A mathematical study of human intracrani 001

Annals of Biomedical Engineering, Vol. 16, pp. 379-401, 1988
Printed in the USA. All rights reserved.

0090-6964/88 $3.00 + .00
Copyright 9 1988 Pergamon Press plc

A MATHEMATICAL STUDY OF HUMAN
INTRACRANIAL HYDRODYNAMICS
PART I-THE CEREBROSPINAL FLUID
PULSE PRESSURE
Mauro Ursino
Department of Electronics, Informatics and Systemistics
University of Bologna
Italy
(Received 7/6/87; Revised 1/5/88)

A n original mathematical model o f human intracranial hydrodynamics is proposed. Equations able to mimic the behavior o f the intracranial arterial vascular bed,
intracranial venous vascular bed, cerebrospinal fluid absorption and production processes, and the constancy o f overall intracranial volume are separately presented and
discussed. The model parameters were given normal values computed using physiological considerations and recent anatomical data. In this paper the model is used to
simulate the genesis and morphology o f the intracranial pressure pulse wave. In particular, dependence o f the intracranial pressure pulse amplitude on mean intracranial
pressure, obtained from the model, shows excellent agreement with recent experimental findings. The model explains the intracranial pressure pulse wave as the result o f

the pulsating changes in cerebral blood volume (related to cerebrovascular compliance) which occur within a rigid space (i.e., the craniospinal compartmenO. A t low
and medium values o f intracranial pressure, the intracranial pressure pulse amplitude
mainly reflects the cerebral pressure-volume relationship. However, during severe
intracranial hypertension, an abrupt increase in the cerebrovascular compliance
becomes evident, which is reflected in an abrupt increase in the intracranial pressure
pulse wave.
Keywords--CSF dynamics, Intracranial pulsatility, Mathematical simulation.

INTRODUCTION
Knowledge of the major causes affecting the morphology and dynamics of the
intracranial pressure (ICP) pulse wave is of the greatest value both for neurological
practice and physiological research.
From a hemodynamical point of view the ICP directly affects both the transmural
pressure and the perfusion pressure o f the cerebral vascular bed; its changes are comparable to opposite changes in the systemic arterial pressure (SAP) (30). From a clinical point of view, examination of the ICP represents an essential condition for the

Address correspondence to Mauro Ursino, Department of Electronics, Informatics and Systemistics,
Viale Risorgimento 2, 40136 Bologna, Italy.

379

380

M. Ursino

comprehension and treatment of several brain disorders associated with intracranial
hypertension. Finally, observation of ICP dynamics is particularly useful in determining some parameters of clinical value, like intracranial compliance and cerebrospinal fluid (CSF) outflow resistance (21,25,26).
Many studies have appeared in recent years to clarify the origin and clinical significance of ICP dynamics. Most of these studies were experimental, providing
interesting data both on man (2,3,4,11,24,33) and animals (3,9,10,24). A few mathematical models of overall intracranial hydrodynamics have also been proposed in
order to quantitatively describe the CSF pressure and volume changes in several clinical and experimental conditions (20,26,28). However, most authors who developed
mathematical formulas for intracranial hydrodynamics focused their attention only
on a particular aspect of it, like the CSF pressure-volume relationship (2,4,13) CSF
production and absorption kinetics (16) or the collapsibility of the cerebral venous
vascular bed (8).
In the following, an original mathematical model of overall human intracranial
hydrodynamics is proposed in order to elucidate the role of different factors in
determining the morphology and time pattern o f the CSF pulse pressure. Both the
arterial and the venous, as well as the tissue compartments and the CSF production
and absorption mechanisms are separately examined from a common mathematical
point of view and included in the model. Autoregulatory adjustments of arterialarteriolar resistance in order to change flow towards its control value have also been
introduced.

The main original aspect of the present model is its ability to take a great number of different features of intracranial dynamics simultaneously into account. To
our knowledge this characteristic is unique, since no model proposed in the past has
been able to combine so many specific properties of cerebral hydrodynamics. For
instance, many studies do not incorporate the intracranial arterial and venous compliances; collapsibility o f the venous vascular bed is often left out also. The introduction of autoregulatory adjustments in cerebrovascular resistance is also new and
can be considered an original feature of the present study with respect to similar previous works.
The importance of having combined many aspects of intracranial dynamics is
especially stressed at high values of ICP. In fact, in this condition, intracranial
dynamics is the result of many causes (increase in venous pressure due to venous collapse, increase in arterial pulsatility, stiffening of the intracranial compartment, progressive loss of autoregulation, changes in CSF production and absorption rates); all
these aspects should be simultaneously taken into account if correct description of
the phenomenon under study is to be presented.
In a subsequent paper the same model will be used to mimic the results of some
relevant clinical tests concerning the intracranial hydrodynamics of normal man.
S T R U C T U R E OF THE M O D E L

Pressure within the intracranial compartment is the result of the interaction of various major causes. CSF pulsatile pressure is mainly determined by the arterial and
venous pulse pressures and by the rigidity of the intracranial compartment (3,33). On
the other hand, the mean value of CSF pressure depends especially on mean cerebral
blood volume (mainly contained in the venous bed), on the rigidity of the intracranial
compartment and on the CSF production and absorption processes.

