Question

Question 1 (20 pts): (a) Oxygen is consumed in body tissues, or by cells maintained in vitro, at a rate which can often be considered independent of O2 concentration. To model a tissue region or an aggregate of cells, consider steady-state diffusion in a sphere of radius $r_0$. Assume that O2 at the outer surface r = $r_0$ is maintained constant at $C_0$. What is the O2 profile, C(r)? What are your assumptions? (10 pts) (b) If $r_0$ and the rate of O2 consumption are sufficiently large, no O2 will reach the central core which is defined as r < $r_c$. For the central core, O2 is limiting and an assumption of zeroth-order kinetics with respect to O2 is no longer valid. At what value of $r_0$ will an oxygen-free center first exist? Find an expression for $r_c$ in terms of $r_0$, $C_0$, the diffusion of O2 (D), and the rate of O2 consumption. (5 pts) (c) In what disease does an aggregate of cells grow until O2 becomes limiting? In some cases, this lack of O2 diffusion stops cell growth. In other cases, growth continues. What enables growth of an aggregate of cells to continue past this limit? (For this question, you may find it useful to consult Langer and Brem et al., \"Isolation of a Cartilage Factor\" Science 1976, available in Blackboard) (5 pts)

          Question 1 (20 pts):
(a) Oxygen is consumed in body tissues, or by cells maintained in vitro, at a rate which
can often be considered independent of O2 concentration. To model a tissue region
or an aggregate of cells, consider steady-state diffusion in a sphere of radius $r_0$.
Assume that O2 at the outer surface r = $r_0$ is maintained constant at $C_0$. What is the
O2 profile, C(r)? What are your assumptions? (10 pts)
(b) If $r_0$ and the rate of O2 consumption are sufficiently large, no O2 will reach the
central core which is defined as r < $r_c$. For the central core, O2 is limiting and an
assumption of zeroth-order kinetics with respect to O2 is no longer valid. At what
value of $r_0$ will an oxygen-free center first exist? Find an expression for $r_c$ in terms
of $r_0$, $C_0$, the diffusion of O2 (D), and the rate of O2 consumption. (5 pts)
(c) In what disease does an aggregate of cells grow until O2 becomes limiting? In some
cases, this lack of O2 diffusion stops cell growth. In other cases, growth continues.
What enables growth of an aggregate of cells to continue past this limit?
(For this question, you may find it useful to consult Langer and Brem et al.,
\"Isolation of a Cartilage Factor\" Science 1976, available in Blackboard) (5 pts)

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