tion of local cerebral blood flow (LCBF) were examined by using three models, i.e., venous equilibrium, tube, and distributed models. The technique most ...
Journal of Cerebral Blood Flow and Metabolism 7:433-442 © 1987 Raven Press, New York
Tracer Disposition Kinetics in the Determination of Local Cerebral Blood Flow by a Venous Equilibrium Model, Tube Model, and Distributed Model
Y.
Sawada, Y. Sugiyama, T. Iga, and M. Ranano
Faculty of Pharmaceutical Sciences, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, Japan
Summary: Tracer distribution kinetics in the determina tion of local cerebral blood flow (LCBF) were examined by using three models, i.e. , venous equilibrium, tube, and distributed models. The technique most commonly used for measuring LCBF is the tissue uptake method, which was first developed and applied by Kety (1951). The mea surement of LCBF with the 1 4C-iodoantipyrine (lAP) method is calculated by using an equation derived by Kety based on the Fick's principle and a two-compart ment model of blood-tissue exchange and tissue concen tration at a single data point (Sakurada et aI. , 1978). The procedure, in which the tissue is to be in equilibrium with venous blood, will be referred to as the tissue equilibra tion model. In this article, effects of the concentration gradient of tracer along the length of the capillary (tube model) and the transverse heterogeneity in the capillary transit time (distributed model) on the determination of LCBF were theoretically analyzed for the tissue sampling method. Similarities and differences among these models
are explored. The rank order of the LCBF calculated by using arterial blood concentration time courses and the tissue concentration of tracer based on each model were tube model (model II) < distributed model (model III) < venous equilibrium model (model I). Data on 1 4C-IAP ki netics reported by Ohno et al. (1979) were employed. The LCBFs calculated based on model I were larger than those in models II or III. To discriminate among three models, we propose to examine the effect of al tering the venous infusion time of tracer on the apparent tissue-to-blood concentration ratio (Aapp). A range of the ratio of the predicted Aapp in models II or III to that in model I was from to 1.3. In the future, there may be a need to determine which model should be used to calcu late the LCBF based on this discriminator and to develop another discriminator by using multiple data points based on positron emission tomography. Key Words: Capillary transit time-Cerebral blood flow-Distributed model -Tube model-Venous equilibrium model.
There are two methods for measuring local cere bral blood flow ( L CBF), i.e., the tissue uptake and the tissue washout or clearance methods. Of these two methods, the one most commonly used in an imal studies is the tissue uptake method. This method was first developed and applied by Kety (1951). In this method, a highly diffusible radioac tive substance [for example, 14 C-iodoantipyrine (lAP)] at the blood-brain barrier (BBB) is intrave nously administered, blood is continuously sam pled, samples of brain at the last sampling time are obtained, and the radioactivities of the indicator in
all blood and brain samples are measured. Then, the blood flow is calculated using an equation de rived by Kety (1951) and based on Fick's principle and the venous equilibrium model of the blood brain exchange. Kety et ai. ( Landau et aI., 1955; Freygang and Sokoloff, 1958; Sakurada et aI., 1978) originally employed a version of the experimental procedure (quantitative autoradiographic method) and Kety's equation to estimate cerebral blood flow. It is known as the tissue saturation (Lacombe et aI., 1980) or the tissue equilibration technique. Furthermore, there have been recent developments in positron emission tomography, imaging the quantitative measurement of LCBF based on the tissue concentration time profile and the precise implementation of the dy namic tracer model (Raichle et aI., 1983). Assumptions of these analyt ical procedures are that the brain is a single well-
45-260%
0.6
Received September 2, 1986; accepted March 10, 1987. Address correspondence and reprint requests to Dr. T. Iga at Faculty of P harmaceutical Sciences, University of Tokyo, Hongo, Bunkyo-ku, Tokyo 113, Japan. Abbreviations used: BBB, blood-brain barrier; lAP, iodoanti pyrine; LCBF, local cerebral blood flow.
433
434
Y.
SAWADA ET AL.
stirred compartment and that distribution equilib rium is achieved so rapidly that the tracer in the emergent venous blood is in equilibrium with that in the brain. Assuming passive diffusion, it then follows that the concentrations of unbound drug in the venous blood and that in the brain are equal. The tissue equilibration technique has been used by many investigators since its introduction by Kety. On the other hand, Morales and Smith (1948), Schmidt (1953), and Johanson and Wilson (1966) have shown that a complete description of ex change between capillary blood and tissue requires the use of a partial differential equation. Estimation of L CBF based on this analytical method and a comparison between the venous equilibrium model and the capillary perfusion model (tube model) have not yet been done. Recently, Hertz and Paulson (1980) showed that the extraction ratio of glucose increased with time in the BBB perme ability studies using the multiple indicator dilution method. This increasing extraction can be due to the heterogeneity of the cerebral circulation; the higher extraction corresponds to the longer contact with BBB and indicates a longer transit time. How ever, there is no report concerning the estimation of L CBF based on the distributed model by intro ducing the statistical distribution of the transit time among microcapillaries in the brain. In this article, we present the tube model and the distributed model based on the concept of trans verse heterogeneity in the capillary transit time. The purpose of the present article is to derive the equations for these models and the venous equilib rium model (well-stirred model) that relate the de terminations of L CBF and to elucidate the similar ities and differences in the behavior of these models.
