A mathematical model of cerebral blood f

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 36. NO. 2 . FEBRUARY I Y X Y

A Mathematical Model of Cerebral Blood Flow
Chemical Regulation-Part I: Diffusion
Processes
MAURO URSINO. PATRIZIA DI GIAMMARCO.

Absfracf-This paper proposes a mathematical model \I hich describes the production and diffusion of vasoactive chemical factors involved in oxygen-dependent cerebral blood flow (CRF) regulation in
the rat.
Partial differential equations describing the relations between input
and output variables have been replaced with simpler ordinary differential equations by using mathematical approximations of the hyperbolic functions in the Laplace transform domain.
This model is composed of two submodels. In the first, oxygen transport from capillary blood to cerebral tissue is analyzed to link changes
in mean tissue oxygen pressure with CBF and arterial oxygen concentration changes. The second submodel presents equations describing
the production of vasoactive metabolites by cerebral parenchyma, due
to a lack of oxygen, and their diffusion towards pial perivascular space.
These equations have been used to simulate the time dynamics of

mean tissue P,,,,perivascular adenosine concentration, and perivascular pH to changes in CBF. The present simulation points out that
the time dela) introduced by diffusion processes is negligible if compared with the other time constants of the system under study.
In a subsequent work the same equations will be included in a model
of the cerebral vascular bed to clarify the metabolite role in CBF regulation.

AND

ENZO BELARDINELLI

following cerebral ischemia, and during functional vasodilation). In all these conditions, the aim of the mechanism is to achieve tissue homeostasis, i.e., changes in
cerebrovascular resistance and blood flow are the means
by which factors important in tissue metabolism are controlled.
In recent years, evidence has appeared which suggests
that changes in blood and tissue oxygen pressure constitute a relevant stimulus, able to activate mechanisms regulating blood flow and peripheral vascular resistance. It
is known that CBF is greatly increased by a reduction in
arterial O2 pressure, while an increase in arterial oxygen
content causes significant, although less pronounced, vasoconstriction and a reduction in CBF [2], [3]. Fairchild
et al. [4], in an experimental investigation on the
hindlimb of a dog, demonstrated that lack of oxygen plays
a major role in the hyperemic response to long-lasting ischemia.

Two different mechanisms have been proposed to explain the effect of oxygen on vessel diameter and peripheral vascular resistance:
a direct mechanism, according to which a low Po?
value would directly affect the contractile activity of
smooth muscle, thus causing muscle relaxation and vasodilation,
an indirect mechunism, mediated by the release of
vasoactive substances from hypoxic tissue.
Both mechanisms are probably involved in the active
response of peripheral blood vessels to oxygen changes.
Nevertheless, some recent experimental findings [2], [5]
demonstrate that, at least for pial arteries and arterioles,
vasodilation is mainly the result of a tissue hypoxia rather
than the consequence of a direct oxygen effect on vascular
smooth muscle tension. It is generally assumed that during any situation of insufficient oxygen supply to tissue,
vasoactive metabolites accumulate in neural parenchyma.
These metabolites subsequently diffuse toward perivascular space where they cause vascular smooth muscle relaxation, thus contributing to raising CBF.
Several substances normally involved in cerebral metabolism have recently been proposed as possible mediators between tissue oxygen need and CBF ( K + , COz, H',
CaL+, osmolality, prostaglandins, and adenosine) [3],
[6], [7]. The most important of these substances seem, at
present. to be adenosine and H + . It has been experimen-

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INTRODUCTION
HE earliest doctrine concerning mechanisms regulating cerebral blood flow was proposed by Monro and
Kellie in the 18th century [ l ] . Since then many studies
have appeared on this subject, but the exact feedback
mechanism involved in cerebral regulation has not yet
been completely understood. Both myogenic and neurogenic, as well as chemical mechanisms have been proposed to explain the active changes in cerebral vessel diameter and the consequent regulation of cerebrovascular
resistance (CVR) and cerebral blood flow (CBF).
In particular, the chemical theory of blood-flow regulation suggests that the caliber of resistive vessels (i.e.,
small arteries and arterioles) is actively controlled by the
concentration of vasoactive substances in the perivascular
space. This mechanism is probably involved in the regulatory response of cerebral circulation when the equilibrium between the blood-flow oxygen supply and tissue
metibolic nced is altered (i.e., during autoregulation,
changes in blood oxygen content, the reactive hyperemia

T

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Manuscript received October 23. 1987: revised August 12. 1988.
The authors are with the Department of Electronics, Informatics. and
System Science. University of Bologna. 1-40136 Bologna, Italy.
IEEE Log Number 882.5092.

