A mathematical study of human intracrani
Annals of Biomedical Engineering, Vol. 16, pp. 403-416, 1988
Printed in the USA. All rights reserved.
0090-6964/88 $3.00 + .00
Copyright 9 1988PergamonPress plc
A MATHEMATICAL STUDY OF HUMAN
INTRACRANIAL HYDRODYNAMICS
PART 2-SIMULATION OF CLINICAL TESTS
Mauro Ursino
Department of Electronics, Informatics and Systemistics
University of Bologna
Italy
(Received 7/6/87; Revised 1/5/88)
The mathematical model o f human intracranial hydrodynamics proposed in a previous paper is here used to simulate the results o f some dynamical tests o f great clinical and physiological value and to analyze the blood f l o w pattern in the in tracranial
human basal arteries (especially in the internal carotid artery). Peak to peak amplitude o f the blood flow waveform in the intracranial basal arteries, computed through
the model, shows a significant increase at intracranial pressure levels above 50-60
mmHg, in accordance with recent experimental data. Moreover, diastolic blood flow
appears to be largely sensitive to intracranial pressure changes during severe intracranial hypertension, whereas systolic blood f l o w is only slightly affected in this
condition. The response o f intracranial pressure to typical saline injection (volumepressure response, steady state infusion and bolus injection tests) and to an abrupt
obstruction in the extrancranial venous drainage pathway is also well reproduced by
the model. Finally, alterations in these responses, due to changes in some significant
intracranial hydrodynamical parameters (i.e., the intracranial elastance coefficient and
CSF outflow resistance) are presented.
Keywords-Intracranial basal artery blood flow, Mathematical simulation, Neurological tests.
INTRODUCTION
In a previous paper (27) an original mathematical model of h u m a n hydrodynamics
has been proposed. The model has been able to simulate the i n t r a c r a n i a l pressure
(ICP) waveshape very well.
In this study further simulations are performed o n the same model in order to verify
its b e h a v i o r in some c o n d i t i o n s o f great neurological a n d physiological significance.
T h e e x p e r i m e n t a l c o n d i t i o n s simulated with the model are:
Address correspondenceto Mauro Ursino, Dipartimentodi Elettronica, lnformatica e Sistemistica,Viale
Risorgimento 2, 40136 Bologna, Italy.
403
404
M. Ursino
1. Analysis of lCP During Liquid Injection Maneuvers
Subarachnoid injection of mock cerebrospinal fluid (CSF) is largely used in neurological practice to gain information on major aspects of intracranial dynamics and
determine the value of some parameters of great clinical and physiological importance, like intracranial compliance and CSF outflow resistance. In particular, the following tests are adopted:
Steady State Infusion Test. This test, first proposed by Katzman and Hussey (15) and
further developed by Ekstedt (9,10) consists of injecting saline into the cranial space
at a constant rate. The value of CSF outflow resistance is obtained from knowledge
of the final steady state value of ICP.
Bolus Injection Test. This method, first proposed by Marmarou et al. (19,20) consists in rapidly, almost instantaneously injecting an amount o f CSF into the craniospinal space and measuring the consequent ICP time pattern. The method allows
computation both of CSF outflow resistance and intracranial compliance, on the
assumption that the craniospinal compartment shows a monoexponential pressurevolume relationship.
Volume-Pressure Response. The term volume-pressure response (VPR) denotes the
immediate ICP rise consequent on the instantaneous injection of a uniform, well
known volume of liquid into the craniospinal cavity, performed at different ICP
starting levels (3).
VPR is a measure of intracranial elastance (i.e., it is inversely related to the compliance of the craniospinal compartment).
2. Correlations Between the Basal Artery Flow Profile and ICP
The transcranial Doppler ultrasound technique (1,12) actually allows the cerebral
blood flow velocity pattern to be noninvasively monitored at the level of the major
cerebral basal arteries (internal carotid artery (ICA), middle, anterior and posterior
cerebral arteries). For this reason, knowledge of the relationship between the basal
artery blood flow profile and intracranial pressure has become of the greatest value
in clinical practice to gain information on cerebrovascular disorders without the need
for invasive ICP measurement maneuvers. In particular, mathematical simulation
may be useful to clarify the exact significance and limitations of several indexes (like
maximum systolic blood flow velocity, diastolic velocity and peak to peak pulsatility index) usually extracted from Doppler velocity waveforms.
3. Obstruction in the Cerebral Venous Return
The increase in the cerebral blood volume (CBV) following an obstruction in the
extracranial cerebral venous drainage pathway (venous impedance increase) represents
a relevant perturbation of intracranial hemodynamics, usually associated with an
increase in ICP, a reduction in cerebral perfusion pressure (CPP) and the activation
of mechanisms regulating both CBF and CBV (5,21,22). Study of the effects of this
perturbation is of great significance, both for the interpretation of clinical tests
(Queckenstedt test) and the treatment of pathological disorders (jugular vein thrombosis) as well as the analysis of physiological experiment results (jugular vein ligature and vena cava congestion maneuvers).
Human lntracranial Hydronamics 2
405
In this study all the experimental conditions described above are separately simulated by making use of the original mathematical model proposed in the previous
paper (27).
In a first stage of the present simulation, parameters were given the normal set of
values as previously assigned (27). The purpose of this study was to verify the capability of the model to reproduce time pattern of the major intracranial quantities in
different conditions, starting with a unique set of parameters.
Finally, in a second stage of the simulation, some curves (VPR, bolus injection
response) are presented with an alteration in the value of a few parameters of the
model, as a proof of the model's capability to adapt to different subjective and clinical situations.
S T R U C T U R E OF T H E M O D E L
A detailed description of model mathematical equations, together with normal values of its parameters, has been reported in the previous work (27). In the following,
only the major qualitative aspects of the model are briefly represented.
The model assumes that the overall cranial cavity volume (equal to the sum of the
cerebral arterial blood volume, cerebral venous brood volume, tissue volume and CSF
volume) is constant (Monro-Kellie principle).
Time dynamics of the cerebral arterial and venous blood volumes are taken into
account with the two lumped parameters, arterial compliance and venous compliance.
According to the exponential nature of pressure-volume relationship in blood vessels,
these parameters are inversely proportional to the transmural pressure in large basal
arteries and large cerebral veins, respectively. This implies that cerebrovascular pulsatility progressively increases at high values of ICP.
A modified monoexponential pressure-volume relationship has also been adopted
to describe cerebral tissue elasticity. Tissue compliance is inversely proportional to
the product of ICP and the tissue elastance coefficient (KE). As a result, the tissue
compartment becomes more rigid during intracranial hypertension.
The time dynamics of model ICP depends on tissue compliance, the CSF production and absorption rates and, during clinical maneuvers, on the rate at which mock
CSF is artificially injected into the cranial cavity. The CSF production rate has been
taken directly proportional to the difference between capillary and intracranial pressures and inversely proportional to the resistance that the choroid plexi offer to CSF
secretion. The CSF absorption rate is directly proportional to the difference between
intracranial and dural sinus pressures and inversely proportional to the resistance that
the arachnoid villi offer to CSF reabsorption (Ro).
The pressure decrease from arterial to venous side of the cerebral vascular bed has
been described using the series of three lumped resistances. These resistances mimic
the pressure losses occurring in the arterial-arteriolar vascular bed, the cerebral veins
and the terminal portion of the intracranial venous pathway (lateral lacunae and
bridge veins), respectively.
Arterial-arteriolar resistance has been assumed to depend on cerebral perfusion
pressure, so as to mimic cerebral blood flow autoregulation (4,13,16). Percentual
changes of arterial-arteriolar conductance approximately reproduce percentual changes
in perfusion pressure between the lower and upper limits of autoregulation.