Human lntracranial Hydrodynamics 1

381

To take all these factors correctly into account, the general model of intracranial
hydrodynamics has been divided into four submodels.
1. The Cerebral Arterial Compartment

In the following, the expression "cerebral arterial compartment" is used to denote
that portion of the cerebral vascular bed which extends from the point where the two
vertebral and the two internal carotid arteries penetrate the dura mater, up to the
level of the smallest intracerebral arterioles and capillaries.
Since the aim of the present model is not to describe propagation of pressure and
blood flow along the cerebral arterial vascular bed, but only to synthesize the role
played by arterial elasticity and arterial hydraulic resistance in the genesis and morphology of ICP, the following simplifications have been adopted: (a) compliance of
the cerebral arterial vascular bed is mainly due to the large basal arteries. Consequently, a unique, lumped arterial compliance, Cai, has been included in the model.
This compliance has been assumed equal to the sum of the compliances of all the
large intracranial arteries; (b) small cerebral arteries downstream of the circle of
Willis have been considered as purely resistive vessels.
With these simplifications the arterial intracranial compartment is described with

the electric analog of Fig. 1. P~, Pc and Pic denote the arterial pressure at the point
where the arteries penetrate the dura mater, the capillary pressure and the ICP,
respectively. Cai is the intracranial arterial compliance, R~i the resistance of the
intracranial arterial vascular bed and, finally, q is the cerebral blood flow (CBF).
In order to get a normal value for arterial resistance, Rai, it is necessary to assign
normal values for arterial pressure, Pa, capillary pressure, Pc, and CBF, q. These
values have been taken as
/3a, = 100 mmHg = 13.3 KPa
Pcn = 25 mmHg = 3.33 kPa
t~, = 12.5 cm3/sec (22) .

Q
---zz~-

Ra i

9

0

Pc

Cai I
0

o

7

FIGURE 1. Electric analog of the cerebral arterial vascular bed. Pa: arterial pressure; Pc: capillary
pressure; Pic: intracranial pressure; Cai: intracranial arterial compliance; Rai: intracranial arterial resistance.

382

M. Ursino

In the following, the subscript, n, is always used to denote a quantity in the normal basal condition; the bar denotes mean value.
From these values one obtains

Rain = 6 mmHg sec/cm 3.

The value of Rai changes due to autoregulatory adjustments in the caliber of cerebral arteries and arterioles. It has been demonstrated (see, among others, 18,23,29,30)
that arterial resistance adjustments maintain cerebral blood flow rather close to normal, despite changes in cerebral perfusion pressure, provided this last does not exceed
a lower and an upper limit (about 60-70 and 130-140 mmHg, respectively).
This phenomenon has been simulated by making use of the following equations

dx
dt

-

Gai =

1
rx

+

1 ( Pa-- Pv-- Pan + Pvn)
~=-P-7~
r

-

Gain( 1

7rl arctg(xTr))

(1)

(2)

where Pv is the cerebral venous pressure and G~g the cerebral arterial conductance
(i.e., Gag = l / g a i ).
According to differential Eq. 1, the autoregulatory action, x, depends on perfusion pressure percent changes. Moreover, its dynamics has been characterized by a
a single real pole, with time constant r. In order to reproduce the time pattern of
autoregulatory responses reported in Kontos et al. (23) r was given the value 2 sec.
The arctan in Eq. 2 simulates the existence of an upper and a lower limit of
autoregulation. When changes in perfusion pressure are very small, the result is
Gai = Gain [ 1 - x]

that is, the percent changes in arterial conductance are equal to the percent changes
in perfusion pressure. However, when perfusion pressure exceeds the lower or upper
limit of autoregulation, Gai settles at a value 50~ greater or less than normal,
respectively.
The dependence of arterial intracranial compliance, Cag, on pressure has been
described by assuming a monoexponential relationship between arterial blood volume, V~, and arterial transmural pressure, Pa - P i c (6,17); that is

Pa - Pi~ = Ploe K'(V'-V'~

(3)

where Plo, Va0 and K~ are constant parameters. According to Eq. 3 any increase in
arterial blood volume causes a stiffening of the vessel wall. Equation 3 implies that
arterial intracranial compliance is given by
dV~

1

Cai - d(Pa - Pic) - Ka(Pa - Pic)"

(4)

The constant parameter, Ka in Eq. 4 denotes the so-called "elastance coefficient"
o f the intracranial arterial vascular bed. Its value has been computed starting from

Human lntracranial Hydrodynamics 1

383

normal values of arterial and intracranial pressures and directly evaluating a normal
value for arterial intracranial compliance, C~,. To this end the arterial intracranial
compliance has been assumed equal to the sum of the compliances of all the large
brain arteries. The following formula has been used (6,31)
C~i = ~ 4a'r2(1 - ~
+ rk/hk)2
t,
k
Ek(1 + 2rk/hk)

