Mechanistic Modeling of Longitudinal Shape Changes: Equations of Motion and Inverse Problems

Area of Science:

• Computational anatomy
• Mathematical modeling
• Biophysics

Background:

• Understanding biological tissue shape changes (growth, atrophy) is crucial.
• Existing models may not fully capture the dynamics of morphological transformations.

Purpose of the Study:

• To develop and analyze a longitudinal shape evolution model for 3D volumes.
• To investigate the role of internal forces ('yank') in driving morphological changes.
• To establish conditions for modeling anatomical structure evolution.

Main Methods:

• Formulation of a shape evolution model based on elastic equilibria.
• Inclusion of a time-derivative of internal force ('yank') to drive transformations.
• Application of regularization for diffeomorphic transformations.
• Analysis of existence and uniqueness of solutions for equations of motion.
• Derivation of conditions for an optimal yank.

Main Results:

• Two distinct models of 'yank' were considered.
• Long-time existence and uniqueness of solutions were addressed for both models.
• Sufficient conditions for the existence of an optimal yank were derived.
• Preliminary results illustrate the model's ability to capture event attributes.

Conclusions:

• The proposed model provides a mathematical framework for understanding longitudinal shape evolution in biological tissues.
• The 'yank' concept offers a novel way to interpret metabolic or triggering events in morphological changes.
• The model shows promise for analyzing and potentially predicting anatomical structure dynamics.