Analysis of diffusion delay in a layered medium. Application to heat measurements from muscle

Abstract

An analysis is presented of diffusional delays in one-dimensional heat flow through a medium consisting of several layers of different materials. The model specifically addresses the measurement of heat production by muscle, but diffusion of solute or conduction of charge through a layered medium will obey the same equations. The model consists of a semi-infinite medium, the muscle, in which heat production is spacially uniform but time varying. The heat diffuses through layers of solution and insulation to the center of the thermal element where heat flow is zero. Using Laplace transforms, transfer functions are derived for the temperature change in the center of the thermopile as a function of the temperature at any interface between differing materials or as a function of heat production in the muscle. From these transfer functions, approximate analytical expressions are derived for the time constants which scale the early and late changes in the central temperature. We find that the earliest temperature changes are limited by the diffusivities of the materials, whereas the approach to steady state depends on the total heat capacity of the system and the diffusivity of muscle. Hill (1937) analyzed a similar geometry by modeling the layered medium as a homogeneous system with an equivalent half thickness. We show that his analysis was accurate for the materials in his system. In general, however, and specifically with regard to modern thermopiles, a homogeneous approximation will lead to significant errors. We compare responses of different thermopiles to establish the limits of time resolution in muscle heat records and to correct them for diffusional delays. Using numerical techniques, we invert the Laplace transforms and show the time course of the temperature changes recorded by different instruments in response to different patterns of heat production.

Dynamics of lattice random walk within regions composed of different media and interfaces

“…Similar expansion of derivatives of the above functions with respect to z yields L (r) ′ µ,ν (n, z|n 0 ) ≡ l M ′ µ (n, z) ≡ m (1) µ (n)(1 − z) −1/2 + m (2) µ (n)(1 − z) 0 + m (3) µ (n)(1 − z) 1/2 + . .…”

“…To compute the limit in equation (E.1), we again expand the relevant functions in a power series of (1 − z), and the results may generally be written as L (r) µ,ν (n, z|n 0 ) ≡ l (1) µ,ν (n, n 0 )(1 − z) −1/2 + l (2) µ,ν (n, n 0 )(1 − z) 0 + l (3) µ,ν (n, n 0 )(1 − z) 1/2 + . .…”

“…. , (E.6) R ν (n 0 , z) ≡ r (1) ν (n 0 )(1 − z) −1/2 + r (2) ν (n 0 )(1 − z) 0 + r (3) ν (n 0 )(1 − z) 1/2 + . .…”

“…+ r (1) ν (n 0 ) c (1) l (1) µ,µ (n, n) + 2m (2) µ (n)r (1) µ (n) + m (1) µ (n)r (2) µ (n) , (E.15) and C µ,ν (n, n 0 ) ≡ m (1) µ (n) r (1) µ (n) c (2) l…”

“…Diffusion in heterogeneous space is a ubiquitous transport phenomenon observed in diverse natural and engineering processes from heat conduction in solids [1] and muscles [2], drug-delivery in tissues [3,4] and invasion of cancerous cells [5], to synthesis of materials [6][7][8], electrical performance in electrodes [9,10] and semiconductors [11,12], and molecular exchanges in biological systems [3,4]. Given such a wide range of applications it is expected that many different methodologies have been used over the years to understand what role heterogeneities have on the random movement of particles.…”