Characterizing the shape of anatomical structures with Poisson's equation

Area of Science:

• Computational Physics
• Medical Imaging
• Shape Analysis

Background:

• Poisson's equation is a fundamental tool in classical physics with unique properties relevant to shape analysis.
• Equipotential sets in Poisson's equation solutions exhibit increasing smoothness with potential.
• Quantifying shape complexity is crucial for detailed anatomical and pathological studies.

Purpose of the Study:

• To introduce and validate a new shape descriptor, the 'shape characteristic', based on Poisson's equation.
• To measure and analyze the complexity of equipotential sets using the displacement map.
• To apply this novel shape analysis technique to neuroanatomical data from magnetic resonance (MR) images.

Main Methods:

• Solving Poisson's equation to obtain potential fields.
• Calculating the displacement map (streamline length of the gradient field) to quantify equipotential set complexity.
• Defining the 'shape characteristic' as a function of potential and displacement map.
• Developing robust algorithms for computing the solution, displacement map, and shape characteristic.
• Illustrating the technique on 2D synthetic and natural shapes.
• Applying the method to 3D neuroanatomical data (caudate nucleus) from MR images.

Main Results:

• The shape characteristic effectively quantifies the complexity and smoothness of equipotential sets.
• The method successfully identified structural shape differences in the caudate nucleus of individuals with Schizotypal Personality Disorder (SPD).
• Novel shape differences were detected in the caudate nuclei of premature infants with asymmetric white matter injury, surpassing the sensitivity of volumetric measurements.

Conclusions:

• The shape characteristic offers a natural and powerful new way to express and analyze shape.
• This Poisson's equation-based approach provides a sensitive tool for neuroanatomical shape analysis, particularly in detecting subtle abnormalities.
• The technique demonstrates potential for advancing the understanding and diagnosis of neurological disorders through detailed shape quantification.