We introduce the concept of the current dipole and spherically symmetric conductor into the Biot-Savart law to reach the Sarvas formula (Forward solution), which is the analytical solution of the Biot-Savart law.
The magnetic field is expressed as \boldsymbol{b} when outputted by the sensor array, or as \boldsymbol{\hat{b}} when estimated by the Sarvas equation. Since the estimated magnetic field \boldsymbol{\hat{b}} from the Sarvas equation is linear to the current moment \boldsymbol{q}, the magnetic field calculation matrix Lead Field Matrix (LFM) is expressed as \boldsymbol{L} and the linear relationship is written as \boldsymbol{\hat{b}}=\boldsymbol{L}\boldsymbol{q}. Now we are ready to solve the inverse problem to compare the output of the sensor array \boldsymbol{b} with the estimated \boldsymbol{\hat{b}}.
The relationship between the Forward solution and the Inverse problem is shown in the figure below. Goodness of fit (GOF) is an important concept in overdetermined systems [1].
Let’s start discussing the forward problem in the next page.
(Reference)
- E Kaukoranta, M Hämäläinen, J Sarvas, R Hari: Mixed and sensory nerve stimulations activate different cytoarchitectonic areas in the human primary somatosensory cortex SI. Neuromagnetic recordings and statistical considerations. Exp Brain Res. 1986;63(1):60-6.