1. Btomwhanm
1975. Vol
8. pp. 351-356.
Pergamon
Press
Printed
m Greatt
Bntain
MATHEMATICAL SIMULATION OF GLIDING CONTUSIONS* PETERL~~WENHIELM Department of Forensic Medicine. University of Lund, Lund, Sweden gliding contusion in the acute phase is characterized by a streaklike hemorrhage of venous origin situated subcortically in a paramedial convolution. In later stages perivascular necrosis may develop. This type of injury is caused by head angular acceleration and is often seen in traffic casualties
Abstract-A
when the head has hit the steering wheel, the dashboard or the windshield. The deformation of the brain matter close to the superior sagittal sinus has been simulated by means of a mathematical viscoelastic model in order to clarify the genesis of the gliding contusions. The blood vessels in the brain matter will be strained as a consequence of the brain deformation which results from head angular acceleration. The highest values of the strain occur subcortically where the blood vessels are injured first. The tolerance levels for gliding contusions have been determined. The calculations which were based on experiments regarding the dynamic properties of the superior cerebral veins and on two alternative injury criteria proposed, indicate that a gliding contusion is not likely to arise if the maximal angular acceleration does not exceed 4500 rad/sec’ or the change in angular -velocity does not exceed 70_rad/sec.
INTRODUCHON
PROBLEMFORMULATIONAND METHODS
Lindenberg and Freytag (1960) have described extensive hemorrhages of traumatic origin which were situated subcortically in the paramedial parts of the cerebral hemispheres. The hemorrhages were of venous nature and were considered to be caused by deformation of the brain matter when the head was subjected to angular acceleration. The lesions were designated as gliding contusions. For consecutive autopsy material Voigt and Lawenhielm (1974) have demonstrated similar injuries in 25% of traffic casualties. Also in these cases a head angular acceleration had occurred. The hemorrhages were however not always as extensive as in the description by Lindenberg and Freytag (1960). They appeared ‘as streaklike perivascular bleedings mostly found in the back aspect of the frontal superior gyrus and in the upper part of the anterior central convolution. The hemorrhages seemed to originate from intracerebral portions of the superior cerebral veins. In victims surviving more than one week areas with necrosis or perivascular glial scars often running deeply into the white matter could be found. Remarkably the peripheral layers of the brain cortex remained intact in spite of sometimes extensive injuries. Brain injuries due to head impact can be very complex. The gliding contusion may be part of such an injury pattern but may as well not be present. On the other hand gliding contusions can occur alone. Different mechanisms of origin probably account for these differences. Although lesions of the gliding contusion type are quite common in traffic accidents they have attracted little attention in the literature. Therefore an investigation to clarify the genesis of the gliding contusion and thereby make it possible to predict threshold levels was needed.
The fact that a gliding contusion can occur irrespective of other traumatic brain injuries suggests that this type of injury can be studied separately. A typical gliding contusion in the acute phase consists of a streaklike perivascular hemorrhage situated in the white matter in a paramedial convolution (Fig. 1). Fortunately a mathematical model needed to study the gliding contusion can be made quite accurate because of this location. In the model used the bleeding is assumed to have emerged from an intracerebral portion of a superior cerebral vein, that radiates into the brain matter forming a 45” angle with the cerebral falx. The rigid structures encasing the brain in this region are the spherically shaped skull and the flat cerebral falx. Suppose therefore that the brain can be described by a half sphere of viscoelastic material contained in a hemispherical rigid shell (the skull) with a rigid wall (the falx) according to Fig. 2. This configuration is in good agreement with a mathematical model formulated by Ljung (1975). which gives the deformation of the brain when the head assumes a sudden change in rotational velocity. The movement is assumed to by symmetrical. i.e. the axis of
* Received 15 Jffrzuury1975.
Superior sogillol sir& Cerebral duro Subcwtical pcrivascular
,
Lcntiform nucleus
Fig. 1. Frontal section of the brain showing the site of a typical gliding contusion hemorrhage.
