A mathematical model of brain injuries due to angular accelerations

Morante Belleni, Aldo (1990) A mathematical model of brain injuries due to angular accelerations. Memorie di fisica matematica in omaggio a Giovanni Carini nel suo 70 compleanno, 1. pp. 381-402.

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Abstract

We consider a simple mathematical model to study brain injuries due to angular accelerations of the head. Since it is known that centrifugal and Coriolis forces due to rotations of the skull can be disregarded, we assume that the skull is simulated by the two rigid walls y= -a and Y= +a, which translate with velocity V(t)= V(t)e1. Note that V(t) is parallel to the x-axis and that V(t) is a given continuous function of time, with V(0)=0 and with V(t)>=0 at any t>=0.
The cerebrospinal fluid is a viscid liquid that fills the two thin layers -a<y<-b and b<y<a, whereas the brain tissue is a viscoelastic material contained in the region -b<y<b, with (a-b)<<b.
By using Laplace transform technique, we show that a crucial parameter to study brain injuries is the product GT, where G is the maximum value of the acceleration of the skull and T is the time duration of the acceleration V(t). Another important parameter is the time spent by a viscoelastic wave to travel from the skull to the center of the brain (and this seems to be some kind of resonance phenomenon between the input acceleration and the physical and geometric properties of the brain).

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