1 Tangential Dipole in Homogeneous Sphere
In order to calculate potentials generated by a tangential dipole in head models consisting of concentric spherical shells, we first require the solution for the case of a single homogeneous sphere in terms of a spherical harmonic expansion. The following derivation is rather cumbersome, but has the advantage of illustrating several subtle issues in mathematical physics. The potential Φ (r, θ, φ) in a homogeneous, isotropic conducting sphere of conductivity σ due to (point) monopolar current sources of strength +I located at (r _{1}, θ_{1}, φ_{2}) and −I located at (r _{1}, θ_{1}, φ_{1}) is a solution of Poisson’s equation (4.12):
(H.1.1)
$${\nabla}^{2}\Phi =\frac{-1}{\sigma {r}^{2}}\delta \left(r-{r}_{1}\right)\delta \left(\mathrm{cos}\theta -\mathrm{cos}{\theta}_{1}\right)\left[\delta \left(\phi -{\phi}_{2}\right)-\delta \left(\phi -{\phi}_{1}\right)\right]$$
Figure
C-1 (appendix
C) shows the appropriate spherical coordinates. When the monopoles are close together, they form a dipole. In contrast to the (mathematically) equivalent electrostatics problem involving charges in a dielectric, we require a balance of sources and sinks because of the physical requirement of current conservation. To find the solution to (
H.1.1), note the expression for the Laplacian of Φ (r, θ, φ) in spherical coordinates
(H.1.2)
$${\nabla}^{2}\Phi =\frac{1}{{r}^{2}}\frac{\partial}{\partial r}\left({r}^{2}\frac{\partial \Phi}{\partial r}\right)+{\nabla}_{S}^{2}\Phi $$
(p.569)
where the surface Laplacian is given by
(H.1.3)
$${\nabla}_{S}^{2}\Phi =\frac{1}{{r}^{2}}\left[\frac{1}{\mathrm{sin}\theta}\frac{\partial}{\partial \theta}\left(\mathrm{sin}\theta \frac{\partial \Phi}{\partial \theta}\right)+\frac{1}{{\mathrm{sin}}^{2}\theta}\frac{{\partial}^{2}\Phi}{\partial {\phi}^{2}}\right]$$
The spherical harmonic functions
Y _{nm}(θ, φ) have the property (Jackson 1975)
(H.1.4)
$${\nabla}_{S}^{2}{Y}_{nm}\left(\theta ,\phi \right)=-\frac{n\left(n+1\right){Y}_{nm}\left(\theta ,\phi \right)}{{r}^{2}}$$
Any well-behaved function
h(θ, φ) may be expanded in the generalized Fourier series (spherical harmonic expansion):
(H.1.5)
$$h\left(\theta ,\phi \right)={\displaystyle \sum _{n=0}^{\infty}{\displaystyle \sum _{m=-n}^{n}{A}_{nm}}}{Y}_{nm}\left(\theta ,\phi \right)$$
The expansion coefficients follow directly from the orthogonal property of the spherical harmonic functions and are given by the surface integral
(H.1.6)
$${A}_{nm}={\displaystyle \int {\displaystyle \underset{S}{\int}h\left(\theta ,\phi \right)}}{Y}_{nm}^{*}\left(\theta ,\phi \right)d\Omega $$
Thus, we expand the angular function on the right-hand side of (
H.1.1), that is
(H.1.7)
$$\delta \left(\mathrm{cos}\theta -\mathrm{cos}{\theta}_{1}\right)\left[\delta \left(\phi -{\phi}_{2}\right)\right]={\displaystyle \sum _{n=0}^{\infty}{\displaystyle \sum _{m=-n}^{n}{A}_{nm}}{Y}_{nm}}\left(\theta ,\phi \right)$$
(H.1.8)
$${A}_{nm}={Y}_{nm}^{*}\left({\theta}_{1},{\phi}_{2}\right)-{Y}_{nm}^{*}\left({\theta}_{1},{\phi}_{1}\right)$$
Note that the spherical harmonics are defined in terms of the associated Legendre functions by (Jackson 1975)
(H.1.9)
$${Y}_{nm}\left(\theta ,\phi \right)\equiv \sqrt{\frac{\left(2n+1\right)\left(n-m\right)!}{4\pi \left(n+m\right)!}}{P}_{n}^{m}\left(\mathrm{cos}\theta \right){e}^{jm\phi}$$
The associated Legendre functions are defined in terms of the Legendre polynomials
(H.1.10)
$${P}_{n}^{m}\left(x\right)={\left(-1\right)}^{m}{\left(1-{x}^{2}\right)}^{m/2}\frac{{d}^{m}}{d{x}^{m}}{P}_{n}\left(x\right)$$
Suppose we place the two monopoles close together in the
y–
z plane such that θ
_{1} is small and (φ
_{1}, φ
_{2}) = (−π/2, +π/2). These sources create a dipole with axis directed in the positive
y-direction (see fig.
