Three phase flow dynamics in tumor growth

Abstract

Existing tumor models generally consider only a single pressure for all the cell phases. Here, a three-fluid model originally proposed by the authors is further developed to allow for different pressures in the host cells (HC), the tumor cells (TC) and the interstitial fluid (IF) phases. Unlike traditional mixture theory models, this model developed within the thermodynamically constrained averaging theory contains all the necessary interfaces. Appropriate constitutive relationships for the pressure difference among the three fluid phases are introduced with respect to their relative wettability and fluid–fluid interfacial tensions, resulting in a more realistic modeling of cell adhesion and invasion. Five different tumor cases are studied by changing the interfacial tension between the three liquid phases, adhesion and dynamic viscosity. Since these parameters govern the relative velocities of the fluid phases and the adhesion of the phases to the extracellular matrix significant changes in tumor growth are observed. High interfacial tensions at the TC–IF and TC–HC interface support the lateral displacement of the healthy tissue in favor of a rapid growth of the malignant mass, with a relevant amount of HC which cannot be pushed out by TC and remain in place. On the other hand, lower TC–IF and TC–HC interfacial tensions tend to originate a more compact and dense tumor mass with a slower growth rate of the overall size. This novel computational model emphasizes the importance of characterizing the TC–HC interfacial properties to properly predict the temporal and spatial pattern evolution of tumor.

Appendices

Appendix A: General forms of mass and momentum balance equations

The mass balance equation for any phase $\alpha$ using averaging theorems is

$$\begin{aligned} \frac{\partial \left( {\varepsilon ^{\alpha }\rho ^{\alpha }} \right) }{\partial t}+\nabla \cdot \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\mathbf{v}^{\overline{\alpha }}} \right) -\sum _{\kappa \in \mathfrak {I}_{c\alpha } } {\mathop {M}\limits ^{\kappa \rightarrow \alpha }} =0 \end{aligned}$$

(32)

where $\varepsilon ^{\alpha }$ is the volume fraction; $\rho ^{\alpha }$ is the density, $\mathbf{v}^{\overline{\alpha }}$ is the local velocity vector,$\mathop {M}\limits ^{\kappa \rightarrow \alpha }$ are the mass exchange terms accounting for transport of mass at the interface between the phases $\kappa$ and $\alpha$, and $\sum _{\kappa \in \mathfrak {I}_{c\alpha }}$ is the summation over all the phases exchanging mass at the interfaces with the phase $\alpha$. However, if the interface is treated as massless, the transfer is to the adjacent phases, designated as $\kappa$.

Any species $i$ dispersed within the phase $\alpha$ has to satisfy mass conservation too, and therefore the following equation is derived by averaging

$$\begin{aligned}&\frac{\partial \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\omega ^{\overline{i\alpha } }} \right) }{\partial t}+\nabla \cdot \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\omega ^{\overline{i\alpha } }\mathbf{v}^{\overline{\alpha }}} \right) \nonumber \\&\quad +\nabla \cdot \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\omega ^{\overline{i\alpha } }\mathbf{u}^{\overline{\overline{i\alpha }} }} \right) -\varepsilon ^{\alpha }r^{i\alpha }+\sum _\kappa {\mathop {M}\limits ^{i\alpha \rightarrow i\kappa } } =0 \end{aligned}$$

(33)

where $\omega ^{\overline{i\alpha }}$ identifies the mass fraction of the species $i$ dispersed with the phase $\alpha$, $\varepsilon ^{\alpha }r^{i\alpha }$ is a reaction term that takes into account the reactions between the species $i$ and the other chemical species dispersed in phase $\alpha$, and $\mathbf{u}^{\overline{\overline{i\alpha }}}$ is the diffusive velocity of species $i$.

The momentum equation for the phase $\alpha$, including multiple species $i$, is

$$\begin{aligned}&\frac{\partial \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\mathbf{v}^{\overline{\alpha }}} \right) }{\partial t}+\nabla \cdot \left( {\varepsilon ^{\alpha }\rho ^{\alpha }\mathbf{v}^{\overline{\alpha }}\mathbf{v}^{\overline{\alpha }}} \right) -\nabla \cdot \left( {\varepsilon ^{\alpha }\mathbf{t}^{\overline{\overline{\alpha }} }} \right) \nonumber \\&\quad -\varepsilon ^{\alpha }\rho ^{\alpha }\mathbf{g}^{\overline{\alpha } }-\sum _{\kappa \in \mathfrak {I}_{c\alpha } } {\left( {\sum _{i\in \mathfrak {I}_s} {\mathop {M_{v}}\limits ^{i\kappa \rightarrow i\alpha } \mathbf{v}^{\overline{\alpha }}} +\mathop {\mathbf{T}}\limits ^{\kappa \rightarrow \alpha } } \right) } =0 \end{aligned}$$

