Model concept and assumptions
The equation of unsteady state diffusion with a chemical reaction was exploited to describe the release process of a chemotherapeutic drug from the drops of an emulsion, its transport by diffusion to glioblastoma cells and drug consumption/elimination by cells, Fig 1. The mechanisms and steps in the emulsion system and cancer cells environment:
(i) diffusional transport of a drug (doxorubicin-DOX) from internal droplets of W1/O/W2 emulsions to the membrane phase drops through the interface of W1/O;
(ii) diffusional transport of a drug from the membrane phase drops to the external continuous phase through the interface of O/W2;
(iii) diffusional transport of a drug in an external environment representing the interstitial fluid of the tissue in which the glioblastoma cells are located;
(iv) elimination/consumption of a drug by the glioblastoma cells (the biological system) via a first-order chemical reaction.
Model assumptions:
The model consists of partial differential equations describing drug release and transport coupled with a drug elimination by cancer cells and initial and boundary conditions. Governing equations for a drug mass balance in the emulsion phases and the tumour environment for the spherical coordinate system:
1. Changes in the concentration of a drug (chemotherapeutic agent – doxorubicin (DOX)) in the membrane phase drops of emulsion for 0 ≤ r ≤ R
\(\frac{\partial C\left(r,t\right)}{\partial t}=D_{e}\bullet\left(\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial C(r,t)}{\partial r}\right)\right)+\ \kappa\bullet\left(\left(\frac{\varphi}{1-\varphi}\right)\bullet(m\bullet C_{S}\left(r,\ t\right)-\ C\left(r,\ t\right))\right)\)(1)
\(\kappa=k_{L}a=\ \frac{3\bullet D_{e}}{r_{w}^{2}}\bullet\frac{\phi^{1/3}}{\left(1-\phi^{1/3}\right)}\)(2)
2. Changes in the concentration of the drug (DOX) in the internal phase droplets for 0 ≤ r ≤ rw
\(\frac{\partial C_{S}\left(r,t\right)}{\partial t}=\kappa\ \bullet(\ C\left(r,\ t\right)-m\bullet C_{S}\left(r,\ t\right))\)(3)
\(m=\ \frac{C^{*}\text{\ \ }}{C_{s}^{*}\ }\) (4)
3. Changes in the concentration of the drug (DOX) in the continuous phase, outside the large drops, i.e. in the environment of cancer cells, for r> R
\(\frac{\partial C_{z}\left(r,t\right)}{\partial t}=D_{e,\ z}\bullet\left(\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial C_{z}(r,t)}{\partial r}\right)\right)-k\bullet C_{z}\left(r,t\right)\)(5)
where \(k\bullet C_{z}\left(r,t\right)\) represents the rate of drug elimination by cancer cells
Solution of the coupled diffusion/elimination equations requires an initial and several boundary conditions for a spherical coordinate system.
The initial conditions:
\(C\left(r,\ t\right)=\ C\left(r,\ 0\right)=\ C_{S}\left(r,\ 0\right)\ \bullet m=\ C_{S,\ 0}\ \bullet\ \phi\ \bullet m\ \)for \(0\leq r\leq R\) (6)
\(C_{z}\left(r,0\right)=0\) for r > R (7)
\(C_{S}(r,\ 0)=0\) for r > R (8)
The boundary conditions at the interface:
for r = R
\(D_{e}\left.\ \frac{\partial C\left(r,t\right)}{\partial r}\right\rceil_{r=R}=D_{e,\ z}\left.\ \frac{\partial C_{z}\left(r,t\right)}{\partial r}\right\rceil_{r=R}\)(9)
\(C\left(R,\ t\right)=\ C_{z}\left(R,\ t\right)\ \bullet n\) (10)
\(n=\ \frac{C^{*}}{C_{z}^{*}}\) (11)
for r=0\(D_{e}\left.\ \frac{\partial C\left(r,t\right)}{\partial r}\right\rceil_{r=0}=0\)(12)
for r→∞ \(C_{z}=0\) (13)
C (r, t) – spatiotemporal drug (DOX) concentration function of change in the membrane phase drops of multiple emulsion, r - radius, t - release time, Cz (r, t) – spatiotemporal drug concentration function in the external phase of multiple emulsion, Cs (r, t) – spatiotemporal drug concentration function equivalent to the concentration in the internal droplets of multiple emulsion, CS, 0 – the initial concentration of drug in the internal droplets, De – effective diffusion coefficient of drug in the membrane phase drops, De, z– effective diffusion coefficient of drug in the external phase, k – the first order elimination rate constant of the drug by GBM cells, kLa – volumetric mass transfer coefficient of drug in the membrane phase drops (kLa = κ), m – equilibrium drug partition coefficient between the membrane phase and the internal phase of emulsion, n - equilibrium drug partition coefficient between the membrane phase and the external phase of emulsion, C* – equilibrium concentration, rw – radius of the internal emulsion droplets, R – radius of the membrane phase drop of emulsion, φ – volume fraction of the internal droplets in the membrane phase drops of multiple emulsion (packing volume fraction).