Electromagnetism in Biology & Medicine

Derivation: Volume Conductor Model and the EEG Forward Problem

In a volume conductor with conductivity $\sigma$, the extracellular potential$\phi(\mathbf{r})$ satisfies Poisson's equation with a source term from the impressed current density $\mathbf{J}_i$ (membrane current):

$$\nabla \cdot (\sigma \nabla \phi) = \nabla \cdot \mathbf{J}_i$$

For a homogeneous infinite medium ($\sigma$ constant), this reduces to Poisson's equation. The solution using the Green's function is:

$$\phi(\mathbf{r}) = \frac{1}{4\pi\sigma} \int_V \frac{\mathbf{J}_i(\mathbf{r}') \cdot \hat{\mathbf{R}}}{R^2} \, dV'$$

where $\mathbf{R} = \mathbf{r} - \mathbf{r}'$ and $R = |\mathbf{R}|$. This is the fundamental equation of bioelectricity. For a localized source, we can approximate using the current dipole:

$$\mathbf{p} = \int_V \mathbf{J}_i(\mathbf{r}') \, dV' \quad \text{(current dipole moment, units: A·m)}$$

The dipole approximation for the potential at distance $r \gg$ source size is:

$$\phi(\mathbf{r}) = \frac{1}{4\pi\sigma} \frac{\mathbf{p} \cdot \hat{\mathbf{r}}}{r^2} = \frac{p\cos\theta}{4\pi\sigma r^2}$$

For the EEG forward problem, we need to compute scalp potentials from cortical dipole sources. In a spherical head model with concentric shells (brain, CSF, skull, scalp) of different conductivities $\sigma_1, \sigma_2, \sigma_3, \sigma_4$:

$$\phi(\mathbf{r}) = \frac{1}{4\pi\sigma_1} \sum_{n=1}^{\infty} \frac{(2n+1)^3}{n} \frac{f_n(r_d)}{g_n} P_n(\cos\theta)$$

where $P_n$ are Legendre polynomials, $r_d$ is the dipole depth, and $f_n, g_n$ depend on the shell radii and conductivities. The skull has very low conductivity ($\sigma_{\text{skull}} \approx 0.004$ S/m vs$\sigma_{\text{brain}} \approx 0.33$ S/m), which strongly attenuates and spatially smooths the cortical potentials. A typical cortical dipole has moment$p \sim 10$ nA·m and produces scalp potentials of ~10–100 $\mu$V.

For a radial dipole at depth $d$ below the scalp in a simplified model:$$\phi_{\text{scalp}} \approx \frac{p}{4\pi\sigma_{\text{eff}}} \frac{1}{d^2}$$The effective conductivity $\sigma_{\text{eff}}$ includes the attenuation by the skull. The $1/d^2$ falloff means deeper sources are harder to detect, limiting EEG to cortical activity within ~3 cm of the surface.

Derivation: Einthoven's Triangle and the Cardiac Vector

The heart can be modeled as a single time-varying current dipole $\mathbf{p}(t)$located at the center of the chest (the cardiac vector). Einthoven placed electrodes on the right arm (RA), left arm (LA), and left leg (LL), forming an equilateral triangle in the frontal plane.

The three standard limb leads measure potential differences:

$$\text{Lead I} = V_{LA} - V_{RA} = \mathbf{p} \cdot \hat{\mathbf{l}}_I$$

$$\text{Lead II} = V_{LL} - V_{RA} = \mathbf{p} \cdot \hat{\mathbf{l}}_{II}$$

$$\text{Lead III} = V_{LL} - V_{LA} = \mathbf{p} \cdot \hat{\mathbf{l}}_{III}$$

where $\hat{\mathbf{l}}_I$, $\hat{\mathbf{l}}_{II}$, $\hat{\mathbf{l}}_{III}$are lead vectors pointing from negative to positive electrode. By the properties of the equilateral triangle:

$$\text{Lead I} + \text{Lead III} = \text{Lead II} \quad \text{(Einthoven's law)}$$

Lead field theory (Helmholtz reciprocity) provides a more general formulation. The voltage measured by a lead is:

$$V_{\text{lead}} = \int_V \mathbf{J}_{\text{lead}}(\mathbf{r}) \cdot \mathbf{J}_{\text{source}}(\mathbf{r}) \, dV$$

where $\mathbf{J}_{\text{lead}}$ is the current density field that would exist if unit current were injected through the recording electrodes. This elegant reciprocal formulation shows that the measured signal depends on the overlap between the source and the lead field.

ECG waveform interpretation: The cardiac vector traces a loop in the frontal plane during each heartbeat:

• P wave: Atrial depolarization. Vector points leftward and inferiorly. Duration ~80 ms, amplitude ~0.25 mV.

• QRS complex: Ventricular depolarization. The vector sweeps from right to left as the depolarization wave spreads through the ventricles. Duration ~80–120 ms. The large amplitude (~1 mV) reflects the massive ventricular muscle.

