Numerical investigation of biomechanically coupled growth in cortical folding

Abstract

Cortical folding—the process of forming the characteristic gyri (hills) and sulci (valleys) of the cortex—is a highly dynamic process that results from the interaction between gene expression, cellular mechanisms, and mechanical forces. Like many other cells, neurons are sensitive to their mechanical environment. Because of this, cortical growth may not happen uniformly throughout gyri and sulci after the onset of cortical folding, which is accompanied by patterns of tension and compression in the surrounding tissue. Here, as an extension of our previous work, we introduce a biomechanically coupled growth model to investigate the importance of interaction between biological growth and mechanical cues during brain development. Our earlier simulations of cortical growth consisted of a homogeneous growing cortex attached to an elastic subcortex. Here, we let the evolution of cortical growth depend on a geometrical quantity—the mean curvature of the cortex—to achieve preferential growth in either gyri or sulci. As opposed to the popular pre-patterning hypothesis, our model treats inhomogeneous cortical growth as the result of folding rather than the cause. The model is implemented numerically in a commercial finite element software Abaqus/Explicit in Abaqus reference manuals, Dassault Systemes Simulia, Providence (2019) by writing user-defined material subroutine (VUMAT). Our simulations show that gyral–sulcal thickness variations are a phenomenon particular to low stiffness ratios. In comparison with cortical thickness measurements of \(N=28\) human brains via a consistent sampling scheme, our simulations with similar cortical and subcortical stiffnesses suggest that cortical growth is higher in gyri than in sulci.

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References

  1. Abaqus/Explicit (2019) Abaqus reference manuals. Dassault Systemes Simulia, Providence

    Google Scholar 

  2. Anava S, Greenbaum A, Jacob EB, Hanein Y, Ayali A (2009) The regulative role of neurite mechanical tension in network development. Biophys J 96(4):1661–1670

    Article  Google Scholar 

  3. Barkovich AJ, Hevner R, Guerrini R (1999) Syndromes of bilateral symmetrical polymicrogyria. Am J Neuroradiol 20(10):1814–1821

    Google Scholar 

  4. Barron DH (1950) An experimental analysis of some factors involved in the development of the fissure pattern of the cerebral cortex. J Exp Zool 113(3):553–581

    Article  Google Scholar 

  5. Bayly P, Okamoto R, Xu G, Shi Y, Taber L (2013) A cortical folding model incorporating stress-dependent growth explains gyral wavelengths and stress patterns in the developing brain. Phys Biol 10(1):016005

    Article  Google Scholar 

  6. Borrell V, Götz M (2014) Role of radial glial cells in cerebral cortex folding. Curr Opin Neurobiol 27:39–46

    Article  Google Scholar 

  7. Budday S, Raybaud C, Kuhl E (2014) A mechanical model predicts morphological abnormalities in the developing human brain. Sci Rep 4:5644

    Article  Google Scholar 

  8. Budday S, Kuhl E, Hutchinson JW (2015a) Period-doubling and period-tripling in growing bilayered systems. Phil Mag 95(28–30):3208–3224

    Article  Google Scholar 

  9. Budday S, Nay R, de Rooij R, Steinmann P, Wyrobek T, Ovaert TC, Kuhl E (2015b) Mechanical properties of gray and white matter brain tissue by indentation. J Mech Behav Biomed Mater 46:318–330

    Article  Google Scholar 

  10. Craddock C, Benhajali Y, Chu C, Chouinard F, Evans A, Jakab A, Khundrakpam BS, Lewis JD, Li Q, Milham M et al (2013) The neuro bureau preprocessing initiative: open sharing of preprocessed neuroimaging data and derivatives. Neuroinformatics 41:37–53

    Google Scholar 

  11. Dale AM, Fischl B, Sereno MI (1999) Cortical surface-based analysis: I. Segmentation and surface reconstruction. Neuroimage 9(2):179–194