Moreover, the model assumes that the last portion of the cerebral venous vascu-
406
M. Ursino
lar bed partially collapses during intracranial hypertension. The terminal venous
intracranial resistance has been made to depend on cerebral venous, intracranial and
dural sinus pressures, so as to reproduce the behavior of a collapsing Starling resistor. This mechanism serves to keep large cerebral veins always open, despite intracranial hypertension.
Finally, as in the traditional Windkessel models, cerebral blood flow is given by
the sum of a term proportional to cerebral perfusion pressure and one proportional
to the arterial pressure time derivative.
All the model parameters have been given normal values, taken from recent literature or computed starting from anatomical and physiological data (27).
RESULTS
Correlations Between the ICA Flow Profile and I C P
According to data reported in (14) and (26) the instantaneous blood flow in a single ICA can be approximately computed as ~ of the total CBF, on the assumption
that pulsatilities in the vertebral and the internal carotid arteries are similar.
In the present simulation, ICP has been assigned several values ranging between
10 and 93 mmHg, by mimicking the injection of a liquid bolus of increasing amplitude into the cranial cavity. At each level of pressure, the ICA blood flow waveform
has been computed over three cardiac periods after the injection. Two examples of
waveform are reported in Figs. la and lb, which refer to moderate intracranial
hypertension (diastolic ICP of about 11 mmHg) and very severe intracranial hypertension (brain tamponade, diastolic ICP of about 70 mmHg), respectively.
In the same graphs the maximum (systolic) and minimum (diastolic) blood flow,
deduced from tracings obtained by Nornes et al. (24) on human ICA, at similar values of cerebral perfusion pressure, are reported. The dependence of model systolic
and diastolic blood flow on cerebral perfusion pressure is in rather good agreement
with the measured one.
Flow wave shape at normal values of CPP looks very similar to pressure shape.
In this condition, arterial compliance is low and the model input impedance is of the
resistive type. However, when C P P tends to zero, arterial compliance becomes extremely high. In this last situation the model input impedance becomes of the capacitive type and the ICA flow profile looks quite different from the arterial one.
This model's result is supported by flow velocity tracings obtained with the Doppler technique by Lindegaard et al. (18) in the human ICA. Two of these tracings are
reported in Fig. 2. Phase difference between flow and pressure is negligible at normal CPP values, but it becomes great during brain tamponade, which confirms that
a shift from resistive to capacitive input impedance occurs when C P P is greatly
reduced.
Finally, from this simulation the ICA blood flow amplitude (systolic minus diastolic blood flow) has been evaluated and plotted vs. the ICP mean value. This plot
is shown in Fig. 3, in which experimental data obtained by Nornes et al. (24) on
patients are also reported.
Constant Infusion Test
The effect of a constant infusion of liquid into the CSF space has been simulated
by giving the mock CSF injection rate (Ii), in paper (27), the constant value 0.06
Human Intracranial Hydronamics 2
407
Qic(Cm3/sec)
6.0
4,0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D
2.0
o.o
0.0
I
I
I
0.8
1.6
2.4
T{sec)
(a)
Q i c(em3/see)
4.0
3.0
2.0
1.0
0.0
-1.0
-z.o
0.0
I
I
I
0.8
1.6
2.4
T(see)
(b)
FIGURES l a and b. Internal carotid artery blood flow profile evaluated with the model in the case
of a diastolic ICP of about 11 mmHg (normality, case a) and a diastolic ICP of about 70 mmHg (brain
tamponade, case b). S and D denote systolic and diastolic blood f l o w values deduced from records
reported in Nornes et al, (24) at the same values of CPP.
408
M. Ursino
J
J
FIGURE 2. Intemal carotid artery flow velocity (upper tracing) and common carotid artery blood pressure (lower tracing) obtained on a normal man (case a) and a girl with brain tamponade (case b).
Curves have been redesigned starting from those reported in Lindegaard et aL (18).
cm3/sec. Subsequently, the simulation has been repeated with an infusion rate o f
0.12 cm3/sec.
Time patterns of the diastolic ICP, in response to these two constant infusion
rates, are plotted in Fig. 4.
Bolus Injection
Time decay of the ICP, resulting from the injection of a 4.5 cm 3 liquid bolus into
the CSF space has been computed by giving the mock CSF injection rate an impulsive pattern. The impulse was given a duration of 1.5 sec and an amplitude of
3 cm3/sec. Simulation was subsequently repeated by giving the injected volume the
value 7.5 cm 3. The resulting diastolic ICP curves are reported in Fig. 5.
,AQ (cm3/sec)
O
@
5.0
3.75
po
2.5
9
o
9
1.25
2
0.0
0
I
20
Pic(mmHg)
I
40
I
60
I
80
',~--100
FIGURE 3 Plot of the ICA blood flow amplitude (~q) vs. the mean ICP (~c). Experimental points
obtained by Nornes et aL (24) (symbol o) and points computed through the present simulation (symbol * )
409
Human Intracranial Hydronamics 2
Pic(mmHg)
60,0
50.0
40.0
30,0
20.0
10.0
0.0
0,0
I
I
I
40.0
80,0
120.0
i
160.0
T(sec)
FIGURE 4. Time pattern of the model diastolic ICP resulting from the constant infusion of 0 . 0 6
cm3/sec (curve I) and 0.12 cm3/sec (curve II) of saline into the craniospinal compartment.
Venous Obstruction
The effect of an obstruction in extracranial venous outflow has been mimicked by
giving the extracranial conductance (Gve), in paper (27), a value 10-fold less than the
normal one. Such a simulation reproduces the effect of a jugular vein ligature, a
Queckenstedt test or a vena cava congestion. The plots of diastolic I C P (Pie) and
diastolic cerebral venous pressure (Pv) during the first 40 sec following obstruction
are reported in Fig. 6. As a comparison, Fig. 7 shows the increase in torcular venous
and cerebral tissue pressures (this last assumed as representative of ICP) obtained by
Cuypers et al. (6) following vena cava congestion in the cat. Since these curves do
not refer to man, only qualitative agreement with the model results has been checked.
Volume Pressure Response
The volume pressure response (i.e., the immediate ICP rise which results from an
instantaneous, uniform increase in CSF volume) has been studied by simulating the
bolus injection o f 1,2 . . . . . 17 cm 3 of liquid into the CSF space, each performed
starting from the normal ICP value.
If p X and Pi~ +L denote the ICP mean value immediately resulting from the injection of K and K + 1 cm 3 of liquid into the cranial space, respectively, then the difference p~+l _ p~r can be considered representative of the VPR obtained from the
injection of 1 cm 3, performed at the starting ICP level PicK.
The relationship between the VPR and the starting ICP mean level, achieved as
410
M. Ursino
Pic(mmHg)
80.0
60.0
40.0
20.0
o.o
0.0
t
I
I
40.0
80.0
120.O
I
160.0
T{sec)
FIGURE 5. Time pattern of the model diastolic ICP resulting from the bolus injection of 4.5 cm 3
(curve I) and 7 . 5 cm 3 (curve II) into the craniospinal c o m p a r t m e n t ,
described above, is illustrated in Fig. 8. In order to reproduce the results (marked
with asterisks in Fig. 8) obtained on four patients by Avezaat and Eijndhoven (2),
the tissue elastance in this simulation was only half that used before (KE = 0.13
cm-3; P0J = 11.7 m m H g in Eq. 20 of paper (27)). This value is still in the range of
normality for man. Model VPR shows a fairly linear increase up to an ICP mean
value of about 50 m m H g ; thereafter, the VPR decreases with pressure.
Sensitivity of the Model to Parameter Changes
Two of the parameters most c o m m o n l y referred to in neurological practice (i.e.,
the intracranial elastance coefficient (Ke) and the CSF outflow resistance (Ro =
1/Go)) have been separately modified to test the effect o f their alteration on model
response. The resulting curves, with reference to a bolus injection test, are reported
in Fig. 9. This figure shows the time pattern of the diastolic ICP resulting from a
bolus injection of about 3.0 cm 3, computed in those cases where the elastance coefficient KE and the CSF outflow resistance, Ro, had basal values (Case l: Ke 0.26
cm -3 Ro = 8.75 m m H g m i n / m l ) Ke had a value three times greater than normal
(case 2) and Ro had a value two times greater than normal (case 3).