(5)

where rk, hk, Ik and Ek denote the inner radius, thickness, length and Young modulus of the generic kth vessel, respectively, o is the Poisson ratio, assumed equal to
0.5 for all the vessels.
The values of inner radius and the length o f basal human cerebral arteries in normal conditions have been taken from recent literature (14,15,37,38,43).
Normal values of the other parameters (Young modulus and radius-thickness
ratio) which appear in Eq. 5 have been taken from (46).
Starting from these data and by making use of Eq. 5 a compliance value of
2.5.10 -3 c m 3 / m m H g has been obtained. However, this value underestimates overall intracranial arterial compliance, since the contribution of terminal branches of the
anterior, posterior and middle cerebral arteries has not been included. Consequently,
human intracranial arterial compliance in normal condition has been given the value
C a i n ~-

3.0" 10 -3 c m 3 / m m H g

(6)

which is slightly higher than that computed above.
Finally, from Eqs. 6 and 4, and by making use of the normal values of Pa and P~c
(Pie, = 9.5 mmHg) we have
Ka =

1

Cal.(Pan - #it.)

= 3.68 cm -3.

2. The CSF Production and Absorption Mechanisms
As is well known, CSF shows a typical circulation from the place where it is produced up to where it is reabsorbed. Even though minor routes also contribute to this
process, CSF is mainly formed by the choroid plexi of the cerebral ventricles and is
mainly reabsorbed by the arachnoid villi of the dural sinuses (especially of the sagittal sinus). Mathematical formulas able to mimic this process are proposed in the
following.
CSF Production. It is thought (40) that CSF originates at the choroid plexi of the
lateral ventricles through an active or energy requiring process. Nevertheless, evidence
that the CSF production rate decreases by increasing the ICP (44) or decreasing the
CBF (19) has been reported.
In the present model it is assumed that the CSF formation rate, q/, is proportional to the transmural pressure at the choroid capillary level; that is

qJ-

Pc

-

Pic

"~r

(7)

384

M. Ursino

where Rf denotes the resistance that the choroid capillary system offers to CSF
secretion.
The assumption that qf linearly depends on transmural pressure represents a simplification of the active processes involved in CSF production. However, this assumption seems reasonable according to Starling's hypothesis of water transport in the
periphery. Moreover, Eq. (7) allows the major results on CSF production to be well
reproduced by the present model.
In fact, the venous vascular bed collapses during intracranial hypertension and
thus, in accordance with the collapse mechanism in a flexible tube, CBF linearly
depends on the difference between the capillary pressure, Pc, and the ICP, Pic (see
below, especially Fig. 2 where Rp~ and R'~s are constant). Consequently, Eq. 7
implies that qf is proportional to CBF: when the ICP does not exceed 30-40 mmHg,
ql remains rather constant due to autoregulation, as observed by Higashi et al. (19)
and Guess et al. (16). However, at high values of ICP, the CSF production rate
progressively declines, as observed by Sahar (44).

CSF Reabsorption. The process by which CSF is reabsorbed by the arachnoid villi
o f the sagittal sinus and other minor routes seems to be purely passive, depending
on the difference between CSF pressure and dural sinus pressure and on the resistance
that the arachnoid villi offer to liquid outflow. Consequently, the following relation
has been adopted
q~ -

P/c - Pus

Ro

(8)

where qa is the rate of CSF absorption, Pus the venous sinus pressure and Ro the
resistance to CSF outflow.
Values for the CSF outflow resistance have recently been obtained, both in man
and animals, using two different methods: the constant infusion technique (21) and
the bolus injection technique (25).
Values of Ro for the normal man lie in the range 3-15 mmHg. min/ml (11,27,45).
Accordingly, in the present simulation it has been assumed that

Ro = 8.75 mmHg min/ml.
The normal mean value of the venous sinus pressure has been taken as
P~s~ = 6.5 mmHg.
Since, in the normal basal condition, the CSF production rate must equalize the
CSF absorption rate, one has
qa, = qf~ = 400 #l/min

Rf = 38.75 mmHg min/ml.
The values of CSF outflow resistance, Ro and CSF production resistance, R f,
have been maintained constant and equal to the normal value whenever the corre-

Human Intracranial Hydrodynamics 1

385

sponding transmural pressure was positive. Where the corresponding transmural pressure was negative, the resistance value was assumed as infinite.

3. The Cerebral Venous Compartment
Pressure within the cerebral veins is determined by many factors, that is, the values
of central venous pressure (Pc~), CBF and CSF pressure (the intracranial pressure).
This last factor is especially influential in conditions associated with intracranial
hypertension, when a collapse of the cerebral venous bed is expected to occur.
In normal conditions the value of dural sinus pressure is lower than that of CSF
pressure; therefore, dural sinuses are maintained open only by virtue of the rigidity
of their walls. However, pressure within the major cerebral veins is normally higher
than CSF pressure and the cerebral veins do not collapse since they are subject to a
positive transmural pressure. Accordingly, the following relations normally hold