• T wave: Ventricular repolarization. The vector points in roughly the same direction as QRS (not opposite!) because repolarization proceeds from epicardium to endocardium (opposite direction to depolarization), and the sign reversal of repolarization cancels the direction reversal.

Derivation: Specific Absorption Rate and Skin Depth

The specific absorption rate (SAR) quantifies the rate of energy absorbed per unit mass of tissue:

$$\text{SAR} = \frac{\sigma |\mathbf{E}|^2}{2\rho} \quad \text{(W/kg)}$$

where $\sigma$ is the tissue conductivity (S/m), $|\mathbf{E}|$ is the electric field amplitude (V/m), and $\rho$ is the tissue density (kg/m$^3$). For thermal equilibrium, the temperature rise is governed by the bioheat equation:

$$\rho c \frac{\partial T}{\partial t} = k\nabla^2 T + \rho \cdot \text{SAR} - \rho_b c_b w_b(T - T_b)$$

where the last term represents blood perfusion cooling. The penetration depth (skin depth) describes how far EM waves penetrate into tissue:

$$\delta = \sqrt{\frac{2}{\omega\mu\sigma}} = \frac{1}{\sqrt{\pi f \mu \sigma}}$$

where $\omega = 2\pi f$ is the angular frequency and $\mu \approx \mu_0$for biological tissue. The field amplitude decays as $E(z) = E_0 e^{-z/\delta}$.

For muscle tissue ($\sigma \approx 0.7$ S/m at 1 GHz):

$$\delta(1 \text{ GHz}) = \frac{1}{\sqrt{\pi \times 10^9 \times 4\pi \times 10^{-7} \times 0.7}} \approx 0.019 \text{ m} = 1.9 \text{ cm}$$

At lower frequencies ($\sigma \approx 0.3$ S/m at 100 MHz):

$$\delta(100 \text{ MHz}) \approx 0.092 \text{ m} = 9.2 \text{ cm}$$

This explains why microwave frequencies (GHz) deposit energy in a thin surface layer (heating), while RF frequencies (MHz) penetrate deeply with less energy deposition per unit volume. Safety limits: whole-body SAR $< 0.4$ W/kg (occupational), $< 0.08$ W/kg (public), with localized limits of 10 W/kg for head and trunk.

Derivation: Bloch Equations and the Spin Echo

The Bloch equations describe the time evolution of the nuclear magnetization vector $\mathbf{M} = (M_x, M_y, M_z)$ in an external field $\mathbf{B}_0 = B_0\hat{z}$:

$$\frac{dM_x}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_x - \frac{M_x}{T_2}$$

$$\frac{dM_y}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_y - \frac{M_y}{T_2}$$

$$\frac{dM_z}{dt} = \gamma(\mathbf{M} \times \mathbf{B})_z - \frac{M_z - M_0}{T_1}$$

In the rotating frame at the Larmor frequency$\omega_0 = \gamma B_0$, after a 90° pulse tips the magnetization into the transverse plane, the free induction decay is:

$$M_{xy}(t) = M_0 e^{-t/T_2^*} e^{i\Delta\omega t}$$

where $T_2^* \leq T_2$ includes field inhomogeneity effects. Thespin echo (Hahn, 1950) refocuses the dephasing from static field inhomogeneities:

• At $t = 0$: 90° pulse tips $M_z \to M_{xy}$

• Free precession: spins dephase due to local field variations. Fast spins advance, slow spins lag.

• At $t = \tau$: 180° pulse flips the transverse magnetization. Now fast spins are behind, slow spins are ahead.

• At $t = 2\tau$: Echo forms as spins rephase. The echo amplitude is $M_0 e^{-2\tau/T_2}$ (only true $T_2$ decay, not $T_2^*$).

The MRI signal intensity for a spin-echo sequence with repetition time TR and echo time TE is:

$$S = \rho_H \left(1 - e^{-\text{TR}/T_1}\right) e^{-\text{TE}/T_2}$$

where $\rho_H$ is the proton density. This equation reveals two contrast mechanisms:

• T$_1$-weighted: Short TR (~500 ms), short TE (~20 ms). The $(1 - e^{-\text{TR}/T_1})$ term dominates — tissues with short T$_1$ (e.g., fat) appear bright.

• T$_2$-weighted: Long TR (~3000 ms), long TE (~80 ms). The $e^{-\text{TE}/T_2}$ term dominates — tissues with long T$_2$ (e.g., CSF) appear bright.

At 1.5 T, typical values: grey matter T$_1$ $\approx 920$ ms, T$_2$ $\approx 100$ ms; white matter T$_1$ $\approx 780$ ms, T$_2$ $\approx 90$ ms; CSF T$_1$ $\approx 4000$ ms, T$_2$ $\approx 2000$ ms; fat T$_1$ $\approx 260$ ms, T$_2$ $\approx 80$ ms.