Human Intracranial Hydronamics 2
411
p(mmHg)
30.0
Pv
25.0
20.0
Pic
15.0
//J J f
....
I0.0
5.0
o.0
0.0
I
I
I
I
10.0
20.0
30.0
40.0
T(secl
FIGURE 6. Time pattern of the diastolic ICP (Pat) and of the diastolic cerebral venous pressure (Pv)
evaluated from the model in response to a lO-fold reduction in the conductance of the extracranial
venous drainage pathway.
P{mmHg)
45.
30"
~
,
~
I
20
I
40
Pv
150
I
0
i
60
40-
20-
~-i
O.
Pic
--
--
f
I
0
I
20
!
40
I
60
T(sec)
FIGURE 7. Time pattern of venous torcular pressure and cerebral tissue pressure, measured following vena cava congestion in the cat. Curves have been redesigned from those reported in Cuypers
e t al. (6).
412
M, Ursino
VPR(mmSg)
12
10
/
/
/
/
8 -
9
.~.
9
~0 0
I
O9
6 9
4-
~
/
.%
9
9
*
/0
0
o
PIC (mmHg)
~'
0
0
l
20
I
40
I
60
I
80
i
100
FIGURE 8. Relationship between model volume-pressure response (VPR) and mean ICP (PIC). The
VPR is referred to the injection of 1 cm 3 of saline into the craniospinal compartment. Asterisks are
data obtained by Avezaat and van Eijndhoven (2) on four patients. The regression line computed
by the same authors is also reported. Circles are model results. This simulation has been performed
by giving the tissue elastance a value only half that used in the previous simulations.
From Fig. 9 it is evident that an increase in the elastance coefficient is associated
with an increase in peak response to the bolus injection. An increase in the CSF outflow resistance is mainly associated with an increase in the final steady state value
of ICP.
DISCUSSION
The mathematical model presented in this and the previous paper has been successful in reproducing most of the known hydrodynamical phenomena of the cerebral circulation. Agreement between model and experimental results is stressed by the
fact that all the parameters of the model were given normal values drawn from the
clinical literature, or computed starting with anatomical and physiological data (see
(27)).
C B F Mean Value and Pulse A m p l i t u d e
Mean CBF in the present model remains almost constant, due to autoregulation,
as long as ICP does not exceed 20-25 mmHg, and slightly decreases when ICP lies
in the range 25-40 mmHg. Above this value, changes in CBF passively follow the
perfusion pressure changes. Cases of more complete or more defective autoregula-
413
Human Intracranial Hydronamics 2
Pic(mmHg)
50.0
40.0
30.0
III
20.0
I0.0
o.0
0.0
I
I
I
40.0
80.0
120.0
I
160.0
T{sec)
FIGURE 9. Time patterns of the diastolic ICP evaluated from the model in response to a bolus injection of about 3.0 cm 3 into the craniospinal compartment. Cases in which the elastance coefficient,
KE, and the CSF outflow resistance, Ro, have "normal value" (curve I), KE has a value three times
the normal (curve II) and R o has a value t w o times the normal (curve III).
tion can also be simulated with the model by simply changing the arterial conductance reactivity to cerebral perfusion pressure variations.
The ICA flow amplitude (Fig. 3), computed through the present simulation, remains almost constant as long as ICP does not exceed a value of about 50-60 mmHg;
thereafter, it shows an evident increase, due to an abrupt increase in the cerebrovascular compliance (especially that of cerebral arteries).
A similar phenomenon of an increasing artery flow profile during severe intracranial hypertension has recently been observed in the vertebral artery of dogs by
using electromagnetic flowmeters (2,7,8) in the ICA of patients by employing the
extracranial Doppler technique (17,18,24) and in the human middle cerebral artery
with the transcranial Doppler technique (12,25). In this condition, blood flow can
also become negative during a large portion of the cardiac cycle (phenomenon of
reverberating blood flow, (18)) as in the case of the simulated flow profile of Figs.
l b and 2b.
The two flow shapes o f Figs. lb and 2b share the same main peculiarities. Both
shapes exhibit a great positive peak, followed by a time period in which flow is negative, a second, smaller positive peak and a final time period of almost no flow.
Nevertheless, the flow profile of l~ig. lb is more irregular. This is a consequence
of having sampled the arterial pressure waveform (32 samples per cardiac period have
been used). Moreover, the arterial pressure waveform has been maintained un-
414
M. Ursino
changed throughout the present simulation (i.e., the arterial vascular bed upstream
of the cerebral circulation has been schematized as an ideal pressure generator).
However, during extreme intracranial hypertension, it is expected that the cerebral
arterial pressure profile is smoothed as a consequence of the high value of cerebrovascular compliance (which behaves like a low-pass filter). This, in its turn, is reflected
in an attenuation of the highest frequencies of the ICA blood flow waveform.
Systolic blood flow in the ICA does not decrease during severe intracranial hypertension. In fact, the effect of the decrease in CBF, which would produce a reduction
in systolic blood flow, is counteracted by the abrupt increase in vascular pulsatility.
However, diastolic blood flow appears to be largely sensitive to ICP changes during
severe intracranial hypertension since both the reduction in mean blood flow and the
increase in cerebrovascular pulsatility contribute to lower its value.
Volume Pressure Response
The volume pressure response is an index of the elastance (i.e., the rigidity) of the
craniospinal compartment. At ICP values below 50 mmHg, elastance linearly rises
with ICP as a consequence of the monoexponential pressure-volume relationship of
brain tissue. This behavior is reflected in a linear increase in the VPR. However, at
ICP levels above 50-60 mmHg, elastance of the craniospinal compartment remains
constant or even decreases; in fact, the reduction in tissue compliance which occurs
in this condition is first compensated and subsequently overcome by the relevant
increase in compliance of the cerebrovascular bed. During severe intracranial hypertension, the cerebral vascular bed becomes a highly compliant structure, thus
causing a progressive reduction in the VPR, as is well reproduced by the present
mathematical model (Fig. 8) and experimentally observed by Avezaat et al. (2,3).
Steady State Infusion and Bolus Injection Tests
The typical time patterns of ICP in response to a bolus injection or a steady state
infusion of liquid into the CSF compartment have been mimicked with the model.
These time patterns are the result of a balance between the CSF production rate, the
CSF absorption rate and the storage capacity of the craniospinal system.
Bolus injection and steady state infusion tests are often used in neurological practice to determine several important parameters like intracranial compliance and CSF
outflow resistance. Alterations in these parameters cause an evident typical modification in the simulated responses, as shown by the examples of Fig. 9.
An increase in elastance coefficient Ke is associated with an increase in the peak
of the bolus injection response, together with an increase in ICP pulsatility. In particular, the reduction in the final value of the diastolic ICP, which is evident from
curve II of Fig. 9, after the increase in the elastance coefficient is only imputable to
the increase in ICP pulse amplitude, while the mean value of ICP remains unchanged.
On the contrary, an increase in CSF outflow resistance (Fig. 9, curve III) is associated with a change in the final mean value of ICP.
Venous Obstruction
After a sudden increase in the resistance of the extracranial venous pathway, both
cerebral venous pressure and ICP rise. The system settles at a new equilibrium con-
Human lntracranial Hydronamics 2
415
dition in a time of about 20 seconds. The ICP increase is about ] of that of the
cerebral venous pressure.
The extracranial venous conductance has been given a value 10-fold less than normal on the assumption that alternative venous drainage pathways (emissary veins,
paravertebral venous plexi) do exist, able to carry a certain amount of blood despite
the venous obstruction.
