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Research Papers

A New Method to Measure Cortical Growth in the Developing Brain

[+] Author and Article Information
Andrew K. Knutsen1

Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, 1 Brookings Drive, P.O. Box 1185, Saint Louis, MO 63130akknutsen@gmail.com

Yulin V. Chang

Department of Mechanical Engineering and Materials Science, Washington University in St. Louis, 1 Brookings Drive, P.O. Box 1185, Saint Louis, MO 63130

Cindy M. Grimm, Ly Phan

Department of Computer Science and Engineering, Washington University in St. Louis, 1 Brookings Drive, P.O. Box 1185, Saint Louis, MO 63130

Larry A. Taber, Philip V. Bayly

Department of Mechanical Engineering and Materials Science and Department of Biomedical Engineering, Washington University in St. Louis, 1 Brookings Drive, P.O. Box 1185, Saint Louis, MO 63130

1

Corresponding author.

J Biomech Eng 132(10), 101004 (Oct 01, 2010) (11 pages) doi:10.1115/1.4002430 History: Received June 18, 2010; Revised August 04, 2010; Posted August 24, 2010; Published October 01, 2010; Online October 01, 2010

Folding of the cerebral cortex is a critical phase of brain development in higher mammals but the biomechanics of folding remain incompletely understood. During folding, the growth of the cortical surface is heterogeneous and anisotropic. We developed and applied a new technique to measure spatial and directional variations in surface growth from longitudinal magnetic resonance imaging (MRI) studies of a single animal or human subject. MRI provides high resolution 3D image volumes of the brain at different stages of development. Surface representations of the cerebral cortex are obtained by segmentation of these volumes. Estimation of local surface growth between two times requires establishment of a point-to-point correspondence (“registration”) between surfaces measured at those times. Here we present a novel approach for the registration of two surfaces in which an energy function is minimized by solving a partial differential equation on a spherical surface. The energy function includes a strain-energy term due to distortion and an “error energy” term due to mismatch between surface features. This algorithm, implemented with the finite element method, brings surface features into approximate alignment while minimizing deformation in regions without explicit matching criteria. The method was validated by application to three simulated test cases and applied to characterize growth of the ferret cortex during folding. Cortical surfaces were created from MRI data acquired in vivo at 14 days, 21 days, and 28 days of life. Deformation gradient and Lagrangian strain tensors describe the kinematics of growth over this interval. These quantitative results illuminate the spatial, temporal, and directional patterns of growth during cortical folding.

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Copyright © 2010 by American Society of Mechanical Engineers
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Figures

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Figure 1

Surfaces and surface relationships. YAS is the younger anatomical surface, OAS is the older anatomical surface, ROAS is the relaxed older anatomical surface, OSS is the older spherical surface and ROSS is the relaxed older spherical surface. F0 is the deformation gradient between OAS and YAS, H is the deformation gradient between OSS and OAS (as well as DOSS and DOAS), and G1 is the deformation gradient between ROSS and OSS. The deformation gradient between ROAS and YAS is given by the tensor product F=H⋅G1⋅H−1⋅F0. The deformation gradient tensor G1 characterizes the displacement between ROSS and OSS. Displacements are calculated by using the finite element method to solve the equation of motion of an elastic membrane subject to viscous body force. The solution minimizes distortions between YAS and ROAS while aligning features on the two surfaces.

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Figure 2

The LACROSS registration algorithm: An initial correspondence is determined between YAS and OAS. The matching terms f1 and f2 along with the deformation gradients F0 and H are calculated. The finite element method is used to solve Eq. 27 for displacements on OSS. Two measures of convergence are checked. If the convergence criterion is not met, then the coordinates are updated, and steps 3–5 are repeated. Once the convergence criterion is met, the coordinates are updated and the surfaces are analyzed.

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Figure 5

Registration of a sphere to a pumpkin: The deformed surface was created by growth and folding of the reference surface. (a) A sphere of radius one was used for the YAS. (b) The OAS surface was created by adjusting the radius as a function of the spherical coordinates θ and ϕ. (c) Initial distortions are visualized by areal expansion (dilatation ratio, J) and the eigenvectors associated with first (blue) and second (green) principal strains. The eigenvectors are scaled by their corresponding principal stretches. (d) After relaxation, the resulting areal expansion is uniform over the surface but the eigenvectors associated with stretch show the necessary anisotropic expansion required for the sphere to grow into the lobed surface.

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Figure 6

Smoothing of a local region on P14 cortical surface: The red region was identified on the P14 cortical surface of kit A.2 and was iteratively smoothed for 500 iterations in CARET software. The smoothed region mimics a less mature cortex in the selected region. (a) The smoothed surface was set to be YAS. (b) The original surface is set to be OAS. (c,d) The feature matching terms f1 and f2 were created from the border that was used to bind the region that was smoothed. (e) The initial difference between the two matching terms was small because the landmark was at the same spatial location on each surface. (f) After applying the registration approach, the difference between the matching terms remains small.

