Friday, April 8, 2016

Darcy’s Law

Table 4.3 of Intermediate Physics for Medicine and Biology contains five transport equations. Each has the form “flux density equals a coefficient times the negative of a gradient of some quantity.” The table includes the flux of particles with the coefficient being the diffusion constant, the flux of heat with the coefficient being the thermal conductivity, the flux of momentum with the coefficient being the viscosity, and the flux of charge with the coefficient being the electrical conductivity. Are there other examples of transport equations important in biology and medicine? Yes. For instance, consider Darcy’s law.

Darcy’s law governs the flow of fluid through a porous medium. It is used to model the movement of groundwater through sedimentary rock, but it also describes the flow of water in tissue's extracellular space. Using a notation consistent with Table 4.3, we can write Darcy’s law as

jv = - K dp/dx

where jv is the flux density of fluid volume, p is the pressure, and K is the hydraulic conductivity. The units for jv are m3 m-2 s−1, or m s−1; therefore jv corresponds to the speed of flow. Pressure has units of pascals, so dp/dx is expressed in Pa m−1. Therefore, the units of hydraulic conductivity are m2 Pa−1 s−1. Hydraulic conductivity is analogous to electrical conductivity or thermal conductivity; it specifies how well a material permits the transport of a quantity (flow of water) caused by some driving force (pressure gradient).

Russ Hobbie and I don’t discuss Darcy’s law in IPMB, but we come close. In Chapter 5 we analyze the flow of water across a membrane, and define the relationship

jv = Lp Δp ,    (5.9)

where jv again is the speed of flow, Δp is the pressure difference across the membrane, and Lp is the hydraulic permeability. If the membrane has a thickness Δx, then we can multiply and divide by Δx and obtain jv = (Lp Δx) (Δp/Δx). The equation looks just like Darcy’s law (except for a minus sign), where the hydraulic conductivity is the hydraulic permeability times the membrane thickness:

K = Lp Δx.

I first encountered Darcy’s law when reading my friend Peter Basser’s paper about “Interstitial Volume, Pressure, and Flow During Infusion into the Brain” (Microvascular Research, 44:143–165, 1992). He derived a model of swelling in the brain that occurs during infusion of a drug. When Basser combined Darcy’s law with the equations of elasticity, he derived a diffusion equation for volume change of the tissue caused by accumulation of interstitial fluid (swelling), in which the diffusion constant is approximately the hydraulic conductivity times the bulk modulus.

Darcy’s law plays a key role in governing fluid flow in many tissues. A nice summary can be found in “Interstitial Flow and Its Effects in Solft Tissues” by Melody Swartz and Mark Fleury (Annual Review of Biomedical Engineering, 9:229–256, 2007). Below is the abstract to their review.
Interstitial flow plays important roles in the morphogenesis, function, and pathogenesis of tissues. To investigate these roles and exploit them for tissue engineering or to overcome barriers to drug delivery, a comprehensive consideration of the interstitial space and how it controls and affects such processes is critical. Here we attempt to review the many physical and mathematical correlations that describe fluid and mass transport in the tissue interstitium; the factors that control and affect them; and the importance of interstitial transport on cell biology, tissue morphogenesis, and tissue engineering. Finally, we end with some discussion of interstitial transport issues in drug delivery, cell mechanobiology, and cell homing toward draining lymphatics.