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00 18-9294/89/0200-0183$01 .OO 0 1989 IEEE

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 36, NO. 2, FEBRUARY 1989

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tally demonstrated that a significant reduction in cerebral
pH and a significant increase in cerebral adenosine concentration take place when oxygen delivery to tissue is
insufficient; this is due to the decrease in the oxidative
phosphorylation rate and simultaneous accumulation of
lactic acid [8]-[ 141. Moreover, experimental evidence reveals that an increase in H + and adenosine perivascular
concentration have a strong vasodilating effect for both
pial arteries [ 151-[20] and intracerebral arterioles [21].
In order to elucidate the relevance of chemical factors
in the control of cerebral circulation, we have developed
an original mathematical model of chemical, oxygen-dependent CBF regulation in the rat which is based on the
release of adenosine and H+ from cerebral tissue. The rat
has been chosen as it is the most frequently used animal
in physiological experiments; indeed a great number of
original results that have appeared in recent years on cerebral chemical regulation refer to this animal.
Some mathematical models of local blood-flow regulation where the chemical and the myogenic mechanisms

are described in detail have been presented in recent years
with reference to organs other than the brain, such as the
leg’s skeletal muscle [22], [23] or the kidney [24]. A few
mathematical models describing autoregulation in the
brain have also been published [25], [26]. However, in
these last models, changes in the caliber of blood vessels
and cerebrovascular resistance are described only empirically, independently of the mechanism responsible for
these changes.
In the present paper, we present mathematical equations which describe oxygen diffusion from capillary to
tissue, the accumulation of vasoactive metabolites in cerebral parenchyma, and their diffusion towards perivascular space. In a subsequent paper, the reactivity of the
cerebral vascular bed to vasoactive metabolites will be
analyzed.
We think that mathematical simulation may be very
useful to synthesize the large number of different experimental results on this subject in recent years, and to gain
a deeper knowledge of the mechanisms involved in cerebral regulation.

OXYGEN TRANSPORT
TO TISSUE
We assumed that oxygen diffuses from blood to tissue
only through the capillary wall: in other words, the small

amount of oxygen that begins to diffuse at the arteriolar
level [27], [28], [29] has been neglected. Since the aim
of the submodel is not to describe exactly the spatial distribution of tissue oxygen pressure around a cylindrical
capillary, but to reproduce the time pattern of average
P , tissue in response to a change in blood oxygen supply,
the classical cylindrical configuration of capillary exchange (Krogh’s cylinder) has been substituted with the
simpler configuration of parallel plane layers. The cylindrical symmetry of Krogh’s model is based on the assumption that blood flow in the capillaries is parallel and
unidirectional. However, the capillary structure in the
brain is probably more complex than that [7]. For in-

stance, if a section plane is used, the capillaries show random distribution of orientation [30]. In general, the capillary geometry is too complex to formulate an exact
mathematical model for it throughout the tissue. Parallel
plane layers offer the advantage of mathematical simplicity, and furthermore, the order of magnitude of the delay
introduced by diffusion processes can be estimated quite
accurately using these layers. Fig. 1 shows two of these
capillary plane layers. It is assumed that each of the two
exchanges oxygen with surrounding tissue from both surfaces. It is also schematized that the whole cerebral parenchyma is crossed by similar plane layers, with a distance 2d from each other.
The oxygen exchange from blood to tissue is properly
described by a system of partial differential equations.
These have been approximated with a system of ordinary

differential equations in order to have a model with a finite
number of state variables.
From the mass balance in a single capillary layer, the
following equation can be written:
qc(f)

cb(x,

t , - qc(f)

cb(x

+ dr, t ,

(1)
where q, ( t) denotes blood flow in a single capillary layer,
cb ( x , t ) and P b ( x , t ) are oxygen blood concentration and
pressure at the generic capillary section of coordinate x
(see Fig. l ) , and c,(x, t ) and P,,(x, t ) are oxygen concentration and pressure in the tissue adjoining the capillary wall. The other symbols may be defined as follows:
Ko2 = Krogh’s oxygen diffusion coefficient in the capillary wall (ml/min/cm/atm),

h, = capillary thickness,
dS = infinitesimal capillary exchange surface,
dV = infinitesimal capillary volume.
In the present work, oxygen concentration and pressure
are related as follows:

