Page 114 - YORAM RUDY BOOK FINAL

P. 114
4.2 The Field of an Elongated Cylindrical Fiber

Typical dimensions of cardiac ventricular myocytes are 100 microns in length and 15

microns in diameter. For modeling purposes, their shape can be idealized as an elongated circular
cylinder 246 (Figure 4.3). We define on the surface S the function V as:
f

Z
− σ i V f ( ) σ= 0 φ ( ) σ− i φ ( ) (4.11)
Z
Z
i
φ
Substitution in (4.8) provides:

1 σ  1  
)
φ = − i ∫ V ( Z ∇  •  s d (4.12)
o
f
4 σπ o s  r 
We extend the V function into the volume of the cylinder v by defining its value on each cross
f
section as a constant, equal to its value on the surface:

Z
V ( ,ρ Z ) V= ( Za, ) V= ( )
f f f (4.13)

With this extension, Gauss theorem 245 can be applied to convert the surface integral (4.12) into a
volume integral:

1 σ   1  
φ = − i ∫ ∇ •  V ρ , )    dv (4.14)
( Z ∇
dv
f
o
4 σπ o v   r  
Considering that V is constant on every cross section A and varies only with Z, (4.14) becomes:
f

 1 
 ∂

∞
Z
Z
1 σ  1  ∂V f ( ) 1 σ i ∂ V f ( )  r 
φ = − i ∫ ∇  •  a ˆ z dv = − ∫ dA dA ∫ dZ dZ (4.15)
dv

o

π
4 σπ o v  r  ∂Z 4 σ o A − ∞ ∂ Z ∂ Z
For a cell in an extensive medium ϕ << ϕ , ϕ ~V and
i
0
i
m
 1 
1 σ ∞ ∂ V ( )  ∂  r 
Z

φ = i ∫ dA ∫ − m dZ (4.16)
dZ
dA
o

π
4 σ o ∂ Z ∂ Z
A − ∞