AMERICANJOURNAL
Vol.
OF PHYSIOLOGY 1975. Print&
229, No. 4, October
in U.S.A.
Measurements
of dog blood-brain
by ventriculocisternal C. S. PATLAK Theoretical Transport
St&tics Section,
AND
J. D. FENSTERMACHER of Cancer
Branch,
Biometry
Treatment,
N&onal
PATLAK, C. S., AND J. D. FENSTERMACHER. Measurements of dog blood-brain fransfer constunfs by uentriculocisternal perfusion. Am. J. PhysioL 229(4) : 877-884. 1975.-Ventriculocisternal perfusions of mongrel dogs were performed for 1-6 h with solutions containing isotopically labeled compounds. At the conclusion of the perfusion period, serial brain samples were taken from the caudate nucleus and analyzed for radioactivity. Tissue concentration profiles were constructed from the data, and apparent tissue diffusion and capillary exchange coefficients were determined. The tissue diffusion constant of sucrose was 3 X 1tY6 cm2/s, which is approximately 45y0 of its free-water V&E. The permeability of the brain capillary complex to creatinine, sodium, and mannitol was so Xow that it could not be accurately measured by this technique. Capillary transfer coefficients, expressed as half-times, were determined for water, urea, and ethylene glycol; the fl/z values were 1.5, 15, and 17 min, respectively. These numbers were converted to PS products and compared to other published values. This work suggests that the exchange of these compounds between blood and brain is partially (water) or nearly completely (urea and ethylene glycol) limited by membrane permeability.
blood-brain barrier; glycol; sucrose
capillary
permeability;
constants
perfusion
and Mathematzcs Division
transfer
water;
urea;
ethylene
FOUR EXPERIMENTAL APPROACHES-the single injection, the osmotic transient, the tissue uptake, and the tissue clearance -are commonly used to measure the permeability of capillaries to solutes and water (4). All these techniques have been employed to study the permeability of the brain capillary complex, the so-called blood-brain barrier. In the pioneering work with the single-injection or indicatordiffusion method, Crone (2, 3) examined the permeability of the blood-brain barrier to a series of organic nonelectrolytes. Other investigators, e.g., Yudilevich and DeRose (29), have continued the study of the transport of material between blood and brain with this technique. Recently, Oldendorf (19, 20) reversed the rule of the indicator molecule, using a highly diffusible material (tritiated water) in place of a nondiffusible marker (e.g., Evans blue-tagged albumin), in his application of the single-injection approach to the assessment of blood-brain exchange. By measuring the osmotic flows of water generated by various polar compounds infused into the vascular system (the osmotic transient approach), Fenstermacher and Johnson (8) determined the reflection coefficients of five organic nonelectrolytes and the filtration coefficient for the rabbit
Division, Cancer
National
Institute
Imtitute,
Bethesda,
of Mental
Maryland
Health,
and Membrane
ZOO14
brain capillary complex. A third methodology, tissue uptake, has been used extensively to examine the permeability of the blood-brain barrier. Among the early experimentalists who employed this technique are Bering (1) with DZO, Katzman and Leiderman (12) with 42K, and Olsen and Rudolph (2 1) with 24Na and 82Br. Other tissue uptake studies include those of Levin and Patlak (18) with 24Na and Davson and Welch (5) with 24Na, 36C1, and [““S]thiourea. The tissue clearance method has been used by Eichling et al. (6) in a recent study on cerebral blood flow and brain-water exchange. In the present work a new approach, which resembles the tissue clearance method, to measuring the exchangeability of various compounds between blood and brain extracellular fluid (ECF) is reported. The material which is to be studied is perfused through the brain ventricular system and, if appropriately diffusible, passes into the brain parenchyma. As the test molecules or ions move through the brain, some of them are lost from the tissue to the Providing that the perfusion is blood via the capillaries. continued for an appropriate period of time, a tissue steady state- a condition in which the ventricular fluid(CSF + perfusate) to-brain exchange is balanced by the brainto-blood loss-is established for the substance in question The steady-state tissue concentration profile (the concentration at various points in the tissue as a function of the distance from the perfused surface) can be determined by serial sampling and used to calculate the transfer constant of the brain capillary complex to the test material. The experimental and analytical procedures of this technique plus the results from studies with the dog will be presented. METHODS
The ventriculocisternal perfusions and tissue sampling were performed as previously described for dogs (7). Mongrel dogs, weighing from 15 to 25 kg, were anesthetized by 30 mg/kg intravenous pentobarbital and maintained in a state of light anesthesia by subsequent infusion of this drug when necessary. Inflow needles were stereotaxically placed in each lateral ventricle using the following parameters; 10 mm posterior from the bregma, 6 mm lateral to the midline, and 15-19 mm down from the dura. A single outflow needle was positioned in the cisterna magna. When the three needles were properly placed, a constant-speed infusion pump (Harvard Apparatus Co.) was turned OIL
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878
C. S. PATLAK
For the initial 13 min, the pump speed was 0.9 1 ml/min; for the remainder of the experimental period, a rate of about 0.21 ml/min was used. The perfusion fluid was an ionically and osmotically balanced artificial CSF which contained two or three radioactively labeled materials (l-6 &i/ml) pl us a trace of trypan blue dye. One of the labeled compounds was an extracellular marker; the markers used in this study were [‘“Cl- and [3H]sucrose and [14C]EDTA, which has been chelated with Ca++. (In several of the earlier experiments, [p-3H]aminohippurate was employed as an extracellular marker; however, as shown by Fenstermacher et al. (lo), this compound does not distribute solely in the brain extracelMar Auid and was not used further.) At the end of the perfusion time, the animal was killed and the brain was rapidly removed (time 5 1.5 min). The brain was split through the midiine, the lateral ventricles exposed, and the hemispheres chilled to medium hardness in liquid nitrogen. The time that elapsed between the death of the animal and the freezing or near freezing of the brain wz1s about 3 min. Control studies demonstrated that no detectable movement of the labeled materials occurred once the tissue was frozen or semifrozen; therefore, all of the tissue samples were kept frozen by regular dipping in liquid nitrogen during the subsequent handling Procedure. The chilled hemispheres were cut into a series of coronal sections. Well-perfused (as indicated by the bluestaining) flat surfaces of the caudate nucleus were trimmed from the coronal sections. From each experiment, two to four blocks of caudate tissue were obtained at the time of sampling. Subsequently, these blocks were sectioned into series of tissue samples of uniform thickness by a special multibladed knife or a freezing-microtome. From 6 to 10 samples were obtained per series; the thickness of the individual samples throughout a series was either 0.4, 0.5, 0.7, or 1.0 mm. The tissue samples were placed in tared liquid scintillation vials. Samples of