Mechanistic modeling of vascular tumor growth: an extension of Biot’s theory to hierarchical bi-compartment porous medium systems

Abstract

Existing continuum multiphase tumor growth models typically do not include microvasculature, or if present, this is modeled as a non-deformable network of vessels. Vasculature behavior and blood flow are usually non-coupled with the underlying tumor phenomenology from the mechanical viewpoint; hence, phenomena like vessel compression/occlusion modifying microcirculation and oxygen supply cannot be taken into account. Here, the tumor tissue is modeled as a reactive bi-compartment porous medium: the extracellular matrix constitutes the solid scaffold; blood flows in the vascular porosity, whereas the extravascular porous compartment is saturated by two cell phases and interstitial fluid (mixture of water and nutrient species). The pressure difference between blood and the extravascular overall pressure is sustained by vessel walls and drives shrinkage or dilatation of the vascular porosity. Model closure is achieved thanks to a consistent non-conventional definition of the Biot’s effective stress tensor. Angiogenesis is modeled by introducing a vascularization state variable and accounting for tumor angiogenic factors and endothelial cells. Closure relationships and mass exchange terms related to vessel formation are detailed in a numerical example reproducing the principal features of angiogenesis. This example is preceded by a first pedagogical numerical study on one-dimensional bio-consolidation. Results demonstrate that the bi-compartment poromechanical model is fully coupled (the external loads impact fluid flow in both porous compartments) and that it can serve as a basis for further applications like modeling of drug delivery and tissue ulceration.

Appendix: Chemotactic-Fickian model for EC diffusive velocity

Let us assume that all non-endothelial cell species (e.g., non-endothelial host cells, chemical species, etc.) can be modeled jointly as a species named H, the dominant species in h. Consequently let us treat species E as a diluted species in h. For such a situation, neglecting body forces potential and assuming isothermal condition, TCAT [11] provides from simplified entropy inequality the following force–flux pair:

$$\begin{aligned} \underbrace{\frac{1}{\theta }{\rho ^h}{\varepsilon ^h}{\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }}}_{{\mathrm{flux}}} \cdot \underbrace{\nabla \left( {{\mu ^{\overline{Eh} }} - {\mu ^{\overline{Hh} }}} \right) }_{{\mathrm{force}}} \ge 0, \end{aligned}$$

(91)

where $\theta$ is the temperature and $\nabla {\mu ^{\overline{Eh}}}$ and $\nabla {\mu ^{\overline{Hh}}}$ are the chemical potentials of endothelial and non-endothelial species, respectively. At equilibrium, both force and flux terms are zero. Conversely, near equilibrium a first-order closure approximation for the force–flux relationship reads

$$\begin{aligned} {\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }} = - {\chi ^{\overline{\overline{Eh}} }}{\chi ^{\overline{\overline{Hh}} }}{{\mathbf{D}}^{Eh}} \cdot \nabla \left( {{\mu ^{\overline{Eh} }} - {\mu ^{\overline{Hh} }}} \right), \end{aligned}$$

(92)

where ${{\mathbf{D}}^{Eh}}$ is a second-order symmetric tensor, and ${\chi ^{\overline{\overline{ih}}}}$ is the molar fraction of species i. The Gibbs–Duhem equation for this binary mixture reads

$$\begin{aligned} \eta \overline{\overline{^h}} \nabla \theta \overline{\overline{^h}} - \nabla {p^h} + \sum \nolimits _i {{\rho ^h}{\omega ^{\overline{ih} }}\nabla {\mu ^{\overline{ih} }}} = {\mathbf{0}}. \end{aligned}$$

(93)

The expected pressure gradient in the phase h is relatively weak. Hence, for the isothermal case considered, Eq. (93) reduces for a binary mixture to

$$\begin{aligned} {\rho ^h}{\omega ^{\overline{Eh} }}\nabla {\mu ^{\overline{Eh} }} + {\rho ^h}{\omega ^{\overline{Hh} }}\nabla {\mu ^{\overline{Hh} }} = {\mathbf{0}}. \end{aligned}$$

(94)

Equation (94) allows us to obtain the gradient of the chemical potential of species H as

$$\begin{aligned} \nabla {\mu ^{\overline{Hh} }} = - \frac{{{\omega ^{\overline{Eh} }}}}{{{\omega ^{\overline{Hh} }}}}\nabla {\mu ^{\overline{Eh} }}. \end{aligned}$$