LCBF can be determined in experimental animals by employing the chemically inert diffusible tracer and ap plying the mathematical principles of Kety to analyze the blood-brain exchange (Kety, Freygang and Sokoloff, Sakurada et aI. , Sakurada et al. showed that lAP, which has a higher oil/water par tition coefficient than antipyrine and also is more diffu sible in the cerebrovasculature, provides values of LCBF with the Kety method in the conscious rat that are com parable to those obtained using 1 3I1-trifluoriodmethane. The experimental design in rats based on the lAP method by Ohno et al. was as follows. The femoral vein was infused for s with isotonic saline containing J.LCi/ ml of lAP. Periodically during the infusion, samples of arterial blood were collected through a femoral arterial cannula. Animals were decapitated s after infusion started. The skull was opened, and the brain was re moved. The blood concentration dpm/ml) time
1951,1978). 1960;
45(I979)
9
45 (Cblood,
J Cereb Blood Flow
Metab, Vol, 7, No. 4, 1987
(1) Cblood = Be-Rt De-St where B, D0, (timeandwhen S are the constants and A B D 0 at t the blood concentration starts to rise). Ohno et al. (1979) obtained the following result. Cblood = 13,630 14,925e-o.227t-28,555e-O.121t (2) where 0 t 37 s. The apparent tissue-to-blood concentration ratio Aapp is defined by the following equation. (3) where CTiss and Cblood are the brain parenchyma (ex cluding intravascular concentration) and the arterial +
A +
=
A,
R,
=
+
+
+
�
�
blood concentration of tracer at time (n after decapita tion of animals. We used the following assumptions, which are common to the models: there is no drug metabolism and excretion in the brain; the binding in blood and the brain parenchyma is linear; and there is no diffusional barrier between the tracer in blood and that in the brain parenchyma.
(1) (2)
(3)
Venous equilibrium model (model I) Assumptions of model I, illustrated in Fig. lA are: each brain region is a single homogeneous compartment, and the tracer in the blood leaving the brain is in equi librium with that in the brain parenchyma. The change of the tracer amount in each region is given by the following mass-balance equation.
(1)
(2)
dCT�;' s(t) [Cblood CTiss(t)/A] (4) where and A are the weight (g) in each brain region, the blood flow (mUmin) in each brain region, and the W
=
F·
-
W, F,
A.
Model I
B.
Model II and III Blood
" I �
THEORETICAL
(1978) 1958;
curve and the local brain concentration of tracer were determined. The arterial blood concentration that was measured during i. v. infusion of lAP was plotted against time and was fitted by nonlinear least squares method, according to the following equation.
I I
•
E;�h
+
�;�
Is
ECC +Cell\
I
�
I
FIG. 1. Compartment model for local brain region. (A) The venous equilibrium model (model I). The tracer concentra tion in the brain and capillary blood is homogeneous and is also equal to that in the emergent venous blood. (8) The tube model and distributed model (models II and III). There is no mixing in the brain and capillary blood along the tube in the direction of blood flow, and tracer concentration is de creasing along the tube, as shown in the profile. Model III considers the transverse heterogeneity in the capillary blood flow (f1' f2' ... , fn)' ECF and F are the extracellular fluid compartment and total brain blood flow, respectively.
DETERMINATION OF CEREBRAL BLOOD FLOW
435
tissue-to-blood concentration ratio of tracer at steady state, respectively. The solution of this equation is:
Based on these assumptions, the mass-balance equa tion of the tracer amount in a cylinder is given by
CTiss(1) = 'A .K r {Cblood. exp[-K(T - t)]}dt (5) where CTiss(T) equals the tissue concentration of the tracer at a given time, T, after the introduction of the tracer into the circulation. The constant K is defined as follows (Kety, 1951). F m K (6) W In Eq. 6, FIW is the rate of blood flow per unit mass of tissue, and m is a constant between 0 and I that repre sents the extent to which diffusional equilibrium between
Wtube. OCTiotss(X,t) = -itube' VuM . OCblooxod(X,t) (9) where CTiss(X,t) and Cblood(X,t) are the tracer concentra tions in the cylinder and the tracer concentration in the blood at point x (the cumulative capillary volume) and time t, respectively. As 'A is the tissue-to-blood concen tration ratio at steady state, we can obtain the following
= -'-
'A
blood and tissue is achieved by the marker material during its passage from the arterial to the venous end of the capillary. Based on the work of Sakurada et al. and 'A was taken to was taken to be equal to be equal to for lAP in the normal brain. Substituting Eq. into Eq. yields,
(1978), m 0.8 I 5
1.0,
=
A . 'A +
A
-- +
K
+
B·'A·K
.
exp(-R T) K- R D·'A·K exp(-S' T) K- S B +
D
---
K- R
--
K- S
. 'A
.
(7)
.K .exp(-K T)
7 into Eq. 3 yields, D·K 'Aapp ( B'K -= A exp(-R . T) exp(-S. T) K- S 'A K- R A B D .K .exp(-K . T) ) K- S K- R K D . exp(-S . T)] (8) [A B .exp(-R . T) Substituting Eq. +
+
--
- - +
--
+
--
--
+
+
-I
where 'Aapp is the apparent tissue-to-blood concentration ratio at time T.
