Consider the physical system shown in the figure on page 532. which represents a simplified scenario for the treatment of tumor (cancer) tissue. In this experiment, a slab of living tissue is in contact with liquid medium bearing a constant concentration of anti-tumor drug A, which serves as a constant source for A. Unfortunately, drug A will encounter difficulty in actually reaching the tumor. First, drug A is not very soluble in the tissue, where its solubility is given by C_Ao= K · C_Aˆž, where K is the partitioning constant for the drug into the healthy tissue, and c_Aˆžis the bulk concentration of the drug in the well-mixed liquid medium. Second, as drug A diffuses through healthy tissue layer down to the tumor tissue, it partially decomposes to by-product B via A †’ B by a first-order homogeneous reaction with rate equation R_A= -k₁C_A. The net flux of drug A that actually reaches the tumor surface at z = L is called the constant therapeutic flux rate, N_As. In independent experiments, it was determined that N_As= 2.0 × 10^-6mg/cm²· s for the drug to be effective, with the requirement that C_As> 0 at z = L. We are interested in developing a model to predict the total flux rate of drug A into the surface of the healthy tissue at z = 0(N_Ao) to achieve the required N_Asat z = L. Towards this end, a model must first be developed to predict the concentration profile of drug A as it diffuses through the healthy tissue layer. 

a. Define the system for mass transfer of drug A, the source for drug A, and the sink(s) for drug A. Propose five reasonable assumptions for this process. 

b. Based on a material balance on a differential volume element of the system for drug A. develop a differential equation for C_A(z) in the healthy tissue layer. 

c. State the boundary conditions that accurately describe the physical system and allow for a mathematical analysis of the diffusion process developed in part (b). 

d. Develop an analytical model, in final integrated form, to predict C_A(z). Your final model should have the following terms in it: C_Aˆž, N_A.s, D_Ae, _K, k₁, L, z; C_Ao and C_As should not be terms in the final model. 

e. Refer the process model input parameters below. What is the total therapeutic drug delivery rate delivered to the tissue (N_A,o at z = 0) in units of mg/cm²-day? 

Process input parameters: Concentration of drug A in liquid medium. c_Aˆž = 3.0 mg/cm³; partitioning constant between liquid medium and healthy tissue, K = 0.1 cm³  tissue/cm³ liquid; diffusion coefficient of drug A in healthy tissue, D_Ae = 1.0 · 10^-5 cm²/s; rate constant for degradation of drug A in healthy tissue, k₁ = 4.0 · 10^-5 s^-1; thickness of healthy tissue layer, L = 0.5 cm; target therapeutic flux rate of drug A into tumor at z = L, N_A,s = 2.0 · 10^-6 mg/cm²-s.

Transcribed Image Text:

Total drug flux Bulk liquid, CA. NAO CA= CAo Net drug Healthy tissue A → B flux RA = -k,CA As -- CA= CAS Tumor tissue