New Numerical Solution of the Laplace Equation for Tissue Thickness Measurement in Three-Dimensional MRI

  • Haissam Haidar
  • Svetlana Egorova
  • Janet S. Soul


Previous work has shown the importance of thickness measurement in vivo using three-dimensional magnetic resonance imaging (3D MRI). Thickness is defined as the length of trajectories, also called streamlines, which follow the gradient of the solution of the Laplace equation solved between the inner and the outer surfaces of the tissue using Dirichlet conditions.

We present a new numerical solution of the Laplace equation for 3D MRI. Our method is accurate and computationally fast. High accuracy is obtained by solving the Laplace equation for anisotropic 3D MRI.

We present also an fast and accurate algorithm for calculation of the length of the streamlines. This algorithm is based on a 26 voxels neighbors method and consists of the summation of the Euclidean distance between different voxels neighbors on the same streamline.

Our approach was tested on set of synthetic images and several medical applications including knee cartilage, cerebral cortex of a normal adult and cerebral cortex of a newborn. We compare the results with the Euclidean distance measured normal to one boundary along a path between the two boundaries. Numerical validation was performed on set of magnetic resonance images of the knee cartilage. It shows that the 3D PDE approach provides a better result than the Euclidean distance.


Laplace equation MRI thickness streamline 


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Copyright information

© Springer 2005

Authors and Affiliations

  • Haissam Haidar
    • 1
  • Svetlana Egorova
    • 2
  • Janet S. Soul
    • 1
  1. 1.Department of NeurologyChildren’s Hospital and Harvard Medical SchoolBostonUSA
  2. 2.Center for Neurologic ImagingBrigham and Women’s Hospital, Harvard Medical SchoolBostonUSA

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