1. Oxygen transport from a single capillary into an organoid. A serious challenge in engineering tissues and large organoids is delivery of nutrients and removal of waste. Vascularizing the tissue with living or artificial blood vessels has been proposed as a potential solution. A simple system to test feasibility of this idea is illustrated below. The system consist of a cylindrical organoid (tissue) of radius " R " and length "L" with a single artificial capillary of radius " r " running through the center. An oxygenated media solution is pumped along the capillary tube at a constant flow rate Q providing small molecule nutrients to the organoid and removing waste at the same time by diffusion across the capillary membrane. (4) a) Supply oxygen enters the capillary tube at flow rate " Q " with concentration Civ dissolved in media. Using the membrane capillary oxygen permeability and other needed parameters, derive a differential equation showing how the oxygen concentration C1(x,t) inside the capillary tube changes with time and distance along the tube. Assume the flow rate Q is constant (velocity, V=Q/(r2) ), diffusion in the " x " direction is negligible, and the concentration in the tissue just outside the capillary is C2W. Neglect water transport through the membrane. (2) b) After a start-up period, suppose the system reaches steady state (concentrations no longer change with time). Based on your answer to "1a", sketch a graph showing how you would expect the oxygen concentration to change inside the capillary tube as a function of distance " x " from the entrance under steady state conditions. Label your graph. (4) c) Now think about oxygen delivery to the cells in the organoid. A serious challenge with a "single capillary" approach is that cells near the capillary consume the oxygen and deprive cells further away. The situation is "diffusion with consumption". What partial differential equation would you use to analyze/quantify oxygen diffusion in the tissue with consumption? Define ALL terms and operators in your equation and describe how they relate to the capillary-organoid question. (Do not attempt to solve the equations.) (2) d) Sketch a graph showing how you would expect oxygen concentration to change with location in the tissue after steady state has been reached..

1. 1. Oxygen transport from a single capillary into an organoid. A serious challenge in engineering
tissues and large organoids is delivery of nutrients and removal of waste. Vascularizing the tissue
with living or artificial blood vessels has been proposed as a potential solution. A simple system
to test feasibility of this idea is illustrated below. The system consist of a cylindrical organoid
(tissue) of radius " R " and length "L" with a single artificial capillary of radius " r " running
through the center. An oxygenated media solution is pumped along the capillary tube at a
constant flow rate Q providing small molecule nutrients to the organoid and removing waste at
the same time by diffusion across the capillary membrane. (4) a) Supply oxygen enters the
capillary tube at flow rate " Q " with concentration Civ dissolved in media. Using the membrane
capillary oxygen permeability and other needed parameters, derive a differential equation
showing how the oxygen concentration C1(x,t) inside the capillary tube changes with time and
distance along the tube. Assume the flow rate Q is constant (velocity, V=Q/(r2) ), diffusion in the
" x " direction is negligible, and the concentration in the tissue just outside the capillary is C2W.
Neglect water transport through the membrane. (2) b) After a start-up period, suppose the system
reaches steady state (concentrations no longer change with time). Based on your answer to "1a",
sketch a graph showing how you would expect the oxygen concentration to change inside the
capillary tube as a function of distance " x " from the entrance under steady state conditions.
Label your graph. (4) c) Now think about oxygen delivery to the cells in the organoid. A serious
challenge with a "single capillary" approach is that cells near the capillary consume the oxygen
and deprive cells further away. The situation is "diffusion with consumption". What partial
differential equation would you use to analyze/quantify oxygen diffusion in the tissue with
consumption? Define ALL terms and operators in your equation and describe how they relate to
the capillary-organoid question. (Do not attempt to solve the equations.) (2) d) Sketch a graph
showing how you would expect oxygen concentration to change with location in the tissue after
steady state has been reached.