The normal mean value of pressure in the major cerebral veins (P~) and value of
normal central venous pressure (Pcv) have been assumed to be
/3on = 14.0 mmHg
/5c~~ = 4.0 mmHg.
Since the transmural pressure in the cerebral venous vascular bed (Iv -Pic) is very
close to zero, collapse of these veins can easily occur. It is generally thought (8,30,41)
that the cerebral venous vascular bed does collapse during intracranial hypertension,
through a mechanism similar to that of the Starling resistor. This hypothesis is supported by in vivo observations (32,39,42,47). In particular Yada et al. (47) showed
that, in the case of intracranial hypertension, mean intraluminal pressure in the
bridge veins and lateral lacunae is always a few mmHg higher than mean ICP, independently of the ICP value. At the same time, in accordance with the mechanism of
the Starling resistor, an abrupt drop in intraluminal pressure does occur at a point
1-2 mm proximal to the junction between the lateral lacunae and the sagittal sinus.
In order to mimic this behavior with the present mathematical model, the cerebral
venous vascular bed has been subdivided into a series of two segments (see Fig. 2).
It is assumed that the first segment (whose hydraulic resistance is denoted by the symbol Ruv) extends from the capillary venules up to and including the large cerebral
veins. The second segment (whose hydraulic resistance is denoted by the symbol Rvs)
extends from the final portion of the large cerebral veins up to the dural sinuses.
Since, as demonstrated by Nakagawa et aL (32), pressure in the large cerebral veins
is always higher than the ICP, the proximal portion of the venous vascular bed does
not collapse. Therefore, the value of Rpv has been kept constant throughout the
present simulation (the effect of small changes in the venous lumen diameter due to
transmural pressure changes has been neglected). On the contrary, the distal segment
of the venous vascular bed is assumed to collapse immediately before entering the
dural sinuses. The basic assumption is that, at a section immediately upstream of the
point o f collapse, intraluminal pressure rises to a value equal to that of ICP.

386

M. Ursino

icv
!

I:lpv

I!

Rvs

Rvs

I

I

I

i

'II
\

I
1
I

FIGURE 2. Electric analog of the cerebral venous vascular bed. Rpv: resistance of the proximal
intracranial venous vascular bed; R~,s: resistance of the distal intracranial venous vascular bed (portion upstream of the collapse); R~s: resistance of the distal intracranial venous vascular bed (portion downstream of the collapse); Cvi: intracranial venous compliance.

With notations of Fig. 2:
R~s = resistance o f the distal portion o f the cerebral venous vascular bed
upstream o f the collapse;
R~;s = resistance of the distal portion of the venous vascular bed downstream of
the collapse;
u

Ro~ = R'os + Ro,.

(9)

By imposing that, at a point immediately upstream of the collapse, the intraluminal
pressure equalizes the extravascular pressure, one obtains
Pic =

Ro"~P~ + R'~P~
R;, + R~,

(10)

From Eqs. 9 and 10 one gets
Rvs-

P ~ - P~s

- Rvs.
P ~ - P~c

(ll)

Since the length of the noncollapsed portion decreases only slightly during collapse
(1-2 mm according to Yada et al. (47)), R~s has been maintained constant. Therefore, all the variability of Rvs has been imputed to Rv"~. Equation 11 has revealed

H u m a n Intracranial Hydrodynamics 1

387

itself particularly useful in reproducing the relationship between cerebral venous and
intracranial pressures in several conditions associated with intracranial hypertension.
The values of the two constant parameters R~s and Rpo have been computed starting from the normal values of C1, Pc, P~, Pic a n d / 3 reported above.
The cerebral venous vascular bed is also characterized by a high value o f compliance which, being mainly due to the large veins, has been schematized in the present mathematical model with a unique, lumped parameter, C~, undergoing the
transmural pressure P~ - Pic of the large cerebral veins. This simplification is similar to that already used for the cerebral arterial vascular bed.
The relationship between cerebral venous volume and transmural pressure has
been assumed to be of the monoexponential type with the addition o f a constant
term; that is
P~ - Pic = Pvoe r~cv~-v~~ + P ~

(12)

where Pro, Po1, K~ and V~o are constant parameters. Expression of the cerebral
venous compliance thus becomes

C~i -

dVo
1
d ( P o - Pie) - K~(P~ - Psc - P~I)

(13)

where Kv denotes the elastance coefficient o f the venous vascular bed, while the
parameter Pvl represents the transmural pressure value at which the large cerebral
veins would collapse and, therefore, venous compliance would tend to become infinite. Taking data reported in (31,36) the value of P~I has been assumed slightly
negative; that is
Pv~ = - 2 . 5 mmHg.
The collapse mechanism of the venous vascular bed may also contribute to compliance. This problem has been extensively studied by Burattini et al. (7). The authors
demonstrated that, when the inflow pressure in a tube is reduced below the external
pressure, a complete collapse in the distal tube occurs during part of the cycle and,
in this condition, compliance of the collapsed portion becomes extremely high. However, this is not the case of the present simulation where the venous pressure always
exceeds extravascular pressure, the venous flow never stops and the distal portion of
the venous vascular bed is only partially collapsed. Compliance of the distal partially
collapsed section is not easily identifiable in this case; the appropriate lengths of the
collapsed (R~s) and noncollapsed portion (Rpv + R~s) should be known (7). Since
Yada et al. (47) experimentally observed that collapse in the cerebral venous vascular bed is limited to a very short segment (1-2 mm before entering the dural sinuses)
its contribution to overall venous compliance can reasonably be assumed to be
negligible.
The value of the constant parameter Kv can be computed starting from the normal values of P~, Pic and Cvi. Unfortunately, it is very difficult to attribute a normal value to the intracranial venous compliance, especially because of the lack of
data in literature on radii and the lengths of human cerebral veins. A normal value
for Cvi has therefore been computed indirectly.

