relation

$$\begin{aligned} P_{\text{c}}=P_{\text{w}} + \Delta P(\alpha _{\text{c}}) + \Lambda (C). \end{aligned}$$
(21)

This means that the stress \(P_{\text{c}}\) associated with the cancer cells differs from the IF pressure \(P_{\text{w}}\) because of the cell–cell stress term \(\Delta P\) and the chemotaxis stress term \(\Lambda \). Similar to Ref. 6 we use

$$\begin{aligned} \Delta P(\alpha _{\text{c}})=\gamma J(1-\alpha _{\text{c}}). \end{aligned}$$
(22)

Herein, \(\gamma >0\) is a coefficient (unit Pa) that depends linearly on the surface tension (unit Pa m) whereas \(J(\alpha _{\text{w}})\) is a monotonic decreasing dimensionless function with respect to the fluid volume fraction \(\alpha _{\text{w}}\). The ability of the cancer cells to generate a force is expressed through the potential function \(\Lambda (C)\)

$$\begin{aligned} \Lambda (C)=\Lambda _0 - \frac{\Lambda _1}{1 + \exp (-\xi _1(C-C_{\text{M}}))}, \end{aligned}$$
(23)

where \(\Lambda _0,\Lambda _1,\xi _1\) are constant parameters with units, respectively, as \([\Lambda _0,\Lambda _1]=\text {Pa}\) and \([\xi _1]=\text {m}^3/\text {kg}\).

There is a drag force (that acts opposite of the direction of movement of the fluid) between the extracellular fluid represented by the fluid velocity \({\mathbf {u}}_{\text{w}}\) and the ECM structure (fibers). We use the following expression for this force (motivated by general multiphase flow modeling)

$$\begin{aligned} {\hat{\zeta }}_{\text{w}} = I_{\text{w}}{\hat{k}}_{\text{w}} \alpha _{\text{w}}^{r_{\text{w}}}, \qquad {\hat{k}}_{\text{w}}>0, \quad r_{\text{w}}<2, \end{aligned}$$
(24)

with \(I_{\text{w}}=\frac{\mu _{\text{w}}}{K}>0\) and K is the permeability of the porous media and \(\mu _{\text{w}}\) the fluid viscosity. Similarly, there is a drag force between the cells and the ECM (fibers) that acts opposite of the direction of the movement of the cells represented by the cell fluid velocity \({\mathbf {u}}_{\text{c}}\),

$$\begin{aligned} {\hat{\zeta }}_{\text{c}} = I_{\text{c}}{\hat{k}}_{\text{c}} \alpha _{\text{c}}^{r_{\text{c}}}, \qquad I_{\text{c}},{\hat{k}}_{\text{c}}>0, \quad r_{\text{c}}<2, \end{aligned}$$
(25)

where \(I_{\text{c}}\) (\({\text {Pa s}}/{\text {m}^2}\)), \({\hat{k}}_{\text{c}}\) and \(r_{\text{c}}\) must be specified (the two last are dimensionless).

Finally, there is also a drag force between the cell phase and the fluid which is caused by pressure (isotropic) and shear stress forces on the surface of the cell phase. This effect is accounted for through the term \({\hat{\zeta }}({\mathbf {u}}_{\text{w}}-{\mathbf {u}}_{\text{c}})\), see (19)\(_{3,4}\) where

$$\begin{aligned} {\hat{\zeta }}= I{\hat{k}}\alpha _{\text{w}}\alpha _{\text{c}}^{1+r_{\text{cw}}}, \qquad {\hat{k}}>0, \quad r_{\text{cw}}>0, \end{aligned}$$
(26)

where I (\({\text {Pa s}}/{\text {m}^2}\)) remains to be determined as well as the dimensionless \({\hat{k}}\) and coefficient \(r_{\text{cw}}\).

