### Mathematical Modeling of Delivery of Chemotherapeutic Drugs to Human Tissues: An Analytical Approach

Dwaipayan Mukherjee and Saikat Chakraborty. Department of Chemical Engineering, Indian Institute of Technology - Kharagpur, Kharagpur 721302, India

This work presents a mathematical model for delivery and uptake of anionic anti-tumor drugs to tumor cells using a novel, implantable, biocompatible, electro-polymeric membrane-based drug release device. The rate of diffusion-mediated transport of the drug from the drug reservoir to the affected tissue at an oncologist-prescribed therapeutic concentration is regulated by applying an electric potential across the membrane with a set of batteries. The transport of an ionic drug through the membrane is further facilitated by its reversible reaction with a cationic dopant that the membrane is impregnated with. Positive and negative voltage scans are applied alternately to drive the reversible reaction in the forward direction so as to form the dopant-drug complex and followed by in the backward direction to release the drug ion from the complex to the tumor.

The delivery device is implanted close to the tumor such that a region of normal tissue separates the device from the affected region. The system is split into four zones, namely, the drug reservoir, the dopant-impregnated membrane through which the drug diffuses, the normal tissue which surrounds the capsule and the tumor-affected region where the drug is required, the regions being connected to each other through the boundary conditions.We use 1-D unsteady state Diffusion-Reaction models in a hollow cylindrical reservoir of radius R, length L and membrane thickness h for the four different regions.

We use the Poisson-Boltzmann equation to model the potential distribution in the reservoir while the Nernst-Planck equation is used to quantify the transport of the ions under influence of electric potential in the reservoir solution and the polymeric membrane. The simultaneous transport and uptake of the drug by the tissue cells is described by the Diffusion-Reaction equation, with the reactive uptake of the drug by cells being given by the Michaelis-Menten kinetics. During the negative scan, there is also a reverse flux of drug anions from the membrane back into the reservoir. This leads to the formation of an electric double layer, the potential of which is estimated by using the Stern theory.

Analytical solutions of the above-described four zone model are obtained to quantify the drug delivery rate along the radial coordinate and with time. We use three different chemotherapeutic drugs namely, Doxorubicin, Chlorambucil and MitomycinC to study the dynamics and efficacy of the drug delivery process. A parametric study has been performed to identify the controlling parameters and the behavior of various drugs and operating conditions. The effect of two operating parameters, namely voltage and scan times and two system parameters namely membrane thickness and therapeutic concentration have been quantified.

The above analysis reveals a distinctly different drug flux from the reservoir to the membrane during the two opposite voltage scans, the flux during the negative scan being diminished due to the back flux. The immobilization of drug anions inside the polymeric membrane results in a zero flux at the membrane-tissue interface during the positive scan. Our analysis however shows the opposite effect in the tumor, the flux being higher in the positive scan than in the negative. This can be attributed to a delay of half a cycle in the diffusional transport of the drug between the reservoir and the tumor. The parametric study of the different drugs shows that the drug with the highest kinetic constant depletes faster, MitomycinC being almost non-existent in the tumor due to its highest rate constant. Therapeutic concentration is predetermined by the oncologist, has a one-to-one correspondence with the applied voltage. Membrane thickness also has a bearing on the rate of transport and has to be decided prior to fabrication. Based on our analysis, we suggest design strategies that regulate the rate and maximize the efficacy of drug delivery by controlling the applied voltage and the time-periods of positive and negative scans and by tuning the kinetic parameters of the drug's reaction with the dopant and the tumor through appropriate choice of drug type.