C-1). Note that the
(p.570)
P _{n}(
x) are polynomials with powers ranging from
x ^{0} to
x ^{n}. With
x → cosθ
_{1}, the term in front of the derivative is (sinθ
_{1})
^{m}. Thus, if we expand the coefficients
A _{nm}(θ
_{1}) in (
H.1.8) in a Taylor series about θ
_{1} = 0, the only first-order contributions in θ
_{1} occur for
m = ±1. (This step is easier using Mathematica). For the case (φ
_{1}, φ
_{2}) = (−π/2, +π/2), we expand the coefficients
A _{n1} about θ
_{1} = 0 to obtain
(H.1.11)
$$\begin{array}{c}{A}_{n1}=\frac{j}{2\sqrt{\pi}}\sqrt{n\left(n+1\right)\left(2n+1\right){\theta}_{1}+\vartheta \left({\theta}_{1}^{3}\right)}\\ {A}_{n,-1}={A}_{n1}\\ {A}_{nm}=\vartheta \left({\theta}_{1}^{3}\right)\text{}m=\pm 3,5,7\cdots \\ {A}_{nm}=0\text{}m=\pm 0,2,4\cdots \end{array}$$
The last result in (
H.1.11) occurs because
A _{nm} is proportional to sin(
mπ/2). The double sum in (
H.1.7) may be reduced to a single sum using (
H.1.10), (
H.1.11), and the following relation:
(H.1.12)
$${Y}_{n,-m}\left(\theta ,\phi \right)={\left(-1\right)}^{m}{Y}_{nm}^{*}\left(\theta ,\phi \right)$$
thereby reducing the double sum in (
H.1.7) to
(H.1.13)
$$\begin{array}{c}{\displaystyle \sum _{n=0}^{\infty}}{\displaystyle \sum _{m=-n}^{n}}{A}_{nm}{Y}_{nm}\left(\theta ,\phi \right)\cong {\displaystyle \sum _{n=1}^{\infty}}\left[{A}_{n1}{Y}_{n1}\left(\theta ,\phi \right)+{A}_{n,-1}{Y}_{n,1}\left(\theta ,\phi \right)\right]\\ =\frac{j2\mathrm{sin}\phi}{\sqrt{4\pi}}{\displaystyle \sum _{n=1}^{\infty}}{A}_{n1}\sqrt{\frac{\left(2n+1\right)\left(n-1\right)!}{\left(n+1\right)!}}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)\\ =\frac{-{\theta}_{1}\mathrm{sin}\phi}{2\pi}{\displaystyle \sum _{n=1}^{\infty}}\left(2n+1\right){P}_{n}^{1}\left(\mathrm{cos}\theta \right)\end{array}$$
We seek a solution to Poisson’s equation of the form
(H.1.14)
$$\Phi \left(r,\theta ,\phi \right)={\displaystyle \sum _{n=0}^{\infty}}{\displaystyle \sum _{m=-n}^{n}}{g}_{nm}\left(r\right){Y}_{nm}\left(\theta ,\phi \right)$$
Substitute (
H.1.14) into (
H.1.1), make use of (
H.1.4) and (
H.1.12), and equate terms with
m = ±1, that is
(H.1.15)
$$\begin{array}{l}{\displaystyle \sum _{n=0}^{\infty}\left[\frac{d}{dr}\left({r}^{2}\frac{d{g}_{n1}}{dr}\right)=n\left(n+1\right){g}_{n1}\right]}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)\\ =\frac{I{\theta}_{1}}{2\pi \sigma}\delta \left(r-{r}_{1}\right){\displaystyle \sum _{n=1}^{\infty}\left(2n+1\right)}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)\end{array}$$
(p.571)
From (
H.1.15) obtain an ordinary differential equation for the radial functions
g _{n1}(
r):
(H.1.16)
$$\frac{d}{dr}\left({r}^{2}\frac{d{g}_{n1}}{dr}\right)-n\left(n+1\right){g}_{n1}=\frac{I{\theta}_{1}\left(2n+1\right)}{2\pi \sigma}\delta \left(r-{r}_{1}\right)$$
Because of the equation discontinuity at
r =