Lymphatic flow is an important component of the circulation. In nearly all tissues, plasma leaks out of blood capillaries, flows through the interstitium and drains into lymphatic vessels, where it passes through lymph nodes before being returned to the venous blood.16 This circulation is expressed in the right hand side of (19)2 through the source terms \(Q=Q_{\text{v}}-Q_{\text{l}}\). The driving forces for interstitial flow \(Q_{\text{v}}\) are hydrostatic and osmotic pressure gradients between the vascular and interstitial space. Starling’s Law is used for the flow of fluid in the interstitium given by

$$\begin{aligned} \begin{aligned} Q_{\text{v}}&= T_{\text{v}}\left (P_{\text{v}}^*-P_{\text{w}} - \sigma _{\text{T}}(\pi _{\text{v}}^*-\pi _{\text{w}}) \right )\\ {}&= T_{\text{v}}\left (\widetilde{P_{\text{v}}}^*-P_{\text{w}} \right ), \qquad T_{\text{v}}=L_{\text{v}}\frac{S_{\text{v}}}{V}, \end{aligned} \end{aligned}$$
(27)

with \(\widetilde{P_{\text{v}}}^*=P_{\text{v}}^*- \sigma _{\text{T}}(\pi _{\text{v}}^*-\pi _{\text{w}})\). Here \(L_{\text{v}}\) is the hydraulic conductivity (\(\text {m}^2 {\text{s}}/\text {kg}=\text {m}/\text {Pa s}\)) of the vessel wall, \(S_{\text{v}}/V\) (\(\text {m}^{-1})\) the exchange area of blood vessels per unit volume of tissues V, \(P_{\text{v}}^*\) and \(P_{\text{w}}\) the vascular and interstitial fluid pressure, \(\pi _{\text{v}}^*\) and \(\pi _{\text{w}}\) the osmotic pressure in the vascular and interstitial space and \(\sigma _{\text{T}}\) the osmotic reflection coefficient for plasma proteins.

The lymphatic network drains excessive fluid from the interstitial space and returns it back to the blood circulation, as expressed by \(Q_{\text{l}}\). By doing so, it regulates the fluid balance in tissues and prevents formation of edema. Tumor lymphatics have two characteristics, common in many cancers. They are not functional in the intratumoral region, and they are hyperplastic and exhibit increased flow at the periphery.16 The loss of functionality is attributed to compressive solid stress that is developed in tumors. This stress has been shown to collapse intratumoral lymphatic vessels, and thus eliminates lymph flow. Similar to (27) we use an expression of the following form to express the fluid adsorption through lymphatics

$$\begin{aligned} Q_{\text{l}}&= T_{\text{l}}(P_{\text{w}} - P_{\text{l}}^*), \qquad T_{\text{l}} = L_{\text{l}}\frac{S_{\text{l}}}{V}. \end{aligned}$$
(28)

Here \(L_{\text{l}}\) is the hydraulic conductivity of the lymphatic vessel walls whereas \(S_{\text{l}}/V\) is the surface area of the lymphatic vessel per volume unit of tissues V and \(P_{\text{l}}^*\) is the effective lymphatic pressure.

Rewritten Form of the Swartz–Kamm Model

We rewrite the model (19) to make it more transparent. In particular, we can obtain explicit expressions for cell velocity \({\mathbf {u}}_{\text{c}}\) and IF velocity \({\mathbf {u}}_{\text{w}}\). Following35,36 we assume incompressible fluids (cell population and fluid). From (19), after we have made it dimensionless (see Ref. 35 for details) and also used (21)