the outflow solution were collected regularly throughout the period of perfusion; 50- or 100-~1 samples of these plus the inflow were pipetted into liquid scintillation vials. All tissue and fluid sampled were digested in 0.5 ml NCS (Nuclear-Chicago soIubiIizer, Amersham/Searle Corp.). To each vial was added 18 ml of a toluene-based counting solution. All samples were counted in a three-channel liquid scintillation counter with automatic external standard (Packard Instrument Co.) fur 100,000 counts or 20 min. Standard procedures were used for isotope separation and quench corrections. The individual corrected counts were divided by sample weight (tissues) or volume (fluids), yielding concentrations of radioactivity (counts/min per mg or counts/min per ~1). From these data, inverse complimentary error function (erfc-l) and semilogarithmic plots of the tissue concentration profiles were made and tissue distribution spaces, brain diffusion coefficients, and capillary transfer constants were calculated. (These procedures will be presented in the next section.) Theory and analysis of datd. A particular compound or ion which is perfused through the ventricular system may remain in the ventricular fluid and circulate with it or, if appropriately permeable, enter the surrounding tissue. To
AND
J. Da FENSTERMACHIZR
move into the brain, the material must cross the fluidtissue interface (the ependyma); subsequently the substance may diffuse through the extracellular fluid (ECF), be taken up by c&s, be bound to various tissue elements, cross the capillary wall, or some combination of these steps. Assuming that I) the concentration of th e solute in the Auid perfusing a specific surface of the bra rema ms constan t with time, 2) the portion of the brain being examined is uniform in composition, and 3) the transport geometry of the perfused tissue is effectively rectangular and one dimensional, the equation for the system is
-ac = 1) a”c d(4 ax2 - ax dt -
(V
(k,iC
-
kpoCJ
-
&
($) e
In this and the following equations, C, Cf, and C, are the effective concentrations of the substance of interest in ECF, ventricular fluid, and arterial plasma, respectively; D is the apparent diffusion constant of the solute through the brain ECF; f is time; x is the distance from the ependyma to the tissue site; v is the velocity of the bulk flow of ECF, if any occurs, through the brain; k,i and k,, are the transfer coefficients of the brain capillary complex for the movement of the substance from brain ECF to plasma and plasma to brain ECF, respectively; V, is the magnitude of the extracellular space (ECS); B is the amount of the solute in the brain cells and/or bound in the tissue per unit mass; and h is the permeability constant of the ependymal layer for the material. Providing that the perfusion is continued until a tissue steady state is reached, the time-dependent terms drop out and equation I becomes
0 = D d25 dx2
dbC> -
dx
-
kpi c + k,o G
(2)
With the assumptions that the brain is, in effect, infinitely long and that v is a constant (if v is not a constant, the solution of equation 2 will not be a simple exponential) and the bou ndary condition = h cc
f
- w)3
0
equation 2 can be solved
to yield 2h
v +
1/v2
+
4kpiD
(4)
Kpi
exp
1 20 ( v
-
4~~
+
4kpiD)x
>
Of the three concentration terms which appear on the left-hand side of equution 4, only C, the concentration of the material in the brain ECF, is not measured during the course of the experiment. The serial sampling of the brain at the end of the perfusion period yields the total (extracellular plus intracellular) amount of the material in the if tracer concentrations of labeled tissue (C,) ; however, materials are used and a steady state is reached, the amount
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BLOOD-BRAIN
TRANSFER
of intracellular
or bound
COEFFICIENTS tracer,
879
B, is proportional
to C
(5)
I3 = KC where
K is the proportionality
constant,
CT =v,c+I3 = C(Ve
- (V, + K) gO c, Di Cf
-
kpoc,
Y--Kpi
Cc = (Ca (6)
+- K)
Rearranging equation 6 to solve this value into equation 4 yields CT
and thus
for
C and
substituting
2we + a v
1( *exp20u
+
1/u”
+
4kpiD
~vZ + 4kpi D)x
(7)
>
Experimental evidence which will be presented in this paper suggest that v, the rate of the bulk flow of ECF through the caudate nucleus of the dog, is virtually zero the ECS exclusively or nearly ( i.e., solutes move through exclusively by diffusion) and thus, equation 7 can be simplified to
CT- we+ K) $Pic, cf--
(8)
hWe +K) = h + 4kpiD
exp (-dk,ilDx)
-uAT
- cxp[-g-J)+;c
(11)
in which N is the number of The following equations, capillaries per gram of brain, FiS and ps are the total permeability of the capillaries per gram of brain for the movement of a material from ECF to plasma and plasma to ECF, respectively, and F is the flow of either blood or plasma per gram of tissue, relate capillary permeability and bulk flow velocity for the single-capillary model to a more operational system
F cc = rc, 0 (
pi/N
= PiS
p,lN
= P,S
uAN
= F
P*S - ks c >
Under the assumptions the equation previously (equation I> becomes dC = D -a2c __ ax2 at
W)
inserting equations 12 into equation II for the mean capillary concentration
Y(]
During the time course of the ventriculocisternal perfusions performed in the present studies, the concentration of the tracer in the arterial plasma, C,, was negligible for most substance. Providing that the arterial plasma concentration is effectively zero and that a tissue steady state has been that CT is an exponential reached, equation 8 indicates therefore, a function of kpi, D, and X; in this situation, semilogarithmic plot of C, verus x should give a straight transfer constant, line from which kpi, the ECF-to-plasma can be evaluated. Following the approach and associated assumptions which were initially developed by Kety (14), a more detailed evaluation of kpi can be made. Assuming that the capillary is homogeneous in structure along its length, that the radial and longitudinal diffusion of the solute within the capillary is negligible, and that the concentration of the solute in the ECF around each capillary is uniform, conservation of mass considerations yield the following differential equation for the steady state
(10)
The average concentration along the capillary, c,, may be evaluated by integrating equation IQ from 0 to I, the length of the capillary, and dividing by I
Rearranging and yields an equation
k ca kPi
concentration,
-$+exp(-p$)+p$2
cc = (ca - p)!$(1
= 2h -
of the material at y = 0 equals the arterial C,, equation 9 may be solved to give
_ exp[-y])
+ %C
(13)
used to develop equations 9 and 1.2, written for the experimental system
-8 w> ax -- G (PiSC e
and thus, since equations I and lowing relationship is apparent kpiC-k,,C~=‘i;;C-p$C~
14 are equivalent,
PiS
the fol-
(w
e
e
Substituting equation 13 into equation X5 and rearranging yields two expressions which relate the capillary permeability constants, P,S and P$, to the experimentally determined k’s
c =
PoCc
-
PiC
(16) where u is the velocity of the blood flow in the capillary, A is the cross-sectional area of the capillary; C, is the concentration of the solute in capillary plasma at each point; y is the distance along the capillary; and pi and pO are the permeabilities of the capillary wall per unit length for the movement from ECF into the plasma and from the plasma into the ECF, respectively. Since the concentration
pis
=
kpi -GF(n(l
-“e”E>
If the term (k,, V,/F) is small (values tions 16 may be approximated as
-
.