Tube model (model II) Assumptions of model II, illustrated in Fig. IB are: the local brain region is composed of a large number of identical tubes, arranged in parallel; blood flows uni directionally along the tubes; blood flows are the same among capillaries = II = fz . . = In); there is no gradient in the tracer concentration from the center to surface within the cylindrical tubes; and at any point along the cylinders, binding equilibrium exists between the tracer at the binding site and that in the cylindrical tube. The release of the tracer from a brain region that might be caused by the blood flow occurs gradually along the cylinder in this nonhomogeneous model. In order to rep resent the mass balance in this model, we should con sider one segment of cylinder of the brain region. Then, if the brain region is composed of such cylinders, which the blood flow rate in have the capillary volume of the cylinder, and the brain parenchyma volume per cylinder, are defined as FIM and W IM, respec tively.
(1)
(ftube
Wtubite,ube,
(2)
(3)
.
M VuIM,
relation.
(10) CTis.(X,t) 'A. Cblood(X,t) Substituting Eq. 10 into Eq. 9: OCblood(X,t) -itube' Vu . OCblood(X,t) 'A .Wtube . ot M ox (11) and OCblood(X,t) = -K'OCblood(X,t) ---(12) ot oX where K = j;ube/'AIWtube = FI'AIW and X (dimensionless capillary volume) = xl(Vulm). The boundary condition is (13) CbloOd(O,t) = Cblood(t) The solution for Cblood(X,t) is as follows. Cblood(X,t) = CbIOOd( 0, t -�) . u (t -�) (14) where u(y) is a step function, i.e., u(y) = 1 at 0, u(y) = 0 at 0 . The averaged brain and u(y) = 0.5 at , O tissue concentration [C\iss(t)] from inlet to outlet is con structed by integration from 0 to 1 as follows. CTiss(t) = f CTiss(X,t)dx = f 'A'Cblood(X,t)dx f 'A'{A B .exp[-R. (t XIK)] D .exp[-S'(t-XIK)]} (15) ·u(t-XIK)dX At time T of the brain sampling, if T IIK, C\iss(T) and 'Aapp/'A are expressed by the following equations, respec =
----
y
=
+
-
+
�
tively.
f {A 'A'
(4) (5)
+
B·exp[-R .(T - XIK)]
XIK)]}dX . T) I]}
+ D'exp[-S'(T 'A' + B .(KIR) .exp(-R + D . (KIS) ·[exp(RIK) ·exp(-S [exp(SIK)
{A
.T).1]
-
(16)
and B·(KIR) .exp(-R . T) D .(KIS) . [exp(RIK) - 1] [exp(-S'T). exp(SIK) - 1] 'Aapp A DB. .exp( exp(-R . T) 'A -S . T) (17) If 11K, then ('Tiss(T) and 'AapJ'A are expressed by the following equations, respectively. A +
+
+ +
T �
J
Cereb Blood Flow Melab, Vol. 7, No.4, 1987
Y.
436
(K
SAWADA ET AL.
A'{A+B'exp[-R .(T - XIK)]
.
X dX
+D'exp[-S (T - IK)]} A'{A'T'K+B'(KIR) .[1 - exp(-R .1)] +D·(KIS) .[1 - exp(-S'1)]}
1).
(18)
and Aapp
=
[A'T·K+B·(KIR)' [1 - exp(-R .1)] + D .(KIS) .[1 - exp(-S .1)] [A + B·exp(-R·1)+D·exp(-S'1)]-1
The distribution scheme is independent of the total number of capillaries. Model II is a special case of model III (n = The mass-balance equation of the tracer amount in an i-th cylinder is given by
(19)
Vu . OCblood,i(x,t) . fi+ W 3CTiss,i(x,t) = 0 (25) M' ot ox M and (26) CTis"i(x,t) A. Cblood,i(x,t) (27) K H(A . W IM) and X = xl(VulM) where Vu is the total volume of M capillaries, x is the cumulative volume of each capillary, X is the dimension =
=
Distributed model (model III) Assumption of model III, illustrated in Fig. IB, is as follows. The brain capillary array comprises many iden tical units whose brain cells all operate with the same tissue binding of tracers, but the fraction of cerebral blood flow assigned to individual capillaries has been given a normal distribution to late simulate the shape of indicator dilution curves (Forker and Luxon, As a distribution of capillary flows, we used an arbitrary prob ability function. Consider N identical capillaries with a mean flow per capillary of] and a range of flows between o and Grouping the capillaries into equally spaced flow classes (n), we define
1978).
less cumulative volume of each capillary and is the tissue-to-blood concentration ratio. Substituting Eqs. and into Eq. yields
26 27 25 OCTiss,i(X,t) = K . OCblood,i(X,t) (28) ot 3X The averaged brain tissue concentration (CTiss,i(x,t)) from inlet to outlet at single capillary integration from 0 toI as _
follows.
2J
fi
=
class midpoint class width =
2=.fin(2 . i - 1). fln
(20)
(Xi
CTis"i(t) = f CTiss,i(X,t)dX
The averaged concentration in a local brain region is ex pressed by
The fraction of capillaries in the i-th class is calculated from the normal density function as approximately the prediction of midpoint frequency and class width. Thus,
I (2 . i - 1).] - fJ 1(J } 2 .] (21) n . ai = (J (27T)Vl n where (J is the standard deviation. To ensure that the ap proximate integration represented by Eq. 21 is consistent with the requirement that 2:�(Xi 1, each approximation of (Xi is divided by the sum of the approximations. With exp{-
_
2" [
2
2
n
CTiss(t) = �1 (Xi . CTiss,i(t) The boundary condition is
Cblood(O,t)
=
+D .exp(-S
(Xi =
[ -1 (2 i 2 [ 1 (2. i
. -I
n
� 1
-
n'c
exp - -
2
n
2
)
]
-I - n ) n
.