(34)

where $\mathbf{g}^{\overline{\alpha }}$ is the body force, $\mathop {M_v}\limits ^{i\kappa \rightarrow i\alpha } \mathbf{v}^{\overline{\alpha }}$ represents the momentum exchange from the phase $\kappa$ to the phase $\alpha$ due to mass exchange of species $i,\, \mathbf{t}^{\overline{\overline{\alpha }}}$ is the stress tensor and $\mathop {\mathbf{T}}\limits ^{\kappa \rightarrow \alpha }$ is the interaction force between phase $\alpha$ and the adjacent interfaces. Given the characteristic times scales (hours and days) of the problem and the small difference in density between cells and aqueous solutions, inertial forces as well as the force due to mass exchange are neglected so that the momentum equation simplifies to

$$\begin{aligned} -\nabla \cdot \left( {\varepsilon ^{\alpha }\mathbf{t}^{\overline{\overline{\alpha }} }} \right) -\varepsilon ^{\alpha }\rho ^{\alpha }\mathbf{g}^{\overline{\alpha }}-\sum _{\kappa \in \mathfrak {I}_{c\alpha }} {\mathop {\mathbf{T}}\limits ^{\kappa \rightarrow \alpha }} =0 \end{aligned}$$

(35)

From TCAT [32] it can be shown that the stress tensor for a fluid phase is of the form $\mathbf{t}^{\overline{\overline{\alpha }} }=-p^{\alpha }\mathbf{1}$, with $p^{\alpha }$ being the averaged fluid pressure and 1 the unit tensor, and that the momentum balance equation can be simplified to

$$\begin{aligned} \varepsilon ^{\alpha }\nabla p^{\alpha }+\mathbf{R}^{\alpha }\cdot (\mathbf{v}^{\overline{\alpha }}-\mathbf{v}^{\overline{s}})=0 \end{aligned}$$

(36)

where $\mathbf{R}^{\upalpha }$ is the resistance tensor.

The form of $\left( {\mathbf{R}^{\alpha }} \right) ^{-1}$ follows the modeling of multiphase flow in porous media [19, 30]; that is to say

$$\begin{aligned} \left( {\mathbf{R}^{\alpha }} \right) ^{-1}=\frac{k_{rel}^\alpha \mathbf{k}}{\mu ^{\alpha }\left( {\varepsilon ^{\alpha }} \right) ^{2}}\qquad \qquad \left( {\alpha =h,t,l} \right) \end{aligned}$$

(37)

where $\mathbf{k}$ and $\mu ^{\alpha }$ are the intrinsic permeability tensor of the ECM and the dynamic viscosity of the of phase $\alpha$, respectively, and $k_{rel}^\alpha$ is the relative permeability.

By introducing (37) in (36), the relative velocity of phase $\alpha$ is given by

$$\begin{aligned} \mathbf{v}^{\overline{\alpha }}-\mathbf{v}^{\overline{s} }=-\frac{k_{rel}^\alpha \mathbf{k}}{\mu ^{\alpha }\varepsilon ^{\alpha }}\cdot \nabla p^{\alpha }\qquad \qquad \left( {\alpha =h,t,l} \right) \end{aligned}$$

(38)

Appendix B: Coefficients of the matrices appearing in equation (30)

In the following equations $K^{\mathrm{s}}$ is the Bulk modulus of the solid skeleton and

$$\begin{aligned} \frac{\partial \mathbf{e}_{sw}^s }{\partial t}=\frac{\mathbf{1}}{3K^{s}}\frac{\partial p^{s}}{\partial t}. \end{aligned}$$

$$\begin{aligned}&\mathbf{C}_{nn} =\int \limits _\Omega {\mathbf{N}_n^T \left( {\varepsilon S^{l}\mathbf{N}_n} \right) } d\Omega \end{aligned}$$

(39)

$$\begin{aligned}&\mathbf{C}_{tt} =\int \limits _\Omega {\mathbf{N}_t^T \left[ {\varepsilon \mathbf{N}_t^ +\frac{S^{t}}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial S^{t}}+S^{h}\frac{\partial p^{h}}{\partial S^{t}}+p^{t}-p^{l}} \right) \mathbf{N}_{t}} \right] } d\Omega \nonumber \\\end{aligned}$$

