Unequal Anisotropy Ratios

In Chapter 7 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the bidomain model, a mathematical description of the anisotropic electrical properties of cardiac tissue.

Anisotropy plays an important role in the bidomain model. To see why, consider a solution to Laplace’s equation in a monodomain—a two-dimensional sheet of homogeneous, anisotropic tissue with straight fibers. If the x direction is chosen to be along the fiber direction (the direction of greatest conductivity), then Laplace’s equation becomes

 Now define a new set of coordinates

and

 You can show that in these new coordinates Laplace’s equation becomes

We have removed the effect of anisotropy by rescaling distance in the direction perpendicular to the fibers. If you try a similar trick with the bidomain model … you can find a new coordinate system that removes the effect of anisotropy in either the intracellular space or the extracellular space, but in general you cannot find a coordinate system that removes the anisotropy in both spaces simultaneously (Roth 1992).

My 1992 paper (Journal of Mathematical Biology, Volume 30, Pages 633–646) contains one of my favorite figures, which illustrates the importance of bidomain anisotropy visually. It is equivalent to an old concept from mechanics called the simultaneous diagonalization of two quadratic forms.

Russ and I continue,

Only in the special case of equal anisotropy ratios (σ_ix/σ_iy = σ_ox/σ_oy ) will the equations simplify dramatically. But the anisotropy ratios in the heart are not equal. In the intracellular space the ratio of conductivities parallel and perpendicular to the fibers is about 10:1, while in the extracellular space this ratio is about 4:1 (Roth 1997). Anisotropy plays an essential role in the electrical behavior of the heart, especially during electrical stimulation.

My 1997 paper (IEEE Transactions on Biomedical Engineering, Volume 44, Pages 326–328), published 20 years ago this month, contains an estimate of the bidomain conductivities. After surveying much of the available experimental data, I found

• Intracellular Longitudinal Conductivity     σ_iL     0.2 S/m 
  
• Intracellular Transverse Conductivity        σ_iT     0.02 S/m
• Extracellular Longitudinal Conductivity    σ_eL    0.2 S/m 
• Extracellular Transverse Conductivity       σ_eT    0.08 S/m

This means that the ratio of the conductivity along the fibers to the conductivity across the fibers is 10:1 in the intracellular space, while only 5:2 in the extracellular space.

… Hold on. In IPMB, Russ and I said σ_eL:σ_eT = 4:1, but now I am saying σ_eL:σ_eT = 5:2. Drat! IPMB is wrong. Another entry for the errata…

The 1997 paper is highly cited: Google Scholar lists 178 citations. This perplexes me, because I have many papers that are more significant and innovative, but have far fewer citations. I guess usefulness can sometimes be as important as significance and innovation.