Question

5. Macrophages entrapped in extracellular matrix release chemokine that diffuse through the matrix to attract other macrophages. Consider a small cluster of macrophages or radius Ri at the center of a 3D spherical matrix of radius R0. The macrophages release chemokine at concentration Ci. Blood flow around the spheres clears the chemokines at a steady rate Γ (mol/sec).

   a. Write down steady state shell balance for radial chemokine transport in the extracellular matrix sphere.

   2. Write down the boundary conditions at r = Ri (concentration condition) and r = R0 (flux condition)

   3. Obtain an expression for the steady concentration profile C(r) through the sphere.

   4. What is the concentration of chemokine at the periphery of the sphere? Sketch the

      profile.

Answer 1

a) Considering a differential element of volume dv at a radius r from center

differential volume 8r dr

Making steady state mass balance over the volume

Rate of mass in - Rate of mass out= 0 ( steady state assumption)

Rate of transport= concentration at r* volume

Shell balance equation is

(Ci)r*8pie r dr-(Ci)r+dr*8pie r dr=0

On solving we get

rCi=K where K is constant

b)at r=Ri ,c=Ci max

at r=R0,c=Cb where Cb is bulk concentration in blood which can be assumed zero.

c) Concentration equation is C(r)=Ci max/r

d) Concentration is inversely related to radial distance ie C(r) is proportional to 1/r