A Biomechanical Model of Cortical Folding

Abstract

A principal characteristic of the geometry of the brain is its folding pattern which is composed of gyri (outward hills) and sulci (inward valleys). We present a preliminary two-dimensional biomechanical model of cortical folding that is implemented computationally using finite elements. This model uses mechanical properties such as stress, strain, and body forces, corresponding to axonal tension, to model the shape of the brain during early cortical development. Despite its simplicity, the proposed model can be used to demonstrate the plausibility of tension generating cortical folds, as has been suggested in Van Essen (Nature 385(6614):313–318, 1997). In addition, this model is used to investigate folding patterns on different domain sizes.

Appendix

4.1.1 Linear Quadrilateral Element

In the non-dimensional coordinate system (ξ, η) (see Fig. 4.7), the four shape functions can be expressed as

$$\displaystyle\begin{array}{rcl} N_{1} = \frac{1} {4}(1-\xi )(1-\eta )\;,\;\,N_{2} = \frac{1} {4}(1+\xi )(1-\eta )\;,& & {}\\ N_{3} = \frac{1} {4}(1+\xi )(1+\eta )\;,\;N_{4} = \frac{1} {4}(1-\xi )(1+\eta )\;.& & {}\\ \end{array}$$

At any point inside the element, $\sum _{i=1}^{4}N_{i} = 1$. The displacement field is expressed as

$$\displaystyle\begin{array}{rcl} u =\sum _{ i=1}^{4}N_{ i}u_{i}\;,& & v =\sum _{ i=1}^{4}N_{ i}v_{i}\;. {}\\ \end{array}$$

The shapes of the interpolation functions N _i are twisted planes whose height is 1 at i-th corner of the element and 0 at the other corners. The partial derivatives with respect to the variables are linear functions [4].

Fig. 4.7

Linear quadrilateral element