388

M. Ursino

If a vascular bed is roughly approximated with the parallel of n equal blood vessels of length l, Young modulus E, internal radius r and diameter thickness ratio d/h,
the result is

C =-

2 7rnr31
Vd
Eh
Eh

(14)

where a simplified thin-walled expression has been used for the compliance (obtainable, for instance, from Eq. 5 when h/r is low and ~ = 0) and V = 7rr2nl denotes
the overall internal volume of the vascular bed. Equation 14 shows that, in a first
approximation, compliance of a vascular bed can be considered as directly proportional to the blood volume and the diameter-thickness ratio and inversely proportional to the Young modulus.
Consequently the ratio between the venous and arterial intracranial compliances
is given by

C,,~
Cai

V,, E~(d/h)o
V~ Eo(d/h)~"

(15)

Since the cerebral venous blood volume in normal conditions is approximately
7 0 % - 8 0 % of the total brain blood volume (1) it is:

Vo/ V~ _~ 2.5.
The values of the Young modulus and the diameter thickness-ratio for compliant
cerebral arteries in normal condition have been taken from (46) lEa = 8.10 6
dyn/cm2; (d/h)a = 8].
The values of venous Young modulus and diameter-thickness ratio have been
taken from (36) lEo = 1.1-10 6 dyn/cm2; (d/h)v = 66].
By inserting the values reported above into Eq. 15 one obtains

Coin = 0.46 c m 3 / m m H g
Kv = 0.31 cm -3.
The validity of the venous compliance value obtained in this way is supported by
the observation that, with such a value, the present mathematical model can well
reproduce the ICP rise following an obstruction in the extracranial venous return
(further publication, part 2).
Finally, the extracranial venous pathway, from the dural sinuses to the heart, has
been schematized with a simple Windkessel element (i.e., with the parallel o f a compliance, Coe, and a resistance, R~e) in series with the central venous pressure, PerThe value of the resistance Rye has been computed using the normal values of P ~ ,
Pcv and 0 reported above. Compliance C~e was given the value 2.35 c m 3 / m m H g ,
that is, about 4~ of the total compliance of head and arm veins (5).

Human Intracranial Hydrodynamics 1

389

4. The Dynamics o f the Craniospinal Compartment

Since the cranial cavity is a closed space, its volume must always remain constant.
The volume of the cranial cavity is the sum of four major contributions: the arterial
and the venous blood volumes (V~ and V~, respectively), the CSF volume (VcsF) and
the volume of cerebral tissue (Vtiss). Constancy of the total intracranial volume is
therefore expressed by the following differential equation, which reflects the original Monro-Kellie principle (29)
d V~
dt

- -

+

d Vo
dt

+

d Vcsr
dt

-

+

d V~i~
dt

-O.

(16)

Since the CSF can be considered as an incompressible liquid, the time derivative
of CSF volume is due only to the difference between the CSF entering and leaving
the cranial space in unit time: that is
dVcsv
dt
- qf - q~ + Ii.

(17)

The term Ig denotes the rate at which saline is artificially injected into the CSF
space. This term has been introduced into the model in order to simulate the results
of some tests largely in use in neurological practice (see part 2).
CSF production rate, qy and CSF absorption rate, qo, have been given the expression 7 and 8, respectively.
Since Cat and Cvg are the compliances of the intracranial arterial and venous vascular beds, respectively,
dV. =Cai d
~ (P~ - P,c)

(18)

dVv
d
d-~- = Cvi ~ (Po - Pie).

(19)

Both Cag and Cog are dependent on the local value of transmural pressure, according
to Eqs. 4 and 13.
Finally, cerebral tissue volume can be assumed as a nonlinear function of the
intracranial pressure: when ICP rises, the cerebral tissue is compressed and, therefore, its volume is reduced. Several authors (4,25,26,35) have observed that the
pressure-volume relationship of the craniospinal compartment, both in man and animals, has approximately a monoexponential form, at least for a segment above and
below the resting pressure, as revealed by the fact that elastance linearly increases
with ICP. Particularly in man (12) the increase in craniospinal elastance is fairly linear up to a CSF pressure of about 3-4 KPa (22.5-30 mmHg); thereafter, the elastance
increase becomes steeper.
In order to reproduce the pattern of the human craniospinal pressure-volume relationship obtained by Friden and Ekstedt (12), the following expression has been
adopted in the present mathematical model for the elastance of cerebral tissue

390

M. Ursmo

E.~s = Cti-~Is =

~

- g ~ Pi~ + \ eo~ / ]

(20)

where K e and PoL are constant parameters.
If ICP is low, the quadratic term in Eq. 20 is negligible and the dependence of tissue elastance on pressure is fairly linear, as in the traditional monoexponential model.
However, when ICP rises, the quadratic term becomes relevant, which is reflected
in a steeper increase in tissue elastance.
Equation 20 implies that
dVtiss
-

-

dt

d P ic
tit

= -Ctiss - -

(21)

Values for Are and P01 have been given in order to best fit experimental data (see
after). We have:
Ke = 0.26 cm -3

Po~ = 7.5 mmHg (1 KPa).