of 0.3 or less), equa-
Rs = V&p, PiS = V,kpi
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(17)
880
C. S. PATLAK
AND
J. D. FENSTERMACHER
with an error of less than 5 %. Therefore, if the flow of fluid in which the solute is entrained-either plasma or blood-is large relative to the plasma-ECF transfer constant (k,,), the k’s in equations 1-8 are proportional to the brain capillary permeability coefficients. If the transport (= I&) and P,S process is passive, k,, and k,i are identical equals PJ, regardless of the value of (R,,V,/F). In all the experiments, a compound that remains extracellular was included in the perfusate+ Providing that such materials do not appreciably cross or bind to the cehlar and capillary membranes of the brain tissue, and that little or no bulk Aow of brain Auid plus entrained solute occurs, equation 1 reduces to
ac =Da’c at ax2
(18)
Employing the boundary conditions, given by equation3, equation 18 can be solved to yield, when h is very large A Exp.723; l
where erfc stands for the complimentary error function* These two equations and supporting experimental results with extracellular markers have been presented in studies with ventriculocisternal perfusion (9, 13, 17, 22, 26) and intravascular infusions (2, 3, 27). All of these studies plus the results given herein indicate that extracellular materials, such as sucrose, move through the brain ECF by diffusion and that equatians 18 and 19 are valid expressions of these particular compound’s movements in this system. Because of this body of evidence about the distribution kinetics and spaces of extracellular materials in the brain, extracellular markers were used as experimental controls in our work. Inverse complimentary error function (erfc-I) plots, as originally developed and reported by Schantz and Lauffer (28) and subsequently employed by the aforementioned authors, were made and used to calculate tissue diffusion coefficients and distributions spaces. The value of C, in &quation 19, the concentration of the marker at the point x = 0 in the tissue, was obtained by a computer program which systematically changed C, and generated the respective series of quotients, (C,/C,), for the data points until the “best-fit” estimates, as determined by linear regression analysis of the erfc-l plots, was found. Using this C,, tissue diffusion coefficients (Ok) and distribution spaces (C, divided by the concentration in the inflowing perfusate times 100 %) were calculated from equation 19.
Perfusion
Time,
h
I 2 3 4 5 1-5
* Mean 3t SE.
Space,
14.8 18.7 16,7 16.5 18.4 17.5
%*
* 1.1 xt 0.9 & 1.2 jz 1.2 zt 0.6
Diff Co&,
X
10gcm2/S*
n
3.4 2.95 3.03 2.95 3.45 3.1
It rt zt * *
0.1 0.3 0.2 0.2 0.1
1 11 8 11 8 39
#
Exp.705:
I .os
time= 131 min t& 12 min timeE247min t ~=11.6min
I .f6
I
i
.24
-32
-40
DISTANCE (X) FROM VENTRICULAR SURFACE IN CM FIG. 1. Semi1ogarithn-k plots of [14C]urea tissue concentration profiles in dog caudate nucleus after 2 different periods of perfusion. Line was drawn to yield a reasonable fit to all of points. RESULTS
For the major portion of the experiments in the study (37 out of 54), sucrose was used as the extracellular marker. Table 1 presents the distribution spaces and the apparent tissue diffusion coefficients which were found for this compound. No significant changes in the spaces or the diffusion coefficients with time are indicated by these results. As mentioned in the introduction, a ventricular Auidto-tissue-to-blood steady state must be reached to measure a capillary transfer coefficient by this technique. In this situation, the amount of labeled material which enters the brain tissue from the ventricular fluid (perfusate and CSF) equals the amount of material that is lost from the tissue ECF to blood. Figure 1 presents data obtained from two experiments in which [14C]urea was the test molecule. The perfusion times were approximately 2 h and 4 h, respectively. Despite the differences in the duration of these two experiments, nearly identical tissue concentration profiles were’ found; therefore, it seems that a tissue steady state for labeled urea is reached at least by 2 h. For some of the other compounds used in this study, no definite tissue steady states were observed. Figure 2 shows tissue concentration profiles obtained with [14C]creatinine in two experiments of differing, approximately 2 h and 4 h, durations. The differences in the two slopes are significant, indicating that a steady state had not been reached by 2 h. From those experiments in which a tissue steady state appeared to be reached, a capillary exchange transfer
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BLOOD-BRAIN
TRANSFER
881
COEFFZCIENTS
constant (k,J and half-time (11/z = ln2/k,i) could be calculated from the semilog plots of the data. Table 2 presents these results for the three compounds which seemed to satisfy the steady-state criteria. As can be seen from these data, the calculated exchange t1&s for water were constant from 1 to 4 h; whereas the results with ethylene glycol and urea indicated constant values from 2 to 4 h. The diffusion coefficients which were used for the calculation of these t&s (or the kPi as in equation 8) were the values for free diffusion in water (D,) at 37”C, i.e., 30 X lo-” cm*/s for water and 15 X lo-” cm2/s for urea and ethylene glycol. As will be discussed later, corrections of the diffusion path length (x) f or tortuosity were applied to equation 8. For three of the test materials, adequate tissue steady states were not found. Complimentary error function plots of these data indicated that the slopes of the tissue concentration profiles increased with time as shown in Fig. 2.
TABLE 3. Apparent tissue dz&sion coeficients (I)) and capillary exchange t&s found af various limes ~-~~----~__l__l-----~ -LLLL---Approximate --__1_1~
Material 2 I
-------..
CreatinineD tll:!
n Sodium
Kmnitol
--
D 2.5 + 0.3 t1/2 38 LIZ 5 n 4
35
--- --
5.5
;;;y;
I_
1.8 It 0.1 1.9zto.22*0+ 29 It 3 41 * 2 6 4 4.1 86 1
b
5
“--_~-.-.-l--l”
2,2 & 0.1 25 zt 1 3
Times,
4
D 6,3ztO.5 h/z 56 xt 3 n 4
I
Perfusion
3
h
117~ 4
0.21.9*0.1 91 dz 3 2 15
0.8
5.1
18
175 dz37 3
+
0.85.6
zt 0.4
12
1.8
-.-
1
Diffusion coefficient V alues are means (X l@ cm2/s) Capillary exchange tl/2 values are means (min) % SE.
+
SE.
From such plots apparent tissue diffusion coefficients were calculated. The results are presented in Table 3. The constancy or near constancy of each particular material’s Dt with time indicates that the tissue concentration profiles continued to change as a function of time and that no tissue steady state was achieved for them within the perfusion periods used in the present work.
A Exp.
J33: time= D=2.2
+ Exp. 716:
,005: ,005 .OOZ.002 .oor .OOll
240min x IO-‘cm’kec
time= 129 min D= 2.2 x IO-’ cm’kec
\
\
r F ’
’ .08
’
’ .I6
DISTANCE
(Xl
’ FROM
’ .24
’
’ -32
’
VENTRICULAR
’ .40
’
SURFACE
’ .48
’
’
IN CM
2. Inverse complimentary error function graph of [14C]creatinine tissue concentration profiles in dog caudate nucleus after 2 different periods of perfusion. Lines are regression analysis fits to separate experiments : ex@riment 133-broken lines, experiment 716-continuous line. FIG.
2. Apparent capillary mchange half-times various perfusion times
(min) found at
TABLE
Approximate
Molecuk. 1
3HzQ
2
Time,
are means
IIZ SE:.