C
2
]
T �
M·fi ) exp(-R'1) CTiss(1) i1 (Xi ' {A+(B' R· W'A R A 1] . [exp ( ��� ) D'M . fi ) (32) exp( _ S .1) +(
20 0.1, 0.3, 0.5,200.7, 0.9,
(Xi. M. fi (Xi. M. (2 . i-I). ]In (Xi . (2. i - 1) F n f'M (23) where M . fi = M·(2 . i - 1). (fin) (2 .i -I) .(FIn)(24)
J
Cereb Blood Flow Metab, Vol. 7, No.4, 1987
=
(22)
After dividing this range into equally spaced flow classes, the fraction of capillaries with a particular flow was calculated from the normal density function using a coefficient of variation equal to and respectively. In our experience, taking n = will define the distribution of blood flows with sufficient precision to estimate LCBF. The fraction of total flow associated with the i-th class of capillaries is
. t) (31)
At time T of brain sampling, if W ' A/(M . fi), then and Aapp/A are expressed by the following equa Ctions, Tiss(1)respectively.
this refinement and on defining the coefficient of varia tion as C = (JI], we have exp -
(30)
. t)
A+B .exp(-R
•
=
(29)
S· W'A R . [exp (
���
and B . (2 A+ ( . [exp D +(
A pp :1\
=
A )
I]}
. i - 1). K) exp(-R .1) R·n
C2. in� �) .)
. (2. i - 1). K S'n
K
) -1 ]
(33)
exp(-S'1)
) 1] (2 . i - l)K ' (Xi it A+B .exp(-R .1)+D .exp(-S .1) . [exp (
n .S
_
437
DETERMINATION OF CEREBRAL BLOOD FLOW
.
. J;),
If T � W A/(M then CTiss(n and Aapp/A are ex pressed by the following equations, respectively. M . J; ) CTissm = 2:I (Xi' {A·T (-W'A n
-
+
}
. [1 - exp(-R . n] . [1 - exp(-S' n]
and
�
A p
=
(
+
B .M
. J;)
R· W'A M·J;I
(D, ) S· W' A (34)
[(2 . i � 1) . K ] i - 1). K] [1 - exp(-R· n] [B. (2.R'n [D' (2 �i.� 1). K] [1 exp(-S'n]) D·exp(-S' n] - l B·exp(- R' n [A (35)
� (Xi (
. A . T'
A case of linear increasing in arterial blood concentration of tracer Simulation experiments for the estimation of the blood flow in the case of linear increase of the arterial blood concentration of tracer were carried out (Blasberg et aI. ,
1984).
a
_
+
+
Indicator dilution method In model III, the coefficient of variation (CV) in the normal density function (Eq. has to be determined. Lucignani et aL showed the concentration of vas cular reference substance in cerebral venous blood as a function of time normalized to the relative concentrations of the tracers injected into the carotid artery. We carried out the pharmacokinetic analysis for the indicator dilu tion curves and determined the CV for Eq. Mass-balance equation and solution to the distributed model are as follows.
22)
(1985)
22.
(:;). (8Cb10;;i(X,t») (:;). (8Cb10;:i(X,t»)'J; = 0 (36) ( ) u J; / . Cblood,i(x,t) = Cblood(O,t-x/J;) t - x (37) The boundary condition is Cblood(O,t) = ( Do ) exp(-F· tIVin) (38) where Do, Vinj, and F are the dose of the tracer, the injec tion volume (ml/g brain), and the total cerebral blood flow (mllmin/g brain), respectively. The time course of the output concentration (Cout) frac tion of the vascular reference substance can be expressed +
VmJ
as follows.
Coutm = ± { (M. J;. (Xi). (_1 ) V'" Do l ' F F ( . exp l _ t _ Mv�J;.)] (39) . u(t - �)} M 'J; where M . J;lF = (2 . i - l)/n and Cou/D o is the outflow fraction (ml-I). The outflow fraction (Cou/D o) was calcu lated from Eq. 39 using Villi' Vu, and F equal to 0.038 ml/g brain (Lucignani et aI., 1985), 0.057 ml/g brain (Cremer and Seville, 1983), and 0.8-2.5 ml/min/g brain, respec VmJ
tively.
(40)
where is a slope of the blood concentration time course. The tissue concentration of tracer (CTis,) and Aapp/A values at time T in three models can be expressed as follows: In model I:
+ +
Cblood = a . t
and
CTiss = a . A. T
+
(a� ) A
K. n - 1] (41)
[exp(-
1 _) [exp(-K. n - 1] (42) 1 (_ K'T In model II: if T 11K, (43) CTiss = a . A . T - (� 2·K) +
�
and
(44) If T
�
11K, CTiss =
and
(45)
( 1)
Aa
pp � = 2'K'T
=
In Model III: if T � W' A/(M . J;)
and
(46)
i
n/(2 . - 1 )/K,
CTiss = � [(Xi ' ( a . A .T _ a / ��� A)] (47)
] (48) ) 2 . T· (2 . i 1). K I If T = W' A/(M. J;) = n/(2'i - 1 )/K, (49) CTiss = � (Xi. ( a . A2 . PW'. MA . J;) and 1 (2. i-I) . K (50) Aapp = 2:(X.' (-·T·----) A 2 Aapp
�
=
± [(Xi ' (1 -
_
n
n
•
n
I
I
n
RESULTS Effect of coefficient of variation (CV) on estimation of K value in the distributed model (model III) Figure 2 illustrates the relationship between the CV values and a fraction of the m capillaries to each flow class (Ui) calculated from Eq. 22. When J Cereb Blood Flow Metab,
Vol. 7,
No.4,
1987
Y.