    Article  Google Scholar 

  12. Fischl B, Dale AM (2000) Measuring the thickness of the human cerebral cortex from magnetic resonance images. Proc Nat Acad Sci 97(20):11050–11055

    Article  Google Scholar 

  13. Garcia K, Kroenke C, Bayly P (2018) Mechanics of cortical folding: Stress, growth and stability. Philos Trans R Soc B Biol Sci 373(1759):20170321

    Article  Google Scholar 

  14. Gómez-Skarmeta JL, Campuzano S, Modolell J (2003) Half a century of neural prepatterning: the story of a few bristles and many genes. Nat Rev Neurosci 4(7):587–598

    Article  Google Scholar 

  15. Henann DL, Anand L (2010) Surface tension-driven shape-recovery of micro/nanometer-scale surface features in a pt57. 5ni5. 3cu14. 7p22. 5 metallic glass in the supercooled liquid region: a numerical modeling capability. J Mech Phys Solids 58(11):1947–1962

  16. Holland MA, Miller KE, Kuhl E (2015) Emerging brain morphologies from axonal elongation. Ann Biomed Eng 43(7):1640–1653

    Article  Google Scholar 

  17. Holland M, Budday S, Goriely A, Kuhl E (2018) Symmetry breaking in wrinkling patterns: gyri are universally thicker than sulci. Phys Rev Lett 121(22):228002

    Article  Google Scholar 

  18. Kaster T, Sack I, Samani A (2011) Measurement of the hyperelastic properties of ex vivo brain tissue slices. J Biomech 44(6):1158–1163

    Article  Google Scholar 

  19. Koser DE, Thompson AJ, Foster SK, Dwivedy A, Pillai EK, Sheridan GK, Svoboda H, Viana M, da F Costa L, Guck J, et al (2016) Mechanosensing is critical for axon growth in the developing brain. Nat Neurosci 19(12):1592

    Article  Google Scholar 

  20. Kriegstein A, Noctor S, Martínez-Cerdeño V (2006) Patterns of neural stem and progenitor cell division may underlie evolutionary cortical expansion. Nat Rev Neurosci 7(11):883–890

    Article  Google Scholar 

  21. Kroenke CD, Bayly PV (2018) How forces fold the cerebral cortex. J Neurosci 38(4):767–775

    Article  Google Scholar 

  22. Nordahl CW, Dierker D, Mostafavi I, Schumann CM, Rivera SM, Amaral DG, Van Essen DC (2007) Cortical folding abnormalities in autism revealed by surface-based morphometry. J Neurosci 27(43):11725–11735

    Article  Google Scholar 

  23. Pfister BJ, Iwata A, Meaney DF, Smith DH (2004) Extreme stretch growth of integrated axons. J Neurosci 24(36):7978–7983

    Article  Google Scholar 

  24. Rakic P (2009) Evolution of the neocortex: a perspective from developmental biology. Nat Rev Neurosci 10(10):724–735

    Article  Google Scholar 

  25. Reillo I, de Juan Romero C, García-Cabezas MÁ, Borrell V (2011) A role for intermediate radial glia in the tangential expansion of the mammalian cerebral cortex. Cereb Cortex 21(7):1674–1694

    Article  Google Scholar 

  26. Rodriguez EK, Hoger A, McCulloch AD (1994) Stress-dependent finite growth in soft elastic tissues. J Biomech 27(4):455–467

    Article  Google Scholar 

  27. Sejnowski TJ, Koch C, Churchland PS (1988) Computational neuroscience. Science 241(4871):1299–1306

    Article  Google Scholar 

  28. Shaw P, Lerch J, Greenstein D, Sharp W, Clasen L, Evans A, Giedd J, Castellanos FX, Rapoport J (2006) Longitudinal mapping of cortical thickness and clinical outcome in children and adolescents with attention-deficit/hyperactivity disorder. Arch Gen Psychiatry 63(5):540–549