19 (n=
Average of all Times
4
1.4
17 * 3 (n = 4) 16 zt 3 (n=3)
[W]ethylene glycol
h
3
1.6 & 0.4 (n = 3)
[“C]urea
Values
Perfusion
1)
(n 14 (n 17 (n
It = zk = * =
0.2 3) 2 5) 2 3)
1.5 (n 15 (n 17 (n
=t = * = & =
0.2 6) 2 9) 1 7)
After completion of the preliminary group of experiments in this study, the need for some objective or relatively objective way to evaluate the acceptability of the data for a given tissue series became apparent. The criterion chosen for this purpose was the form of the inverse complimentary error function plot of the extracellular marker’s data+ If the points yielded a line which either bent up or down, the results with both the extracellular and the test marker were not included in the reportable data; if, on the other hand, the extracellular material’s points gave a straight line, the data were considered acceptable and included. Initially, about 50 % of our data were eliminated by this test. In our latest set of experiments, a group of 20 which were run in an identical manner over a 2-mo period, 50 out of the 60 tissue series gave linear erfc-l graphs of the extracellular marker data. The experimental points of the companion test materials in this group all yielded either good erfc-1 or semilog plots. For the graphs of the test material’s results in the remaining 10 tissue series, approximately onehalf produced straight-line erfc-l or semilog plots; whereas the other half did not. The application of this criterion, therefore, did not eliminate a very large portion of our results (s of this series) once the experimental technique was perfected and may have, if anything, excluded some “good” tissue data. Although in the previous paragraph we have emphasized the use of the graphs as a means of analyzing the transport mechanism involved in the exchange of material between blood, brain, and CSF during a ventriculocisternal perfusion, such plots in themselves are not clear proof that the material is moving through the tissue ECF solely by diffusion (an erfc-l plot) or that a tissue steady state has been reached (a semilog plot). Because of the scatter which
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882
C. S. PATLAK
is found in the experimental results and the ability of the computer program to force a fit, the same group of data points often yielded fairly good straight lines on both types of graphs and two sets of transport numbers could be calculated. More definitive evidence of the mechanism or mechanisms involved emerges from an examination of the apparent tissue diffusion coefficients at various times (the Dt’s should be constant if diffusion through the ECF is the principal transport process) or the capillary exchange halftimes determined at different times (the 11,2’s should be approximately equal for two different perfusion periods if a true tissue steady state has been reached). Perfusions of varying lengths of time were performed and reported (Tables 1, 2, and 3) in an attempt to more clearly establish and quantitate the transport phenomena involved. The apparent diffusion coefficients and the distribution spaces of sucrose in the caudate nucleus were constant with time. The indicated independence of sucrose’s Dt and distribution space from time is consistent with the assumptions that: I) little or no bulk flow of brain ECF occurs within the caudate nucleus of the dog; 2) extracellular materials move through the ECF of this brain region primarily by diffusion; 3) sucrose remains extracellular in its distribution within the caudate nucleus for up to 5 h, and 4) sucrose exchange between ECF and blood in this portion of the CNS is small. The average (1-5 h) Dt reported in Table l is roughly equal to 45 % of sucrose’s diffusion coefficient in water at 37°C (D, = 7 X 1W6 cm2/s; ref. 17 and this study). A similar reduction in the tissue digusitivity of two other extracellular markers, inulin (9) and EDTA (this study; mean Dt it SE = 3.3 & 0.3 X lo+ cm2/s, N = 7;D, = 6.8 X 10e6 cm2/s) have been found for dog caudate nucleus. This lowering of the apparent D probably is the result of the tortuosity of the extracellular space as has been previously suggested by Levin et al. (17), and not of the restriction to free diffusion imposed by “tight” extracellular channels. The tortuosity factor (X), a measure of the increase in the path length for diffusion, is given by x=dmt.U’gh sin t is equation and a D,/Dt ratio of 2.2 (7.0/3.1), the calculated X for sucrose in the dog caudate nucleus is 1.5. Estimates of X from the previously mentioned inulin (9) and EDTA (this study) data yield values of x = 1.5 and 1.45, respectively, for these two extracellulartype molecules, indicating that the microscopic increase in the extracellular diffusion path in this CNS region is around 50 %. For the calculations of the /cp’s by equation 8, either the D or the x term must be adjusted for the path length. In this study the appropriate correction was applied to the distance factor, x. Equation I indicates that the tissue concentration profile at any time is a function, in part, of the materials’ exchangeability across the capillary complex. To successfully evaluate the transfer constant, k,, during the transient period, all of the other terms in equation I, notably the cellular exchange and/or binding, must also be measured. As the number of constants which are to be determined increases, the number of experiments which must be performed to evaluate them also increases. The tissue steady state provides a way of avoiding this problem in the measurement of k, for materials of moderate-to-high exchangeability. To properly ascertain that a constant tissue con-
AND
J. De FENSTERMACHER
centration profile has been reached and that a capillary transfer coefficient can be measured, the similarity of the slopes and the calculated f 1,~‘s for two relatively divergent times must be clearly demonstrated. This is the essential point in the results presented in Table 2. Computer modeling of quution I indicates that the capillary transfer coefficients which are determined before a true tissue steady state has been reached will be erroneously high, and the related capillary exchange times will be too low. In agreement with this, the results presented in Table 3 show steadily increasing f 1,2’s, which result from changing tissue concentration profiles, for creatinine, sodium, and mannitol. In order to quantitate capillary exchange phenomena, various assumptions are made to take care of unmeasurable or unknown components in the system. (For a comprehensive review of the standard assumptions, see ref* 16.) In the present work the assumption that a steady-state concentration profile exists within the capillary from the arterial to the venous end was made. This assumption would appear to be satisfied in our system since two factors-a steady state and low (l-4 times background) concentrations of test material in the arterial blood-would combine to give constant exchange rates along the capillary and an intracapillary steady state. Such an assumption is a significant problem for the tissue clearance technique (e.g., 6) and would be a problem for single-injection experiments (e.g., 3, 20) i f si g ni fi cant backflux of material from tissue fluid to blood occurs during their respective experimental periods. Radial mixing of the tracer material within the capillaries, another assumption made in this study, is likely to be complete, since the capillary diameter is small, the diffusion coefficients are relatively high (D > 1 .O X lo-” cm2/s), and the blood cells serve as plasma stirrers. In connection with this, Taylor diffusion, which often complicates the single-injection technique by causing the intravascular separation of test and reference substance (15, 28), does not affect the determination of the transfer constant in this study. A uniform concentration of test material in the extracellular fluid around a capillary was also assumed. In the steady-state condition of our system, there is a concentration gradient in the ECF in the x direction; however, since the capillaries are randomly oriented, the diffusion coefficient is of reasonable magnitude (1 .O X 1Om5 cm2/s), and the steady-state gradient is not too steep (exchange 1112> 1 min), the change in ECF concentration across a capillary-tissue unit would not be appreciable for most capillaries. The two terms used in equation 8 to calculate the capillary exchange constant are the diffusion coefficient and the distance from the ventricular surface to a point in the tissue. The D that was used in this study to obtain the values of kpi was the D for the compound in water at 37°C; however, the appropriate value of the distance, x, was harder to ascertain. Since molecules such as urea and ethylene glycol would be expected to move into deeper tissue sites primarily by diffusion through the extracellular space (ECS), the assumption was made that the x should be corrected for the tortuosity of this space and multiplied by 1.5. On the other hand, water enters most cells very rapidly (t I,2 of exchange in human red cells = 4 ms; ref. 23). The movement of this molecule into the caudate nucleus prob-