Both time delay and amplitude of the pressure responses shown in Fig. 6 are in
agreement with the curve obtained by Cuypers et al. (6) (Fig. 7), with reference to
a vena cava congestion experiment in the cat, and with the time pattern of epidural
pressure, monitored in the rat (11) after bilateral external jugular vein ligature.
The present mathematical model provides a possible, quantitative explanation of
the major phenomena concerning human intracranial hydrodynamics. It may represent a valuable tool for correct interpretation o f clinical and physiological tests and
a first step towards a more precise, quantitative diagnosis of neurological diseases.
However, two aspects of the model are to be underlined. First, "normal" or
"basal" condition has been assigned to the model by giving all the quantities and
parameters a value drawn from recent literature or computed starting from physiological and anatomical considerations, in an effort to simulate the cerebral hydrodynamics of a standard, healthy subject. Nevertheless, in a real case, the parameter
values can also be widely different from those used in the present and previous simulations, both because of the wide differences between healthy subjects and the possible influence of cerebral pathologies. Accordingly, parameters should be reassigned
for every particular real case, by best fitting the experimental curves to the simulated
ones.
Second, mechanisms regulating cerebral blood volume (CBV) are thought to operate on cerebral vessels (21,22) besides mechanisms regulating CBF. Their function is
probably that of reducing ICP through elimination of the excess CBV, achieved with
active contraction of the larger arterial and venous intracranial vessels.
During severe intracranial hypertension (which was the most interesting condition
to be simulated with the present model), when cerebral vessels are in a condition o f
so-called vasoparalysis, the influence of control mechanisms on ICP dynamics is
probably negligible. On the contrary, this influence becomes significant within the
range of cerebral vasoactivity.
A more detailed mathematical analysis of mechanisms regulating CBF and CBV
will be the subject of further developments of the present model.
REFERENCES
1. Aaslid, R.; Markwalder, T.M.; Nornes H. Noninvasivetranscranial Doppler ultrasound recording of
flow velocityin basal cerebral arteries. J. Neurosurg. 57: 769-774; 1982.
2. Avezaat, C . J . J . ; van Eijndhoven, J.H.M. The role of the pulsating pressure variations in intracranial
pressure monitoring. Neurosurg. Rev. 9:113-120; 1986.
3. Avezaat, C . J . J . ; v a n Eijndhoven, J.H.M.; Wyper, D.J. Cerebrospinal fluid pulse pressure and
intracranial volume-pressure relationship. J. Neurol. Neurosurg. Psych. 42: 687-700; 1979.
4. Baumbach, G.L.; Heistad, D.D. Regional segmental and temporal heterogeneity of cerebral vascular autoregulation. Ann. Biomed. Eng. 13: 303-310; 1985.
5. Belardinelli, E.; Gnudi, G.; Ursino, M. A simulation study of physiologicalmechanismscontrolling
cerebral blood flow in venous hypertension. IEEE Trans. Biomed. Eng. 32: 806-816; 1985.
6. Cuypers, J.; Matakas, F.; Potolicchio, S.J. Effect of central venous pressure on brain tissue pressure
and brain volume. J. Neurosurg. 45: 89-94; 1976.
416
M. Ursino
7. Eijndhoven van, J.H.M.; Avezaat, C.J.J. The CSF pulse pressure as indicator of intracranial elastance:
The role of the pulsating changes in cerebral blood volume. In: Ishii, S.; Nagai, H.; and Brock, M.
eds. Intracranial Pressure V. Berlin, Heidelberg: Springer-Verlag; 1983: pp. 191-196.
8. Eijndhoven van, J.H.M.; Avezaat, C.J.J. Cerebrospinal fluid pulse pressure and the pulsatile variation in cerebral blood volume: An experimental study in dogs. Neurosurgery 19: 507-522; 1986.
9. Ekstedt, J. CSF hydrodynamic studies in man: Method of constant pressure CSF infusion. J. Neurol. Neurosurg. Psych. 40; 105-119; 1977.
10. Ekstedt, J. CSF hydrodynamic studies in man. 2: Normal hydrodynamic variables related to CSF pressure and flow. J, Neurol. Neurosurg. Psych. 41: 345-353; 1978.
11. Giulioni, M.; Ursino, M.; Gallerani, M.; Cavalcanti, S.; Paolini, F.; Cerisoli, M.; Alvisi, C. Epidural
pressure measurement in the rat. J. Neurosurg. Sci. 30: 177-181; 1986.
12. Harders, A. Neurosurgical applications of transcranial Doppler sonography. New York: SpringerVerlag, Wien; 1986.
13. Harper, S.L.; Bohlen, G.; Rubin, M.J. Arterial and microvascular contributions to cerebral cortical
autoregulation in rats. Am. J. Physiol. 246: H17-H24; 1984.
14. Hillen, B.; Gaasbeek, T.; Hoogstraten, H.W. A mathematical model of the flow in the posterior communicating arteries. J. Biomech. 15: 441-448; 1982.
15. Katzman, R.; Hussey, F. A simple constant infusion manometric test for measurement of CSF absorption. I: Rationale and method. Neurology (Minneapolis) 20: 534-544; 1970.
16. Kontos, H.A.; Wei, E.P.; Novari, R.M.; Levasseur, J.E.; Rosenblum, W.I.; Patterson, J.L. Responses
of cerebral arteries and arterioles to acute hypotension and hypertension. Am. J. Physiol. 234 (4):
H371-H383; 1978.
17. Lindegaard, K.F.; Grip, A.; Nornes, H. Precerebral hemodynamics in brain tamponade. Neurochirurgia
23: 133-142; 1980.
18. Lindegaard, K.F.; Grip, A.; Nornes, H. Precerebral hemodynamics in brain tamponade, part 2: Experimental studies. Neurochirurgia 23: 187-196; 1980.
19. Marmarou, A.; Schulman, K.; LaMorgese, J. Compartmental analysis of compliance and outflow
resistance of the cerebrospinal fluid system. J. Neurosurg. 43: 523-534; 1975.
20. Marmarou, A.; Schulman, K.; Rosende, R.M. A nonlinear analysis of the cerebrospinal fluid system
and intracranial pressure dynamics. J. Neurosurg. 48: 332-344; 1978.
21. Mchedlishvili, G.I. Physiological mechanisms regulating cerebral blood flow. Stroke 11: 240-248; 1980.
22. Mchedlishvili, G.I. Arterial behavior and blood circulation in the brain. New York: Plenum Press;
1986.
23. Miller, J.D.; Stanek, A.; Langfltt, T.W. Concepts of cerebral perfusion pressure and vascular compression during intracranial hypertension. In: Meyer, J.S.; Schade, J.P. eds. Progress in Brain Research. Amsterdam: Elsevier; 1972: pp. 411-432.
24. Nornes, H.; Aaslid, R.; Lindegaard, K.F. Intracranial pulse pressure dynamics in patients with
intracranial hypertension. Acta Neurochirurgica 38: 177-186; 1977.
25. Rungelstein, E.B. Transcranial Doppler monitoring. In: Aaslid, R. ed. Transcranial Doppler Sonography. New York: Springer-Verlag; 1986; pp. 147-163.
26. Uematsu, S.; Yang, A.; Preziosi, T.J.; Kouba, R.; Toung, T.J.K, Measurement of carotid blood flow
in man and its clinical application. Stroke 14: 256-266; 1983
27. Ursino., M. Mathematical simulation of human intracranial hydrodynamics. 1: The cerebrospinal fluid
pulse pressure. Ann. Biomed. Eng. 16: 379-401; 1988.
NOMENCLATURE
Ro = C S F o u t f l o w r e s i s t a n c e
Ii
= mock CSF injection rate
Gve = e x t r a c r a n i a l v e n o u s c o n d u c t a n c e
Ke = c e r e b r a l t i s s u e e l a s t a n c e c o e f f i c i e n t
Printed in the USA. All rights reserved.