    Article  Google Scholar 

  29. Shinmyo Y, Terashita Y, Duong TAD, Horiike T, Kawasumi M, Hosomichi K, Tajima A, Kawasaki H (2017) Folding of the cerebral cortex requires cdk5 in upper-layer neurons in gyrencephalic mammals. Cell Rep 20(9):2131–2143

    Article  Google Scholar 

  30. Shraiman BI (2005) Mechanical feedback as a possible regulator of tissue growth. Proc Nat Acad Sci 102(9):3318–3323

    Article  Google Scholar 

  31. Sun T, Hevner RF (2014) Growth and folding of the mammalian cerebral cortex: from molecules to malformations. Nat Rev Neurosci 15(4):217–232

    Article  Google Scholar 

  32. Tallinen T, Chung JY, Biggins JS, Mahadevan L (2014) Gyrification from constrained cortical expansion. Proc Nat Acad Sci 111(35):12667–12672

    Article  Google Scholar 

  33. Van Dommelen J, Van der Sande T, Hrapko M, Peters G (2010) Mechanical properties of brain tissue by indentation: interregional variation. J Mech Behav Biomed Mater 3(2):158–166

    Article  Google Scholar 

  34. Van Essen DC (1997) A tension-based theory of morphogenesis and compact wiring in the central nervous system. Nature 385(6614):313–318

    Article  Google Scholar 

  35. Walker AE (1942) Lissencephaly. Archiv Neurol Psychiatry 48(1):13–29

    Article  Google Scholar 

  36. Welker W (1990) Why does cerebral cortex fissure and fold? A review of determinants of gyri and sulci. In: Jones E, Peters A (eds) Cerebral cortex. Plenum Press, New York, pp 3–136

    Chapter  Google Scholar 

  37. Xu G, Knutsen AK, Dikranian K, Kroenke CD, Bayly PV, Taber LA (2010) Axons pull on the brain, but tension does not drive cortical folding. J Biomech Eng 132(7):071013

    Article  Google Scholar 

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Acknowledgements

MAH acknowledges support from the National Science Foundation under Grant No. (IIS-1850102).

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Correspondence to Maria A. Holland.

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Appendix 1: Curvature calculation

Appendix 1: Curvature calculation

In this section, we elaborate on the details of the curvature calculation in Eq. (13), starting with the 2-D case first and then extending it to a full 3-D case. The method we adopt here follows the work from Henann and Anand (2010), where they first utilized FORTRAN’s global module along with VUMAT to obtain the mean curvature, which requires non-local information.

Consider meshing the cortex into four-node quadrilateral plane-strain elements (shown in Fig. 10a). Unlike Henann and Anand (2010), we calculate the curvature at the centroid point of each element . The current coordinates of each centroid point are calculated based on the current coordinates of four integration points (\(\times\)) provided by Abaqus. To obtain the mean curvature at the centroid point A, we fit a parabola through point A and centroid points B and C from two adjacent elements, \(y^\prime = a x^{\prime 2} + b x^\prime + c\). The fitting is based on a local coordinate \(A x^\prime y^\prime\), where the outward normal \(\hat{{\mathbf{n}}}\) is perpendicular to the dashed line connecting both points B and C. Note that both the slope and value of the parabola should be zero at the origin of the local coordinates, which makes the function reduce to \(y^\prime = a x^{\prime 2}\) (in terms of the local coordinates). Finally, by definition, the mean curvature is \(\kappa = - 1/2 (\partial ^2 y^\prime / \partial x^{\prime 2}) = - a\).

Given the global coordinates of the centroid point \(A (x_0, y_0)\), and its two adjacent centroid points \(B(x_1,y_1)\) and \(C(x_2,y_2)\), the calculation of mean curvature at point A in 2-D is summarized as follows:

  1. 1.