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BLOOD-BRAIN
TRANSFER
883
COEFFICIENTS
ably occurs at fairly similar rates through both extracellular and intracellular pathways; therefore, the correct value of X lies somewhere between 1.0 and 1.5. For the water results presented in Table 2, the tortuosity factor was assumed to be 1. If in fact the x should be fully corrected for the tortuosity of the ECS (X = 1.5) in the calculation of tritiated water’s k,i (a situation which seems highly unlikely), the tl/ 2 reported in Table 2 would be 2.5 times larger. For metabolically inert substances, which were used in this study, it is likely that the transfer constants are equal and hence the relationship between the transfer constant, k,, and the usual permeability-surface area product, PS, may be found from equation 16. The transfer constant is dependent not only on PS but also F and V,. For this study F is the flow rate of that particular volume of the capillary blood in which the test material distributes during its initial passage through the capillary. Based on Freygang and Sokoloff’s (11) measurement of blood flow to the caudate nucleus in unanesthetized cats, the 1ikeIy values of F range from 0.55 (the plasma water flow rate) to 1.0 ml/g-min (the blood water flow rate). The volume term, V,, refers to the tissue volume or space from which the material exchanges directly with the intracapillary fluid. In most cases V, equals the ECS; however, for substances which move in and out of some or all of the brain cells extremely rapidly, the functional exchanging space would be larger and approach the tissue water space in size. For the caudate nucleus the likely range of the V./F ratio extends from (X3-0.8 min. According to equation 16 and the related discussion, k, equals PS/V, if k,,V,/F < 0.1. Using values of V./F = 0.3 and 0.8, the limits Of ~I/Z which satisfies this condition are 2.1 and 5.6 min, respectively. For either V./F ratio, a much larger tl/ 2 (much smaller kp) was found with urea and ethylene glycol; therefore, for these two compounds, the approximation of PS = k,V, is valid and yields PS g 1.5 X Mm4 cm3/g-s. On the other hand, the t l/2 of [3H]water is below these limits and equation 16 must be used to calculate PS. Taking the water t I/ 2 from Table 2 (E 1.5 min) and assuming that V, = 0.8 cm3/g and F = LO cm3/g-min, the PS product for water is about 0.46 cm3/g-min (7.7 X 1W3 cm3/g-s) . A comparison of the results in this study to other works
on blood-brain exchange can be made for several of the substances. By using the working equation given by Crone (3) for the calculation of P and our equation 16, the following relationship between the fractional loss or extraction (E) as determined in single-injection experiments and k, emerges: E = k&/F. For urea, Crone reported an E of 0.11 in dogs (3) ; in the present study a value of E around one-fifth that of Crone’s was determined. Subsequent work by Yudilevich and DeRose (29) suggested that the E measured by Crone for urea was probably artifactual and mainly caused by Taylor diffusion. Oldendorf’s work (19) also indicates that the permeability of the rat blood-brain barrier to urea is very low (BUI < 2 o/I,; E _< 0.02). Bering (1) reported extremely large exchange rates of water between blood and brain in dogs; using compartmental analysis, gray matter appearance half-times around 12 s were calculated from the data. The sharp fall in the vascular ID20 levels during the first several minutes of the experimental period may account, in part, for the higher permeability figures obtained by Bering for water. Raichle et al. (25) and Eichling et al. (6) have recently reported for rhesus monkeys that water does not freely equilibrate between blood and brain tissue at normal rates of cerebral blood flow and that the uptake of labeled water by the brain is partly limited by the permeability of the bloodbrain barrier. Eichling et al.% estimate of brain capillary PS for H2150 in the monkey (0.019 cm3/g-s) is in fair agreement with the one we calculated for dog caudate nucleus. The relatively low permeability of the brain capillary complex to mannitol and sodium found by ventriculocisternal perfusion is supported by the study of Yudilevich and DeRose (29) with the indicator dilution or single-injection technique in dogs. We are grateful to Ernest S. Owens and Roosevelt Hyman for technical assistance, Qrs. Victor A. Levin and David P, Rall for aid in developing the technique, and Ms. Mary Tyler for secretarial support. Preliminary reports of portions of this work have been presented to the Fifty-fourth Annual Meeting of the Federation of American Societies for Experimental Biology, Atlantic City, N. J., April 1970 and the Sixteenth Annual Meeting of the Biophysical Society, Toronto, Canada, February 1972. Received
for publication
5 November
1974.
REFERENCES BERING, E. A. Water exchange of central nervous system and cerebrospinal fluid. J. Newosurg. 9 : 275-287, 1952. CRONE, C. The permeability of capillaries in various organs as determined by the use of the indicator diffusion method. Acta Physiol. &and. 58: 292-305, 1963. CRONE, C. The permeability of brain capillaries to non-electrolytes. Acta Physiol. Stand. 64 : 407-417, 1965. CRONE, C. Capillary permeability-techniques and problems. In : Capillary Permeability, edited by C. Crone and N. A. Lassen. Copenhagen : Munksgaard, 1970, p. 15-3 1. DAVSON, I-I., AND K. WELCH. The permeation of several materials into the fluids of the rabbit’s brain. J. Physz’ol., Lo&n 218 : 337351, 1971. EICHLING, J . , M. RALCHLE, R. GRUBB, AND M. TER-POGOWAN. Evidence of the limitations of water as a freely diffusable tracer in brain of the rhesus monkey. Circulation Res. 35 : 358-364, 1974. FENSTERMACHER, J. Ventriculocisternal perfusion as a technique for studvinE transDort and metabolism within the brain. In:
in Neurochemistry, edited by N. Marks and R. York : Plenum, 1972, vol. 1, p. 165-l 78. J., AND J. A. JOHNSON. Filtration and reflection 8. FENSTERMACHER, coefficients of the rabbit blood-brain barrier. Am. J. psiol. 211 : 341-346, 1966. C. PATLAK, AND V. LEVXN. Ventri9. FENSTERMACHER, J., D. RALL, culocisternal perfusion as a technique for analysis of brain capillary permeability and extracellular transport. In : Capillary Permeability, edited by C. Crone and N. Lassen. Copenhagen: Munksgaard, 1970, p. 483-490. 10. FENSTERMACHER, J., C. PATLAK, AND R. BLASBERG. Transport of material between brain extracellular fluid, brain cells and blood. Research
Rodnight.
Federation
11. FREYGANG, of regional radioactive 12. KATZMAN,
Methods
New
Proc.
33:
2070-2074,
1974.
W., AND I,. SOKOL,OFF. Quantitative measurements circulation in the central nervous system by the use of inert gas. Adwn. Biol. Med. Phys. 6 : 263-279, 1958. R., AND P. H. LEIDERMAN. Brain potassium exchange
Downloaded from www.physiology.org/journal/ajplegacy at Univ of Texas Dallas (129.110.242.050) on February 12, 2019.
C. S. PATLAK
884 in normal
adult
and immature
1953. 13. KATZMAN,
R., I-I. SCHIMMEL, as a measure of extracellular
Virchow 14. KETY,
35.
16.
17.
18.
19.
20.
21.
Med.
Sot.
City
NY
rats. Am. J. Physiol. AND
fluid 26 : 254-280,
I75
C. WILSON. Diffusion space in brain.
: 263-270,
of inulin Proc.
Rudolf
1968.
