A numerical study on the effect of osmotic pressure on stress and strain in intercellular structures of tumor tissue in the poro-elastic model

Abstract

Understanding the biomechanical phenomena governing the intercellular space can be considered as an effective tool for the treatment of tumors and cancerous tissues. One of the factors that plays a significant role in the quality and efficiency of the drug delivery process to tumors and cancerous tissues as one of the influential biomechanical factors is the osmotic pressure in the interstitial fluid in the intercellular space of tumors. In this study, the effect of osmotic pressure on the distribution of stress and strain in a tumor tissue is investigated in two modes using a poro-elastic and two-dimensional model. The results of this study show that the effect of osmotic pressure and change in the circumference of tumor tissue causes a decrease or increase in mechanical stress and strain, which can be used to control the tumor growth for treatment. In fact, according to the results, by isolating the tumor tissue (This refers to a condition in which tumor cells have grown in rigid containers) and considering the osmotic pressure, the stress created in the tissue is reduced, but if the tumor tissue is surrounded by healthy tissue and osmotic pressure is applied to it, the stress to increase by about 30 Pa compared to when these two assumptions are not considered.

Appendix

We start the solution from Eq. (6):

$$ \mathop \sigma \limits^{ = } _{{total}} = \mathop \sigma \limits^{ = } _{t} + \mathop \sigma \limits^{ = } _{i} $$

(13)

Using Eqs. (7) and (8) we will have:

(14)

Using Eq. (9) and assuming that the porosity value is constant, we will have:

(15)

Assuming the theory of infinitesimal strain, Eq. (11), we will have:

(16)

Now we take divergence from Eq. (16) and so we will have:

(17)

Considering the constant and equal volumetric values in both fluid and solid spaces for porous media, i.e.

$$ \left\{ {\begin{array}{*{20}c} {\varepsilon _{i} = \varepsilon _{t} = \varepsilon } \\ {\varepsilon _{i} + \varepsilon _{t} = 1} \\ \end{array} } \right.~~ \to 2\varepsilon = 1~~ \to \varepsilon = \frac{1}{2} $$

(18)

and assuming the process of tissue production and conversion is ignored, i.e.

$$ S = 0 $$

(19)

and also, by taking divergence from Eq. (4) and then using Eq. (10), we have:

$$ \begin{gathered} \nabla \cdot \left( { - \nabla P_{i} - \left( {\frac{{\mu _{i} }}{k}} \right)\varepsilon _{i} \left( {\overrightarrow {{V_{i} }} - \overrightarrow {{V_{t} }} } \right)} \right) = - \nabla ^{2} P_{i} - \left( {\frac{{\mu _{i} }}{k}} \right)\nabla \cdot \left( {\varepsilon _{i} \overrightarrow {{V_{i} }} - \varepsilon _{i} \overrightarrow {{V_{t} }} } \right) \hfill \\ \;\;\;\;\;\;\;\,\,\,\,\;\;\,\,\,\;\;\,\,\,\,\;\,\,\;\,\,\,\;\,\,\;\;\,\,\,\,\;\,\;\;\,\,\;\,\,\,\,\;\, = - \nabla ^{2} P_{i} - \left( {\frac{{\mu _{i} }}{k}} \right)\left( {\nabla \cdot \left( {\varepsilon _{i} \overrightarrow {{V_{i} }} } \right) - \nabla \cdot \left( {\varepsilon _{t} \overrightarrow {{V_{t} }} } \right)} \right) \hfill \\ \,\,\,\,\,\,\;\,\,\,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = - \nabla ^{2} P_{i} - \left( {\frac{{\mu _{i} }}{k}} \right)\left( {\left( {\left( {\varphi _{B} - \varphi _{L} } \right) - \nabla \cdot \left( {\varepsilon _{t} \overrightarrow {{V_{t} }} } \right)} \right) - \nabla \cdot \left( {\varepsilon _{t} \overrightarrow {{V_{t} }} } \right)} \right) = 0~~~ \hfill \\ \end{gathered} $$

(20)

According to:

$$ \varepsilon _{i} = \varepsilon _{t} = \varepsilon = \frac{1}{2} $$

(21)

we will have:

$$ - \nabla ^{2} P_{i} - \left( {\frac{{\mu _{i} }}{k}} \right)\left( {\varphi _{B} - \varphi _{L} } \right) + \left( {\frac{{\mu _{i} }}{k}} \right)\nabla \cdot \left( {\overrightarrow {{V_{t} }} } \right) = 0~ $$

(22)

Now considering Eq. (12) and using two Eqs. (17) and (22) we will have:

(23)

Now, using Eqs. (2) and (3), we rewrite Eq. (23):

(24)