r _{1}, we seek separate solutions to the homogeneous version of (
H.1.16) in two regions, that is
(H.1.17)
$${g}_{n1}^{a}\left(r\right)={a}_{n}{r}^{n}+{b}_{n}{r}^{-\left(n+1\right)}\text{}r<{r}_{1}$$
(H.1.18)
$${g}_{n1}^{d}\left(r\right)={c}_{n}{r}^{n}+{d}_{n}{r}^{-\left(n+1\right)}\text{}r>{r}_{1}$$
In order to find relations between the coefficients in (
H.1.17) and (
H.1.18), we apply boundary conditions at the surface
r =
r _{1}; however, the presence of the delta function makes this procedure a little tricky in this case. Application of continuous tangential electric field (
4.8) to (
H.1.16) and (
H.1.17) is identical to forcing a continuous potential across the boundary. However, the presence of the dipole source on the boundary means that the normal derivative of potential is discontinuous even though conductivity is constant, thus (
4.7) does not apply at
r =
r _{1}. With this in mind, boundary conditions are applied as follows:
(H.1.19)
$$\underset{{r}_{1}-\epsilon}{\overset{{r}_{1}+\epsilon}{\int}}\left[\frac{d}{dr}\left({r}^{2}\frac{d{g}_{n1}}{dr}\right)-n\left(n+1\right){g}_{n1}\right]}dr={K}_{n}{\displaystyle \underset{{r}_{1}-\epsilon}{\overset{{r}_{1}+\epsilon}{\int}}\delta \left(r-{r}_{1}\right)dr$$
For a dipole with pole separation
d, located at radial location
r _{1}, the angular separation of the poles is θ
_{1} =
d/2
_{r1}. The terms
K _{n} are defined by (
H.1.15), that is
(H.1.20)
$${K}_{n}=\frac{Id\left(2n+1\right)}{4\pi \sigma {r}_{1}}$$
Relation (
H.1.19) then yields
(H.1.21)
$$n{r}_{1}^{2n+1}\left({c}_{n}-{a}_{n}\right)-\left(n+1\right){d}_{n}={K}_{n}{r}_{1}^{n}$$
Combine (
H.1.21) with (ii) to obtain
(H.1.22)
$${d}_{n}=\frac{-{K}_{n}{r}_{1}^{n}}{2n+1}=\frac{-I{d}_{1}^{n-1}}{4\pi \sigma}$$
(p.572)
From (H.11.14), (H.11.15), and (H.11.18):
(H.1.23)
$$\Phi \left(r,\theta ,\phi \right)=\mathrm{sin}\phi {\displaystyle \sum _{n=0}^{\infty}\left({c}_{n}{r}^{n}+\frac{{d}_{n}}{{r}^{n+1}}\right)}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)$$
where the
d _{n} are given by (
H.1.22) and the
c _{n} are determined by the boundary condition on the surface of the outer sphere
r =
R. Consider first the case of a dipole located at the origin (
r _{1}→0). A finite solution then requires that
d _{0} = 0. The constant term involving
c _{0} may be subtracted from (
H.1.23) without loss of generality since only potential gradients have physical meaning. We can generally drop the
n = 0 terms in (
H.1.14) for all problems in which current is conserved inside the volume conductor (in contrast to the mathematically equivalent electrostatics problem involving charges). Apply boundary condition (
4.9) at the surface
r =
R indicating that all current is confined inside to obtain
(H.1.24)
$${c}_{n}=\frac{n+1}{n{R}^{2n+1}}{d}_{n}$$
Thus, the potential due to a
y-directed tangential dipole in a homogeneous sphere is