Pex = atis&
P b = abcb.
(2)
1/ a b and 1/utiss = oxygen solubility coefficient in blood
and cerebral tissue, respectively. 1/ab must be considered as an apparent solubility coefficient [31] as it takes
oxygen transport by hemoglobin into account. However,
the relationship between blood oxygen pressure and blood
oxygen concentration is more complex than (2) since it is
expressed by the nonlinear oxyhemoglobin dissociation
curve. A more accurate description of this curve may represent a subsequent improvement of the present model.
Finally, from (1) and (2) we have

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qc(f)

c b ( x , f,

- qc(f)

= dV-acb(x7 t ,

at

cb(x

+ dx, t >

2

+ d S - s : ; [ - Cb(X,

1

t ) - c,,(x, t )

(3)
where De is the oxygen diffusion coefficient of the capillary wall, expressed in cm2/s.

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URSINO e! a l . : MODEL OF CEREBRAL BLOOD FLOW REGULATION-I

/zyxwvutsrqp
I

I

I
I
I

in which

--+---

-

c ( z , t ) = mean oxygen concentration in cerebral tissue

I

I

at distance

+(z, t)

I

I

- -L

Ob, = oxygen diffusion coefficient in cerebral tissue
/s 1,
K,?(z, t ) = tissue oxygen consumption per unit volume in unit time.
From the previous equations we obtain

zyxwvu
I

(

/

/

,/

d

z from capillary plane layer,

= oxygen flow per tissue unit surface,

-z

0'

Fig. 1. Geometrical configuration (parallel plane layers) used to describe
oxygen exchange between capillary and tissue.

If (3) is integrated between the arterial and venous capillary sides (x = 0 and x = L, respectively) and the contribution of all capillary layers which supply cerebral tissue unit weight is added, we have

Equation ( 5 ) is linear and, therefore, can be studied in the
Laplace transform domain. In the following, all Laplace
transforms refer to changes of different quantities with respect to a hypothetical basal equilibrium condition. The
boundary conditions of ( 5 ) have been assumed as follows:

zyxwvuts

in which
q / W = blood flow per cerebral tissue unit weight,
c, = cb ( 0 , t ) = oxygen concentration in arterial blood,
c1,= cb ( L , t ) = oxygen concentration in venous blood,
VI / W = capillary blood volume per cerebral tissue unit
weight,
S / W = surface of capillary exchange per cerebral tissue unit weight,
1
cb = jk c b ( x , t ) dr = mean oxygen concentration in
capillary blood,
1

c,

=

ac(d, s ) -0

zyxwvuts
zyxwvutsr
L ji c,(x,

t ) dr = mean oxygen concentration

in the tissue adjoining capillary wall.
Equation (3') describes the mean oxygen exchange
through the capillary wall as a function of time. The dependence of oxygen venous concentration c, on mean oxygen blood concentration ?(,has been assumed as follows:

c, = h?b
h < 1.
(4)
It is remarkable that, owing to relationship (4),(3') becomes nonlinear since the input variable q / W is multiplied by the state variable Cb.
Equations (3') and (4) describe oxygen transport
through the capillary wall. Subsequently, oxygen diffuses
from the capillary wall towards the surrounding tissue.
According to the previous simplifications, the dependence
of tissue oxygen concentration on the coordinate x has
been neglected; in other words, the model refers only to
the mean values of the different quantities along this coordinate.
The oxygen diffusion across tissue is described by the
two following partial differential equations:

11) at z = d

(7)

~

az

where C(z, s ) denotes the Laplace transform of tissue oxygen concentration changes at a distance z from the capillary layer and ?b (s ) is the Laplace transform of capillary
blood oxygen concentration changes. Equation (6) describes oxygen diffusion through the capillary wall. Equation (7) is a consequence of the hypothesized symmetry
of capillary layers (Fig. 1).
By solving ( 5 ) in the Laplace transform domain, taking
into account (6) and (7), we have

