### Abstract

A surface-based diffeomorphic algorithm to generate 3D coordinate grids in the cortical ribbon is described. In the grid, normal coordinate lines are generated by the diffeomorphic evolution from the gray/white (inner) surface to the gray/csf (outer) surface. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries. Conceptually, the inner surface sits on top of the white matter structure and the outer on top of the gray matter. It is assumed that the cortical ribbon consists of cortical columns which are orthogonal to the white matter surface. This might be viewed as a consequence of the development of the columns in the embryo. It is also assumed that the columns are orthogonal to the outer surface so that the resultant vector field is orthogonal to the evolving surface. Then the distance of the normal lines from the vector field such that the inner surface evolves diffeomorphically towards the outer one can be construed as a measure of thickness. Applications are described for the auditory cortices in human adults and cats with normal hearing or hearing loss. The approach offers great potential for cortical morphometry.

Language | English |
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Title of host publication | Mathematics of shapes and applications |

Editors | Sergey Kushnarev, Anqi Qiu, Laurent Younes |

Place of Publication | Singapore |

Publisher | World Scientific Publishing |

Chapter | 7 |

Pages | 167-179 |

Number of pages | 13 |

ISBN (Electronic) | 9789811200144 |

ISBN (Print) | 9789811200120 |

DOIs | |

Publication status | Published - 2020 |

Externally published | Yes |

### Publication series

Name | Lecture Notes Series, Institute for Mathematical Sciences |
---|---|

Publisher | World Scientific Publishing |

Volume | 37 |

ISSN (Print) | 1793-0758 |

### Fingerprint

### Bibliographical note

Copyright the Author(s) 2020. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.### Cite this

*Mathematics of shapes and applications*(pp. 167-179). (Lecture Notes Series, Institute for Mathematical Sciences; Vol. 37). Singapore: World Scientific Publishing. https://doi.org/10.1142/9789811200137_0007

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*Mathematics of shapes and applications.*Lecture Notes Series, Institute for Mathematical Sciences, vol. 37, World Scientific Publishing, Singapore, pp. 167-179. https://doi.org/10.1142/9789811200137_0007

**3D normal coordinate systems for cortical areas.** / Ratnanather, J. Tilak; Arguillère, Sylvain; Kutten, Kwame S.; Hubka, Peter; Kral, Andrej; Younes, Laurent.

Research output: Chapter in Book/Report/Conference proceeding › Chapter › Research › peer-review

TY - CHAP

T1 - 3D normal coordinate systems for cortical areas

AU - Ratnanather, J. Tilak

AU - Arguillère, Sylvain

AU - Kutten, Kwame S.

AU - Hubka, Peter

AU - Kral, Andrej

AU - Younes, Laurent

N1 - Copyright the Author(s) 2020. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.

PY - 2020

Y1 - 2020

N2 - A surface-based diffeomorphic algorithm to generate 3D coordinate grids in the cortical ribbon is described. In the grid, normal coordinate lines are generated by the diffeomorphic evolution from the gray/white (inner) surface to the gray/csf (outer) surface. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries. Conceptually, the inner surface sits on top of the white matter structure and the outer on top of the gray matter. It is assumed that the cortical ribbon consists of cortical columns which are orthogonal to the white matter surface. This might be viewed as a consequence of the development of the columns in the embryo. It is also assumed that the columns are orthogonal to the outer surface so that the resultant vector field is orthogonal to the evolving surface. Then the distance of the normal lines from the vector field such that the inner surface evolves diffeomorphically towards the outer one can be construed as a measure of thickness. Applications are described for the auditory cortices in human adults and cats with normal hearing or hearing loss. The approach offers great potential for cortical morphometry.

AB - A surface-based diffeomorphic algorithm to generate 3D coordinate grids in the cortical ribbon is described. In the grid, normal coordinate lines are generated by the diffeomorphic evolution from the gray/white (inner) surface to the gray/csf (outer) surface. Specifically, the cortical ribbon is described by two triangulated surfaces with open boundaries. Conceptually, the inner surface sits on top of the white matter structure and the outer on top of the gray matter. It is assumed that the cortical ribbon consists of cortical columns which are orthogonal to the white matter surface. This might be viewed as a consequence of the development of the columns in the embryo. It is also assumed that the columns are orthogonal to the outer surface so that the resultant vector field is orthogonal to the evolving surface. Then the distance of the normal lines from the vector field such that the inner surface evolves diffeomorphically towards the outer one can be construed as a measure of thickness. Applications are described for the auditory cortices in human adults and cats with normal hearing or hearing loss. The approach offers great potential for cortical morphometry.

UR - http://www.scopus.com/inward/record.url?scp=85075755688&partnerID=8YFLogxK

U2 - 10.1142/9789811200137_0007

DO - 10.1142/9789811200137_0007

M3 - Chapter

SN - 9789811200120

T3 - Lecture Notes Series, Institute for Mathematical Sciences

SP - 167

EP - 179

BT - Mathematics of shapes and applications

A2 - Kushnarev, Sergey

A2 - Qiu, Anqi

A2 - Younes, Laurent

PB - World Scientific Publishing

CY - Singapore

ER -