(40)

$$\begin{aligned}&\mathbf{C}_{th} =\int \limits _\Omega {\mathbf{N}_t^T \left[ {\frac{S^{t}}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial S^{h}}+S^{h}\frac{\partial p^{h}}{\partial S^{h}}+p^{h}-p^{l}} \right) \mathbf{N}_h} \right] } d\Omega \end{aligned}$$

(41)

$$\begin{aligned}&\mathbf{C}_{tl} =\int \limits _\Omega {\mathbf{N}_t^T \left[ {\frac{S^{t}}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial p^{l}}+S^{h}\frac{\partial p^{h}}{\partial p^{l}}+S^{l}} \right) \mathbf{N}_l} \right] } d\Omega \end{aligned}$$

(42)

$$\begin{aligned}&\mathbf{C}_{ht} =\int \limits _\Omega {\mathbf{N}_h^T \left[ {\frac{S^{h}}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial S^{t}}+S^{h}\frac{\partial p^{h}}{\partial S^{t}}+p^{t}-p^{l}} \right) \mathbf{N}_t} \right] } d\Omega \end{aligned}$$

(43)

$$\begin{aligned}&\mathbf{C}_{hh} =\int \limits _\Omega \mathbf{N}_h^T \left[ \varepsilon \mathbf{N}_h^ +\frac{S^{h}}{K^{s}}\left( S^{t}\frac{\partial p^{t}}{\partial S^{h}}+S^{h}\frac{\partial p^{h}}{\partial S^{h}}+p^{h}-p^{l} \right) \mathbf{N}_{h} \right] d\Omega \nonumber \\ \end{aligned}$$

(44)

$$\begin{aligned}&\mathbf{C}_{hl} =\int \limits _\Omega {\mathbf{N}_h^T \left[ {\frac{S^{h}}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial p^{l}}+S^{h}\frac{\partial p^{h}}{\partial p^{l}}+S^{l}} \right) \mathbf{N}_l} \right] } d\Omega \end{aligned}$$

(45)

$$\begin{aligned}&\mathbf{C}_{lt} =\int \limits _\Omega {\mathbf{N}_l^T \left[ {\frac{1}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial S^{t}}+S^{h}\frac{\partial p^{h}}{\partial S^{t}}+p^{t}-p^{l}} \right) \mathbf{N}_t} \right] } d\Omega \end{aligned}$$

(46)

$$\begin{aligned}&\mathbf{C}_{lh} =\int \limits _\Omega {\mathbf{N}_l^T \left[ {\frac{1}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial S^{h}}+S^{h}\frac{\partial p^{h}}{\partial S^{h}}+p^{h}-p^{l}} \right) \mathbf{N}_h} \right] } d\Omega \end{aligned}$$

(47)

$$\begin{aligned}&\mathbf{C}_{ll} =\int \limits _\Omega {\mathbf{N}_l^T \left[ {\frac{1}{K^{s}}\left( {S^{t}\frac{\partial p^{t}}{\partial p^{l}}+S^{h}\frac{\partial p^{h}}{\partial p^{l}}+S^{l}} \right) \mathbf{N}_l} \right] } d\Omega \end{aligned}$$

(48)

$$\begin{aligned}&\left( {\mathbf{C}_{uu} } \right) _{ij} =-\int \limits _\Omega {\mathbf{B}^{\mathrm{T}}\mathbf{D}_s \mathbf{B}} d\Omega \end{aligned}$$

(49)

$$\begin{aligned}&\mathbf{K}_{nn} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_n } \right) ^\mathrm{T}\left( {\varepsilon S^{l}D_{eff}^{\overline{nl} } \nabla \mathbf{N}_n } \right) d\Omega } \end{aligned}$$

(50)

$$\begin{aligned}&\mathbf{K}_{tt} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_t } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial S^{t}}\nabla \mathbf{N}_t } \right) d\Omega } \end{aligned}$$

(51)

$$\begin{aligned}&\mathbf{K}_{th} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_t } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial S^{h}}\nabla \mathbf{N}_h } \right) d\Omega } \end{aligned}$$

(52)

$$\begin{aligned}&\mathbf{K}_{tl} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_t } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial p^{l}}\nabla \mathbf{N}_l } \right) d\Omega } \end{aligned}$$

(53)

$$\begin{aligned}&\mathbf{K}_{ht} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_h } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial S^{t}}\nabla \mathbf{N}_t } \right) d\Omega } \end{aligned}$$

