$($Welty $25\mathrm{.}9)$ The design of "artificial organs" for growing transplantable human tissues requires

that the cell type of interest, for example pancreatic cells, be grown and sustained within a three$-$

dimensional array where the cells are closely packed together to form a continuous living tissue.

Furthermore, engineered systems for maintaining the tissues within the three$-$dimensional

array require that oxygen be delivered to the living tissue for respiration processes, which is a

major design challenge. One way to get oxygen to the tissue is by a porous scaffold containing

capillary ducts organized into a rectangular tissue monolith, as shown in the figure below. Pure

oxygen gas flows through the capillary ducts and contacts the tissue, where it dissolves into the

tissue by Henry's law $($i$.$e$.P_A=H{c}_A).$ Two sides of the monolith are sealed, but the other two sides

are composed of a porous mesh$-$like sheath, which contains the tissue, but also exposes it to the

surrounding liquid medium, which contains dissolved oxygen at a constant bulk concentration

of $c_A.$ The tissue and the liquid medium surrounding the

monolith approximate the physical properties of liquid water.

Once dissolved in the tissue, the tissue consumes the dissolved

oxygen by a first$-$order reaction process defined by rate

constant $k_1.$ We are interested in predicting the concentration

profile of dissolved oxygen within the tissue monolith.

a$)$ Develop the differential model for the concentration

profiles of dissolved oxygen $(c_A)$ within quadrant IV of the

tissue monolith. State all reasonable assumptions and

define the system$($s$),$ likely source$($s$),$ and likely sink$(s)$ for

$O_2($species $A)$ within the tissue monolith.

b$)$ Specify appropriate boundary conditions.