Basic mathematical and electromagnetic concepts of the biomagnetic inverse problem.

Area of Science:

• Biophysics
• Biomagnetism
• Mathematical Modeling

Background:

• The biomagnetic inverse problem involves reconstructing internal sources from external magnetic field measurements.
• Understanding the forward problem (field generation) is crucial for solving the inverse problem.

Purpose of the Study:

• To review mathematical and physical concepts of the biomagnetic inverse problem.
• To present new approaches for solving the biomagnetic inverse problem.
• To analyze challenges like solution non-uniqueness and electric potential contributions.

Main Methods:

• Discussion of forward problem modeling for homogeneous and inhomogeneous media.
• Application of Geselowitz' formulae and surface integral equations for piecewise homogeneous conductors.
• Detailed analysis of spherically symmetric and horizontally layered media.

Main Results:

• The non-uniqueness of solutions to the magnetic inverse problem is highlighted.
• The influence of electric potential on external magnetic fields is investigated.
• Practical methods like weighted least-squares and minimum norm estimates are discussed.

Conclusions:

• Effective methods for solving the biomagnetic inverse problem are presented.
• Challenges in biomagnetic inverse problem solutions are identified and analyzed.
• The study contributes to advancing biomagnetic inverse problem methodologies.