Phase 2 · In Silico Physics Workbench

Biophysical Contrast Simulation Sandbox

Interactive computational workbench for exploring the physical governing equations of neural contrast. Simulate spin magnetization relaxation (Bloch equations), optical photon diffusion (Beer-Lambert transport), and electromagnetic dipole volume conduction in real time.

Nuclear Magnetic Resonance (Bloch Equation Simulator)

Simulate how magnetic field strength $B_0$, radiofrequency (RF) pulse flip angle $\alpha$, and tissue-specific $T_1 / T_2^*$ relaxation times dictate transverse decay $M_{xy}(t)$ and longitudinal recovery $M_z(t)$.

Tissue Compartment Presets

Main Magnetic Field ($B_0$) 3.0 Tesla

RF Flip Angle ($\alpha$) 90°

Spin-Lattice Relaxation ($T_1$) 1,300 ms

Transverse Relaxation ($T_2^*$) 45 ms

Echo Time (TE) Marker 30 ms

Larmor Frequency ($\omega_0 / 2\pi$): 127.7 MHz

Signal at Echo Time $M_{xy}(\text{TE})$: 51.3%

Equilibrium Magnetization $M_0 \propto B_0$: 3.00 a.u.

Bloch Governing Equations: $$\frac{dM_x}{dt} = \gamma (\mathbf{M} \times \mathbf{B})_x - \frac{M_x}{T_2^*}, \quad \frac{dM_y}{dt} = \gamma (\mathbf{M} \times \mathbf{B})_y - \frac{M_y}{T_2^*}, \quad \frac{dM_z}{dt} = \gamma (\mathbf{M} \times \mathbf{B})_z - \frac{M_z - M_0}{T_1}$$ In a standard Gradient Recalled Echo (GRE) pulse sequence, initial transverse magnetization is $M_{xy}(0) = M_0 \sin(\alpha)$. At time $t = \text{TE}$, the recorded free induction decay (FID) signal is $S(\text{TE}) = M_0 \sin(\alpha) e^{-\text{TE} / T_2^*}$. BOLD functional MRI relies upon localized microvascular deoxygenation shifting $\Delta T_2^*$.

Diffuse Optical Spectroscopy & Photon Migration (fNIRS)

Explore photon banana trajectories through scalp, skull, CSF, and cerebral cortex under the diffusion approximation of radiative transport.

Source-Detector Distance ($d$) 30 mm

Scalp & Skull Thickness 12 mm

Cortical $\Delta[\text{HbO}]$ Activation +4.5 μM

Cortical $\Delta[\text{HbR}]$ Deactivation -1.8 μM

Mean Photon Penetration Depth ($\approx d/2$): 15.0 mm

Cortical Sensitivity Fraction: 18.4%

Optical Density Shift $\Delta \text{OD}_{850\text{nm}}$: +0.042

Modified Beer-Lambert Law (MBLL): $$\Delta \text{OD}(\lambda) = \left[ \epsilon_{\text{HbO}}(\lambda) \Delta [\text{HbO}] + \epsilon_{\text{HbR}}(\lambda) \Delta [\text{HbR}] \right] \cdot d \cdot \text{DPF}(\lambda) + G$$ Because near-infrared light ($700\text{–}900\text{ nm}$) undergoes thousands of forward scattering events ($\mu_s' \gg \mu_a$), photons travel in an arcuate "banana-shaped" probability volume. Light only probes the cerebral cortex if the source-detector distance $d$ exceeds approximately $2.5 \times$ the skull-scalp thickness.

Neuronal Dipole Volume Conduction (EEG vs. MEG)

Compare scalp electrical potential $V(\mathbf{r})$ and external magnetic flux $B(\mathbf{r})$ as a function of dipole depth, orientation (radial vs tangential), and volume conductor conductivity.

Dipole Depth from Inner Skull ($z$) 15 mm

Dipole Orientation Angle ($\theta$) 90° (Tangential)

Dipole Moment ($Q$) 25 nA·m

Skull Conductivity Ratio ($\sigma_{\text{brain}} / \sigma_{\text{skull}}$) 30:1 (Normal)

Peak Scalp Electric Potential ($V_{\text{max}}$): 3.4 μV

Peak External Magnetic Field ($B_{\text{max}}$): 148.2 fT

Radial vs. Tangential Field Cancellation: 0% (Pure Tangential)

Volume Conduction & Dipolar Symmetries: In a spherically symmetric volume conductor, a purely radial current dipole produces an external magnetic field of exactly $\mathbf{B} = 0$ everywhere outside the sphere, because primary source fields are perfectly cancelled by volume return currents! In contrast, a tangential dipole produces symmetric bipolar magnetic flux loops with minimal skull smearing. EEG detects both radial and tangential dipoles but suffers severe spatial smearing from the low conductivity of the skull ($\sigma \approx 0.01\text{ S/m}$).