438
SAWADA ET AL.
'" o
C
2;
0.3
0.3
.....
V> V>
'r-
m S-
...0
"'
0.2
0.2
,--C.V.= 0.1
ro
I
01 ...... C
'r-
E
C.V.=- 0.3
C
I
...... 0.1
0.1
E :::L
0-
"'
U
..... o
c:: o
10
:;:;
12
14
U
"' \.. u..
Flow class number
16
18
20
(i)
FIG. 2. Transverse distribution of capillary flows. Fraction of the capillary in the ith flow class. The CV is the coefficient of variation in normal probable density function.
the CV value is large, the pattern of (Xi is gently sloping. However, the concentration of the density at the midarea of the flow class number is caused by small CV value. Figure 3 illustrates the relation ship between the Aap/A. and the K value, with CV value ranging from 0.3 to 0.9 calculated from Eqs. 33 and 35. The K values in the high CV value ranges were larger than those in the low CV value ranges. The simulation curves based on the distrib uted model lay midway between those of the venous equilibrium model (model I, Eq. 8) and the tube model (model II, Eqs. 17 and 19), with Aapp/A ranging from 0.4 to 0.9. No remarkable difference was observed between the distributed model (model III) and the tube model (model Il), with
r-;
18
a) 0.80 ml/min/g brain
18
FIG. 3. Relationship between the Aapp/A and K value in three models. The CV and K are the coefficient of variation in normal probable density function and FI(W' A), respectively. The number next to each curve is the CV value, which varied from 0.3 to 0.9.
Aap/A ranging from 0 to 0.4, whereas the simula tion curve of model I was larger than those of model II and model III. Relationship between A.appA. and the K values at various infusion times of tracer and comparison among three models Figure 4 shows effects of the cerebral blood flow and the CV value in normal probability density function on indicator dilution curves of vascular reference substance based on the intracarotid injec tion method of the blood containing the tracer. As the whole brain blood flow (F, ml/min/g brain), which is required for calculation of indicator dilu-
b) 1.50 m1/min/g brain
18
E
16
C
14
14
14
r-
12
12
12
U
m S4-
10
10
10
3: o
,
.
o
+>
4+> ::l o
c) 2.50 m1/min/g brain
C.V.·O.IO C.V.=O.30 C.V.-O.50 C.V.�O.70 C.V.-O.90
8
2 10
12
14
}6
10
12
14
16
10
12
14
16
Time, sec FIG. 4. Simulated venous outflow curves for a nonremovable reference solute such as [57Co]DTPA. Distribution of capillary
transit times represented by the curves correspond to Gaussian distribution of the capillary flow rates. The number next to the lines is the CV value, which varied from 0.1 to 0.9. Total brain blood flows are 0.8, 1.5, and 2.5 ml/min/g brain in A, e, and C, respectively.
J
Cereb Blood Flow Metab, Vol. 7, No.4, 1987
DETERMINATION OF CEREBRAL BLOOD FLOW tion profile in vivo, is not reported by Lucignani et al. (1985), it is difficult to determine the reasonable CV value from comparison between the observed indicator dilution profile by Lucignani et al. (1985) and the calculated dilution profiles in Fig. 4 of this study. Accordingly, we accepted 0.3 as the CV value, jUdging from the dilution curves in the average value of LCBF (1.0-1.5 mllmin/g brain). Figure 5 illustrates the relationships between the Aapp/A and the K values at various infusion times of the tracer calculated by using Eqs. 8, 17, 19, 33, and 35, based on models, I, II, and III (CV 0.3) in the Theoretical section. The K values in model I are larger than those in model II at each infusion time. The simulation curves based on the distrib uted model (model III) lay midway between those of the venous equilibrium model (model I) and tube model (model II) at each infusion time. Figure 6 shows the intermodel differences in the K values calculated, based on the experimental design by Ohno et al. (1979) (infusion time 0.617 min). A remarkable difference was observed in a larger Aapp/A value between model I and model II (KmodelIlKmodelII is 2-3 times larger than 1.0 at Aap iA > 0.7), whereas a small difference in the simulation curve was observed between model II and model III ( CV 0.3).
�
KModel I / KModel I I
2
\
' o
'\ KMode 1 I I I / KMode 1 I I
=
=
Comparison among three models in the case of linear increasing in arterial blood concentration of tracer As shown in Fig. 7, the simulation curve in model I (Eq. 42) was larger than that in model II (Eqs. 44 and 46) at both [decapitation times T (1.0 and 0.5 min).] Furthermore, the simulation curve based on model III (Eqs. 48 and 50) lay midway between those of model I and model II. A. Gray matter, parietal 1.6 w � 0
:;: � � � � �
� 0
::: w � 0
:;:
� � � � �
::'-
B. Olfactory bulb
439
0.4
0.2
0.6
0.8
1.0
Aapp / A FIG. 6. The comparison of the K values predicted by three models.