351
352
PETERL~WENHIELM tnfraccrcbrol aspecr superior
Of
w
cerebral
vein
R
o(,)
-1
Fig. 2. Outline of the model studied. rotation is perpendicular to the falx. In a condensed form the brain motion can be written: u(t, r, 0) = vo.G(t, I, 6), u(t,r,Q
is the displacement of the point (r,B) and t is the time lag since application of a step v0 in the tangential velocity of the shell (the skull bone) at the edge of the hemisphere. G(t, r, 0) is an expression governing the brain motion. Thus G(r, R, 1~12)= r. In this model the material properties of the brain are described by a Kelvin-Voigt rheological model, i.e. a spring (shear modulus) parallel with a dashpot (kinematic viscosity). Ljung (1975) has determined these parameters from model tests on fresh cadaver brain. If many velocity steps u(tl). . . . . u(t,,) are applied at different times t, . . . _.tn the displacement u(t) can be written:
pulses, for example head acceleration pulses arising in traffic accidents. In order to produce a hemorrhage the deformation of the brain matter must be great enough to compromise the solidity of the blood vessel. The dynamic properties of the portion of the superior cerebral veins which bridges over the subdural space to the superior sagithl sinus have been investigated by Liiwenhielm (1974). Under uniaxial loading conditions with constant strain rates the ultimate strain of these bridging veins were found to be strongly time dependent. When the strain rate was kept low the order of magnitude of the ultimate strain was 100%. With increasing strain rate however the ultimate strain was reduced to the 20% level. Consequently the dynamic tolerance can be expressed as a function of strain E and strain rate.&. These properties are assumed to be valid for the intracerebral portions of the superior cerebral veins too. Within the brain matter it is reasonable to assume that these delicate blood vessels passively follow the movement of the brain matter. When the brain matter undergoes shear, the blood vessel will therefore be strained to a corresponding degree. The strain can be written as
c(t)
=
J0 1+
2
g
-
1)
where s is a coordinate along the blood vesd and au/as denotes the shear. As the position vectors (r,0) approximately form a 45” angle with the blood vessel in the area studied, i.e. ds = dr .,,,?I we obtain:
u(t) = v(t,).G(t - t1) + . . . . . . + u(tJ.G(t - t,) = i o(tJ.G(ti=l
ti).
Now the strain rate can be calculated:
Passing to the limit in the sum above we obtain: u(t) =
J G(t
I’
=
1
t').dv(t')
a(t). at
As u(t) and au(t)/& are proportional to the amplitude A of the acceleration pulse, A can be chosen such that the corresponding values for the strain and the strain rate agree with some tolerance level for the actual veins. Two methods have been used to establish an injury criterion.
0
=
i(t)
G(t - r’).a(t’).dt’
=st 0
a(t - t’). G(t). dt’ ,
0
Method
where a(t) is the tangential acceleration of the shell (the skull bone). In the following calculations an analytical expression has been used to describe the acceleration pulse:
u(t) = i
0
; t
to.
Here to denotes the length of the acceleration pulse and A its amplitude. This pulse shape may be regarded as representative for physically occurring
1
A curve is fitted to the data valid for bridging vein disruption given by Liiwenhielm (1974) see Fig. 3. In a lin-log representation a straight line is fitted by polynomial regression to the data for which the strain rate is less than loO/sec. The reason for this restriction will be discussed later. Now, for any head angular acceleration the corresponding strain and strain rate of the studied veins can be calculated for any number of time steps. If these data are plotted in an ei-diagram different curves are obtained according to the choice of acceleration amplitude. The acceleration amplitude can be chosen such that the simulated
353
Simulation of gliding contusions
t
holds even in cases where the strain is not constant during the total loading time. This expression can therefore be used for a rupture criterion. The critical acceleration for a simulated gliding contusion is thus obtained by evaluating the integral:
Elongation of bridging vein W.1
s fL
k.c.i" dt.
0
Elongation rate f see-’ I 1
10
100
1000
Fig. 3. Tolerance data for disruption of the superior cerebral veins in the subdural space (Lowenhielm, 1974). The straight line fitted to the data for which 1 < lOO/sec obeys . the equation: C = exp[ -(e - 0809/0~108)].
curve will fall tangent to the exponential curve fitted to the bridging vein data. This acceleration is now considered as critical. Head angular acceleration pulses with shorter duration than Smsec can be considered as rare in real traffic accidents. If a 5 msec pulse is studied for a critical acceleration the resulting strain rate will not exceed lOO/sec. This is the reason why data with < > lOO/sec were omitted at the fitting. Method 2
The &-relationship for disruption of the bridging cerebral veins given by Liiwenhielm (1974) were determined from tests at constant strain rates. To every such constant strain rate a corresponding life length L can be related, i.e. the time till the blood vessel is disrupted. Assume that during a part dt of the total loading time L a certain part I+,:) of the life length is consumed, then obviously
s
where E and i are simulated values and t, the time for which E = E,_. If the integral exceeds unity rupture should have occurred. Decreasing strain values are not considered to consume life length. Furthermore the assumption for CIhas no physical meaning if i is negative.