(H.1.25)
$$\begin{array}{c}\Phi \left(r,\theta ,\phi \right)=\frac{-Id\mathrm{sin}\phi}{4\pi \sigma {r}^{2}}\\ \times {\displaystyle \sum _{n=1}^{\infty}\left[\frac{n+1}{n}{\left(\frac{r}{R}\right)}^{n+2}+{\left(\frac{R}{r}\right)}^{n-1}\right]}{\left(\frac{{r}_{1}}{R}\right)}^{n-1}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)\end{array}$$
We can check (
H.1.25) against the expression for a radial dipole in the center of the sphere by letting
r _{1} → 0. In this limit only the
n = 1 term in the sum is nonzero. The associated Legendre function follows from (
H.1.10),
${P}_{1}^{1}\left(\mathrm{cos}\theta \right)=-\mathrm{sin}\left(\theta \right)$, so the potential due to a central dipole with axis along the
y axis is given by
(H.1.26)
$$\Phi \left(r,\theta ,\phi \right)=\frac{Id\mathrm{sin}\theta \mathrm{sin}\phi}{4\pi \sigma {r}^{2}}\left[1+2{\left(\frac{r}{R}\right)}^{3}\right]$$
Compare (
H.1.26) to the expression (
6.18), noting that the dipole axis is aligned with the
z axis. Replace θ in (
6.18) by the angle γ between the
y axis and the vector
r:
(H.1.27)
$$\Phi \left(r,\theta ,\phi \right)=\frac{Id\mathrm{cos}\gamma}{4\pi \sigma {r}^{2}}\left[1+2{\left(\frac{r}{R}\right)}^{3}\right]$$
(p.573)
Use (
C.5) and fig.
C-1 to find that cosγ = sinθ sinφ so that (
H.1.26) and (
H.1.27) agree. In the case of an infinite homogeneous medium (
R → ∞), both equations reduce to the basic dipole formula (
1.7).
2 Tangential Dipole in Concentric Spheres Model
Equation (H.1.23) may be used as the basis for constructing solutions for head models consisting of concentric spherical shells. Formal solutions are constructed for each shell and interface boundary conditions used to find the unknown coefficients in the expansions as described in appendix G. Rewrite (H.1.23) using (H.1.22) and the notation of appendix G:
(H.2.1)
$$\Phi \left(r,\theta ,\phi \right)=-\mathrm{sin}\phi {\displaystyle \sum _{n=1}^{\infty}\left[{C}_{n}^{1}{r}^{n}+\frac{Id}{4\pi {\sigma}_{1}{r}_{4}^{2}}{\left(\frac{{r}_{4}}{{r}_{z}}\right)}^{2}{\left(\frac{{r}_{z}}{r}\right)}^{n+1}\right]}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)$$
Here we have changed the label for the radial location of the dipole
r _{1} to
r _{z} to be consistent with appendix
G. In order to simplify the following expressions, we normalize potentials with respect to
(G.4.4)
$${\Phi}_{0}=\frac{Id}{4\pi {\sigma}_{1}{r}_{4}^{2}}$$
and normalize sphere radii with respect to outer sphere radius by setting
r _{4} = 1. CSF and skull conductivities (σ
_{2}, σ
_{3}) are normalized with respect to brain (or scalp) conductivity by setting σ
_{1} = σ
_{4} = 1. Thus, (
H.2.1) expressed in nondimensional form is
(H.2.2)
$$\frac{\Phi \left(r,\theta ,\phi \right)}{{\Phi}_{0}}=-\mathrm{sin}\phi {\displaystyle \sum _{n=1}^{\infty}\left[{C}_{n}^{1}{r}^{n}+{r}_{z}^{-2}{\left(\frac{{r}_{z}}{r}\right)}^{n+1}\right]}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)$$