Derivation: Radiation Dose-Response and Transcranial Magnetic Stimulation

The absorbed dose in radiotherapy is:

$$D = \frac{E_{\text{absorbed}}}{m} \quad \text{(Gray, 1 Gy = 1 J/kg)}$$

Cell survival after irradiation follows the linear-quadratic (LQ) model:

$$S(D) = \exp(-\alpha D - \beta D^2)$$

where $\alpha$ represents single-hit lethal damage (double-strand breaks from a single ionization track) and $\beta$ represents two-hit lethal damage (interaction of two sub-lethal lesions). The $\alpha/\beta$ ratio characterizes tissue radiosensitivity:

• High $\alpha/\beta \approx 10$ Gy: Rapidly dividing tissues (tumors, mucosa, bone marrow). The linear term dominates — less sensitive to fractionation.

• Low $\alpha/\beta \approx 3$ Gy: Late-responding tissues (spinal cord, lung, kidney). The quadratic term is relatively more important — more sensitive to dose per fraction.

The therapeutic ratio is the ratio of tumor control dose to normal tissue complication dose. For $n$ fractions of dose $d$:

$$\text{BED} = nd\left(1 + \frac{d}{\alpha/\beta}\right)$$

This biologically effective dose (BED) allows comparison of different fractionation schemes. Since tumors have high $\alpha/\beta$ and normal tissue has low $\alpha/\beta$, smaller fractions spare normal tissue more than tumor — the biological basis of fractionated radiotherapy.

Transcranial magnetic stimulation (TMS): A time-varying magnetic field induces an electric field in the brain via Faraday's law:

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

For a circular coil of radius $a$ with $N$ turns carrying current$I(t)$, the induced electric field at depth $z$ below the coil center is approximately:

$$E(r,z) \approx -\frac{\mu_0 N a^2}{2} \frac{r}{(a^2 + z^2)^{3/2}} \frac{dI}{dt}$$

where $r$ is the radial distance from the coil axis. Typical TMS parameters: peak $dI/dt \approx 10^8$ A/s, inducing $E \approx 100$ V/m in cortex (sufficient to depolarize neurons). The figure-of-eight coil configuration provides better spatial focality, with the maximum E-field at the intersection of the two loops. TMS is FDA-approved for treatment-resistant depression (2008) and obsessive-compulsive disorder (2018).

Derivation: Spatial Encoding with Magnetic Field Gradients

MRI creates images by spatially encoding the NMR signal using magnetic field gradients. A gradient $G_x$ in the $x$-direction makes the Larmor frequency position-dependent:

$$\omega(x) = \gamma(B_0 + G_x \cdot x)$$

Slice selection: A frequency-selective RF pulse applied during a gradient $G_z$ excites only spins within a thin slice. The slice thickness is:

$$\Delta z = \frac{\Delta\omega}{\gamma G_z} = \frac{2\pi\Delta f}{\gamma G_z}$$

where $\Delta\omega$ is the bandwidth of the RF pulse. Typical values:$G_z = 10$ mT/m, $\Delta f = 2$ kHz gives$\Delta z = 2\pi \times 2000/(2.675 \times 10^8 \times 0.01) \approx 4.7$ mm.

Frequency encoding: During signal readout, a gradient$G_x$ maps spatial position to frequency. The signal is:

$$s(t) = \int \rho(x) e^{-i\gamma G_x x t} dx$$

This is a Fourier transform! The image is reconstructed by inverse FT. The spatial resolution is determined by the gradient strength and readout duration:$\Delta x = 2\pi/(\gamma G_x T_{\text{read}})$. For $G_x = 20$ mT/m and$T_{\text{read}} = 5$ ms: $\Delta x \approx 0.74$ mm.

Phase encoding: The second spatial dimension uses a gradient pulse $G_y$ of varying amplitude before readout, imparting a position-dependent phase: $\phi(y) = \gamma G_y y \tau$. A full image requires$N_y$ phase-encoding steps (typically 128–512), which determines the scan time. The total imaging time for a spin-echo sequence is$T = N_y \times \text{TR}$. For $N_y = 256$ and TR = 500 ms, this gives $T = 128$ seconds — a major limitation that drove the development of fast imaging sequences like EPI and FLASH.

Functional MRI (fMRI) and the BOLD Effect

Functional MRI exploits the blood oxygenation level-dependent (BOLD) effect discovered by Ogawa (1990). Deoxyhemoglobin is paramagnetic while oxyhemoglobin is diamagnetic. When neurons are active, local blood flow increases (neurovascular coupling), overcompensating for oxygen consumption, thus decreasing the local deoxyhemoglobin concentration. This reduces the local magnetic field inhomogeneity, increasing $T_2^*$:

$$\frac{1}{T_2^*} = \frac{1}{T_2} + \gamma \Delta B_{\text{inhomog}}$$

The BOLD signal change is typically only 1–5% above baseline, requiring statistical analysis over many repeated trials. The hemodynamic response function (HRF) peaks at about 5–6 seconds after neural activation, with a spatial resolution of ~2–3 mm at 3T. At 7T ultra-high field, submillimeter resolution can resolve cortical columns and layers.