Voltages Inside the Chest: Laplace on a Disk

Electrical impedance tomography wraps a ring of electrodes around a patient's chest to watch the lungs. For uniform tissue the voltage inside obeys Laplace's equation, so the boundary pattern alone fixes the interior: each wavy part of the pattern fades toward the center, and the center sits at the boundary's average.

How it works

Laplace's equation on a disk and the mean value property

In one sentence

Laplace's equation in polar coordinates, $u_{rr} + \frac1r u_r + \frac{1}{r^2}u_{\theta\theta} = 0$ , on the disk $r < R$ with $u(R, \theta) = g(\theta)$ has the solution $u = A_0 + \sum (r/R)^n(A_n\cos n\theta + B_n\sin n\theta)$ , where $A_n$ and $B_n$ are the Fourier coefficients of $g$ . In particular $u(0) = A_0 = \frac{1}{2\pi}\int_0^{2\pi} g\,d\theta$ .

General formula

u({\htmlData{sym=1}{r}}, {\htmlData{sym=3}{\theta}}) = {\htmlData{sym=0}{A_0}} + \sum_{{\htmlData{sym=6}{n}}=1}^{\infty}\left(\frac{\htmlData{sym=1}{{r}}}{\htmlData{sym=2}{{R}}}\right)^{\htmlData{sym=6}{n}}\left({\htmlData{sym=4}{A_n}}\cos {\htmlData{sym=6}{n}}{\htmlData{sym=3}{\theta}} + {\htmlData{sym=5}{B_n}}\sin {\htmlData{sym=6}{n}}{\htmlData{sym=3}{\theta}}\right)

Each boundary wiggle reaches inward but fades, faster the wigglier it is. The center hears only the average.

What the symbols mean

$A_0$
Average boundary voltage
The mean of the electrode voltages, which is also the voltage at the center.
$r$
Distance from the center
From 0 at the center to R at the chest wall.
$R$
Disk radius
The radius of the electrode ring (1 in the graph).
$\theta$
Angle
Position around the chest.
$A_n$
Cosine coefficients
How much of each cos nθ wave is in the boundary pattern.
$B_n$
Sine coefficients
How much of each sin nθ wave is in it.
$n$
Wave number
How many waves fit around the boundary.

With uniform tissue and no sources inside, the voltage $u$ satisfies Laplace's equation $\nabla^2u = 0$ in the disk, with the electrode pattern as boundary data.
In polar coordinates, separation of variables gives $r^n\cos n\theta$ and $r^n\sin n\theta$ : the solution is the boundary's Fourier series with each term faded by $(r/R)^n$ .
The wiggly high- $n$ terms fade fastest toward the center, so the inside is smoother than the boundary. At the center only $A_0$ survives: the center value is the boundary's average (the mean value property).
No point inside can be higher or lower than the boundary's extremes (the maximum principle). Real EIT solves the inverse problem: from boundary measurements, infer the conductivity that bends these curves.

Learn the math

Laplace's Equation on a DiskPDE · Lesson 13, Laplace's Equation Poisson's and Laplace's EquationsE&M · Lesson 12, Electric Potential

Voltages Inside the Chest: Laplace on a Disk