Po,

=

zyxw

v,

Do* ab c,(s)

hc

= ?(O, s) =

(s

+ K,)

cb(s)

atiss

V2 tanh a
--

s

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a

+ -Doz
hc

(9)

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 36. NO. 2, FEBRUARY 1989

in which c, ( s ) is the Laplace transform of oxygen concentration changes in tissue adjoining the capillary wall,
whose antitransform is used in ( 3 ' ) , and S / V 2 is the capillary exchange surface per cerebral tissue unit volume.
Po, is the Laplace transform of the average Po, tissue
changes, that is, the output variable of the present submodel.
The oxygen exchange from blood to tissue is thus completely described by ( 3 ' ) and the antitransforms of (8) and
(9). The parameter values are reported in Table I, in which
subscript n is used to denote a quantity in basal equilibrium condition, and the term (CMR,)
= ( q / W ) , (c, ct,) denotes the normal oxygen consumption rate per unit
weight of cerebral tissue. The value of parameter K, has
been computed by assuming a normal equilibrium mean
tissue Po2 of 32 mmHg, deduced from curves reported by
Thews [ 3 11. Most parameters refer to the rat. For the other
parameters in Table I (diffusion coefficients, solubility
coefficients, specific weight), changes from one animal to
another do not seem to be relevant.
Equations (8) and (9) include the meromorphic function
tanh a / a . This function has been approximated with a
rational function in order to achieve a model with a finite
number of state variables. The approximation has been
achieved by using the continued fraction expansion [ 3 7 ] ,
truncated after a finite number of elements, i.e.,
1
tanh
- a(10)
CY
CY2
1 +
CY2
3+5
* * *
The number of elements in the continued fraction (10) has
been chosen so as to reproduce the tissue Po2 time pattern
in response to a step change in cerebral blood flow, without significant errors. In particular, if we take tanh C Y / C Y
= 1, this is equivalent to assuming that there is instantaneous oxygen diffusion across cerebral tissue. On the contrary, if one takes

TABLE I
PARAMETER
VALUESFOR THE OXYGEN
DIFFUSION
* IO-* cmZ/s
1 . 6 . 10-5cmZ/s

= 1.3

=

= 1 pm
=

1.5 ml,/mlblood/atm

= 0.024 ml,/ml,,,,/atm

0.076 mla/g/min
1.04 g/cm3
= 1.84 . lo-'
= 1.415 .
mm
= 75 ml/min/100 g
= 26 pm
= 1.32 s - '
= 0.16 ml,/mlblood
= 0 . 0 6 mle/mlblmd
= 0.8219
=
=

zy

the following ordinary differential equations in time domain:
r
1
3-

ab

atiss

3 + yKad 2
Do2

L

zyxwvutsrq
+

-tanh
= - -a
CY

3

3

s+K,

~ + C Y ' - -

3+-

ob,

"

.,

(11)

d'

the dynamics of tissue Po, is delayed as a consequence of
the period necessary for oxygen to diffuse across the cerebral tissue. The addition of further terms in (10) does
not cause significant changes in the model response. Consequently, (1 1) has been taken in the present work.
With the parameter values reported in Table I, and using
(1 l), it is demonstrable that the two transfer functions (8)
and (9) can be further simplified without any significant
alteration in the model dynamics, as follows:
a) capillary wall permeability, Doz/ h c , can be assumed
as infinite,
b) the pole in (11) has a time constant ( T ~=
d2/(3Db,
K, d 2 ) = 0.12s), negligible if compared
with the time delays of the system under study.
With the previous simplifications, and using ( 3 ' ) and
the antitransforms of (8), (9), and ( l l ) , we finally reached

+

F,(t)

=

3ab

3 + yK ad 2

z

306,

(t

= 306,

+ K,d2 p

(t);

U 0 2

(13)

c, = X c b

in which p b = a b c b is the average oxygen pressure in capillary blood. The term 3 0 & / ( 3 D &
K, d 2 ) = 0.84
takes into account the mean drop in oxygen pressure from
capillary blood to tissue.
As is evident from these equations, the oxygen supply