The value of K e is in the range of the elastance coefficients normally reported for
man (2,34,45) and it is equal to the mean value found by Avezaat and van Eijnd~
hoven (2).
Figure 3 shows a comparison between the value of tissue compliance [Ctiss =
E~s~], evaluated through formula 20 and that of arterial compliance (Cog) resulting
from formula 4. The plot of C~i has been obtained by giving arterial pressure a normal value (100 mmHg). As shown in Fig. 3, the model implies that the effect of
arterial compliance is negligible at ICP values below 60-70 mmHg. On the contrary,
when ICP rises above 60-70 mmHg, compliance of the arterial intracranial vascular bed becomes relevant, thus producing an increase in CSF pulse pressure and a
flattening in the volume pressure response, as experiment observed by Avezaat et al.
(3,4) on dogs and patients.
M O D E L SIMULATION
According to the general considerations developed above, human intracranial
hydrodynamics is described by the electric analog of Fig. 4. The model is completed
by Eqs. 1 and 2, which mimic autoregulation, Eqs. 4, 13 and 20, which represent the
dependence of the arterial, venous and tissue compliance from pressure, and by Eq.
11, which reproduces the collapsibility of the venous vascular bed.
The model is of the fourth order, with the autoregulatory action (x), the cerebral
venous, intracranial and dural sinus pressures as state variables. The input variables
of the system are the arterial pressure and the mock CSF injection rate. The parameters of the model were given the set of values reported in Table 1. Symbol G in
Table 1 is used to denote hydraulic conductances (i.e., the inverse of the corresponding hydraulic resistances). The system has been numerically integrated on a
VAX/VMS computer, by using the 4th order Runge-Kutta method.
In this paper the model has been used to simulate the genesis of the intracranial
pulse pressure and to study the relationship between cerebral arterial, capillary,
venous and intracranial pressure waveforms at various levels of ICP. In a subsequent

Human lntracranial Hydrodynamics 1

391

C(cm3/mmHg)
_

-i

0.3

i

i

0.2

\
Ctiss

0.1

j

Cai

""\.,

0.0

~

0

I

I

1

I

20

40

60

80

i
I

_

100
Pic(mmHg)

FIGURE 3. Plots of the tissue cornpliance (C~;,s) and of the intracranlal arterial compliance
ICP. computed for an arterial pressure value of 1 0 0 mrnHg.

(C,i)

vs.

paper (part 2) the model's capability in reproducing human hydrodynamics will be
further tested in several conditions of clinical significance.
Arterial pressure has been given an unchanged pulsating waveform throughout the
present simulation (diastolic pressure 75 mmHg, systolic pressure 125 mmHg, period
0.8 sec, Fig. 5), independently of the ICP mean value. This means that the system
formed by the heart and the extracranial arterial vascular bed (extending from the
aortic valve up to the point where arteries penetrate the dura mater) has been schema-

TABLE 1. Parameter values used during the present simulation.
G~i = 0 . 1 6 6

cm 3 sec -~ m m H g -'~

G~,= = 2 . 7 7 c r n 3 sec -~ mrnHg -~
G f = 0 . 4 2 . 1 0 - 3 cm 3 sec -~ m m H g -1
C ~ , = 2.3~, c r n 3 m m H g - ~
K~ = 0 . 3 1

Pvl = - 2 . 5

crn -3

rnmHg

Gpv =

1 . 1 3 6 c m 3 sec -1 mmHg -1
Go = 1.90- 10 -3 cm 3 s e c - ~ m m H g -~

G~ = 6.25 crn 3 sec -~ mrnHg -1
K= = 3 . 6 8 c r n - 3
KE = 0 . 2 6

cm -3

Pot = 8 . 3 r n r n H g

392

M. Ursino
Rai

7~

ai

Pc

I

~ •

Rpv

~f

Pv

-~ ~vi

t"~ ''"

Rvs

Ro Rvel

Pcv

Cve

"

FIGURE 4. Electric analog of overall human hydrodynamics, q: cerebral blood flow; Pa: arterial
pressure; Pc: capillary pressure; Pv: cerebral venous pressure; Pic: intracranial pressure; Pv~; venous
sinus pressure; Cai: intracranial arterial compliance; Ram:intracranial arterial resistance; Rp,,: resistance of the proximal venous vascular bed; Rv=: resistance of the distal venous vascular bed (lateral
lacunae and dural sinuses); C~;: intracranial venous compliance; R( resistance to CSF formation; Ro:
resistance to CSF outflow; Ct~: cerebral tissue compliance;/~: rate of saline infusion; Rve: resistance of the extracranial venous pathway; Cv,: extracranial venous compliance; P~: central venous
pressure.