S. The theory and application of the exchange of inert gas at the lungs and tissues. PharmacoZ. Reu. 3 : l-41, 1951. LASSEN, N., J, TRAP-JENSEN, S. C. ALEXANDER, J. OLBSEN, AND 0. PAULSON. Blood-bran barrier studies in man using the doubleindicator method. Am. J. Physiol. 220: 1627-1633, 1971. LEONARD, E., AND S. B. JORGENSEN. The analysis of convection and diffusion in capillary beds. Ann. Rev. Biophys. Bioeng. 3: 293-339, 1974. LEVIN, V.,J. FENSTERMACHER, AND C. PATLAK. Sucrose andinulin space measurements of cerebral cortex in four mammalian species. Am. J. Pltysiol. 219: 1528-1533, 1970. LEVIN, V., AND C. PATLAK. A compartmental analysis of 24Na kinetics in rat cerebrum, sciatic nerve and cerebrospinal fluid. J. Physiol., London 224: 559-581, 1972. OLDENDORF, W. Brain uptake of radiolabeled amino acids, amines, and hexoses after arterial injection. Am. J. Pfzysiol. 221 : 1629-1639, 1971. OLDENDORF, W. Carrier-mediated blood-brain barrier transport of short-chain monocarboxylic organic acids. Am. J. Physiol. 1450-1453, 1973. OLSEN, N., AND G. RWDOLPH. Transfer of sodium and bromide
AND
J.
D. FENSTERMACHER
ions
between blood, cerebrospinal fluid, and brain tissue. Am. J. 183 : 427-432, 1955. 22. W., AND D. RALL. Brain extracellular space as measured by diffusion of various molecules into brain. In: Bruz’n Edema, edited by I. Klatzo and F. Seitelberger. New York: SpringerVerlag, 1967, p. 333-346. 23. PAGANELLI, C., AND A. IS. SOLOMON. The rate of exchange of tritiated water across the human red cell membrane. J. Gen. Physiol. OPPELT,
259-277, 1957. AND R. KAPLAN. Diffusion of non-electrolytes brain tissue. Brain Res. 17 : 407-4 16, 1970. 25. RAICHLE, M,, J* EICHLING, AND R. GRUBB. Brain permeability of water. Arch. Neural. 30 : 319-321, 1974. 26. RALL, D., W. OPPELT, AND C. PATLAK. Extracellular space
24.
physiol. POLLAY,
41:
M.,
in
of brain as determined by diffusion of inulin from the ventricular system. Life Sci. 2: 43-48, 1962. 27. REED, D. J., AND D. M. WOODBURY. Kinetics of movement of iodide, sucrose, inulin, and radio-iodinated serum albumin in the central nervous system and cerebrospinal fluid of the rat. J. Physiol., London 169, 8 16-850, 1963. 28. SCHANTZ, E., AND M. LAUFFER. Diffusion measurements in agar gel. Biochemistry 1 : 658-663, 1962. 29. YUDILEVICH, D., AND N. DEROSE. Blood-brain transfer of glucose and other molecules measured by rapid indicator dilution. Am. J. Physiol. 220 : 84 1-846, 197 1.
Downloaded from www.physiology.org/journal/ajplegacy at Univ of Texas Dallas (129.110.242.050) on February 12, 2019.
Vol.
OF PHYSIOLOGY 1975. Print&
229, No. 4, October
in U.S.A.
Measurements
of dog blood-brain
by ventriculocisternal C. S. PATLAK Theoretical Transport
St&tics Section,
AND
J. D. FENSTERMACHER of Cancer
Branch,
Biometry
Treatment,
N&onal
PATLAK, C. S., AND J. D. FENSTERMACHER. Measurements of dog blood-brain fransfer constunfs by uentriculocisternal perfusion. Am. J. PhysioL 229(4) : 877-884. 1975.-Ventriculocisternal perfusions of mongrel dogs were performed for 1-6 h with solutions containing isotopically labeled compounds. At the conclusion of the perfusion period, serial brain samples were taken from the caudate nucleus and analyzed for radioactivity. Tissue concentration profiles were constructed from the data, and apparent tissue diffusion and capillary exchange coefficients were determined. The tissue diffusion constant of sucrose was 3 X 1tY6 cm2/s, which is approximately 45y0 of its free-water V&E. The permeability of the brain capillary complex to creatinine, sodium, and mannitol was so Xow that it could not be accurately measured by this technique. Capillary transfer coefficients, expressed as half-times, were determined for water, urea, and ethylene glycol; the fl/z values were 1.5, 15, and 17 min, respectively. These numbers were converted to PS products and compared to other published values. This work suggests that the exchange of these compounds between blood and brain is partially (water) or nearly completely (urea and ethylene glycol) limited by membrane permeability.
blood-brain barrier; glycol; sucrose
capillary
permeability;
constants
perfusion
and Mathematzcs Division
transfer
water;
urea;
ethylene
FOUR EXPERIMENTAL APPROACHES-the single injection, the osmotic transient, the tissue uptake, and the tissue clearance -are commonly used to measure the permeability of capillaries to solutes and water (4). All these techniques have been employed to study the permeability of the brain capillary complex, the so-called blood-brain barrier. In the pioneering work with the single-injection or indicatordiffusion method, Crone (2, 3) examined the permeability of the blood-brain barrier to a series of organic nonelectrolytes. Other investigators, e.g., Yudilevich and DeRose (29), have continued the study of the transport of material between blood and brain with this technique. Recently, Oldendorf (19, 20) reversed the rule of the indicator molecule, using a highly diffusible material (tritiated water) in place of a nondiffusible marker (e.g., Evans blue-tagged albumin), in his application of the single-injection approach to the assessment of blood-brain exchange. By measuring the osmotic flows of water generated by various polar compounds infused into the vascular system (the osmotic transient approach), Fenstermacher and Johnson (8) determined the reflection coefficients of five organic nonelectrolytes and the filtration coefficient for the rabbit
Division, Cancer
National
Institute
Imtitute,
Bethesda,
of Mental
Maryland
Health,
and Membrane
ZOO14
brain capillary complex. A third methodology, tissue uptake, has been used extensively to examine the permeability of the blood-brain barrier. Among the early experimentalists who employed this technique are Bering (1) with DZO, Katzman and Leiderman (12) with 42K, and Olsen and Rudolph (2 1) with 24Na and 82Br. Other tissue uptake studies include those of Levin and Patlak (18) with 24Na and Davson and Welch (5) with 24Na, 36C1, and [““S]thiourea. The tissue clearance method has been used by Eichling et al. (6) in a recent study on cerebral blood flow and brain-water exchange. In the present work a new approach, which resembles the tissue clearance method, to measuring the exchangeability of various compounds between blood and brain extracellular fluid (ECF) is reported. The material which is to be studied is perfused through the brain ventricular system and, if appropriately diffusible, passes into the brain parenchyma. As the test molecules or ions move through the brain, some of them are lost from the tissue to the Providing that the perfusion is blood via the capillaries. continued for an appropriate period of time, a tissue steady state- a condition in which the ventricular fluid(CSF + perfusate) to-brain exchange is balanced by the brainto-blood loss-is established for the substance in question The steady-state tissue concentration profile (the concentration at various points in the tissue as a function of the distance from the perfused surface) can be determined by serial sampling and used to calculate the transfer constant of the brain capillary complex to the test material. The experimental and analytical procedures of this technique plus the results from studies with the dog will be presented. METHODS
The ventriculocisternal perfusions and tissue sampling were performed as previously described for dogs (7). Mongrel dogs, weighing from 15 to 25 kg, were anesthetized by 30 mg/kg intravenous pentobarbital and maintained in a state of light anesthesia by subsequent infusion of this drug when necessary. Inflow needles were stereotaxically placed in each lateral ventricle using the following parameters; 10 mm posterior from the bregma, 6 mm lateral to the midline, and 15-19 mm down from the dura. A single outflow needle was positioned in the cisterna magna. When the three needles were properly placed, a constant-speed infusion pump (Harvard Apparatus Co.) was turned OIL
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878
C. S. PATLAK