Practical examples of estimated LCBF based on three models We compared the K values calculated from models I, II, and III using data of lAP by Ohno et al. (1979). The method for calculation based on these models was as follows. The A, A, B, D, Rand S, and FIW (=K . A) values calculated with Eq. 7 and based on model I were conversely substituted into Eq. 8 and the observed values by Ohno et al. (1979), and the AappA values were recalculated and presumed. Then, substituting the estimated AappA into Eqs. 17, 19, 33, and 35 based on models II and III, we can obtain the K (FIW A) values in models II and III. As the K value is a part of multiexponen tial terms in Eqs. 17 and 33, the calculations by these equations were carried out by the Newton Raphson method. As shown in Table 1, the values calculated based on model I were 45% (hippo campus) to 95% (gray matter, parietal) larger than those in model II. Furthermore, the K values in model III were 0-10% larger than those in mod el II. .
1.4
A.
T
=
B.
1. 00 mi n
T
=
0.500 min
1.2 1.0 0.8 0.6
E "'
E
0.4 0.2
Aapp / A
Time, min FIG. 5. Relationship between the }.app/}. and K value in three models based on constant infusion experiments. T is the constant infusion time of tracer. In model III, 0.300 is used as the CV value.
FIG. 7. Relationship between }.app/}. and K value in three models in the case of linear increasing in the arterial blood concentration of tracer. T is the venous infusion time of tracer. In model III, 0.300 is used as the CV value. J
Cereb Blood Flow Metab. Vol. 7, No.4, /987
Y.
440
SAWADA ET AL.
TABLE 1. Local cerebral blood flow (LCBF) calculated with 14C-iodoantipyrine method
based on various models
"ap/"a
Kmodell'c
KmodelIlb,d
KmodelIll,e
Olfactory bulb Caudate nucleus Hippocampus Frontal lobe Occipital lobe Thalamus + hypothalamus Superior colliculus Inferior colliculus Cerebellum P ons
0.407 0.594 0.504 0.570 0.594 0.532 0.570 0.637 0.407 0.474
Medulla Gray matter, parietal White matter, corpus callosum P ituitary gland
0.458 0.731 0.442 0.413
1.28 2.25 1.73 2.10 2.25 1.88 2.10 2.55 1.28 1.58 1.50 3.38 1.43 \.30
0,966 1.41 1.19 1.35 1.41 1.26 1.35 1.51 0.966 1.12 1.09
0.966 1.43 1.20 1.37 1.43 1.27 1.36 1.55 0.968 1.13 1.09 1.89 1.05 0.980
1.73 1.05 0.980
a "app
and" are the apparent tissue-to-blood concentration ratio and the tissue-to-blood concentra tion ratio at steady state. b K = F/(W' ,,) (ml/min/g brain), where F (mllmin) and W (g) are the local cerebral blood flow and the tissue weight, respectively. c Calculated by Eq. 8. d Calculated by Eqs. 17 and 19, e Calculated by Eqs, 33 and 35.
DISCUSSION Physiologically, model II and model III appear to be more appropriate than model I to estimate L CBF. One can easily conceive of a declining tracer concentration along the length of the capil laries with distribution of tracer by brain paren chyma lining the capillaries. This plug flow situa tion is analogous to a steady-state plug flow reactor (Levenspiel, 1972) in which the material balance for a reaction component must be made for a differen tial element of volume or flow. This flow within the reactor is orderly, however, with no mixing occur ring between any element ahead of and after it. There may be lateral mixing, but no mixing or no diffusion along the paths. This pattern of circula tion may be applied to most organs within the body. Furthermore, the potential influences of the trans verse heterogeneity of the capillary transit time on elimination from the liver (Gore sky et aI., 1973; Wolkoff et aI., 1979; Sawada et aI., 1985a), lung (Goresky et aI., 1969), heart (Rose and Goresky, (1976», and kidney (Trainor and Silverman, 1982; Itoh et aI., 1985; Sawada et aI., 1985b) were evalu ated by using the distributed models. For example, Goresky et al. (1973) carried out multiple indicator dilution studies on the hepatic circulation using la beled red blood cells, albumin, and other sub stances in the dog. They concluded that the outflow patterns of these substances could best be ex plained in terms of a distribution of the transit time. Studies in humans of BBB permeability using the indicator dilution method by Hertz and Paulson (1980) revealed that the extraction of the test subJ
Cereb Blood Flow Melab, Vol. 7, No.4, 1987
stance increased during the upslope of the venous (outflow) dilution curve. The increasing extraction can be ascribed to the heterogeneity of the cerebral circulation; the higher extraction corresponds to the longer contact with BBB and indicates a longer transit time. This finding seems to indicate the het erogeneity of the microcirculation in the human brain.
Experimental design for discrimination among the models by varying intravenous infusion time A direct demonstration is not yet carried out for discrimination among the venous equilibrium model (model I), the tube model (model Il), and the distributed model (model III). Theoretical analysis of these three models revealed that one powerful discriminator among them is the effect of changes in the intravenous infusion time on the estimated
Aap/A.