0090-6964/88 $3.00 + .00
Copyright 9 1988PergamonPress plc
A MATHEMATICAL STUDY OF HUMAN
INTRACRANIAL HYDRODYNAMICS
PART 2-SIMULATION OF CLINICAL TESTS
Mauro Ursino
Department of Electronics, Informatics and Systemistics
University of Bologna
Italy
(Received 7/6/87; Revised 1/5/88)
The mathematical model o f human intracranial hydrodynamics proposed in a previous paper is here used to simulate the results o f some dynamical tests o f great clinical and physiological value and to analyze the blood f l o w pattern in the in tracranial
human basal arteries (especially in the internal carotid artery). Peak to peak amplitude o f the blood flow waveform in the intracranial basal arteries, computed through
the model, shows a significant increase at intracranial pressure levels above 50-60
mmHg, in accordance with recent experimental data. Moreover, diastolic blood flow
appears to be largely sensitive to intracranial pressure changes during severe intracranial hypertension, whereas systolic blood f l o w is only slightly affected in this
condition. The response o f intracranial pressure to typical saline injection (volumepressure response, steady state infusion and bolus injection tests) and to an abrupt
obstruction in the extrancranial venous drainage pathway is also well reproduced by
the model. Finally, alterations in these responses, due to changes in some significant
intracranial hydrodynamical parameters (i.e., the intracranial elastance coefficient and
CSF outflow resistance) are presented.
Keywords-Intracranial basal artery blood flow, Mathematical simulation, Neurological tests.
INTRODUCTION
In a previous paper (27) an original mathematical model of h u m a n hydrodynamics
has been proposed. The model has been able to simulate the i n t r a c r a n i a l pressure
(ICP) waveshape very well.
In this study further simulations are performed o n the same model in order to verify
its b e h a v i o r in some c o n d i t i o n s o f great neurological a n d physiological significance.
T h e e x p e r i m e n t a l c o n d i t i o n s simulated with the model are:
Address correspondenceto Mauro Ursino, Dipartimentodi Elettronica, lnformatica e Sistemistica,Viale
Risorgimento 2, 40136 Bologna, Italy.
403
404
M. Ursino
1. Analysis of lCP During Liquid Injection Maneuvers
Subarachnoid injection of mock cerebrospinal fluid (CSF) is largely used in neurological practice to gain information on major aspects of intracranial dynamics and
determine the value of some parameters of great clinical and physiological importance, like intracranial compliance and CSF outflow resistance. In particular, the following tests are adopted:
Steady State Infusion Test. This test, first proposed by Katzman and Hussey (15) and
further developed by Ekstedt (9,10) consists of injecting saline into the cranial space
at a constant rate. The value of CSF outflow resistance is obtained from knowledge
of the final steady state value of ICP.
Bolus Injection Test. This method, first proposed by Marmarou et al. (19,20) consists in rapidly, almost instantaneously injecting an amount o f CSF into the craniospinal space and measuring the consequent ICP time pattern. The method allows
computation both of CSF outflow resistance and intracranial compliance, on the
assumption that the craniospinal compartment shows a monoexponential pressurevolume relationship.
Volume-Pressure Response. The term volume-pressure response (VPR) denotes the
immediate ICP rise consequent on the instantaneous injection of a uniform, well
known volume of liquid into the craniospinal cavity, performed at different ICP
starting levels (3).
VPR is a measure of intracranial elastance (i.e., it is inversely related to the compliance of the craniospinal compartment).
2. Correlations Between the Basal Artery Flow Profile and ICP
The transcranial Doppler ultrasound technique (1,12) actually allows the cerebral
blood flow velocity pattern to be noninvasively monitored at the level of the major
cerebral basal arteries (internal carotid artery (ICA), middle, anterior and posterior
cerebral arteries). For this reason, knowledge of the relationship between the basal
artery blood flow profile and intracranial pressure has become of the greatest value
in clinical practice to gain information on cerebrovascular disorders without the need
for invasive ICP measurement maneuvers. In particular, mathematical simulation
may be useful to clarify the exact significance and limitations of several indexes (like
maximum systolic blood flow velocity, diastolic velocity and peak to peak pulsatility index) usually extracted from Doppler velocity waveforms.
3. Obstruction in the Cerebral Venous Return
The increase in the cerebral blood volume (CBV) following an obstruction in the
extracranial cerebral venous drainage pathway (venous impedance increase) represents
a relevant perturbation of intracranial hemodynamics, usually associated with an
increase in ICP, a reduction in cerebral perfusion pressure (CPP) and the activation
of mechanisms regulating both CBF and CBV (5,21,22). Study of the effects of this
perturbation is of great significance, both for the interpretation of clinical tests
(Queckenstedt test) and the treatment of pathological disorders (jugular vein thrombosis) as well as the analysis of physiological experiment results (jugular vein ligature and vena cava congestion maneuvers).
Human lntracranial Hydronamics 2
405
In this study all the experimental conditions described above are separately simulated by making use of the original mathematical model proposed in the previous
paper (27).
In a first stage of the present simulation, parameters were given the normal set of
values as previously assigned (27). The purpose of this study was to verify the capability of the model to reproduce time pattern of the major intracranial quantities in
different conditions, starting with a unique set of parameters.
Finally, in a second stage of the simulation, some curves (VPR, bolus injection
response) are presented with an alteration in the value of a few parameters of the
model, as a proof of the model's capability to adapt to different subjective and clinical situations.
S T R U C T U R E OF T H E M O D E L
A detailed description of model mathematical equations, together with normal values of its parameters, has been reported in the previous work (27). In the following,
only the major qualitative aspects of the model are briefly represented.
The model assumes that the overall cranial cavity volume (equal to the sum of the
cerebral arterial blood volume, cerebral venous brood volume, tissue volume and CSF
volume) is constant (Monro-Kellie principle).
Time dynamics of the cerebral arterial and venous blood volumes are taken into
account with the two lumped parameters, arterial compliance and venous compliance.
According to the exponential nature of pressure-volume relationship in blood vessels,
these parameters are inversely proportional to the transmural pressure in large basal
arteries and large cerebral veins, respectively. This implies that cerebrovascular pulsatility progressively increases at high values of ICP.
A modified monoexponential pressure-volume relationship has also been adopted
to describe cerebral tissue elasticity. Tissue compliance is inversely proportional to
the product of ICP and the tissue elastance coefficient (KE). As a result, the tissue
compartment becomes more rigid during intracranial hypertension.
The time dynamics of model ICP depends on tissue compliance, the CSF production and absorption rates and, during clinical maneuvers, on the rate at which mock
CSF is artificially injected into the cranial cavity. The CSF production rate has been
taken directly proportional to the difference between capillary and intracranial pressures and inversely proportional to the resistance that the choroid plexi offer to CSF
secretion. The CSF absorption rate is directly proportional to the difference between
intracranial and dural sinus pressures and inversely proportional to the resistance that
the arachnoid villi offer to CSF reabsorption (Ro).
The pressure decrease from arterial to venous side of the cerebral vascular bed has
been described using the series of three lumped resistances. These resistances mimic
the pressure losses occurring in the arterial-arteriolar vascular bed, the cerebral veins
and the terminal portion of the intracranial venous pathway (lateral lacunae and
bridge veins), respectively.
Arterial-arteriolar resistance has been assumed to depend on cerebral perfusion
pressure, so as to mimic cerebral blood flow autoregulation (4,13,16). Percentual
changes of arterial-arteriolar conductance approximately reproduce percentual changes
in perfusion pressure between the lower and upper limits of autoregulation.
