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A numerical study on the effect of static magnetic field on the hemodynamics of magnetic fluid in biological porous media

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Abstract

High interstitial fluid pressure in the tumor is among the most important barriers to drug delivery. The use of the static magnetic field is one of the treatment modalities for cancers and tumors. In this study, the effect of magnetic field on pressure and magnetized interstitial fluid preservation is discussed. The simulation is solved in two-dimensional space using the COMSOL Multiphysics 5.4.0 software. The geometry includes the porous space of healthy tissue and tumor, as well as the parent vessel as the source and the lymphatic system as the sink modeled based on the biology of the body. Governing coupled equations solved in this simulation include fluid flow in the porous space, which are solved using the law of mass and momentum conservation. The magnetic field equations are solved independently and finally the term of the volume force is coupled to the serial flow equations affecting the interstitial fluid pressure. The concentration distribution is the product of Fick’s equations, which is solved in porous media by the magnetic field according to the bolus injection method. Simulation results show that by applying the static magnetic field generated from permanent magnet to the homogenous magnetic fluid, the magnetized interstitial fluid pressure in the tissue and especially in the tumor region decreases, leading to the higher concentration of the drug that reaches the tumor.

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Abbreviations

\(P_{\text{V}}\) :

Intravascular pressure (Pa)

\(P_{\text{i}}\) :

Interstitial pressure (Pa)

\(P_{\text{L}}\) :

Hydrostatic pressure of lymphatic (Pa)

\(V_{\text{i}}\) :

Interstitial fluid velocity (m s−1)

\(L_{\text{P}}\) :

Hydrostatic conductivity of the vessel wall (m Pa−1 s−1)

\(F\) :

Volume force (N)

\(C\) :

Solute Concentration (Kg m−3)

\(C^{*}\) :

Dimensionless concentration

\(P\) :

Diffusive permeability (m s−1)

\(C_{\text{P}}\) :

Plasma concentration of the solute (Kg m−3)

\(\frac{S}{V}\) :

Surface area per unit volume of tissue for transport in the interstitium (1 m−1)

\(\frac{{L_{\text{PL}} S_{\text{L}} }}{V}\) :

The lymphatic filtration coefficient (1 Pa−1 s−1)

\({\text{Pe}}\) :

Peclet number

\(D_{\text{eff}}\) :

Effective diffusion (m2 s−1)

\(H\) :

Magnetic field vector (A m−1)

\(B\) :

Magnetic flux density (T)

\(J\) :

Current density vector (A m−2)

\(B_{\text{rem}}\) :

Remanent magnetic flux density (A m−1)

\(M\) :

Magnetization vector in the blood stream (A m−1)

\(X\) :

Magnetic susceptibility

\(K\) :

Permeability of the porous media (m2)

\(L_{\text{PL}}\) :

Hydrostatic conductivity of the lymphatic vessel wall (m/Pa−1 s−1)

\(r\) :

Length (m)

\(r^{*}\) :

Dimensionless length

\(v_{\text{vas}}\) :

Fluid inlet velocity from blood vascular (m s−1)

\(v_{\text{L}}\) :

Fluid outlet velocity to lymphatic vessel (m s−1)

\(j_{\text{vas}}\) :

Inward flux density of fluid (mol m−2 s−1)

\(j_{\text{L}}\) :

Outward flux density of fluid (mol/m−2 s−1)

J :

Electric current density vector (A m−2)

i:

Interstitial medium

L:

Lymphatic drainage

vas:

Blood vasculature

eff:

Effective

P:

Plasma

\(\rho\) :

Interstitial fluid density (Kg m−3)

\(\mu\) :

Interstitial fluid viscosity (Pa s)

\(\kappa\) :

Hydraulic conductivity of the interstitium (m2 Pa−1 s−1)

\(\sigma_{\text{f}}\) :

Solvent-drag reflection coefficient

\(\pi_{\text{V}}\) :

Osmotic pressure of the plasma (Pa)

\(\pi_{\text{i}}\) :

Osmotic pressure of the interstitial fluid (Pa)

\(\tau\) :

Drug half-life in plasma (h)

\(\mu_{0}\) :

Magnetic permeability of vacuum (N A−2)

\(\mu_{\text{r}}\) :

Relative magnetic permeability of the permanent magnet (N A−2)

\(\sigma\) :

Average osmotic reflection coefficient for the plasma proteins

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Authors and Affiliations

  1. School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    Seyed Mahdi Hosseinikhah, Borhan Beigzadeh, Majid Siavashi & Mahdi Halabian

  2. Applied Multi-Phase Fluid Dynamics Laboratory, School of Mechanical Engineering, Iran University of Science and Technology, Tehran, Iran

    Seyed Mahdi Hosseinikhah & Majid Siavashi

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  1. Seyed Mahdi Hosseinikhah
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Correspondence to Majid Siavashi.

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Hosseinikhah, S.M., Beigzadeh, B., Siavashi, M. et al. A numerical study on the effect of static magnetic field on the hemodynamics of magnetic fluid in biological porous media. J Therm Anal Calorim 141, 1543–1558 (2020). https://doi.org/10.1007/s10973-020-09703-x

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  • DOI: https://doi.org/10.1007/s10973-020-09703-x

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