    Obtain the local outward-normal \(\hat{{\mathbf{n}}} = n_x {\mathbf{e}}_x + n_y {\mathbf{e}}_y\) in the global coordinate system, based on the coordinates of centroid points from two adjacent elements at \((x_1,y_1)\) and \((x_2,y_2)\), with components given by

    $$\begin{aligned} \left. \begin{aligned} n_x&= \dfrac{y_1 - y_2}{\sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}}, \\ n_y&= \dfrac{x_2 - x_1}{\sqrt{(y_2 - y_1)^2 + (x_2 - x_1)^2}}. \end{aligned}\right. \end{aligned}$$
    (17)
  2. 2.

    Obtain the rotation matrix connecting global and local coordinates,

    $$\begin{aligned} \begin{bmatrix} {\mathbf{Q}}\end{bmatrix} = \begin{bmatrix} n_y &{} -n_x \\ n_x &{} n_y \\ \end{bmatrix}. \end{aligned}$$
    (18)
  3. 3.

    Obtain the coordinates of centroid points from adjacent elements in terms of local coordinates,

    $$\begin{aligned} \left. \begin{aligned} \begin{bmatrix} x^{\prime }_1 \\ y^{\prime }_1 \\ \end{bmatrix}&= \begin{bmatrix} {\mathbf{Q}}\end{bmatrix} \begin{bmatrix} x_1 - x_0 \\ y_1 - y_0 \end{bmatrix}, \\ \begin{bmatrix} x^{\prime }_2\\ y^{\prime }_2\\ \end{bmatrix}&= \begin{bmatrix} {\mathbf{Q}}\end{bmatrix} \begin{bmatrix} x_2 - x_0\\ y_2 - y_0 \end{bmatrix}. \end{aligned}\right. \end{aligned}$$
    (19)
  4. 4.

    Perform a linear least squares fit to the system of equations

    $$\begin{aligned} \begin{bmatrix} y^{\prime }_1\\ y^{\prime }_2\\ \end{bmatrix} = a \begin{bmatrix} x^{\prime 2}_1\\ x^{\prime 2}_2\\ \end{bmatrix}, \end{aligned}$$
    (20)

    , which yields

    $$\begin{aligned} a = \dfrac{x^{\prime 2}_1 y^{\prime }_1 + x^{\prime 2}_2 y^{\prime}_2}{x^{\prime 4}_1 + x^{\prime 4}_2}. \end{aligned}$$
    (21)
  5. 5.

    Calculate mean curvature as

    $$\begin{aligned} \kappa = -\dfrac{1}{2} \dfrac{\partial ^2 y^\prime }{\partial x^{\prime 2}} = -a. \end{aligned}$$
    (22)

The procedure for the 3-D case is similar but more tedious. Centroid point A again served as our point of interest. To calculate the mean curvature at point A, we fit a paraboloid through it and its eight nearest centroid points. The details are summarized as follows:

  1. 1.

    Find point A’s eight nearest neighbors by using the insertion sort algorithm.

  2. 2.

    Obtain the local outward-normal \(\hat{{\mathbf{n}}} = n_x {\mathbf{e}}_x + n_y {\mathbf{e}}_y + n_z {\mathbf{e}}_z\) in the global coordinate system by fitting the surface equation \(z = ax + by + c\) to the eight nearest centroid points. The components are given by

    $$\begin{aligned} \left. \begin{aligned} n_x&= \dfrac{-a}{\sqrt{a^2 + b^2 + 1}}, \quad n_y = \dfrac{-b}{\sqrt{a^2 + b^2 + 1}}, \\ n_z&= \dfrac{1}{\sqrt{a^2 + b^2 + 1}}. \end{aligned}\right. \end{aligned}$$
    (23)
  3. 2.

    Obtain the rotation matrix linking global to local coordinates,

    $$\begin{aligned} \begin{bmatrix} {\mathbf{Q}}\end{bmatrix} = \begin{bmatrix} \dfrac{n_x^2 n_z + n_y^2}{n_x^2 + n_y^2} &{} \dfrac{-n_x n_y (1 - n_z)}{n_x^2 + n_y^2} &{} -n_x\\ \dfrac{-n_x n_y (1 - n_z)}{n_x^2 + n_y^2} &{} \dfrac{n_x^2 + n_y^2 n_z}{n_x^2 + n_y^2} &{} -n_y\\ n_x &{} n_y &{} n_z \end{bmatrix}. \end{aligned}$$
    (24)
  4. 3.