tized as an ideal oscillatory pressure source and that the Cushing response has not
been included in the model. The consequences of these simplifications are to be
discussed.
The ICP has been given several different values, ranging between 10 and 93
mmHg, by simulating the injection of various liquid boli into the CSF space. The
time patterns of cerebral capillary pressure, cerebral venous pressure, intracranial
pressure, venous sinus pressure and of lateral lacunae resistance (Rvs) have been
computed over three cardiac periods after the injection. The most significant waveforms among these quantities are reported in Figs. 6, 7 and 8, which refer to a moderate hypertension (diastolic ICP of about 11 mmHg), a mid-high hypertension
(diastolic ICP of about 40 mmHg) and a very severe hypertension (diastolic ICP of
about 75 mmHg), respectively.
From Figs. 7b and 8b it is evident that intracranial hypertension causes partial collapse in the distal cerebral venous vascular bed, which is reflected in a large increase
in the resistance Rvs. The slow return of Rw towards baseline, shown in Figs. 7b and
8b, is especially a consequence of the progressive ICP reduction with time, due to
CSF reabsorption. ICP time reduction is also evident from Fig. 8a.
Partial collapse of the terminal venous vascular bed makes the capillary pressure,
Pc, and the venous pressure, Pv, increase in parallel with ICP, as shown in Figs. 6
and 7a. The difference between venous and intracranial pressure is always only a few
m m H g , in accordance with data reported by Yada et aL (47). Venous sinus pressure
is almost unaffected by intracranial hypertension (Figs. 6 and 7a).
Starting from data obtained through this simulation, the plot of ICP pulse amplitude APic (ICPPA) vs. the mean intracranial pressure,/sic, (MICP) has been drawn.
Since the ICP shows fairly rapid decay towards the baseline, due to the CSF reab-

393

Human lntracranial Hydrodynamics 1
Pa(mmI-Ig)

125.0

100.0

75.0

50.0

0.0

I

I

0.8

1.6

I

2.4
T(sec)

FIGURE 5. Arterial pressure waveform used during the present simulation.

p(aunng)

30.0

~

~
9

\

20.0

10.0
Pvs

o,o

0.0

I

I

0.8

1.6

t

2.4
T(sec)

FIGURE 6. Waveforms of the capillary pressure (P=), cerebral venous pressure (Pv), ICP (Pie) and
the venous sinus pressure (Pvs) evaluated in the case of a diastolic ICP of 11 mmHg.

394

M. Ursino

P(mmHg)
80.0

60.0

40.0
20.0

~s
o.o

0.0

I

I

0.8

1.6

2.4

T(sec)
FIGURE 7a. Waveforms of the capillary pressure (Pc), cerebral venous pressure (Pv), ICP (P;c) and
the venous sinus pressure (Pvs) evaluated in the case of a diastolic ICP of 4 0 mmHg.

Rvs(mmHg. see/era3 )

14.0 l
13,0
12.0
ii.0

\

I

1o.o
0.0

O.B

J-

I

I

1.6

2.4
T(sec)

FIGURE 7b. Resistance of the distal portion of the cerebral venous vascular bed (lateral lacunae and
dural sinuses) evaluated in the case o f a diastolic ICP of 4 0 mmHg.

Human lntracranial Hydrodynamics 1

395

Pic(ramHg)
125.O

I00.0

75.0

50.0

I
0,0

0.8

1.6

I
2.4

T(sec}
FIGURE 8a. ICP waveform evaluated in the case of a diastolic ICP of 75 mmHg.

Rv s(mmHg "sec/cm3 )
200.0

150.O

100.0

50.0

o.o
0.0

I

I

I

0.8

1.6

2.4

T(sec)
FIGURE 8b. Resistance of the distal portion of the cerebral venous vascular bed (dural sinuses and
lateral lacunae) evaluated in the case of a diastolic ICP of 75 mmHg.

396

M. Ursino

sorption process, the following formulas have been used to compute I C P P A and
MICP:

APic =

Pic(tmax) + Pic(tmax + T )
-- Pic(tmin)
2

Pic = [Pic(tmax) + Pic(tmax q- T ) ] / 2 + Pic(trnin)

where T is the cardiac period, tma x and tma• + T are the instant of two successive
maximi in the ICP and tmin is the intermediate instant at which the I C P has a
minimum.
The resulting plot of I C P P A against the M I C P is shown in Fig. 9. In the same figure the experimental curves obtained by Nornes et al. (33) and Avezaat and van
Eijndhoven (2) on man are also reported. The agreement between model and experimental results is excellent.
The I C P P A shows a linear increase with mean intracranial pressure up to a breakpoint lying at about 60 m m H g ; above this point the I C P P A increases more rapidly.

DISCUSSION
CSF pulse pressure is the product of pulsating changes in cerebral blood volume,
synchronous with the cardiac beat, which occur within a rigid space (i.e., the
intracranial compartment). According to this, CSF pulse amplitude is mainly affected
by two factors: the elastance of the craniospinal system, as derived from the pressure-volume relationship (Eq. 20) and the compliance of the cerebral vascular bed.
This last depends on both the intracranial arterial and venous compliances.

~ Pic(rnmHg)

/.

40
/
30
/

f

20

10

I
20

I
40

Pic(mmHg)
I
60

I
80

I
100

~-

FIGURE 9. Plot of ICP pulse amplitude (~PJc) vs. mean ICP (Pjc). Regression lines experimentally
obtained in man by Nornes et al. (33) (dotted line) and by Avezaat and van Eijndhoven (2) (continuous line} and points evaluated through the present simulation*.