$$\begin{aligned}&\alpha _{\text{ct}} + \nabla \cdot (\alpha _{\text{c}} {\mathbf {u}}_{\text{c}})=S_{\text{c}}, \nonumber \\&\alpha _{\text{wt}} + \nabla \cdot (\alpha _{\text{w}} {\mathbf {u}}_{\text{w}})=-S_{\text{c}} + (Q_{\text{v}}-Q_{\text{l}})\nonumber \\&\alpha _{\text{c}}\nabla ( P_{w}+\Delta P (\alpha _{\text{w}}) + \Lambda (C))=-{\hat{\zeta }}_{\text{c}}{\mathbf {u}}_{\text{c}}+ {\hat{\zeta }}({\mathbf {u}}_{\text{w}}- {\mathbf {u}}_{\text{c}})\nonumber \\&\alpha _{\text{w}}\nabla P_{w}=-{\hat{\zeta }}_{\text{w}}{\mathbf {u}}_{\text{w}}- {\hat{\zeta }}({\mathbf {u}}_{\text{w}}- {\mathbf {u}}_{\text{c}}) \nonumber \\&\rho _{\text{t}} ={-\lambda _{21}G \rho } + \rho \left (\lambda _{22} - \lambda _{23}\alpha _{\text{c}} - \lambda _{24}(\frac{\rho }{\rho _{\text{M}}})\right )\nonumber \\&G_{\text{t}}= \nabla \cdot (D_{\text{G}} \nabla G) - \nabla \cdot ({\mathbf {u}}_{\text{w}}G) - \lambda _{31} G \nonumber \\ {}&\quad \quad+\,\, \alpha _{\text{c}}\left ({\lambda }_{32}-\lambda _{33}\left (\frac{G}{G_{\text{M}}}\right )^{\nu _{\text{G}}}\right )\nonumber \\&C_{\text{t}} = \nabla \cdot (D_{\text{C}} \nabla C) - \nabla \cdot ({\mathbf {u}}_{\text{w}}C)-\lambda _{44}\alpha _{\text{c}}\nonumber \\ {}&\quad \quad + \,\, G\rho \left (\lambda _{41}-\lambda _{42}\left (\frac{C}{C_{\text{M}}}\right )^2-\lambda _{43}\left (\frac{C}{C_{\text{M}}}\right )^{\nu _{\text{C}}}\right ), \end{aligned}$$
(29)

with \({\mathbf {u}}_i=(u_i^x,u_i^y)\) for \(i= {\text{c}}, {\text{w}}\). The model is combined with the boundary condition

$$\begin{aligned} P_{\text{w}}{\Big |}_{\partial \Omega }=P_{\text{B}}^*, \,\, \frac{\partial }{\partial \nu } G\Big |_{\partial \Omega }=0,\,\, \frac{\partial }{\partial \nu } C\Big |_{\partial \Omega }=0, \,\, t>0 \end{aligned}$$

where \(\nu \) is the outward normal on \(\partial \Omega \) and \(P_{\text{B}}^*\) is a known pressure. The corresponding initial data are given by (6). Referring to Appendix A (see also Ref. 35 for details) we find the cell velocity \({\mathbf {u}}_{\text{c}}\) as well as the IF velocity \({\mathbf {u}}_{\text{w}}\) expressed as

$$\begin{aligned} \begin{aligned} {\mathbf {u}}_{\text{c}}&= {\mathbf {U}}_{\text{T}} \left [ \frac{{\hat{f}}_{\text{c}}(\alpha _{\text{c}})}{\alpha _{\text{c}}} \right ] - \left [ \frac{{\hat{h}}(\alpha _{\text{c}})}{\alpha _{\text{c}}} \right ] \nabla (\Delta P + \Lambda ), \\ {\mathbf {u}}_{\text{w}}&= {\mathbf {U}}_{\text{T}} \left [ \frac{{\hat{f}}_{\text{w}}(\alpha _{\text{c}})}{\alpha _{\text{w}}} \right ] + \left [ \frac{{\hat{h}}(\alpha _{\text{c}})}{\alpha _{\text{w}}} \right ] \nabla (\Delta P + \Lambda ), \end{aligned} \end{aligned}$$
(30)

with fractional flow functions \({\hat{f}}_{\text{c}}(\alpha _{\text{c}})\) and \({\hat{f}}_{\text{w}}(\alpha _{\text{c}})\), respectively, for the cell and fluid phase given by