    Obtain the position of the eight nearest centroid points in terms of local coordinates,

    $$\begin{aligned} \begin{bmatrix} x_i^\prime \\ y_i^\prime \\ z_i^\prime \end{bmatrix} = \begin{bmatrix} {\mathbf{Q}}\end{bmatrix} \begin{bmatrix} x_i - x_0\\ y_i - y_0 \\ z_i - z_0 \end{bmatrix} \quad \text {for} \quad i = 1 \,\,\text {to} \,\,8. \end{aligned}$$
    (25)
  5. 4.

    Perform a linear least squares fit to the system of equations

    $$\begin{aligned} z_i^\prime = \alpha x^{\prime 2}_i + \beta y^{\prime 2}_i + \gamma x_i^\prime y_i^\prime \quad \text {for} \quad i = 1 \,\,\text {to} \,\,8, \end{aligned}$$
    (26)

    which yields eight equations with three unknowns. Let \([d]^{\scriptscriptstyle \top }\) denote an array of the three unknowns (\(\alpha\), \(\beta\), \(\gamma\)) and \([z]^{\scriptscriptstyle \top }\) denote an array of the eight measured values of \(z_i\), and [A] denote an 8-by-3 matrix storing the values \((x^{\prime 2}_i, y^{\prime 2}_i, x^\prime_i y^\prime_i )\) for \(i=1\) to 8. Thus, equation (26) can be rewritten

    $$\begin{aligned} \begin{bmatrix} A \end{bmatrix} \begin{bmatrix} d \end{bmatrix}^{\scriptscriptstyle \top } = \begin{bmatrix} z \end{bmatrix}^{\scriptscriptstyle \top }, \end{aligned}$$
    (27)

    with the optimal solution given by

    $$\begin{aligned} \begin{bmatrix} d \end{bmatrix}^{\scriptscriptstyle \top } =\big (\begin{bmatrix} A \end{bmatrix}^{\scriptscriptstyle \top } \begin{bmatrix} A \end{bmatrix} \big )^{-1} \begin{bmatrix} A \end{bmatrix}^{\scriptscriptstyle \top } \begin{bmatrix} z \end{bmatrix}^{\scriptscriptstyle \top }. \end{aligned}$$
    (28)
  6. 6.

    Calculate mean curvature as

    $$\begin{aligned} \kappa = - \dfrac{1}{2} \bigg ( \dfrac{\partial ^2 z^\prime }{\partial {x^{\prime 2}}} + \dfrac{\partial ^2 z^\prime }{\partial {y^{\prime 2}}} \bigg ) = - (\alpha + \beta ). \end{aligned}$$
    (29)

The algorithm was implemented in both our user-defined subroutine (VUMAT) and MATLAB. We verified our implementations by comparing the calculated mean curvature values at the middle surface of the wrinkled cortex between the two programs (Fig. 11).

Fig. 10
figure10

Schematics of finite element mesh used in the cortex. a 2-D four-noded equilateral plane-strain elements and b 3-D brick elements. Note integration points are denoted as cross markers, and the centroid points are denoted as red circles

Fig. 11
figure11

Verification of our curvature calculation algorithm. a Profiles of wrinkled cortex with boundary and centroid points denoted as black symbol lines and red circles. b Comparison of mean curvatures at the centroid points between MATLAB and Abaqus

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Wang, S., Demirci, N. & Holland, M.A. Numerical investigation of biomechanically coupled growth in cortical folding. Biomech Model Mechanobiol 20, 555–567 (2021). https://doi.org/10.1007/s10237-020-01400-w

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Keywords

  • Biomechanics
  • Brain development
  • Curvature
  • Finite elements