For the initial 13 min, the pump speed was 0.9 1 ml/min; for the remainder of the experimental period, a rate of about 0.21 ml/min was used. The perfusion fluid was an ionically and osmotically balanced artificial CSF which contained two or three radioactively labeled materials (l-6 &i/ml) pl us a trace of trypan blue dye. One of the labeled compounds was an extracellular marker; the markers used in this study were [‘“Cl- and [3H]sucrose and [14C]EDTA, which has been chelated with Ca++. (In several of the earlier experiments, [p-3H]aminohippurate was employed as an extracellular marker; however, as shown by Fenstermacher et al. (lo), this compound does not distribute solely in the brain extracelMar Auid and was not used further.) At the end of the perfusion time, the animal was killed and the brain was rapidly removed (time 5 1.5 min). The brain was split through the midiine, the lateral ventricles exposed, and the hemispheres chilled to medium hardness in liquid nitrogen. The time that elapsed between the death of the animal and the freezing or near freezing of the brain wz1s about 3 min. Control studies demonstrated that no detectable movement of the labeled materials occurred once the tissue was frozen or semifrozen; therefore, all of the tissue samples were kept frozen by regular dipping in liquid nitrogen during the subsequent handling Procedure. The chilled hemispheres were cut into a series of coronal sections. Well-perfused (as indicated by the bluestaining) flat surfaces of the caudate nucleus were trimmed from the coronal sections. From each experiment, two to four blocks of caudate tissue were obtained at the time of sampling. Subsequently, these blocks were sectioned into series of tissue samples of uniform thickness by a special multibladed knife or a freezing-microtome. From 6 to 10 samples were obtained per series; the thickness of the individual samples throughout a series was either 0.4, 0.5, 0.7, or 1.0 mm. The tissue samples were placed in tared liquid scintillation vials. Samples of the outflow solution were collected regularly throughout the period of perfusion; 50- or 100-~1 samples of these plus the inflow were pipetted into liquid scintillation vials. All tissue and fluid sampled were digested in 0.5 ml NCS (Nuclear-Chicago soIubiIizer, Amersham/Searle Corp.). To each vial was added 18 ml of a toluene-based counting solution. All samples were counted in a three-channel liquid scintillation counter with automatic external standard (Packard Instrument Co.) fur 100,000 counts or 20 min. Standard procedures were used for isotope separation and quench corrections. The individual corrected counts were divided by sample weight (tissues) or volume (fluids), yielding concentrations of radioactivity (counts/min per mg or counts/min per ~1). From these data, inverse complimentary error function (erfc-l) and semilogarithmic plots of the tissue concentration profiles were made and tissue distribution spaces, brain diffusion coefficients, and capillary transfer constants were calculated. (These procedures will be presented in the next section.) Theory and analysis of datd. A particular compound or ion which is perfused through the ventricular system may remain in the ventricular fluid and circulate with it or, if appropriately permeable, enter the surrounding tissue. To
AND
J. Da FENSTERMACHIZR
move into the brain, the material must cross the fluidtissue interface (the ependyma); subsequently the substance may diffuse through the extracellular fluid (ECF), be taken up by c&s, be bound to various tissue elements, cross the capillary wall, or some combination of these steps. Assuming that I) the concentration of th e solute in the Auid perfusing a specific surface of the bra rema ms constan t with time, 2) the portion of the brain being examined is uniform in composition, and 3) the transport geometry of the perfused tissue is effectively rectangular and one dimensional, the equation for the system is
-ac = 1) a”c d(4 ax2 - ax dt -
(V
(k,iC
-
kpoCJ
-
&
($) e
In this and the following equations, C, Cf, and C, are the effective concentrations of the substance of interest in ECF, ventricular fluid, and arterial plasma, respectively; D is the apparent diffusion constant of the solute through the brain ECF; f is time; x is the distance from the ependyma to the tissue site; v is the velocity of the bulk flow of ECF, if any occurs, through the brain; k,i and k,, are the transfer coefficients of the brain capillary complex for the movement of the substance from brain ECF to plasma and plasma to brain ECF, respectively; V, is the magnitude of the extracellular space (ECS); B is the amount of the solute in the brain cells and/or bound in the tissue per unit mass; and h is the permeability constant of the ependymal layer for the material. Providing that the perfusion is continued until a tissue steady state is reached, the time-dependent terms drop out and equation I becomes
0 = D d25 dx2
dbC> -
dx
-
kpi c + k,o G
(2)
With the assumptions that the brain is, in effect, infinitely long and that v is a constant (if v is not a constant, the solution of equation 2 will not be a simple exponential) and the bou ndary condition = h cc
f
- w)3
0
equation 2 can be solved
to yield 2h
v +
1/v2
+
4kpiD
(4)
Kpi
exp
1 20 ( v
-
4~~
+
4kpiD)x
>
Of the three concentration terms which appear on the left-hand side of equution 4, only C, the concentration of the material in the brain ECF, is not measured during the course of the experiment. The serial sampling of the brain at the end of the perfusion period yields the total (extracellular plus intracellular) amount of the material in the if tracer concentrations of labeled tissue (C,) ; however, materials are used and a steady state is reached, the amount
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BLOOD-BRAIN
TRANSFER
of intracellular
or bound
COEFFICIENTS tracer,
879
B, is proportional
to C
(5)
I3 = KC where
K is the proportionality
constant,
CT =v,c+I3 = C(Ve
- (V, + K) gO c, Di Cf
-
kpoc,
Y--Kpi
Cc = (Ca (6)
+- K)
Rearranging equation 6 to solve this value into equation 4 yields CT
and thus
for
C and
substituting
2we + a v
1( *exp20u
+
1/u”
+
4kpiD
~vZ + 4kpi D)x
(7)
>
Experimental evidence which will be presented in this paper suggest that v, the rate of the bulk flow of ECF through the caudate nucleus of the dog, is virtually zero the ECS exclusively or nearly ( i.e., solutes move through exclusively by diffusion) and thus, equation 7 can be simplified to
CT- we+ K) $Pic, cf--
(8)
hWe +K) = h + 4kpiD
exp (-dk,ilDx)
-uAT
- cxp[-g-J)+;c
(11)
in which N is the number of The following equations, capillaries per gram of brain, FiS and ps are the total permeability of the capillaries per gram of brain for the movement of a material from ECF to plasma and plasma to ECF, respectively, and F is the flow of either blood or plasma per gram of tissue, relate capillary permeability and bulk flow velocity for the single-capillary model to a more operational system
F cc = rc, 0 (
pi/N
= PiS
p,lN
= P,S
uAN
= F
P*S - ks c >
Under the assumptions the equation previously (equation I> becomes dC = D -a2c __ ax2 at
W)
inserting equations 12 into equation II for the mean capillary concentration
Y(]
During the time course of the ventriculocisternal perfusions performed in the present studies, the concentration of the tracer in the arterial plasma, C,, was negligible for most substance. Providing that the arterial plasma concentration is effectively zero and that a tissue steady state has been that CT is an exponential reached, equation 8 indicates therefore, a function of kpi, D, and X; in this situation, semilogarithmic plot of C, verus x should give a straight transfer constant, line from which kpi, the ECF-to-plasma can be evaluated. Following the approach and associated assumptions which were initially developed by Kety (14), a more detailed evaluation of kpi can be made. Assuming that the capillary is homogeneous in structure along its length, that the radial and longitudinal diffusion of the solute within the capillary is negligible, and that the concentration of the solute in the ECF around each capillary is uniform, conservation of mass considerations yield the following differential equation for the steady state
(10)
The average concentration along the capillary, c,, may be evaluated by integrating equation IQ from 0 to I, the length of the capillary, and dividing by I
Rearranging and yields an equation
k ca kPi
concentration,
-$+exp(-p$)+p$2
cc = (ca - p)!$(1
= 2h -
of the material at y = 0 equals the arterial C,, equation 9 may be solved to give
_ exp[-y])
+ %C
(13)
used to develop equations 9 and 1.2, written for the experimental system
-8 w> ax -- G (PiSC e
and thus, since equations I and lowing relationship is apparent kpiC-k,,C~=‘i;;C-p$C~
14 are equivalent,
PiS
the fol-
(w
e
e
Substituting equation 13 into equation X5 and rearranging yields two expressions which relate the capillary permeability constants, P,S and P$, to the experimentally determined k’s
c =
PoCc
-
PiC
(16) where u is the velocity of the blood flow in the capillary, A is the cross-sectional area of the capillary; C, is the concentration of the solute in capillary plasma at each point; y is the distance along the capillary; and pi and pO are the permeabilities of the capillary wall per unit length for the movement from ECF into the plasma and from the plasma into the ECF, respectively. Since the concentration
pis
=
kpi -GF(n(l
-“e”E>
If the term (k,, V,/F) is small (values tions 16 may be approximated as
-
.