+

4 (c,
-

- c,) represents the input variable of the present

model, i.e., the variable whose changes are reflected in
blood and tissue oxygen pressure changes, thus triggering
the action of chemical regulatory mechanisms. The normal value of venous oxygen concentration, cVn,has been
computed from data of normal CBF, ( q / W ) , , and normal oxygen consumption rate, ( CMROz),,,per tissue unit
weight reported in Table I. Finaily, a value of X has been
obtained by imposing the normal equilibrium condition to
(12), in which the values of all the other parameters are
known.
Fig. 2 shows the time pattern of mean tissue Po2, computed using (12) and (13), in response to a step change in
CBF from normal to one-half its value. The time constant
of this response (about 1.6-1.7 s ) is in agreement with
that reported in a recent experimental work on the rat [ 3 8 ] .

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URSINO

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MODEL OF CEREBRAL BLOOD FLOW REGULATION-I

9
PIAL ARTERY

’I
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CEREBRAL

’0

zyxwv

L

0

1

3

2

4

5

t (sec)

Fig. 3. Geometrical configuration used to describe diffusion of metabolites
from cerebral parenchyma to pial artery perivascular space.

Fig. 2 . Time pattern of mean cerebral tissue oxygen pressure in response
to a step change in CBF from its normal to half its value.

PRODUCTION
AND DIFFUSION
OF METABOLITES
As described in the previous paragraph, several metabolites accumulate in cerebral parenchyma as a consequence of tissue hypoxia. These metabolites subsequently
diffuse towards perivascular space where they provoke
vasodilation and contribute to raising CBF. In Fig. 3 , the
geometrical configuration used in the present model to describe diffusion of metabolites is shown. According to Wei
and Kontos [39], this configuration is equivalent to the
schematization of the brain as a plane source of vasodilating substances, positioned opposite the pial arteries at
a distance 1.
Diffusion of metabolites towards the vessel wall is described by the following partial differential equation,
equivalent to ( 5 ) :

TISSUE

in which c,( y , s) denotes the Laplace transform of metabolite concentration changes, at a distance - y from tissue plane layer, and Po,(s) is the Laplace transform of
average tissue oxygen pressure changes, obtainable from
the previous submodel. According to (15), the amount of
the generic metabolite m , crossing the unit surface of cerebral parenchyma in unit time, depends on oxygen lack.
Moreover, it is assumed that the metabolic rate of cerebral
tissue does not change during simulation and that adequate oxygen delivery is always expressed by the condition: P & ( s ) = 0. The boundary condition (16) schematizes the effect of the blood-brain barrier, which does not
allow reabsorption into the blood flow of either adenosine
[16], [40] or H + [34].
Equation (14), with the boundary conditions (15) and
(16), gives the following expression for perivascular metabolite concentration:

zyxwvut
zyxwvutsrqpo
zyxwvu
zyxwvutsr
C,(O,

in which
c, = concentration of the generic metabolite m at a distance - y from a pial artery,
Dm = metabolite diffusion coefficient,
K,c, = amount of metabolite reabsorbed or degraded
per unit volume in unit time.
In the following, all the Laplace transforms refer to
changes of different quantities with respect to the normal
equilibrium condition.
It is assumed that at the instant r = 0, the system is in
normal equilibrium condition. Moreover, the following
boundary conditions, in Laplace transform domain, have
been used:
-1, s)
a t y = -1
-D,
= -GmFO2(s)
(15)
ay

aty = 0

-D, ac,(o, s)
ay

Po*(s
sinh
K,) -

G m

s) = -

l(s

+

P

in which

I

The meromorphic transfer function (17) has been approximated by a rational function using the Taylor series of
sinh P I P limited to a finite number of elements. If two
terms of the series are taken, one has

The transfer function (18) has two real simple poles:

=o

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING. VOL. 36. NO. 2. FEBRUARY 1989

The second pole of (18) takes metabolic diffusion process
into account. Unfortunately, it is very difficult to assign
a correct and univocal value to the parameter D m / f 2because of the complex geometrical arrangement of the pial
arterial bed and a lack of data in literature on metabolite
diffusron coefficients. Nevertheless, if one assumes that
the distance between cerebral tissue and the arteriolar wall
does not exceed 15 pm [39] and that the diffusion coefficient of generic metabolite is comparable to that of CO2
(Dco2 2: 0.3
cm2/sec), the result is: 6Dm/12L
8 s-'. This means that the time delay introduced by the
metabolite diffusion process is of a few tenths of a second
(time constant less than 0.125 s) according to what was
also deduced by Wei and Kontos [39]. Since the accumulation of both adenosine and H+ occurs with a time
delay much greater than that introduced by the second
pole, its effect on the model dynamics has been neglected.
Consequently, the following ordinary differential equation in time domain has been reached:

-

20

0

40

60
t (sec)

Fig. 4. Time pattern of rat's perivascular adenosine concentration during
60 s of total ischemia. Curve resulting from the present model (continuous line) and experimental results reported in Winn et al. [lo] (symbol

zy

*).

zyx

in which

G, =

6 e mDm
f(60, + KmZ2)'

The ordinary differential (19) has been used to simulate
the dynamics of adenosine and H perivascular concentration in several conditions associated with an alteration
in oxygen delivery to tissue.
Adenosine: 60 s of total cerebral ischemia in the rat
cause a five-fold increase in brain adenosine concentration
[lo]. If ischemia is protracted for a longer time [16] the
adenosine concentration continues to increase until, in 510 min, it settles at a value 13-14 higher than normal.
These results are clearly reproduced by the model if the
gain Gad,and the constant Kad of (19), relative to adenosine, are given the values

the adenosine concentration is too
- low , the rephosphorylation rate [equal to Gad(P, - P,,,)] must slow down.
In this way, an adenosine dynamics, with an inferior saturation level at cad = 0,is obtained. The normal adenosine concentration value in the rat's brain has been given
[lo] as follows:

zyxwvu

+

Kad = 0.01 s-'

nmoles
0.9 -.
g
A comparison between the model's and the experimental
rat's adenosine concentration [101 in 60 s of total cerebral
ischemia, is reported in Fig. 4. Moreover, the increase in
adenosine concentration which occurs with the model
during several minutes of total cerebral ischemia is shown
in Fig. 5 .
pH: In order to simulate some recent experimental results on cerebral tissue acidosis [6], [12], the parameters
of (19) have been given the values
cad, =

KH+ = 0.001 s-'
GH+ = GH+o = 8.7
These parameters have been kept constant during tissue
hypoxia. On the contrary, during tissue hyperoxia and for
adenosine concentration values less than normal, the gain
of the process must be considered dependent on concentration values. We assumed
if ?jo2 > Pan and

cad

*

As with the adenosine, we have assumed that

ifPo,

-

> P,, and cH+<

CH +

cH+,then GH+= GH+o-

CH+n

otherwise GH+ = GH+o.

cad
< Cad,, then Gad= Gado-

Cadn

Otherwise Gad = Gado.

(20)
Equation (20) is justified since, during hyperoxia, a large
amount of adenosine is rephosphorylated. However, when

moles
mdg'

lo-'' 1 s

(21 )
Normal cerebral pH is [6]
pH, = 7.35.
A comparison between the cerebral tissue pH obtained
with the model and that measured by Betz and Heuser [6]

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URSINO ef al.: MODEL OF CEREBRAL BLOOD FLOW REGULATION-I

TABLE I1
PARAMETER
VALUES
FOR THE METABOLITE
DYNAMICS

Kad = 0.01 s-l

Gad,, = 4.04

K”*

G,+,

= 0.001 S K I

=

. lo-’

8 . 7 . lo-’’

nmoles

nmoles

cad”= 0.9 l?

g . s . mmHg

nmoles

1

pH,,

=

7.35

. s . mmHg

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3

----_____--________________

0

I

I

I
1

I
I

1

2

3

4

I

0

I

0

5

10

15

20

25

I
I

5

(a)

t(min)

Fig. 5 . Time pattern of rat’s perivascular adenosine concentration during
5 min of total cerebral ischemia.

in the dog, during 30 min of total cerebral ischemia, is
presented in Fig. 6(a) and (b). With the parameter values
reported above, the present model may also reproduce a
measured pH time pattern consequent on a long lasting
arterial hypoxia [ 121.