(54)

$$\begin{aligned}&\mathbf{K}_{hh} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_h } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial S^{h}}\nabla \mathbf{N}_h } \right) d\Omega } \end{aligned}$$

(55)

$$\begin{aligned}&\mathbf{K}_{hl} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_h } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial p^{l}}\nabla \mathbf{N}_l } \right) d\Omega } \end{aligned}$$

(56)

$$\begin{aligned}&\mathbf{K}_{lt} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_l } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial S^{t}}\nabla \mathbf{N}_t +\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial S^{t}}\nabla \mathbf{N}_t } \right) d\Omega } \nonumber \\\end{aligned}$$

(57)

$$\begin{aligned}&\mathbf{K}_{lh} =\int \limits _\Omega {\left( {\nabla \mathbf{N}_l } \right) ^{\mathrm{T}}\left( {\frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial S^{h}}\nabla \mathbf{N}_h +\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial S^{h}}\nabla \mathbf{N}_h } \right) d\Omega } \nonumber \\\end{aligned}$$

(58)

$$\begin{aligned}&\mathbf{K}_{ll} =\int \limits _\Omega \left( {\nabla \mathbf{N}_l } \right) ^{\mathrm{T}}\left( \frac{k_{rel}^t \mathbf{k}^{ts}}{\mu ^{t}}\frac{\partial p^{t}}{\partial p^{l}}\nabla \mathbf{N}_l +\frac{k_{rel}^h \mathbf{k}^{hs}}{\mu ^{h}}\frac{\partial p^{h}}{\partial p^{l}}\nabla \mathbf{N}_{l}\right. \nonumber \\&\qquad \quad \!\left. +\frac{k_{rel}^l \mathbf{k}^{ls}}{\mu ^{l}}\nabla \mathbf{N}_{l} \right) d\Omega \end{aligned}$$

(59)

$$\begin{aligned}&\mathbf{f}_n =\int \limits _\Omega {\mathbf{N}_n^{T} \left( {\frac{1}{\rho }\left( {\omega ^{\overline{nl} }\mathop {M}\limits ^{l\rightarrow t} -\mathop {M}\limits ^{nl\rightarrow t} } \right) -\varepsilon S^{l}\mathbf{v}^{\overline{l} }\cdot \nabla \omega ^{\overline{nl} }} \right) } d\Omega \end{aligned}$$

(60)

$$\begin{aligned}&\mathbf{f}_t =\int \limits _\Omega \mathbf{N}_t^T \left[ \frac{1}{\rho }\mathop {M}\limits _{growth}^{l\rightarrow t} -S^{t}\mathrm{tr}\left( {\frac{\partial \mathbf{e}^{s}}{\partial t}-\frac{\partial \mathbf{e}_{sw}^s }{\partial t}} \right) \right. \nonumber \\&\left. \qquad \quad \!-\nabla \left( {S^{t}} \right) \cdot \left( {\varepsilon \frac{\partial \mathbf{u}^{s}}{\partial t}} \right) \right] d\Omega \end{aligned}$$

(61)

$$\begin{aligned}&\mathbf{f}_h =\int \limits _\Omega {\mathbf{N}_h^T \left[ {-S^{h}\mathrm{tr}\left( {\frac{\partial \mathbf{e}^{s}}{\partial t}-\frac{\partial \mathbf{e}_{sw}^s }{\partial t}} \right) -\nabla \left( {S^{h}} \right) \cdot \left( {\varepsilon \frac{\partial \mathbf{u}^{s}}{\partial t}} \right) } \right] } d\Omega \nonumber \\\end{aligned}$$

(62)

$$\begin{aligned}&\mathbf{f}_l =\int \limits _\Omega {\mathbf{N}_l^T \left[ {-\mathrm{tr}\left( {\frac{\partial \mathbf{e}^{s}}{\partial t}-\frac{\partial \mathbf{e}_{sw}^s }{\partial t}} \right) } \right] } d\Omega \end{aligned}$$

(63)

$$\begin{aligned}&\mathbf{f}_u =\int \limits _\Omega {\mathbf{B}^{T}\left( {\mathbf{D}_s \frac{\partial \mathbf{e}_{vp}^s }{\partial t}} \right) } d\Omega +\int \limits _\Omega {\mathbf{B}^{T}\left( {\mathbf{D}_s \frac{\partial \mathbf{e}_{sw}^s }{\partial t}} \right) } d\Omega \end{aligned}$$

(64)