(302d) What Really Happens in Diffusion With Reversible Trapping, and What Are the Implications for Biological Diffusion Models?

A classical question in homogenization/coarse
graining is, what are the effective (average) diffusion coefficient D_eff
and rate coefficient k_eff characterizing the macroscopically
observable transport and reaction of a solute molecules/particle that diffuse
through an array (periodic or random) of perfectly or partially absorbing traps
(reactive or sticky grains)?  Most analyses to date have addressed the
irreversible case where?once trapped?the diffusing solute is effectively
destroyed or bound forever (see representative references^1-12).  This
case is directly relevant to such technological applications as aerosol
deposition.  In biological systems, however, reversibility of binding/trapping
is the rule rather than the exception.  Indeed, any irreversibly bound species
would necessarily accumulate without bound, and spell disease or death in the
absence of a mechanism for removal/clearance.  Interestingly, there is a dearth
of homogenization theory for the reversible case.

This talk develops a comprehensive
answer to the following question.  Suppose a solute diffuses through a medium
with two-phase microstructure, so that its free concentration C(x,y,z,t)
and bound concentration B(x,y,z,t) are
governed by the equations

           
∂ C / ∂ t  =  D_a ∇²C ? [(k_on)_aC ? (k_off)_aB]

           
∂ B / ∂ t  =                   (k_on)_aC ? (k_off)_aB                                                                       
(1a)

in
phase a, and

           
∂ C / ∂ t  =  D_b ∇²C ? [(k_on)_bC ? (k_off)_bB]

           
∂ B / ∂ t  =                   (k_on)_bC ? (k_off)_bB                                                                       
(1b)

in phase b at the microscopic scale.  What is the equation
governing the macroscopically observable behavior?  The answer turns out not
to be of the same form with homogenized coefficients, namely

           
∂ C / ∂ t  =  D_eff ∇²C ? [(k_on)_effC ? (k_off)_effB]

           
∂ B / ∂ t  =                    (k_on)_effC ? (k_off)_effB,                                                               
   (2)

as might be expected.  Rather, the
correct macroscopic description generally involves a free concentration C(x,y,z,t)
and two effective bound concentrations B^[1](x,y,z,t)
and B^[2](x,y,z,t) that evolve collectively
according to the more complex system of transport equations

           
∂ C    / ∂ t  =  D_eff ∇²C ? [(k_on^[1])_effC ? (k_off^[1])_effB^[1]]
? [(k_on^[2])_effC ? (k_off^[2])_effB^[2]]

           
∂ B^[1] / ∂ t  =                     (k_on^[1])_effC ? (k_off^[1])_effB^[1]

           
∂ B^[2] / ∂ t 
=                                                                    (k_on^[2])_effC ? (k_off^[2])_effB^[2].      
(3)

A general prescription is derived to
calculate the effective coefficients (k_on^[1])_eff,
(k_off^[1])_eff, (k_on^[2])_eff
and (k_off^[2])_eff, involving an
eigenvalue problem posed in a unit cell of the microstructure.  Asymptotic
analysis yields an explicit formula that approximates the requisite eigenvalue
very accurately, thereby allowing the effective binding rate constants to be determined
without solving a diffusion problem.  Although the analysis is originally
carried out for spatially periodic microstructures, the asymptotic reveals a
broader applicability of the results, which are verified vis-à-vis a
brute-force numerical calculation.  The analysis is generalized to an N-phase
microstructure for any N ≥ 2.

The homogenization paradigm has
important implications for diffusion models of biological tissues, for which:
(ii) multiphase microstructure invariable produces two or more microscopic sets
of on- and off binding rate constants  [(k_on)_a, (k_off)_a], [(k_on)_b, (k_off)_b], ?; and (ii)
Eq. (2) is invariably assumed to apply at the macroscopic scale.  Practical
application of the analysis is demonstrated within the context of drug/chemical
diffusion through the surface (barrier) layer of skin, the stratum corneum, in
which reversible binding to keratin protein is known to occur.^13-16

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13.
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14.
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