BEH.460/10.449
Homework
Bone, Part 1 (11/13/01)
Due Tuesday, Nov. 27
Consider transplantation of marrow into alloplast devices, and on problems scaling from rat to
human.
(a) Consider a cylindrical porous device for a femoral implant. Calculate the dimensionless oxygen
concentration profiles as a function of dimensionless radius and derive the cylindrical dimensionless
group ( Thiele parameter) as we did for a slab, again for the zero order case. You may assume that
the cylinder is infinitely long (i.e., only radial diffusion)
(b) Using values for cell sizes and oxygen consumption given in the tables in class, estimate the
maximum number of and concentration of marrow cells that can be implanted in a device of suitable
dimensions for a rat (0.5 cm diameter) if the concentration of oxygen falls to zero exactly in the
center of the device. Assume and oxygen concentration of 52 mm Hg. You may assume rat
parameters for metabolism are similar to human parameters. How does your answer change if the
surface oxygen concentration is 45 mm Hg?
(c) Perform the same calculation for a device suitable for humans (4 cm diameter). How much would
the marrow need to be purified in order to have a CTP concentration of ~ 100,000 CTP/cm3?
(d) Consider the data on oxygen concentration in the center of a 1 cm diameter marrow-loaded porous
graft (attached). Assume that the initial cell concentration was comparable to that in a marrow
aspirate, 4 x 107 cells/ml. First, calculate how fast the cells would use up oxygen present if the
graft were initially filled with (i) cells + blood plasma (in equilibrium with 90 mm Hg oxygen as
shown in the graft) and (b) cells plus blood clot including hemoglobin, assuming the only source of
oxygen is what is included at the time of loading. Then, calculate a steady-state oxygen
concentration profile within the graft for diffusion from the surface and metabolism. How close
does the concentration at the center of the graft come to the data reported for day 1 (i.e., 42 mmHg)?
(e) Consider the data presented in the paper by Lennon, Edmison, and Caplan on the cultivation of rat
bone marrow cells at low oxygen. Refer to Figure 4B. Calculate an approximate doubling time for
the cells (Note: dN/dt = kt, where k is a constant; show that the time to increase cell number by a
factor of 2 (ie, the doubling time) is k/ln2. Given the doubling time you estimate, do you think the
change in oxygen concentration seen from Day 1 to Day 2 in the graph of oxygen in the center of the
implant (handout) is consistent with a hypothesis that the stem cells are doubling?
(f) Using the data for oxygen consumption rates given in the table for class as representative for MSC,
calculate the true oxygen concentration at the cell surface for the culture experiments described by
Lennon et al, at the beginning and end of a culture experiment for each other oxygen concentrations
used in the gas phase. You may assume that the cells consume oxygen at a steady state and are in a
monolayer on the bottom of the petri dish, and that the oxygen diffuses through the culture medium
to reach the cells. The authors did not report the total volume of medium used per dish; you may
assume it provided a medium depth of 5 mm.