where all radii are interpreted as fractions of
r _{4} in the following expressions. Solutions are expressed formally in each of the four shells as in appendix
G. For example, the potential in the outer (scalp) shell is given by
(H.2.3)
$$\frac{\Phi \left(r,\theta ,\phi \right)}{{\Phi}_{0}}=-\mathrm{sin}\phi {\displaystyle \sum _{n=1}^{\infty}\left[{C}_{n}^{4}{r}^{n}+{D}_{n}^{4}{r}^{-\left(n+1\right)}\right]}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)$$
Applying the usual boundary condition of zero current flux normal to the scalp at the scalp surface (
r = 1) allows the coefficients
${D}_{n}^{4}$ to be expressed in terms of the
${D}_{n}^{4}$ so that (
H.2.3) simplifies to
(H.2.4)
$$\frac{\Phi \left(r,\theta ,\phi \right)}{{\Phi}_{0}}=-\mathrm{sin}\phi {\displaystyle \sum _{n=1}^{\infty}{C}_{n}^{4}\left(\frac{2n+1}{n+1}\right)}{P}_{n}^{1}\left(\mathrm{cos}\theta \right)$$
(p.574)
Here the coefficients are given in terms of ratios of radii and conductivities by
(H.2.5)
$${C}_{n}^{4}=\frac{-\left(n+1\right){\left(2n+1\right)}^{3}{\sigma}_{2}{\sigma}_{3}{r}_{z}^{n-1}{\left({r}_{2}{r}_{3}\right)}^{n+1}}{GF+J\left[H+{n}^{2}\left({K}_{1}+{K}_{2}+{K}_{3}\right)+n\left({P}_{1}+{P}_{2}\right)\right]}$$
where individual terms in the denominator are defined as follows:
(H.2.6)
$$G=n\left(2n+1\right){r}_{2}^{n+1}{r}_{3}^{-n}\left(n+{r}_{3}^{2n+1}+n{r}_{3}^{2n+1}\right)$$
(H.2.7)
$$F={\sigma}_{3}\left[\left({\sigma}_{2}-1\right)\left({\sigma}_{2}+n{\sigma}_{2}+n{\sigma}_{3}\right){r}_{1}^{2n+1}+\left({\sigma}_{3}-{\sigma}_{2}\right)\left(n+{\sigma}_{2}+n{\sigma}_{2}\right){r}_{2}^{2n+1}\right]$$
(H.2.8)
$$J=-n{\left({r}_{2}{r}_{3}\right)}^{-n}\left[1+\left({\sigma}_{3}-1\right){r}_{3}^{2n+1}+n\left(1+{\sigma}_{3}+\left({\sigma}_{3}-1\right){r}_{3}^{2n+1}\right)\right]$$
(H.2.9)
$$H={\sigma}_{2}{\sigma}_{3}{\left({r}_{2}{r}_{3}\right)}^{2n+1}$$
(H.2.10)
$${K}_{1}=\left({\sigma}_{2}-1\right)\left({\sigma}_{3}-{\sigma}_{2}\right){\left({r}_{1}{r}_{3}\right)}^{2n}$$
(H.2.11)
$${K}_{2}=\left({\sigma}_{2}+1\right)\left({\sigma}_{3}-{\sigma}_{2}\right){r}_{2}^{4n+2}$$
(H.2.12)
$${K}_{3}={r}_{2}\left({\sigma}_{2}+{\sigma}_{3}\right)\left[\left({\sigma}_{2}-1\right){r}_{1}{\left({r}_{1}{r}_{2}\right)}^{2n}+\left({\sigma}_{2}+1\right){r}_{3}{\left({r}_{2}{r}_{3}\right)}^{2n}\right]$$
(H.2.13)
$${P}_{1}={r}_{1}\left({\sigma}_{2}-1\right)\left[{\sigma}_{2}{r}_{2}{\left({r}_{1}{r}_{2}\right)}^{2n}+\left({\sigma}_{3}-{\sigma}_{2}\right){r}_{3}{\left({r}_{1}{r}_{3}\right)}^{2n}\right]$$
(H.2.14)
$${P}_{2}={r}_{2}\left[{\sigma}_{2}\left({\sigma}_{3}-{\sigma}_{2}\right){r}_{2}^{4n+1}+\left({\sigma}_{2}^{2}+{\sigma}_{3}+2{\sigma}_{2}{\sigma}_{3}\right){r}_{3}{\left({r}_{2}{r}_{3}\right)}^{2n}\right]$$