Moreover, the model assumes that the last portion of the cerebral venous vascu-
406
M. Ursino
lar bed partially collapses during intracranial hypertension. The terminal venous
intracranial resistance has been made to depend on cerebral venous, intracranial and
dural sinus pressures, so as to reproduce the behavior of a collapsing Starling resistor. This mechanism serves to keep large cerebral veins always open, despite intracranial hypertension.
Finally, as in the traditional Windkessel models, cerebral blood flow is given by
the sum of a term proportional to cerebral perfusion pressure and one proportional
to the arterial pressure time derivative.
All the model parameters have been given normal values, taken from recent literature or computed starting from anatomical and physiological data (27).
RESULTS
Correlations Between the ICA Flow Profile and I C P
According to data reported in (14) and (26) the instantaneous blood flow in a single ICA can be approximately computed as ~ of the total CBF, on the assumption
that pulsatilities in the vertebral and the internal carotid arteries are similar.
In the present simulation, ICP has been assigned several values ranging between
10 and 93 mmHg, by mimicking the injection of a liquid bolus of increasing amplitude into the cranial cavity. At each level of pressure, the ICA blood flow waveform
has been computed over three cardiac periods after the injection. Two examples of
waveform are reported in Figs. la and lb, which refer to moderate intracranial
hypertension (diastolic ICP of about 11 mmHg) and very severe intracranial hypertension (brain tamponade, diastolic ICP of about 70 mmHg), respectively.
In the same graphs the maximum (systolic) and minimum (diastolic) blood flow,
deduced from tracings obtained by Nornes et al. (24) on human ICA, at similar values of cerebral perfusion pressure, are reported. The dependence of model systolic
and diastolic blood flow on cerebral perfusion pressure is in rather good agreement
with the measured one.
Flow wave shape at normal values of CPP looks very similar to pressure shape.
In this condition, arterial compliance is low and the model input impedance is of the
resistive type. However, when C P P tends to zero, arterial compliance becomes extremely high. In this last situation the model input impedance becomes of the capacitive type and the ICA flow profile looks quite different from the arterial one.
This model's result is supported by flow velocity tracings obtained with the Doppler technique by Lindegaard et al. (18) in the human ICA. Two of these tracings are
reported in Fig. 2. Phase difference between flow and pressure is negligible at normal CPP values, but it becomes great during brain tamponade, which confirms that
a shift from resistive to capacitive input impedance occurs when C P P is greatly
reduced.
Finally, from this simulation the ICA blood flow amplitude (systolic minus diastolic blood flow) has been evaluated and plotted vs. the ICP mean value. This plot
is shown in Fig. 3, in which experimental data obtained by Nornes et al. (24) on
patients are also reported.
Constant Infusion Test
The effect of a constant infusion of liquid into the CSF space has been simulated
by giving the mock CSF injection rate (Ii), in paper (27), the constant value 0.06
Human Intracranial Hydronamics 2
407
Qic(Cm3/sec)
6.0
4,0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
D
2.0
o.o
0.0
I
I
I
0.8
1.6
2.4
T{sec)
(a)
Q i c(em3/see)
4.0
3.0
2.0
1.0
0.0
-1.0
-z.o
0.0
I
I
I
0.8
1.6
2.4
T(see)
(b)
FIGURES l a and b. Internal carotid artery blood flow profile evaluated with the model in the case
of a diastolic ICP of about 11 mmHg (normality, case a) and a diastolic ICP of about 70 mmHg (brain
tamponade, case b). S and D denote systolic and diastolic blood f l o w values deduced from records
reported in Nornes et al, (24) at the same values of CPP.
408
M. Ursino
J
J
FIGURE 2. Intemal carotid artery flow velocity (upper tracing) and common carotid artery blood pressure (lower tracing) obtained on a normal man (case a) and a girl with brain tamponade (case b).
Curves have been redesigned starting from those reported in Lindegaard et aL (18).
cm3/sec. Subsequently, the simulation has been repeated with an infusion rate o f
0.12 cm3/sec.
Time patterns of the diastolic ICP, in response to these two constant infusion
rates, are plotted in Fig. 4.
Bolus Injection
Time decay of the ICP, resulting from the injection of a 4.5 cm 3 liquid bolus into
the CSF space has been computed by giving the mock CSF injection rate an impulsive pattern. The impulse was given a duration of 1.5 sec and an amplitude of
3 cm3/sec. Simulation was subsequently repeated by giving the injected volume the
value 7.5 cm 3. The resulting diastolic ICP curves are reported in Fig. 5.
,AQ (cm3/sec)
O
@
5.0
3.75
po
2.5
9
o
9
1.25
2
0.0
0
I
20
Pic(mmHg)
I
40
I
60
I
80
',~--100
FIGURE 3 Plot of the ICA blood flow amplitude (~q) vs. the mean ICP (~c). Experimental points
obtained by Nornes et aL (24) (symbol o) and points computed through the present simulation (symbol * )
409
Human Intracranial Hydronamics 2
Pic(mmHg)
60,0
50.0
40.0
30,0
20.0
10.0
0.0
0,0
I
I
I
40.0
80,0
120.0
i
160.0
T(sec)
FIGURE 4. Time pattern of the model diastolic ICP resulting from the constant infusion of 0 . 0 6
cm3/sec (curve I) and 0.12 cm3/sec (curve II) of saline into the craniospinal compartment.
Venous Obstruction
The effect of an obstruction in extracranial venous outflow has been mimicked by
giving the extracranial conductance (Gve), in paper (27), a value 10-fold less than the
normal one. Such a simulation reproduces the effect of a jugular vein ligature, a
Queckenstedt test or a vena cava congestion. The plots of diastolic I C P (Pie) and
diastolic cerebral venous pressure (Pv) during the first 40 sec following obstruction
are reported in Fig. 6. As a comparison, Fig. 7 shows the increase in torcular venous
and cerebral tissue pressures (this last assumed as representative of ICP) obtained by
Cuypers et al. (6) following vena cava congestion in the cat. Since these curves do
not refer to man, only qualitative agreement with the model results has been checked.
Volume Pressure Response
The volume pressure response (i.e., the immediate ICP rise which results from an
instantaneous, uniform increase in CSF volume) has been studied by simulating the
bolus injection o f 1,2 . . . . . 17 cm 3 of liquid into the CSF space, each performed
starting from the normal ICP value.
If p X and Pi~ +L denote the ICP mean value immediately resulting from the injection of K and K + 1 cm 3 of liquid into the cranial space, respectively, then the difference p~+l _ p~r can be considered representative of the VPR obtained from the
injection of 1 cm 3, performed at the starting ICP level PicK.
The relationship between the VPR and the starting ICP mean level, achieved as
410
M. Ursino
Pic(mmHg)
80.0
60.0
40.0
20.0
o.o
0.0
t
I
I
40.0
80.0
120.O
I
160.0
T{sec)
FIGURE 5. Time pattern of the model diastolic ICP resulting from the bolus injection of 4.5 cm 3
(curve I) and 7 . 5 cm 3 (curve II) into the craniospinal c o m p a r t m e n t ,
described above, is illustrated in Fig. 8. In order to reproduce the results (marked
with asterisks in Fig. 8) obtained on four patients by Avezaat and Eijndhoven (2),
the tissue elastance in this simulation was only half that used before (KE = 0.13
cm-3; P0J = 11.7 m m H g in Eq. 20 of paper (27)). This value is still in the range of
normality for man. Model VPR shows a fairly linear increase up to an ICP mean
value of about 50 m m H g ; thereafter, the VPR decreases with pressure.
Sensitivity of the Model to Parameter Changes
Two of the parameters most c o m m o n l y referred to in neurological practice (i.e.,
the intracranial elastance coefficient (Ke) and the CSF outflow resistance (Ro =
1/Go)) have been separately modified to test the effect o f their alteration on model
response. The resulting curves, with reference to a bolus injection test, are reported
in Fig. 9. This figure shows the time pattern of the diastolic ICP resulting from a
bolus injection of about 3.0 cm 3, computed in those cases where the elastance coefficient KE and the CSF outflow resistance, Ro, had basal values (Case l: Ke 0.26
cm -3 Ro = 8.75 m m H g m i n / m l ) Ke had a value three times greater than normal
(case 2) and Ro had a value two times greater than normal (case 3).