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Figure 7

Registration of a P14 cortical surface to itself with smoothing applied to a local region: The goal of this test case is to be able to identify local growth on a complex surface. (a,c) Initial distortions are visualized by the strain-energy density function and the areal expansion (dilatation ratio J) between YAS and OAS. Only the coordinates within the patch were adjusted, so W0 and J0 are constant outside of the patch and vary within it. (e,f) Eigenvectors associated with first (blue) and second (black) principal stretches are plotted with the dilatation ratio as an underlay.

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Figure 9

Areal expansion associated with cortical growth in the ferret from P14 to P21 (kit A.2): The dilatation ratio J was calculated before (a) and after (b) the relaxation algorithm was applied. (c) Expansion of the color scale provides more detail into the differences in growth within the cortex (medial and lateral views).

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Figure 10

Areal expansion and principal stretch fields associated with cortical surface growth in the ferret from P14 to P21 (both kits): plotted over the dilation ratio J are the eigenvectors associated with first (blue) and second (black) principal stretches. Vectors are scaled by their corresponding principal stretch value and were plotted for every fifth point on the surface. The primary direction of growth tends to be across gyri (i.e., perpendicular to sulci).

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Figure 11

Surface feature matching terms for registration of P21 to P28 ferret cortical surfaces (kit A.2): Mean curvature and geodesic distance from the medial wall were used to calculate the surface matching terms f1 and f2. (a) Surface feature function f1 on YAS. (b) Function f2 on OAS. (c) The difference between matching terms (f1−f2) at the initial time t0. (d) The difference between matching terms (f1−f2) at the final time tf. The initial differences between f1 and f2 are reduced after the application of the relaxation algorithm. As from P14 to P21, the largest reductions are seen around the medial wall.

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Figure 12

Areal expansion results for P21 to P28 (kit A.2): The dilatation ratio J was calculated before (a) and after (b) the relaxation algorithm was applied. (c) Expansion of the color scale provides more detail into the differences in growth within the cortex. Again, a similar pattern of growth is seen in kit A.1. Also, the pattern of growth is similar to the pattern seen from P14 to P21. However, the total amount of growth and the variability in growth throughout the cortex are reduced from P21 to P28.

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Figure 13

Areal expansion and principal strain fields from P21 to P28 (both kits): plotted over the dilatation ration J are the eigenvectors associated with first (blue) and second (black) principal stretches. Vectors are scaled by their corresponding principal stretch value, and were plotted for every fifth point on the surface. Growth is smaller and more uniform during this period compared with the interval from P14 to P21, and it appears to be more isotropic.

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Figure 4

Registration of a sphere to itself with surface feature matching: (a,b) The surface matching term f1 contains a band of high intensity on the sphere that is offset in the x-direction from f2. (c,d) The initial difference between the surface matching terms is reduced by approximately one order of magnitude after the solution converges. (e) Initial distortions are visualized by areal expansion (the dilatation ratio J) between YAS and OAS. Also, the eigenvectors associated with first (blue) and second (green) principal strains are plotted. The vectors are scaled by the principal stretch magnitudes. Only the vectors at every fourth point on the surface are displayed. After convergence of the algorithm the initial distortions have been relaxed away. For the surface matching terms to align, the coordinates need to shift in the negative x-direction, which causes expansion on the right (x>0) side of the sphere and compression on the left (x<0) side. (f) After relaxation the areal expansion between YAS and OAS is almost uniform in each of three regions: the right (x>0) side of the sphere where expansion has occurred, the left (x<0) side of the sphere where compression has occurred and the center (x≈0) region, which remains relatively undeformed.

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Figure 8

Surface feature matching terms for registration of the P14 and P21 ferret brain (kit A.2): Mean curvature and geodesic distance from the medial wall were used to calculate the surface matching terms f1 and f2. (a) Surface feature term f1 on YAS. (b) Function f2 on OAS. (c) The difference between matching terms (f1−f2) at the initial time t0. (d) The difference between matching terms (f1−f2) at the final time tf. The initial differences between f1 and f2 are reduced after the application of the relaxation algorithm. In particular, marked reductions are seen around the medial wall.

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Figure 3

T2-weighted MR images of the live ferret brain acquired using a spin-echo pulse sequence at postnatal days 14, 21, and 28. Using the CARET software (Van Essen (15)) images were manually segmented at the boundary of GM and CSF. Cortical surface representations were created from the segmentation volumes. Sulci in the ferret brain: coronolateral sulcus (CLS, red); sylvian sulcus/presylvian sulcus (SS/PSS, dark blue); suprasylvian sulcus (SSS, green); cruciate sulcus/splenial sulcus (CS/SpS, cyan); anterior rhinal fissure (aRF, pink); rhinal sulcus (RhS, orange); and ansate sulcus (AS, yellow).

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