The following design may be adopted for the dis criminatory studies. At a control infusion time (e.g., T 0.617 min), the Aap/A. values were cal culated to be 0.73 I for gray matter parietal and 0.407 for olfactory bulb (Table 1) ( Ohno et aI., 1979). By appropriate substitution of the Aapp/A and the infusion time (T) into Eq. 8 for model I, Eqs. 17 and 19 for model II, and Eqs. 33 and 35 for model III, the K values were calculated. Assuming that the value of the K value is constant, the Aap/A of tracer was predicted for each new infusion time by using Eqs. 8, 17, 19, 33, and 35. The predicted values calculated according to models I, II, and III need to be compared with the observed data. As we could not obtain the observed data at each infusion =
DETERMINATION OF CEREBRAL BLOOD FLOW
��- �----------
time, we showed the simulation lines calculated by using Ohno's data as reference point (T 0.617 min) (Fig. 8). Figure 8 illustrates the relationship between the (Aapp/A)model n/(Aap/A)model I or (Aapp/ A)modelII/(AapplA)model I and the infusion time of tracer at gray matter parietal and olfactory bulb calculated using Eqs. 8, 17, 19, 33, and 35, respec tively as shown in Fig. 9. The (Aapp/A)model II orIII/ (Aap/A)modelI is smaller than 1 at T < 0.617 min, whereas it is larger than 1 at T > 0.617 min. As we could not obtain the Aap/A values at each infusion time, the discrimination among these models has not been done. In the future, there may be a need to determine which model should be used to calculate the L CBF based on this discriminator. Analysis for the estimation of L CBF in this study is based on the tissue concentration at a single data point and is limited by actual interstudy variation of K in the same brain region under identical conditions. Re cent developments of positron emission tomog raphy imaging permit the measurement of L CBF based on multitissue concentration. This method will make it possible to analyze the multiple data points with precision and to discriminate these models.
Eq. 8
=
( Aapp!A)Model 1= funcl (T,K) Eqs. 17 and 18 (\app!A)Model
T
=
0.167
min
B.
T
=
O.
T
=
0.617 min
/
c
'"
I.0
Z c c
�"
L-�__����� C.
T
=
1.00 mi n
2.00
min
funcll( T,K )
=
Eqs. 33 and 35 ( 'apph )Mode 1 III
=
funclll (T,K )
_.1
A reference point ( Table I )
T
=
l
0.617 min-
Aapp!\
=
0.731 (Gray matter, parietal )
0.407 (Olifactory bulb )
-1
Calculation of K values (K Model I' KModel II' �ode 1 III) by each equation (Aapph )Model I
=
func (T,K I Model I)
(' app!')Model II = funcII(T,KModel II)
(A app/A )Model III = funcIII(T ,KModel III)
.L
Simulation of relationship between ( i and T Aapp A ) Fig. 8
.L Experiments in different infusion time ( T' )
Perspective Indeed, the brain vasculature is a highly ramified network with cross-anastomosed and throughfare channels, like liver, lung, and kidney ( Chambers and Zweifach, 1944). Accordingly, in this study, the distributed model may be insufficient for a descrip tion of brain microcirculation. Metzger (1969) has considered the oxygen diffusion from two- and three-dimensional capillary structures in the form A.
441
( Aapph)Model I If (Aa p!')observe p
=
func ( , l T' �odel I)
i(Aapp!A)Model
I
=
1. 0,
Mode 1 l is correct If ( 'a p!A )observe p
i( 'appfA)Model
I
" 1.0
Mode 1 II, Mode 1 III or other models may be correct
FIG. 9. Procedure to discriminate among the models.
of square or cubic lattices, the latter intended to simulate capillary structure in the brain. An impor tant feature of this proposed model is the possibility of studying the impact of nonhomogeneous capil lary perfusion on the distribution of oxygen in the tissue. Recently, Metzger (1973, 1976) applied the two-dimensional square lattice model to study on the effect of spatially nonhomogeneous oxygen consumption on oxygen distribution in the tissue. Estimation of cerebral blood flow based on this proposed space-distributed model (the ramified net work model) is in progress.
APPENDIX ).app
! ).
FIG. 8. The comparison of Mpp/h. as predicted by models I, II, and III with perturbation of venous infusion time T. The
simUlation curves were calculated by using Ohno's data based on 14C-iodoantipyrine (lAP) method.
t:
T: x:
X:
time, seconds or minutes time after decapitation (min) cumulative capillary volume (mllg brain) dimensionless capillary volume (0 1) �
J
Cereb Blood Flow Metab. Vol. 7, No.4, 1987
Y.
442
F:
f:
A:
Cblood: fTiss: CTiss: Cout: Do: m:
W: Wtube: Vu : K: Vinj:
a:
CV:
(Xi:
u (y):
n:
M: A, B, D: R, S: a:
SAWADA ET AL.
averaged blood flow of whole brain (mIl min/g brain) capillary blood flow in the ith class (mll min) mean capillary blood flow (mllmin) tissue-to-blood concentration ratio of tracer apparent tissue-to-blood concentration ratio of tracer blood concentration (dpm/ml) tissue concentration (dpm/mt) averaged tissue concentration (dpm/mO tracer concentration in outlet (dpm/mO dose of tracer (dpm) extent to which diffusional equilibrium IS established between tissue and blood brain weight (g) brain parenchyma volume per cylinder capillary volume (mllg brain) m F/(W' A) injection volume (mllg brain) standard deviation coefficient of variation fraction of capillary in the ith class step function at the argument y equally spaced flow class number total capillary number constant (dpm/mO constant (s -I) constant (dpm/ml/min) .