(95)

With ${\omega ^{\overline{Eh} }} \ll {\omega ^{\overline{Hh} }}$ it follows that $\nabla {\mu ^{\overline{Hh} }} \ll \nabla {\mu ^{\overline{Eh} }}$ and consequently Eq. (92) reduces to

$$\begin{aligned} {\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }} = - {\chi ^{\overline{\overline{Eh}} }}{\chi ^{\overline{\overline{Hh}} }}{{\mathbf{D}}^{Eh}} \cdot \nabla {\mu ^{\overline{Eh} }}. \end{aligned}$$

(96)

To gain usefulness of the previous equation, a relationship between the macroscale chemical potential of species E and its mass fraction is needed. The macroscale chemical potential for the species E can be written as

$$\begin{aligned} {\mu ^{\overline{Eh} }} = \mu _0^{\overline{Eh} }\left( {{p^h},\theta } \right) + \frac{{R\theta }}{{{M_E}}}\ln \left( {{\chi ^{\overline{\overline{Eh}} }}{\gamma ^{\overline{\overline{Eh}} }}} \right), \end{aligned}$$

(97)

where $\mu _0^{\overline{Eh} }\left( {{p^h},\theta } \right)$ is a reference chemical potential for species E, R is the ideal gas constant, $M_E$ is the molar mass of species E, and ${\gamma ^{\overline{\overline{Eh}} }}$ is the macroscale activity coefficient. With the system in isothermal condition ($\theta = {\theta _0}$) and the impact of pressure gradient of phase h assumed negligible, differentiating this expression in space gives

$$\begin{aligned} \nabla {\mu ^{\overline{Eh} }} = \frac{{R{\theta _0}}}{{{M_E}}}\frac{1}{{{\chi ^{\overline{\overline{Eh}} }}}}\nabla {\chi ^{\overline{\overline{Eh}} }} + \frac{{R{\theta _0}}}{{{M_E}}}\frac{1}{{{\gamma ^{\overline{\overline{Eh}} }}}}\nabla {\gamma ^{\overline{\overline{Eh}} }}. \end{aligned}$$

(98)

For dilute species, the macroscale activity coefficient is usually assumed constant and equal to 1. To account for chemotaxis we set here an activity coefficient linearly dependent on the mass fraction of TAF in h. We assume here local chemical equilibrium, so despite the mass fraction of TAF in h, ${\omega ^{\overline{Ah}}}$, not being a primary variable of the model, its value can be linearly related (as a first approximation) to the mass fraction of TAF in the adjacent IF phase, ${\omega ^{\overline{Ah}}}\propto {\omega ^{\overline{Al}}}$. This allows us to assume that the activity coefficient, ${\gamma ^{\overline{\overline{Eh}} }}$, is linearly dependent on the mass fraction of TAF in l. Thanks to short-range diffusion and molecular signaling the TAF in the phase l interferes (via the hl interface) with endothelial cells modifying their activity coefficient. The following relationship is assumed, with c the constant chemotactic coefficient and ${\chi ^{\overline{\overline{Al}}}}$ the molar fraction of TAF in IF:

$$\begin{aligned} {\gamma ^{\overline{\overline{Eh}} }} = 1 - c{\chi ^{\overline{\overline{Al}} }}. \end{aligned}$$

(99)

Introducing Eq. (99) into Eq. (98) gives

$$\begin{aligned} \nabla {\mu ^{\overline{Eh} }} = \frac{{R{\theta _0}}}{{{M_E}}}\frac{1}{{{\chi ^{\overline{\overline{Eh}} }}}}\nabla {\chi ^{\overline{\overline{Eh}} }} - \frac{{R{\theta _0}}}{{{M_E}}}\frac{c}{{\left( {1 - c{\chi ^{\overline{\overline{Al}} }}} \right) }}\nabla {\chi ^{\overline{\overline{Al}} }}. \end{aligned}$$

(100)

We reasonably assume here that the molar masses of phases h and l are weakly affected by variation of species concentration. This allows us to assume constant molar masses of phases h and l and to express the molar fraction of the species E, A and H as functions of the respective mass fractions:

$$\begin{aligned} {\chi ^{\overline{\overline{Eh}} }} = \frac{{{M_h}}}{{{M_E}}}{\omega ^{\overline{Eh} }}\mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{}\!\!\!\!\!,\; {\chi ^{\overline{\overline{Al}} }} = \frac{{{M_l}}}{{{M_A}}}{\omega ^{\overline{Al} }}\mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{} \mathop {}\nolimits _{}^{}\!\!\!\!\!,\, {\chi ^{\overline{\overline{Hh}} }} = \frac{{{M_h}}}{{{M_H}}}{\omega ^{\overline{Hh} }}. \end{aligned}$$