$$\begin{aligned} \begin{aligned} {\hat{f}}_{\text{c}}(\alpha _{\text{c}})&:=\frac{{\hat{\lambda }}_{\text{c}}}{{\hat{\lambda }}_{\text{T}}}= \frac{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}]+ \alpha _{\text{c}}{\hat{\zeta }}}{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}] + [\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + {\hat{\zeta }}}, \\ {\hat{f}}_{\text{w}}(\alpha _{\text{c}})&:=\frac{{\hat{\lambda }}_{\text{w}}}{{\hat{\lambda }}_{\text{T}}}= \frac{[\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}]+ \alpha _{\text{w}}{\hat{\zeta }}}{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}] + [\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + {\hat{\zeta }}}, \\ {\hat{h}}(\alpha _{\text{c}})&=\frac{\alpha _{\text{c}}^2\alpha _{\text{w}}^2}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}}, \end{aligned} \end{aligned}$$
(31)

where the coefficients \({\hat{\lambda }}_{\text{c}}\), \({\hat{\lambda }}_{\text{w}}\), and \({\hat{\lambda }}_{\text{T}}\) (so-called mobility functions38) are given as follows:

$$\begin{aligned} \begin{aligned} {\hat{\lambda }}_{\text{c}}&=\frac{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}]+ \alpha _{\text{c}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]},\\ {\hat{\lambda }}_{\text{w}}&=\frac{[\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + \alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}, \end{aligned} \end{aligned}$$
(32)

and

$$\begin{aligned} {\hat{\lambda }}_{\text{T}}={\hat{\lambda }}_{\text{c}}+{\hat{\lambda }}_{\text{w}}=\frac{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}] + [\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + {\hat{\zeta }} }{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}. \end{aligned}$$
(33)

Expressions for Cell Velocity and IF Velocity

We can write the momentum balance Eq. (29)3,4 as

$$\begin{aligned}&\alpha _{\text{w}}\nabla P_{w} = {\hat{\zeta }}{\mathbf {u}}_{\text{c}}- ({\hat{\zeta }}_{\text{w}}+{\hat{\zeta }}){\mathbf {u}}_{\text{w}}\\&\alpha _{c}\nabla \Lambda (C,\rho ) + \alpha _{\text{c}} \nabla (\Delta P) + \alpha _{\text{c}}\nabla P_{w} \nonumber =-({\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}){\mathbf {u}}_{\text{c}}\\ {}&\qquad \qquad \qquad \qquad + {\hat{\zeta }}{\mathbf {u}}_{\text{w}}\nonumber \end{aligned}$$
(34)

We can solve for \({\mathbf {u}}_{\text{w}}\) and \({\mathbf {u}}_{\text{c}}\) from this 2 × 2 linear system and find that

$$\begin{aligned} {\mathbf {u}}_{\text{c}}&=-\frac{[\alpha _{\text{c}} {\hat{\zeta }}_{\text{w}}]+{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla P_{w}\nonumber \\&\quad - \frac{\alpha _{\text{c}}[{\hat{\zeta }}_{\text{w}}+{\hat{\zeta }}]}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) \nonumber \\ {}&\quad - \frac{\alpha _{\text{c}}[{\hat{\zeta }}_{\text{w}}+{\hat{\zeta }}]}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla \Lambda (C) \nonumber \\ {\mathbf {u}}_{\text{w}}&=-\frac{[\alpha _{\text{w}} {\hat{\zeta }}_{\text{c}}]+{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla P_{w} \nonumber \\ {}&\quad - \frac{\alpha _{\text{c}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) \nonumber \\ {}&\quad - \frac{\alpha _{\text{c}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla \Lambda (C). \end{aligned}$$
(35)