RESULTS AND
DISCUSSION
The displacement of a blood vessel at different times since the application of the acceleration pulse is presented in Fig. 4. The displacement close to the surface of the brain g i.e. for /-’2 I is quite small (p is the dimensionless radius r/R) but increases as p decreases. The slope of the different curves reflect the shear in the brain matter. The model is not studied for smaller values of p than 0.82 for a two-fold reason. The gliding contusions appear in the studied region and the results might become erroneous at some distance from the brain surface since the influence of such boundaries as the skull base are not included in the model. The shear in the brain matter has been evaluated by differentiation of u(t) with respect to r: au(t) -=
&
s ’
a(t - t’)
ac(o
dt’
.T
0
If au/& is plotted vs p for different times a set of curves is obtained, the broken curves in Fig. 5. The Brain hylacement
20msec
I.
M
(E,Z) dt = 1.
0
For CIthe following assumption
kmm)
\
is made:
r = k.6.P.
As the tension tests of the bridging veins were carried out at constant strain rates we obtain: k.Z”+‘.
L2 2=
1
tive US
The constants n and k can be determined from the experimental data shown in Fig. 3. It will now be assumed that the expression:
70 msec
-1 1
Y”
“”
Fig. 4. Brain at different times after the ap. displacement ^ phcatron 01 a single acceleration pulse.
354
Shear
I
,---
a
P
ar
‘\
i Relative radius 1.0
-++--I ’Llj , /’
/’
-.
1’
,-
-./
l\,
,4 i iomsecY\ \\\\ /’ \,\t =lOmsec /’
0.8-
0.6 -
't =15msec
/'
/
0.4,
i
5msec
,_
relative
radius
.92
.91
.96
.90
1.0 p
Fig. 5. Shear of the brain matter. The broken curves indicate the shear at different times after the application of a single acceleration pulse (to = 10msec, A = IOoom/set’). The successive maxima of these curves constitute the unbroken curve.
maxima of these curves fall deeper and deeper into the brain matter as time increases. A curve fitted to these maxima (the full-drawn curve in Fig.5) then represents the value and site of the maximal shear as a function of time. Two such curves corresponding to the pulse durations 5 and 1OOmsec respectively are drawn in Fig. 6. These curves indicate that the site of the maximal shear with respect to time is dependent on the pulse duration. This variation of site occurs however within a narrow range of radii (Fig. 7) but nevertheless it has to be considered when calculating the shear. For practical purposes it is sufficient to say that the maximal shear in the brain matter or the strain of a blood vessel occurs at a constant level below the surface of the brain. This distance is approximately 8 mm, which is well below the cortex of the brain and corresponds to the location of the gliding contusions. According to Fig. 8 the peripheral layers of the brain will sustain negligible strain while
20
40
60
,boFz 60
The site of the absolute shear maximum in the brain matter as a function of the pulse length.
Fig. 7.
considerable strain is obtained some 8mm below the brain surface. Thus it is obvious that gliding contusions can be produced by displacements of the brain, induced by head angular acceleration. Due to low shear levels in the cortex it is also evident that the peripheral layers of the brain can be preserved while deeper lying brain tissue can show extensive injuries. According to Fig.6 the maximal shear occurs deeper in the brain as time increases as previously mentioned. When the acceleration pulse is of short duration high strain values are maintained for decreasing radii. This fact suggests that shear levels high enough to produce injury would extend over deeper lying tissue structures. It must be kept in mind though that the results given in Fig.6 are based on a single, unidirectional acceleration pulse. If the duration of the pulse is shorter than 25 msec the maximal shear occurs after the pulse has ceased to act (Fig. 9). The influence of later acceleration pulses can therefore be substantial. This effect on the position and value of
Normalized shear au
t
&
l.O-
0.5 -
__---_----44
.66
.86
.90
.92
.9L
.96
Relative radius .96 1.0 p
Fig. 6. The site and relative value of the shear in the brain matter as a function of time for the pulse lengths 5 and 100 msec. With increasing time the maxima are moved to the left along the curves.