Human Intracranial Hydronamics 2
411
p(mmHg)
30.0
Pv
25.0
20.0
Pic
15.0
//J J f
....
I0.0
5.0
o.0
0.0
I
I
I
I
10.0
20.0
30.0
40.0
T(secl
FIGURE 6. Time pattern of the diastolic ICP (Pat) and of the diastolic cerebral venous pressure (Pv)
evaluated from the model in response to a lO-fold reduction in the conductance of the extracranial
venous drainage pathway.
P{mmHg)
45.
30"
~
,
~
I
20
I
40
Pv
150
I
0
i
60
40-
20-
~-i
O.
Pic
--
--
f
I
0
I
20
!
40
I
60
T(sec)
FIGURE 7. Time pattern of venous torcular pressure and cerebral tissue pressure, measured following vena cava congestion in the cat. Curves have been redesigned from those reported in Cuypers
e t al. (6).
412
M, Ursino
VPR(mmSg)
12
10
/
/
/
/
8 -
9
.~.
9
~0 0
I
O9
6 9
4-
~
/
.%
9
9
*
/0
0
o
PIC (mmHg)
~'
0
0
l
20
I
40
I
60
I
80
i
100
FIGURE 8. Relationship between model volume-pressure response (VPR) and mean ICP (PIC). The
VPR is referred to the injection of 1 cm 3 of saline into the craniospinal compartment. Asterisks are
data obtained by Avezaat and van Eijndhoven (2) on four patients. The regression line computed
by the same authors is also reported. Circles are model results. This simulation has been performed
by giving the tissue elastance a value only half that used in the previous simulations.
From Fig. 9 it is evident that an increase in the elastance coefficient is associated
with an increase in peak response to the bolus injection. An increase in the CSF outflow resistance is mainly associated with an increase in the final steady state value
of ICP.
DISCUSSION
The mathematical model presented in this and the previous paper has been successful in reproducing most of the known hydrodynamical phenomena of the cerebral circulation. Agreement between model and experimental results is stressed by the
fact that all the parameters of the model were given normal values drawn from the
clinical literature, or computed starting with anatomical and physiological data (see
(27)).
C B F Mean Value and Pulse A m p l i t u d e
Mean CBF in the present model remains almost constant, due to autoregulation,
as long as ICP does not exceed 20-25 mmHg, and slightly decreases when ICP lies
in the range 25-40 mmHg. Above this value, changes in CBF passively follow the
perfusion pressure changes. Cases of more complete or more defective autoregula-
413
Human Intracranial Hydronamics 2
Pic(mmHg)
50.0
40.0
30.0
III
20.0
I0.0
o.0
0.0
I
I
I
40.0
80.0
120.0
I
160.0
T{sec)
FIGURE 9. Time patterns of the diastolic ICP evaluated from the model in response to a bolus injection of about 3.0 cm 3 into the craniospinal compartment. Cases in which the elastance coefficient,
KE, and the CSF outflow resistance, Ro, have "normal value" (curve I), KE has a value three times
the normal (curve II) and R o has a value t w o times the normal (curve III).
tion can also be simulated with the model by simply changing the arterial conductance reactivity to cerebral perfusion pressure variations.
The ICA flow amplitude (Fig. 3), computed through the present simulation, remains almost constant as long as ICP does not exceed a value of about 50-60 mmHg;
thereafter, it shows an evident increase, due to an abrupt increase in the cerebrovascular compliance (especially that of cerebral arteries).
A similar phenomenon of an increasing artery flow profile during severe intracranial hypertension has recently been observed in the vertebral artery of dogs by
using electromagnetic flowmeters (2,7,8) in the ICA of patients by employing the
extracranial Doppler technique (17,18,24) and in the human middle cerebral artery
with the transcranial Doppler technique (12,25). In this condition, blood flow can
also become negative during a large portion of the cardiac cycle (phenomenon of
reverberating blood flow, (18)) as in the case of the simulated flow profile of Figs.
l b and 2b.
The two flow shapes o f Figs. lb and 2b share the same main peculiarities. Both
shapes exhibit a great positive peak, followed by a time period in which flow is negative, a second, smaller positive peak and a final time period of almost no flow.
Nevertheless, the flow profile of l~ig. lb is more irregular. This is a consequence
of having sampled the arterial pressure waveform (32 samples per cardiac period have
been used). Moreover, the arterial pressure waveform has been maintained un-
414
M. Ursino
changed throughout the present simulation (i.e., the arterial vascular bed upstream
of the cerebral circulation has been schematized as an ideal pressure generator).
However, during extreme intracranial hypertension, it is expected that the cerebral
arterial pressure profile is smoothed as a consequence of the high value of cerebrovascular compliance (which behaves like a low-pass filter). This, in its turn, is reflected
in an attenuation of the highest frequencies of the ICA blood flow waveform.
Systolic blood flow in the ICA does not decrease during severe intracranial hypertension. In fact, the effect of the decrease in CBF, which would produce a reduction
in systolic blood flow, is counteracted by the abrupt increase in vascular pulsatility.
However, diastolic blood flow appears to be largely sensitive to ICP changes during
severe intracranial hypertension since both the reduction in mean blood flow and the
increase in cerebrovascular pulsatility contribute to lower its value.
Volume Pressure Response
The volume pressure response is an index of the elastance (i.e., the rigidity) of the
craniospinal compartment. At ICP values below 50 mmHg, elastance linearly rises
with ICP as a consequence of the monoexponential pressure-volume relationship of
brain tissue. This behavior is reflected in a linear increase in the VPR. However, at
ICP levels above 50-60 mmHg, elastance of the craniospinal compartment remains
constant or even decreases; in fact, the reduction in tissue compliance which occurs
in this condition is first compensated and subsequently overcome by the relevant
increase in compliance of the cerebrovascular bed. During severe intracranial hypertension, the cerebral vascular bed becomes a highly compliant structure, thus
causing a progressive reduction in the VPR, as is well reproduced by the present
mathematical model (Fig. 8) and experimentally observed by Avezaat et al. (2,3).
Steady State Infusion and Bolus Injection Tests
The typical time patterns of ICP in response to a bolus injection or a steady state
infusion of liquid into the CSF compartment have been mimicked with the model.
These time patterns are the result of a balance between the CSF production rate, the
CSF absorption rate and the storage capacity of the craniospinal system.
Bolus injection and steady state infusion tests are often used in neurological practice to determine several important parameters like intracranial compliance and CSF
outflow resistance. Alterations in these parameters cause an evident typical modification in the simulated responses, as shown by the examples of Fig. 9.
An increase in elastance coefficient Ke is associated with an increase in the peak
of the bolus injection response, together with an increase in ICP pulsatility. In particular, the reduction in the final value of the diastolic ICP, which is evident from
curve II of Fig. 9, after the increase in the elastance coefficient is only imputable to
the increase in ICP pulse amplitude, while the mean value of ICP remains unchanged.
On the contrary, an increase in CSF outflow resistance (Fig. 9, curve III) is associated with a change in the final mean value of ICP.
Venous Obstruction
After a sudden increase in the resistance of the extracranial venous pathway, both
cerebral venous pressure and ICP rise. The system settles at a new equilibrium con-
Human lntracranial Hydronamics 2
415
dition in a time of about 20 seconds. The ICP increase is about ] of that of the
cerebral venous pressure.
The extracranial venous conductance has been given a value 10-fold less than normal on the assumption that alternative venous drainage pathways (emissary veins,
paravertebral venous plexi) do exist, able to carry a certain amount of blood despite
the venous obstruction.
