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 prepatterning 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 userdefined 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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Acknowledgements
MAH acknowledges support from the National Science Foundation under Grant No. (IIS1850102).
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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 2D case first and then extending it to a full 3D 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 nonlocal information.
Consider meshing the cortex into fournode quadrilateral planestrain 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 2D is summarized as follows:

1.
Obtain the local outwardnormal \(\hat{{\mathbf{n}}} = n_x {\mathbf{e}}_x + n_y {\mathbf{e}}_y\) 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.
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.
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.
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.
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 3D 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.
Find point A’s eight nearest neighbors by using the insertion sort algorithm.

2.
Obtain the local outwardnormal \(\hat{{\mathbf{n}}} = n_x {\mathbf{e}}_x + n_y {\mathbf{e}}_y + n_z {\mathbf{e}}_z\) 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) 
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) 
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) 
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 8by3 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.
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 userdefined 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).
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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/s1023702001400w
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Keywords
 Biomechanics
 Brain development
 Curvature
 Finite elements