blood-brain barrier permeability. J Clin Invest 65:1,1451,151
Itoh N, Sawada Y, Sugiyama Y, Iga T, Hanano M (1985) P erme ability of materials in postglomerular capillary bed and dis
tribution to interstitium of kidney in rats. Jpn J Physiol 35:291-299 Johanson JA, Wilson TA (1966) A model for capillary exchange. Am J Physiol 210:1 ,299-1,303 Kety SS (1951) T he theory and applications of the exchange of inert gas at the lungs and tissues. Pharmacol Rev 3:1-41 Kety SS (1960) Measurement of local blood flow by the ex change of an inert, diffusable substance. Methods Med Res 8:228-236 Lacombe P, Meric P, Seylaz J (1980) Validity of cerebral blood flow measurements obtained with quantitative tracer tech niques. Brain Res Rev 2:105-168 Landau W M, Freygang WH Jr, Roland LP, Sokoloff L, Kety SS (1955) T he local circulation of the living brain: values in the unanesthetized and anesthetized cat. Trans Am Neurol Assoc 80:125-129 Levenspiel 0 (1972) Chemical reaction engineering, ed 2. New York, Wiley. Lucignani G, Nehlig A, B1asberg R, Patlak CS, Anderson L, Fieschi C, Fazio F, Sokoloff L (1985) Metabolic and kinetic considerations in the use of 1251 HIP DM for quantitative measurement of regional cerebral blood flow. J Cereb Blood Flow Metabol 5:86-96 Metzger H (1969) Distribution of oxygen partial pressure in a two-dimensional tissue supplied by capillary meshes and concurrent and countercurrent system. Math Biosci 5:143154 Metzger H (1973) Geometric considerations in modeling oxygen transport processes in tissue. Adv Exp Med Bioi 37b:761772 Metzger H (1976) T he influence of space-distributed parameters on the calculation of substrate and gas exchange in micro vascular units. Math Biosci 30:31-45 Morales MF, Smith RE (1948) On the theory of blood tissue ex change of inert gases. VI. Validity of approximate uptake expressions. Bull Math Biophys 10:191-200 Ohno K, P ettigrew KD, Rapoport SI (1979) Local cerebral blood flow in the conscious rat as measured with 14C-antipyrine,
REFERENCES B1ashberg R, Molnar P, Groothuis D, P atlak C, Owens E, Fen � .ermacher J (1984) Concurrent measurements of blood flow and transcapillary transport in Avian sarcoma virus induced experimental brain tumors: implications for chemotherapy. J Pharmacol Exp Therap 231:724-735 Chambers R, Zweifach BW (1944) Topography and function of the mesenteric capillary circulation. Am J Anat 75:173-205 Cremer 1£, Seville MP (1983) Regional brain blood flow, blood volume, and haematocrit values in the adult rat. J Cereb Blood Flow Metabol 3:254-256 Forker EL, Luxon B (1978) Hepatic transport kinetics and plasma disappearance curves: distributed modeling versus conventional approach. Am J PhysioI235:E648-E660 Freygang WH Jr, Sokoloff L (1958) Quantitative measurement of regional circulation in the central nervous system by the use of radioactive inert gas. Adv Bioi Med Phys 6:263-279 Goresky CA, Cronin RFP, Wangel BE (1969) Indicator dilution measurements of extravascular water in the lungs. J Clin Invest 48:487-501 Goresky CA, Bach GG, Nadeau BE (1973) On the uptake of materials by the intact liver. T he transport and net removal of galactose. J Clin Invest 52:991-998 Hertz MM, P aulson OB (1980) Heterogeneity of cerebral capil lary flow in man and its consequences for estimation of
J Cereb Blood Flow Metab.
Vol. 7. No.4. 1987
14C-iodoantipyrine and 3 H-nicotine. Stroke 10:62-67. Raichle ME, Martin WRW, Herscovitch P, Mintum MA, Markham J (1983) Brain blood flow measured with intrave nous H2150 II. Implementation and validation. J Nucl Med 24:790-798 Rose CP, Goresky CA (1976) Vasomotor control of capillary transit time heterogeneity in the canine coronary circula tion. Circ Res 39:541-554 Sakurada 0, Kennedy C, Jehle C, Brawn JD, Carbin GL, Soko loff L (1978) Measurement of local cerebral blood flow with iodo 14C antipyrine. Am J Physiol 234:H59-H66 Sawada Y, Sugiyama Y, Miyamoto Y, Iga T, Hanano M (1985a) Hepatic drug clearance model: comparison among the dis tributed, parallel-tube and well-stirred models. Chem Pharm Bull (Tokyo) 33:319-326 Sawada Y, Itoh N, Sugiyama Y, Iga T, Hanano M (1985b) Anal ysis of multiple indicator dilution curves for estimation of renal tubular transport parameters. Comput Progr Biomed 20:51-61 Schmidt GW (1953) T he time course of capillary exchange. Bull Math Biophys 15:477-488 Trainor C, Silverman M (1982) Transcapillary exchange of mo lecular weight markers in the postglomerular circulation: application of a barrier-limited model. Am J Physiol 242:F436-446 Wolkoff AW, Goresky CA, Sellin J, Gatmaitan Z, Arias 1M (1979) Role of ligand in transfer of bilirubin from plasma into liver. Am J PhysioI236:E638-E648