DISCUSSION
The mathematical model proposed in the present work
allows the dynamics of cerebral metabolite perivascular
concentration to be simulated in any condition associated
with an alteration in oxygen delivery to tissue. Only the
main substances (adenosine and Hf ) which are known to
play a significant role in oxygen-dependent CBF control
have been included in the model. For instance, the potassium ion action, which seems to be effective in only a few
situations [12], [6] and for only a short time [29], has not
been included in this model. However, mathematical
equations similar to those developed in the present work
can also be used to simulate the dynamics of other chemical agents involved in regulating cerebral circulation.
The main conclusion of the present simulation is that,
with respect both to oxygen transport from blood to tissue
and to perivascular metabolite concentration changes, the
time delay introduced by diffusion processes is negligible
if compared with other time constants of the system under
study. This is a consequence of the very small distance
between the cerebral capillaries [31], [41] and between
the brain tissue and pial arterioles [39], [7], which has

30

t(min)

7.0

6.8

zyxwv

6.6

1

o

I

;

T(HIN)

I

d 30
(b)

Fig. 6. Time pattern of cerebral pH during 30 min of total cerebral ischemia. Curve resulting from the present model (a) and experimental
results obtained by Heuser [I31 on the dog (b).

been reported in literature. This result strongly supports
the hypothesis that chemical factors play a significant role
in the control of cerebral circulation. In fact, one of the
major arguments against chemical regulation of CBF is
that the speed of observed regulatory response would be
inconsistent with the time delay necessary for diffusion
and action.
The adenosine is characterized by a rather quick dynamics. A significant increase in perivascular adenosine
concentration, which can cause vasodilation, becomes evident within the first minute of a tissue hypoxia (Fig. 4).
Consequently, adenosine could be involved in short term
regulation of CBF and CVR. On the contrary, a tissue
hypoxia of several minutes is required before deep acidosis, consistent with a sustained vasodilation, develops

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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 36, NO. 2. FEBRUARY 1989

in cerebral tissue. However, it is probable that pH is involved in the active response of cerebral vessels in the
medium period, when the action of other, more rapid
mechanisms has become exhausted or has been revealed
insufficient to ensure an adequate level of CBF. This conclusion is in agreement with experimental results reported
by Zwetnow [42] and [43]. The author states that cerebral
pH changes are almost negligible if the reduced arterial
pressure is contained within the limits of autoregulation.
By contrast, pH alterations become significant if arterial
pressure is diminished below the lower limit of autoregulation as well as during reactive response to cerebral
ischemia.
The present equations, together with an original mathematical model of cerebral vascular reactivity, will be
used in a subsequent work to simulate the overall feedback mechanism responsible for chemical oxygen-dependent CBF regulation.

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URSINO er u l . : MODEL OF CEREBRAL BLOOD FLOW REGULATION-I

[40]

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Mauro Ursino was born in Bologna, Italy, on October 17, 1958. He received the Dr Ing degree in
electronic engineering from the University of Bologna, Bologna, Italy, in 1983 and the Ph.D degree in bioengineering in 1987.
Since 1983 he has been with the Department of
Electronics, Informatics, and System Science of
the University of Bologna. His present research
activities are focused on modeling and analysis of
the cerebrovascular control mechanisms, intracranial hydrodynamics, and wave propagation in
blood vessels.

Patrizia Di Giammarco received the Dr.lng. degree in electronic engineering from the University
of Bologna, Bologna, Italy, in 1986.
Since 1986 she has been with the Department
of Electronics, Informatics, and System Science
of the University of Bologna where she is now a
Ph.D. student in bioengineering. Her main rcsearch interests are focused on modeling and analysis of the cardiovascular control mechanisms.

Enzo Belardinelli was born in Italv on January
12, 1930.
He joined the University of Bologna, Bologna,
Italy, in 1956 as a Researcher and Assistant Professor of Electronics and Computer Science From
1967 to 1982 he was Full Professor of Automatic
Control and, since 1983, he has been Full Professor of Bioengineenng in the Department of Electronics, Informatics, and System Science At the
beginning of the 1970’s he founded a Center ot
Biomedical Research and the School of Bioengineering in the University of Bologna. Bologna, Italy Presently, he IS Director of the National Project on Cardiovascular System He has developed
new methods for analysis of biological systems, applying concepts of sys
tem and control theory to them His scientific interest is particularly tocused on analysis and modeling of the cardiovascular system as a whole,
arterial dynamics, and the cerebrovascular system.

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licensed
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