We are interested analyzing the diffusion processes associated with the treatment of cancer cells. Consider the experimental system shown in the figure on page 524. A hemispherical clump of cancer tissue is surrounded by a hemispherical layer of healthy tissue. Surrounding the healthy tissue is a well-mixed liquid nutrient medium containing a constant concentration of drug species A in it. In order to keep the cancer tissue from growing, the drug must diffuse through the healthy tissue, from r = R₂= 0.1 cm, to the boundary of the cancer tissue and the healthy tissue at r = R₁= 0.05 cm. Once the drug reaches the cancer tissue boundary (r = R₁), it is immediately consumed, and so the drug concentration at this boundary is essentially zero. Independent experiments have also confirmed that (1) the cancer tissue will not grow (i.e., R₁will not change with time) so long as the drug delivery flux reaching the surface of the cancer tissue at r = R₁is N_A= 6.914 × l0^-4mmole A/cm²·day, (2) the drug is not consumed by the healthy tissue, which approximates the properties of water, (3) the diffusion coefficient of the drug (A) through the healthy tissue (B) is D_AB= 2.0 · l0^-7cm²/s, and (4) the drug is not very soluble in the healthy tissue.

a. Describe the system for diffusion mass transfer of the drug (system boundary, source, sink), and state at least three reasonable assumptions in addition to those stated above. 

b. State the boundary conditions that best describe the mass- transfer process based on the system for mass transfer.

c. Simplify the general differential equation for mass transfer that best describes the particular physical system. Then develop a model in final integrated form to describe the flux N_A of the drug to the surface of the cancer cell clump at r= R₁. 

d. Based on the model above, determine the constant surface concentration c_AO at r = R₂ needed to treat the cancer cells, in units of mmole/cm³.

Transcribed Image Text:

Well-mixed bulk fluid (+ drug A) Сдо NA Healthy tissue (B) CAS Clump of cancer cells R, = 0.05 cm R2 = 0.1 cm Inert surface