A mathematical model of cerebral blood flow chemical ...

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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.

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

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.

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

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 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

/ d

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

I / /

,/

-z

z from capillary plane layer,

= oxygen flow per tissue unit surface,

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 (

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

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,

=

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:

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:

11) at z = d

ac(d, s ) -0

(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,

=

v,

Do* ab c,(s)

hc

= ?(O, s) =

(s

+ K,)

cb(s)

atiss

V2 tanh a --

s


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 = =

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

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

+

3 + yKad 2 Do2

L

+

-tanh = - -a

ab

atiss

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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PI U / . :

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MODEL OF CEREBRAL BLOOD FLOW REGULATION-I

9 PIAL ARTERY

’I ’0

CEREBRAL

TISSUE

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 ) :

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: 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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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 *).

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 +

Kad = 0.01 s-'

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: 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+