Hence, the corresponding Darcy velocities \({\mathbf {U}}_{\text{c}}\) and \({\mathbf {U}}_{\text{w}}\) are given by (also often referred to as superficial velocity)

$$\begin{aligned} {\mathbf {U}}_{\text{c}}:=&\alpha _{\text{c}}{\mathbf {u}}_{\text{c}}=-{\hat{\lambda }}_{\text{c}} \nabla P_{w} - {\hat{\lambda }}_{\text{c}} \nabla (\Delta P) \nonumber \\ {}&\qquad \qquad + \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) - {\hat{\lambda }}_{\text{c}}\nabla \Lambda (C,\rho )\nonumber \\&\qquad \qquad + \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla \Lambda (C)\\ {\mathbf {U}}_{\text{w}}:=&\alpha _{\text{w}}{\mathbf {u}}_{\text{w}}=-{\hat{\lambda }}_{\text{w}} \nabla P_{w} - \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) \nonumber \\ {}&\qquad \qquad - \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla \Lambda (C)\nonumber \end{aligned}$$
(36)

with generalized mobility functions of the form (32) and and total mobility \({\hat{\lambda }}_{\text{T}}\) given by (33). Summing the two mass balance Eq. (29)1,2 and making use of (20), we find the following equation

$$\begin{aligned} \nabla \cdot {\mathbf {U}}_{T}=\nabla \cdot ({\mathbf {U}}_{\text{c}}+ {\mathbf {U}}_{\text{w}})=Q_{\text{v}}-Q_{\text{l}}. \end{aligned}$$
(37)

From (36), it follows after a summation that

$$\begin{aligned} \begin{aligned} {\mathbf {U}}_{\text{T}}&={\mathbf {U}}_{\text{c}}+ {\mathbf {U}}_{\text{w}}\\ {}&=-({\hat{\lambda }}_{\text{c}}+{\hat{\lambda }}_{\text{w}}) \nabla P_{w} - {\hat{\lambda }}_{\text{c}} \nabla (\Delta P) - {\hat{\lambda }}_{\text{c}}\nabla \Lambda (C,\rho ). \end{aligned} \end{aligned}$$
(38)

By taking the divergence (\(\nabla \cdot \)) of (38) and referring to (37), we then arrive at

$$\begin{aligned} \begin{aligned} \nabla \cdot ({\hat{\lambda }}_{\text{T}} \nabla P_{w})&=-(Q_{\text{v}}-Q_{\text{l}}) - \nabla \cdot ({\hat{\lambda }}_{\text{c}} \nabla (\Delta P)) \\&\quad - \nabla \cdot ({\hat{\lambda }}_{\text{c}}\nabla \Lambda (C,\rho )). \end{aligned} \end{aligned}$$
(39)

This gives an elliptic equation for \(P_{\text{w}}\) that can be solved subject to the boundary condition \(P_{\text{w}}|_{\partial \Omega }=P^*\). This in turn allows us to compute \({\mathbf {U}}_{\text{T}}\) from (38).

Elimination of Explicit Dependence on IF Pressure \(P_{\text{w}}\)

Next, we observe that we have the following expression for \(\nabla P_{\text{w}}\) [in view of (38)]

$$\begin{aligned} \nabla P_{w}= -\frac{{\mathbf {U}}_{\text{T}}}{{\hat{\lambda }}_{\text{T}}} - \frac{{\hat{\lambda }}_{\text{c}}}{{\hat{\lambda }}_{\text{T}}}\nabla (\Delta P) - \frac{{\hat{\lambda }}_{\text{c}}}{{\hat{\lambda }}_{\text{T}}} \nabla \Lambda (C). \end{aligned}$$
(40)

Combining this with (36) we can derive expressions for the fluid velocities \({\mathbf {U}}_{\text{c}}\) and \({\mathbf {U}}_{\text{w}}\) as follows:

$$\begin{aligned} {\mathbf {U}}_{\text{c}}=&-{\hat{\lambda }}_{\text{c}} \nabla P_{w} - {\hat{\lambda }}_{\text{c}} \nabla (\Delta P)\nonumber \\&\quad + \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) - {\hat{\lambda }}_{\text{c}}\nabla \Lambda (C)\nonumber \\ {}&\quad + \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla \Lambda (C) \nonumber \\&= {\mathbf {U}}_{\text{T}}{\hat{f}}_{\text{c}}(\alpha _{\text{c}}) - {\hat{h}}(\alpha _{\text{c}}) \nabla (\Delta P) - {\hat{h}}(\alpha _{\text{c}}) \nabla \Lambda (C) \end{aligned}$$
(41)

