Problem Statement: In a steady-state system, O₂ diffuses through the tissues (radius, Rₒᵤₜ and length L) from the capillaries (radius, Rₑᵢₙ and length L) and it is consumed within the tissue in a homogeneous reaction (zero order). Assume there is no convection of O₂ through the tissues, and O₂ concentration is independent of axial position. The O₂ concentration at the outer surface of the capillaries is Cap. Derive the particular solution of the O₂ profile through the tissues that are around (or enclose) the capillaries.

Question

Case I) Assume that the O₂ is completely consumed by the time it reaches the outer surface of the tissue.

FIND:
IN - OUT + SOURCE - SINK = ΔC
FIND GENERAL AND PARTICULAR SOLUTION
FIND BOUNDARY CONDITIONS

a) General Solution
b) Particular Solution
c) Initial Boundary Conditions
d) Final Boundary Conditions

Answer 1

a) General Solution:
The general solution for the diffusion of O₂ through the tissues can be expressed as:

ΔC = A ln(r) + B

where A and B are constants to be determined, and r is the distance from the center of the capillaries.

b) Particular Solution:
Since O₂ is completely consumed by the time it reaches the outer surface of the tissue, the O₂ concentration at the outer surface of the tissue is zero. Therefore, the particular solution of the O₂ profile through the tissues can be expressed as:

ΔC = A ln(r)

c) Initial Boundary Conditions:
- O₂ concentration at the center of the tissue, r=0: ΔC = A ln(0) = -∞ (assumed to be constant)
- O₂ concentration at the outer surface of the capillaries, r=Rᵣ: ΔC = A ln(Rᵣ) = Cap

d) Final Boundary Conditions:
- O₂ concentration at the outer surface of the tissue, r=Rₒᵤₜ: ΔC = A ln(Rₒᵤₜ)

By solving the boundary conditions, the constants A and B can be determined and the particular solution for the O₂ profile through the tissues can be obtained.