Human Intracranial Hydrodynamics 1

397

The relative importance of arterial and venous pulsatility in producing CSF pulse
pressure is still a matter of discussion. Generally, it is expected that in those cases in
which the capability of the arterial wall to pulsate is minimal, the CSF pulse wave
is mainly the result of cerebral venous pulsatility. On the contrary, when the capability of the arterial wall to pulsate becomes maximal, the CSF pulse wave reflects
the greater influence of the arterial waveform. This last situation is thought to occur,
for instance, when the arterial transmural pressure is low (i.e., during arterial
hypotension (3) or intracranial hypertension).
Figure 3 shows that, according to the present model, compliance of the intracranial arterial vascular bed is negligible at low and medium values of intracranial pressure. In these conditions, the ICP pulse amplitude appears to be mainly the result of
oscillations in cerebral venous blood volume. However, when ICP is high and, consequently, arterial transmural pressure is reduced, the influence of arterial compliance becomes relevant. Such behavior is reflected both in the morphology of the ICP
waveform and in the pattern of the graphic " I C P P A vs. M I C P . "
Figures 6, 7 and 8 show the morphology of ICP pulsations obtained from the
model at various levels of ICP. When ICP is low (Fig. 6), the ICP pulse wave is
largely different in shape from that of arterial pressure. On the contrary, the morphology of arterial and intracranial pressure pulse waves becomes similar when ICP
tends to arterial pressure (Fig. 8).
A similar result is in agreement with the observations made by Dereymaker et al.
(10). These authors experimentally observed that the morphology of CSF pulsations
is of the "venous type" at normal values of arterial pressure, whereas it is of the
"arterial type" when the arterial pressure is reduced (that is, when the arterial transmural pressure is low).
Dependence of the ICPPA on MICP is also well simulated with the model. Several
authors (2,3,4,33) observed that, both in the dog and man, the I C P P A moderately
increases with CSF pressure up to a breakpoint lying at about 60-70 mmHg. Further
increases in CSF pressure above breakpoint cause an abrupt, disproportional increase
in CSF pulsatility.
The present mathematical model reproduces this pattern very well, as shown by
the curves of Fig. 9. In accordance with the considerations developed above, the linear increase in the I C P P A below breakpoint is explained by the model with the progressive increase in craniospinal elastance, due to the exponential nature of the
pressure-volume relationship (Eq. 20). The sharp increase in pulse amplitude above
breakpoint is imputed to the sudden rise of cerebrovascular compliance, especially
that of the arterial intracranial vascular bed (Eq. 4 and Fig. 3), consequent on the
low value of transmural pressure.
The present mathematical model is able to mimic the genesis and morphology o f
the CSF pulse wave in normal man very well. Nevertheless some differences between
model and experimental responses are expected to occur in some cases as a consequence both of the control mechanisms operating on the cerebral vascular bed and
the variability of SAP. Major limitations of the model in describing the intracranial
pulsatility can be synthetized as follows: (a) Arterial pressure waveform has been kept
constant throughout the present simulation. This assumption, however, may become
false during extreme intracranial hypertension. In fact, when ICP approaches SAP,
a spontaneous increase in the latter, probably of neurogenic origin, occurs in an
effort to restore a minimal value of cerebral perfusion pressure (Cushing response).

398

M. Ursino

Moreover, during severe intracranial hypertension the intracranial arterial compliance
becomes very high (Fig. 3) so the i n p u t i m p e d a n c e o f the cerebral vascular bed
becomes that of a low-pass filter, with the effect o f a t t e n u a t i n g a n d s m o o t h i n g the
arterial pressure waveform; (b) I C P pulsatility is also affected by the v a s o m o t o r tone
o f the cerebral vascular bed which, in its turn, strictly depends o n the action o f mechanisms regulating the cerebral b l o o d flow a n d cerebral b l o o d v o l u m e . It is expected
that those actions which cause a relaxation in vascular muscle tone a n d vasodilation
are associated with a n increase in cerebrovascular pulsatility, a n increase in M I C P
a n d , therefore, also a n increase in I C P P A . However, a n increase in vascular tone is
associated with a decrease in cerebrovascular pulsatility, a n M I C P decrease a n d a
r e d u c t i o n in I C P P A . In particular, h y p e r c a p n i a has been observed to increase the
C S F pulse pressure, whereas hyperoxia has the opposite effect (41,42).
A m o r e detailed m a t h e m a t i c a l description o f regulatory m e c h a n i s m s w o r k i n g o n
the cerebral vascular bed, a n d o f their effect o n vascular s m o o t h muscle t o n e a n d
i n t r a c r a n i a l d y n a m i c s will be the subject o f s u b s e q u e n t models.
The present m a t h e m a t i c a l m o d e l m a y constitute a first step t o w a r d a m o r e complete, quantitative description of h u m a n hydrodynamical p h e n o m e n a , especially valuable in clinical investigation. T o this end, the capability o f the model in r e p r o d u c i n g
the results o f some typical clinical tests, used in n e u r o l o g i c a l practice, will be accurately checked in a s u b s e q u e n t paper.

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