where we use the expressions (32) and (33) in combination with some algebraic manipulations, and where we have defined \({\hat{f}}_{\text{c}}\), \({\hat{f}}_{\text{w}}\) and \({\hat{h}}\) as follows:

$$\begin{aligned} \begin{aligned} {\hat{f}}_{\text{c}}(\alpha _{\text{c}})&= \frac{{\hat{\lambda }}_{\text{c}}}{{\hat{\lambda }}_{\text{T}}}= \frac{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}]+ \alpha _{\text{c}}{\hat{\zeta }}}{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}] + [\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + {\hat{\zeta }}}, \\ {\hat{f}}_{\text{w}}(\alpha _{\text{c}})&= \frac{{\hat{\lambda }}_{\text{w}}}{{\hat{\lambda }}_{\text{T}}}= \frac{[\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}]+ \alpha _{\text{w}}{\hat{\zeta }}}{[\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}}] + [\alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}] + {\hat{\zeta }}}, \\ {\hat{h}}(\alpha _{\text{c}})&=\frac{\alpha _{\text{c}}^2\alpha _{\text{w}}^2}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}}. \end{aligned} \end{aligned}$$
(42)

Similarly, for \({\mathbf {U}}_{\text{w}}\) we find

$$\begin{aligned} {\mathbf {U}}_{\text{w}}&=-{\hat{\lambda }}_{\text{w}} \nabla P_{w} - \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]}\nabla (\Delta P) \nonumber \\ {}&\quad - \frac{\alpha _{\text{c}}\alpha _{\text{w}}{\hat{\zeta }}}{{\hat{\zeta }}_{\text{c}}{\hat{\zeta }}_{\text{w}} + {\hat{\zeta }}[{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}_{\text{w}}]} \nabla \Lambda (C)\\&= {\mathbf {U}}_{\text{T}}{\hat{f}}_{\text{w}}(\alpha _{\text{c}}) + {\hat{h}}(\alpha _{\text{c}}) \nabla (\Delta P) + {\hat{h}}(\alpha _{c})\nabla \Lambda (C).\nonumber \end{aligned}$$
(43)

This amounts to, expressed in terms of interstitial velocity \({\mathbf {u}}_{\text{c}}\) and \({\mathbf {u}}_{\text{w}}\), relations of the form

$$\begin{aligned} {\mathbf {u}}_{\text{c}}=&\,\,{\mathbf {U}}_{\text{T}}\left [ \frac{\alpha _{\text{c}}{\hat{\zeta }}_{\text{w}}+{\hat{\zeta }}}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ] - \left [ \frac{\alpha _{\text{c}}\alpha _{\text{w}}^2}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ] \nabla (\Delta P) \nonumber \\ {}&\quad - \left [ \frac{\alpha _{\text{c}}\alpha _{\text{w}}^2}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ]\nabla \Lambda (C) \nonumber \\ {\mathbf {u}}_{\text{w}}=&\,\,{\mathbf {U}}_{\text{T}} \left [ \frac{\alpha _{\text{w}}{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ] + \left [ \frac{\alpha _{\text{c}}^2\alpha _{\text{w}}}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ] \nabla (\Delta P) \nonumber \\ {}&\quad + \left [ \frac{\alpha _{\text{c}}^2\alpha _{\text{w}}}{\alpha _{\text{c}}^2{\hat{\zeta }}_{\text{w}} + \alpha _{\text{w}}^2{\hat{\zeta }}_{\text{c}}+{\hat{\zeta }}} \right ] \nabla \Lambda (C) \end{aligned}$$
(44)

Appendix B: Model Input Data

We refer to Table 1 for the model input data that is used in this work and which are based on the investigations in Refs. 35 and 36.

Table 1 Model parameters (dimensional) in the model (2) with relevant reference values.

