Gj. Hademenos et al., A NONLINEAR MATHEMATICAL-MODEL FOR THE DEVELOPMENT AND RUPTURE OF INTRACRANIAL SACCULAR ANEURYSMS, Neurological research, 16(5), 1994, pp. 376-384
Mathematical models of aneurysms are typically based on Laplace's law
which defines a linear relation between the circumferential tension an
d the radius. However, since the aneurysm wall is viscoelastic, a nonl
inear model was developed to characterize the development and rupture
of intracranial spherical aneurysms within an arterial bifurcation and
describes the aneurysm in terms of biophysical and geometric variable
s at static equilibrium. A comparison is made between mathematical mod
els of a spherical aneurysm based on linear and nonlinear forms of Lap
lace's law. The first form is the standard Laplace's law which states
that a linear relation exists between the circumferential tension, T,
and the radius, R, of the aneurysm given by T = PR/2t where P is the s
ystolic pressure. The second is a 'modified' laplace's law which descr
ibes a nonlinear power relation between the tension and the radius def
ined by T = AR(P/2At) where A is the elastic modulus for collagen and
t is the wall thickness. Differential expressions of these two relatio
ns were used to describe the critical radius or the radius prior to an
eurysm rupture. Using the standard Laplace's law, the critical radius
was derived to be R(c) = 2Et/P where E is the elastic modulus of the a
neurysm. The critical radius from the modified Laplace's law was R = [
2Et/P](2At/P). Substituting typical values of E = 1.0 MPa, t = 40 mu m
, P = 150 mmHg, and A = 2.8 MPa, the critical radius is 4.0 mm using s
he standard Laplace's law and 4.8 mm for the modified Laplace's law. I
n conclusion, a biomathematical model has been developed based on a no
nlinear expression of Laplace's law which integrates the quantitative
influence of collagen in the tension of the aneurysm wall. The nonline
ar model better describes the influence of biophysical variables on th
e critical radius in comparison to the model based on the standard Lap
lace's law. The critical radius from the modified Laplace's law more a
ccurately predicts aneurysm rupture based on previously published clin
ical observations.