(101)

Introducing the first two relationships of Eq. (101) into Eq. (100) and setting $C = c\frac{{{M_l}}}{{{M_A}}}$ gives

$$\begin{aligned} \nabla {\mu ^{\overline{Eh} }} = \frac{{R{\theta _0}}}{{{M_E}}}\frac{1}{{{\omega ^{\overline{Eh} }}}}\nabla {\omega ^{\overline{Eh} }} - \frac{{R{\theta _0}}}{{{M_E}}}\frac{C}{{\left( {1 + C{\omega ^{\overline{Al} }}} \right) }}\nabla {\omega ^{\overline{Al} }}. \end{aligned}$$

(102)

We now introduce Eq. (102) into Eq. (96) and express ${\chi ^{\overline{\overline{i\alpha }} }}$ as function of ${\omega ^{\overline{i\alpha } }}$. After some calculations, we obtain

$$\begin{aligned} {\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }} = - \underbrace{\frac{{{{\left( {{M_h}} \right) }^2}}}{{{M_E}{M_H}}}{\omega ^{\overline{Hh} }}\frac{{R{\theta _0}}}{{{M_E}}}{{\mathbf{D}}^{Eh}}}_{ \cong {{\text { constant second{-}order tensor}}}} \cdot \nabla {\omega ^{\overline{Eh} }} + \frac{{C{\omega ^{\overline{Eh} }}}}{{\left( {1 + C{\omega ^{\overline{Al} }}} \right) }}\underbrace{\frac{{{{\left( {{M_h}} \right) }^2}}}{{{M_E}{M_H}}}{\omega ^{\overline{Hh} }}\frac{{R{\theta _0}}}{{{M_E}}}{{\mathbf{D}}^{Eh}}}_{ \cong {{\text { constant second{-}order tensor}}}} \cdot \nabla {\omega ^{\overline{Al} }}. \end{aligned}$$

(103)

As shown in the previous equation, some quantities are expected to stay almost constant, resulting in always ${\omega ^{\overline{Hh} }} \cong 1$. This observation allows us to rewrite the previous equation in a simplified form

$$\begin{aligned} {\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }} = - {{{\hat{\mathbf{D}}}}^{Eh}} \cdot \nabla {\omega ^{\overline{Eh} }} + \frac{{C{\omega ^{\overline{Eh} }}}}{{\left( {1 + C{\omega ^{\overline{Al} }}} \right) }}{{{\hat{\mathbf{D}}}}^{Eh}} \cdot \nabla {\omega ^{\overline{Al} }}, \end{aligned}$$

(104)

where the diffusivity tensor ${{{\hat{\mathbf{D}}}}^{Eh}}$ reads

$$\begin{aligned} {{{\hat{\mathbf{D}}}}^{Eh}} = {\left( {\frac{{{M_h}}}{{{M_E}}}} \right) ^2}\frac{{R{\theta _0}}}{{{M_H}}}{\omega ^{\overline{Hh} }}{{\mathbf{D}}^{Eh}}. \end{aligned}$$

(105)

We assume here an isotropic effective diffusivity which linearly increases with the volume fraction of phase h. Therefore, Eq. (104) can be rewritten in the form

$$\begin{aligned} {\omega ^{\overline{Eh} }}{{\mathbf{u}}^{\overline{\overline{Eh}} }} = - D_{\mathrm{{eff}}}^{\overline{Eh} } \cdot \nabla {\omega ^{\overline{Eh} }} + \frac{{C{\omega ^{\overline{Eh} }}}}{{\left( {1 + C{\omega ^{\overline{Al} }}} \right) }}D_{\mathrm{{eff}}}^{\overline{Eh} } \cdot \nabla {\omega ^{\overline{Al} }}, \end{aligned}$$

(106)

where indicating with $D_0^{\overline{Eh}}$ the bulk diffusivity of endothelial cells in h, the effective diffusivity reads

$$\begin{aligned} D_{\mathrm{{eff}}}^{\overline{Eh} } = D_0^{\overline{Eh} }{\varepsilon ^h}. \end{aligned}$$

(107)