Appendix C: Supplementary Simulation

Base Case with the Effect of a Denser ECM with Reduced Conductivity Secondly, we also want to test the model prediction when we take into consideration that the tumor cells sense the increased fluid–ECM resistance force through the change of \({\hat{k}}_{\text{w}}\), which leads to a stronger fluid imposed stress on the tumor cells, and respond to it by doing an adjustment such that the fluid–stress term \({\mathbf {u}}_{c,{\text {fluid}}-{\text{stress}}}\) can resist this stronger stress from the flowing fluid and also go in the upstream migration. More precisely, we make an adjustment of \({\hat{k}}\) in (15) regulating the cell–fluid interaction by changing it from \({\hat{k}}=1\) to \({\hat{k}}=15\). The effect of this adjustment is a fractional flow function \({\hat{f}}_{\text{c}}(\alpha _{\text{c}})\) as shown in Fig. 11b which again contains a large negative dip signaling upstream migration. However, as for the base case it also contains a positive part for sufficiently small volume fractions that will guide tumor cells in the flow direction.

Simulation results are shown in Figs. 13 and 14. Comparing Figs. 9a and 13a, it is observed that in the latter case the tumor cells can go towards chemokine gradients more independent of the relatively strong fluid flow towards the lymphatics (see panel e). In the first case (panel a, Fig. 9) the tumor clusters tend to be guided towards the lymphatics driven by both the \({\mathbf {u}}_{c,\text {chemotaxis}}\) and \({\mathbf {u}}_{c,{\text {fluid}}-{\text{stress}}}\) velocity component. For example, the upmost cluster is split into three clusters where two of them will bend towards nearby lymphatics consistent with a corresponding IF flow field. This is not seen for the same cluster in Fig. 13a. The corresponding cell velocity \({\mathbf {u}}_{\text{c}}\) seen, respectively, in Fig. 10a and Fig. 14a also reflects this difference. Another difference is that the clusters reflected by Fig. 13a that migrate toward the lymphatics, respectively, in the northwest, northeast, southeast, and southwest direction tend to generate a tail that almost connects with the primary tumor. We attribute this behavior to the fluid–stress component \({\mathbf {u}}_{c,{\text {fluid}}-{\text{stress}}}\) which can give rise to upstream migration if the cell volume fraction is larger than 0.035 as reflected by Fig. 11b. See Figs. 14c and 14c*, for an example of this.

Figure 13
figure 13

Dense ECM and reduced hydraulic conductivity gives increased detachment of clusters. The same case as in Fig. 9 but where we have accounted for tumor cell sensitivity to the reduced conductivity by also increasing cell–fluid interaction setting \({\hat{k}}=15\). This gives rise to upstream migration generated by fluid stress through \({\mathbf {u}}_{c,{\text {fluid}}-{\text{stress}}}\) as reflected by the negative dip of \({\hat{f}}_{\text{c}}(\alpha _{\text{c}})\) in Fig. 11b. (a) Cell volume fraction. (b) Protease distribution. (c) Chemokine distribution. (d) IF pressure \(P_{\text{w}}\). (e) IF velocity \({\mathbf {u}}_{\text{w}}\).

Figure 14
figure 14

Dense ECM and reduced hydraulic conductivity gives increased detachment of clusters. (a, a*) Net cell velocity \({\mathbf {u}}_{\text{c}}\). (b, b*) Dispersive cell velocity \({\mathbf {u}}_{c,\text {dispersion}}\). (c, c*) The fluid–stress component \({\mathbf {u}}_{c,{\text {fluid}}-{\text{stress}}}\) contributes mostly in the peritumoral region and is a driving force for the outwardly directed migration but gives rise to upstream migration when \(\alpha _{\text{c}}\) is sufficiently large, see left and right cluster in c* . (d, d*) The chemotactic component \({\mathbf {u}}_{c,\text {chemotaxis}}\) contributes greatly both at the margin of the primary tumor and at the local clusters.

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Evje, S., Waldeland, J.O. How Tumor Cells Can Make Use of Interstitial Fluid Flow in a Strategy for Metastasis. Cel. Mol. Bioeng. 12, 227–254 (2019). https://doi.org/10.1007/s12195-019-00569-0

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