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.

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

• 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)

• 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 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. 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

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

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