Both time delay and amplitude of the pressure responses shown in Fig. 6 are in
agreement with the curve obtained by Cuypers et al. (6) (Fig. 7), with reference to
a vena cava congestion experiment in the cat, and with the time pattern of epidural
pressure, monitored in the rat (11) after bilateral external jugular vein ligature.
The present mathematical model provides a possible, quantitative explanation of
the major phenomena concerning human intracranial hydrodynamics. It may represent a valuable tool for correct interpretation o f clinical and physiological tests and
a first step towards a more precise, quantitative diagnosis of neurological diseases.
However, two aspects of the model are to be underlined. First, "normal" or
"basal" condition has been assigned to the model by giving all the quantities and
parameters a value drawn from recent literature or computed starting from physiological and anatomical considerations, in an effort to simulate the cerebral hydrodynamics of a standard, healthy subject. Nevertheless, in a real case, the parameter
values can also be widely different from those used in the present and previous simulations, both because of the wide differences between healthy subjects and the possible influence of cerebral pathologies. Accordingly, parameters should be reassigned
for every particular real case, by best fitting the experimental curves to the simulated
ones.
Second, mechanisms regulating cerebral blood volume (CBV) are thought to operate on cerebral vessels (21,22) besides mechanisms regulating CBF. Their function is
probably that of reducing ICP through elimination of the excess CBV, achieved with
active contraction of the larger arterial and venous intracranial vessels.
During severe intracranial hypertension (which was the most interesting condition
to be simulated with the present model), when cerebral vessels are in a condition o f
so-called vasoparalysis, the influence of control mechanisms on ICP dynamics is
probably negligible. On the contrary, this influence becomes significant within the
range of cerebral vasoactivity.
A more detailed mathematical analysis of mechanisms regulating CBF and CBV
will be the subject of further developments of the present model.
REFERENCES
1. Aaslid, R.; Markwalder, T.M.; Nornes H. Noninvasivetranscranial Doppler ultrasound recording of
flow velocityin basal cerebral arteries. J. Neurosurg. 57: 769-774; 1982.
2. Avezaat, C . J . J . ; van Eijndhoven, J.H.M. The role of the pulsating pressure variations in intracranial
pressure monitoring. Neurosurg. Rev. 9:113-120; 1986.
3. Avezaat, C . J . J . ; v a n Eijndhoven, J.H.M.; Wyper, D.J. Cerebrospinal fluid pulse pressure and
intracranial volume-pressure relationship. J. Neurol. Neurosurg. Psych. 42: 687-700; 1979.
4. Baumbach, G.L.; Heistad, D.D. Regional segmental and temporal heterogeneity of cerebral vascular autoregulation. Ann. Biomed. Eng. 13: 303-310; 1985.
5. Belardinelli, E.; Gnudi, G.; Ursino, M. A simulation study of physiologicalmechanismscontrolling
cerebral blood flow in venous hypertension. IEEE Trans. Biomed. Eng. 32: 806-816; 1985.
6. Cuypers, J.; Matakas, F.; Potolicchio, S.J. Effect of central venous pressure on brain tissue pressure
and brain volume. J. Neurosurg. 45: 89-94; 1976.
416
M. Ursino
7. Eijndhoven van, J.H.M.; Avezaat, C.J.J. The CSF pulse pressure as indicator of intracranial elastance:
The role of the pulsating changes in cerebral blood volume. In: Ishii, S.; Nagai, H.; and Brock, M.
eds. Intracranial Pressure V. Berlin, Heidelberg: Springer-Verlag; 1983: pp. 191-196.
8. Eijndhoven van, J.H.M.; Avezaat, C.J.J. Cerebrospinal fluid pulse pressure and the pulsatile variation in cerebral blood volume: An experimental study in dogs. Neurosurgery 19: 507-522; 1986.
9. Ekstedt, J. CSF hydrodynamic studies in man: Method of constant pressure CSF infusion. J. Neurol. Neurosurg. Psych. 40; 105-119; 1977.
10. Ekstedt, J. CSF hydrodynamic studies in man. 2: Normal hydrodynamic variables related to CSF pressure and flow. J, Neurol. Neurosurg. Psych. 41: 345-353; 1978.
11. Giulioni, M.; Ursino, M.; Gallerani, M.; Cavalcanti, S.; Paolini, F.; Cerisoli, M.; Alvisi, C. Epidural
pressure measurement in the rat. J. Neurosurg. Sci. 30: 177-181; 1986.
12. Harders, A. Neurosurgical applications of transcranial Doppler sonography. New York: SpringerVerlag, Wien; 1986.
13. Harper, S.L.; Bohlen, G.; Rubin, M.J. Arterial and microvascular contributions to cerebral cortical
autoregulation in rats. Am. J. Physiol. 246: H17-H24; 1984.
14. Hillen, B.; Gaasbeek, T.; Hoogstraten, H.W. A mathematical model of the flow in the posterior communicating arteries. J. Biomech. 15: 441-448; 1982.
15. Katzman, R.; Hussey, F. A simple constant infusion manometric test for measurement of CSF absorption. I: Rationale and method. Neurology (Minneapolis) 20: 534-544; 1970.
16. Kontos, H.A.; Wei, E.P.; Novari, R.M.; Levasseur, J.E.; Rosenblum, W.I.; Patterson, J.L. Responses
of cerebral arteries and arterioles to acute hypotension and hypertension. Am. J. Physiol. 234 (4):
H371-H383; 1978.
17. Lindegaard, K.F.; Grip, A.; Nornes, H. Precerebral hemodynamics in brain tamponade. Neurochirurgia
23: 133-142; 1980.
18. Lindegaard, K.F.; Grip, A.; Nornes, H. Precerebral hemodynamics in brain tamponade, part 2: Experimental studies. Neurochirurgia 23: 187-196; 1980.
19. Marmarou, A.; Schulman, K.; LaMorgese, J. Compartmental analysis of compliance and outflow
resistance of the cerebrospinal fluid system. J. Neurosurg. 43: 523-534; 1975.
20. Marmarou, A.; Schulman, K.; Rosende, R.M. A nonlinear analysis of the cerebrospinal fluid system
and intracranial pressure dynamics. J. Neurosurg. 48: 332-344; 1978.
21. Mchedlishvili, G.I. Physiological mechanisms regulating cerebral blood flow. Stroke 11: 240-248; 1980.
22. Mchedlishvili, G.I. Arterial behavior and blood circulation in the brain. New York: Plenum Press;
1986.
23. Miller, J.D.; Stanek, A.; Langfltt, T.W. Concepts of cerebral perfusion pressure and vascular compression during intracranial hypertension. In: Meyer, J.S.; Schade, J.P. eds. Progress in Brain Research. Amsterdam: Elsevier; 1972: pp. 411-432.
24. Nornes, H.; Aaslid, R.; Lindegaard, K.F. Intracranial pulse pressure dynamics in patients with
intracranial hypertension. Acta Neurochirurgica 38: 177-186; 1977.
25. Rungelstein, E.B. Transcranial Doppler monitoring. In: Aaslid, R. ed. Transcranial Doppler Sonography. New York: Springer-Verlag; 1986; pp. 147-163.
26. Uematsu, S.; Yang, A.; Preziosi, T.J.; Kouba, R.; Toung, T.J.K, Measurement of carotid blood flow
in man and its clinical application. Stroke 14: 256-266; 1983
27. Ursino., M. Mathematical simulation of human intracranial hydrodynamics. 1: The cerebrospinal fluid
pulse pressure. Ann. Biomed. Eng. 16: 379-401; 1988.
NOMENCLATURE
Ro = C S F o u t f l o w r e s i s t a n c e
Ii
= mock CSF injection rate
Gve = e x t r a c r a n i a l v e n o u s c o n d u c t a n c e
Ke = c e r e b r a l t i s s u e e l a s t a n c e c o e f f i c i e n t