355
Simulation of gliding contusions Normalized strain E
t 1
0.5
F!;atii’ I ___--/I. .8L .88 .88 .90 .92 .94 .96 .98
t”
p
Fig. 8. Distribution of the strain in the studied blood vessel at the time of maximal shear in the brain matter. Time to maximal shear
Fig. 11. Simulation results. The broken curve corresponds to the line fitted to the data for disruption of the superior cerebral veins in the subdural space, see Fig. 3. The unbroken curves represent two simulations with the pulse lengths 5 and 1OOmsec respectively. The acceleration amplitudes are chosen critical thereby making the simulated curves fall tangent to the tolerance curve.
60 50 -
10 30 20 . 10
msec lb 2b 3b ItI 50 60 70 eb
90
Fig. 9. The time to maximal shear in the brain matter as a function of the pulse length. When t, < 25 msec the maximal shear occurs after the acceleration pulse has ceased to act.
the maximal strain as a function of time is illustrated in Fig. 10. A short acceleration pulse is followed by an equal pulse in the opposite direction. The maximal shear is reduced and the high shear values for smaller radii are not maintained. This effect should be kept in mind when acceleration levels are to be related to some injury scale.
For the different pulse durations studied the strain and the strain rate have been calculated for the radius which gives the maximal shear. The pulse amplitude has been chosen so that the described injury criterion 1 or 2 is just fulfilled. Consequently these critical accelerations give the minimal possible level for injury since a single pulse has been used. As an example Fig. 11 shows two simulations fulfilling injury criterion 1. Different parameters can be used to describe tolerance thresholds. In this case the angular acceleration and the change in angular velocity have been chosen since these quantities are easily measured both in the real life, and in experimental situations. Using angular acceleration and change in angular velocity, tolerance curves according to the two injury criteria described have been calculated, see Fig. 12.
Normalizedshear
\ au ar t = 1Omsec 1.0.
0.5.
L__________
, .a
Relative radius
\ 36
.88
.90
.92
.9G
.96
.96
1.0
-
p
Fig. IO. The shear in the brain matter for two different accelerations. Note the reduction of the shear when a double pulse is used.
356
PETERUWENHIELM
t
Peak angular acceleration %nox radlsec’
i
,Tolerance curve for injury criterion /Tolerance
I
curve for injury criterion II
disruption
(“length of life”)
f tiwenhielm 1
Fig. 12. Tolerance curves for gliding contusion according to the two injury criteria used. The broken lines indicate the tolerance levels for disruption of the superior cerebral veins in the subdural space according to Liiwenhielm (1974). For a fixed pulse length the simulated results will fall along a straight line according to the choice of acceleration amplitude.
For long acceleration pulses the tolerance depends on the acceleration only, while for short pulses the tolerance level depends on the acceleration and the pulse-duration. In fact for short pulses it is proportional to the change in angular velocity. The critical change in angular velocity attains a constant value as the pulse length increases and the same is valid for the angular acceleration when t becomes small. Thus for practical purposes it is sufficient to say that a gliding contusion will probably not arise if at least one of the following conditions is fulfilled:
angular acceleration, The resulting shear in the brain matter has maximal values subcortically, thus producing gliding contusions there. Such an injury implies considerable strain of the surrounding tissues, which can be injured. This paper has dealt with the purely mechanical aspects of injury. Future studies on possible alteration of the functional state of a nerve cell after a transient strain might add to the knowledge about concussion (reduced or blocked conduction of axon-potentials, influence on the refractory period etc.).
&%X< 4500 rad/sec2
REFERENCES
A$ < 7Qradlsec. In Fig. 12 the tolerance levels for disruption of the superior cerebral veins in the subdural space (bridging veins) are indicated too. This is a different kind of injury but the two types of blood vessel injury are interesting to compare as they at autopsy are found with the same frequency. To sum up, this investigation has shown that the brain can obtain deformations of centimeter magnitude when the head is subjected to nondeforming
Lindenberg, R. and Freytag, E. (1960) The mechanism of cerebral contusions. Arch. Path. 69, 44@469. Ljung, C. B. A. (1975) A model for brain deformation due to rotation of the skull. J. Biomechanics 8, 000. Liiwenhielm, C. G. P. (1974) Dynamic properties of the parasagittal bridging veins. Z. Recht&e&in 74, 55-62. Lowenhielm. C. G. P. (1974) Strain tolerance of the vv. cerebri sup. (bridging veins) calculated from head-on collision tests with cadavers. Z. Rechtsmedizin 75, 131-144. Voigt, G. E. and Lowenhielm, C. G. P. (1974) “Gliding contusions” des Grosshirns. H&e Unfallheilk. 117, 329335.