Download First half of semester notes and more Lecture notes Biotechnology in PDF only on Docsity! Module 10, Lecture 1: Extracellular Fields for Unknown Cardiac Sources, Part 1 Green's identity: fare — BV7A)dV = fare - BVA) «dS
A, B are scalar fields, let A=1/, B=aby
0
1 1 I i,
Vm, by 2= =f (-P0e DoV—)-dS
| (iphev—enaees)av (2700, - 0-72)
1 1
ee a has an interesting property: v = 0 except when r = 0
Divide V into two volumes: one very small sphere around ‘b’ and
the other everywhere else.
dV=| dv
Ve yam] ae)
yo v V-e €
N
(, yl
ra &
aN [av =o because r #0)
V-e
1
So our volume integral reduces to — | ob, 0? (=) av
€
Since € is so small and ®y is well behaved,
®, is constant in € and equal to ®,.
Rewrite using Gauss’s theorem:
-ob, , v2 () dV =-c®, f v (=) aS “| Can be simplified
e f several ways
€
Solid angle:
4 a,
Since V (;) = +
tr) oor
For ‘b’ in center, 2, = +47 = 00,40
Integrate in spherical coordinates:
iH i
—o®, wes, = -o®, 407° = 00,40
€
Achange in a acts as a source;
is called a secondary source.
1) Account for its contribution and 2) exclude it as a source.
OL e
\ @ ds_\
— —_—
' i
ae Le@Brou
Green's theorem with A and B scalar substitutions:
| ee oy? av f Looe, ov dS
pr TON OWS TP PPOs — OSV
Add the following surfaces: 1) lung outer surface and 2) lung inner surface.
1 a a
GF (FFs ~ OPS, 4urV (")) -dSiour © Stour
1 1 —
f oul Psu OL Psy 0 z)} Sin — Stun
Simplifications:
on 6
(1) Siw = Stour and dSyin = dSrour \ 4
ae 1 True
1 1 a
- ($7 ur = 905. 04,7 (-)) Win
1 1\\ —_—
+f Fol Osun — 9,05 yV eI }° Sin
Simplifications:
At a tissue boundary, the following conditions must be met.
* Normal current is continuous
Potential is continuous; not always true (e.g., cell membrane)
d®srin @®siour =
a =o or OV Ps, = VPs ayy
(2) o,
3) ®suin = Psrour
Using simplifications 1, 2, and 3, the first terms in each integral cancel
A. 4
f oP in7 (=) - Sun — f OPsiinV () - Stn
or + $(F — 0) P sul @) dS
or (a, —9)$Ps);ydQyy — Using solid angle notation (changes sign)
one surface multiple surfaces
1 1 1 [VdysdSy (a, -0)
bose body — Tf bydny Lp PE AOL ts
Important points to note
+, can be determined from surface potentials (equivalent sources)
+ Anatomy (geometry) is a critical determinant
* Tissue heterogeneities can contribute to the , (e.g., ECG)
+ Itis possible to calculate ,, from , (inverse solution)