of 0.3 or less), equa-
Rs = V&p, PiS = V,kpi
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(17)
880
C. S. PATLAK
AND
J. D. FENSTERMACHER
with an error of less than 5 %. Therefore, if the flow of fluid in which the solute is entrained-either plasma or blood-is large relative to the plasma-ECF transfer constant (k,,), the k’s in equations 1-8 are proportional to the brain capillary permeability coefficients. If the transport (= I&) and P,S process is passive, k,, and k,i are identical equals PJ, regardless of the value of (R,,V,/F). In all the experiments, a compound that remains extracellular was included in the perfusate+ Providing that such materials do not appreciably cross or bind to the cehlar and capillary membranes of the brain tissue, and that little or no bulk Aow of brain Auid plus entrained solute occurs, equation 1 reduces to
ac =Da’c at ax2
(18)
Employing the boundary conditions, given by equation3, equation 18 can be solved to yield, when h is very large A Exp.723; l
where erfc stands for the complimentary error function* These two equations and supporting experimental results with extracellular markers have been presented in studies with ventriculocisternal perfusion (9, 13, 17, 22, 26) and intravascular infusions (2, 3, 27). All of these studies plus the results given herein indicate that extracellular materials, such as sucrose, move through the brain ECF by diffusion and that equatians 18 and 19 are valid expressions of these particular compound’s movements in this system. Because of this body of evidence about the distribution kinetics and spaces of extracellular materials in the brain, extracellular markers were used as experimental controls in our work. Inverse complimentary error function (erfc-I) plots, as originally developed and reported by Schantz and Lauffer (28) and subsequently employed by the aforementioned authors, were made and used to calculate tissue diffusion coefficients and distributions spaces. The value of C, in &quation 19, the concentration of the marker at the point x = 0 in the tissue, was obtained by a computer program which systematically changed C, and generated the respective series of quotients, (C,/C,), for the data points until the “best-fit” estimates, as determined by linear regression analysis of the erfc-l plots, was found. Using this C,, tissue diffusion coefficients (Ok) and distribution spaces (C, divided by the concentration in the inflowing perfusate times 100 %) were calculated from equation 19.
Perfusion
Time,
h
I 2 3 4 5 1-5
* Mean 3t SE.
Space,
14.8 18.7 16,7 16.5 18.4 17.5
%*
* 1.1 xt 0.9 & 1.2 jz 1.2 zt 0.6
Diff Co&,
X
10gcm2/S*
n
3.4 2.95 3.03 2.95 3.45 3.1
It rt zt * *
0.1 0.3 0.2 0.2 0.1
1 11 8 11 8 39
#
Exp.705:
I .os
time= 131 min t& 12 min timeE247min t ~=11.6min
I .f6
I
i
.24
-32
-40
DISTANCE (X) FROM VENTRICULAR SURFACE IN CM FIG. 1. Semi1ogarithn-k plots of [14C]urea tissue concentration profiles in dog caudate nucleus after 2 different periods of perfusion. Line was drawn to yield a reasonable fit to all of points. RESULTS
For the major portion of the experiments in the study (37 out of 54), sucrose was used as the extracellular marker. Table 1 presents the distribution spaces and the apparent tissue diffusion coefficients which were found for this compound. No significant changes in the spaces or the diffusion coefficients with time are indicated by these results. As mentioned in the introduction, a ventricular Auidto-tissue-to-blood steady state must be reached to measure a capillary transfer coefficient by this technique. In this situation, the amount of labeled material which enters the brain tissue from the ventricular fluid (perfusate and CSF) equals the amount of material that is lost from the tissue ECF to blood. Figure 1 presents data obtained from two experiments in which [14C]urea was the test molecule. The perfusion times were approximately 2 h and 4 h, respectively. Despite the differences in the duration of these two experiments, nearly identical tissue concentration profiles were’ found; therefore, it seems that a tissue steady state for labeled urea is reached at least by 2 h. For some of the other compounds used in this study, no definite tissue steady states were observed. Figure 2 shows tissue concentration profiles obtained with [14C]creatinine in two experiments of differing, approximately 2 h and 4 h, durations. The differences in the two slopes are significant, indicating that a steady state had not been reached by 2 h. From those experiments in which a tissue steady state appeared to be reached, a capillary exchange transfer
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BLOOD-BRAIN
TRANSFER
881
COEFFZCIENTS
constant (k,J and half-time (11/z = ln2/k,i) could be calculated from the semilog plots of the data. Table 2 presents these results for the three compounds which seemed to satisfy the steady-state criteria. As can be seen from these data, the calculated exchange t1&s for water were constant from 1 to 4 h; whereas the results with ethylene glycol and urea indicated constant values from 2 to 4 h. The diffusion coefficients which were used for the calculation of these t&s (or the kPi as in equation 8) were the values for free diffusion in water (D,) at 37”C, i.e., 30 X lo-” cm*/s for water and 15 X lo-” cm2/s for urea and ethylene glycol. As will be discussed later, corrections of the diffusion path length (x) f or tortuosity were applied to equation 8. For three of the test materials, adequate tissue steady states were not found. Complimentary error function plots of these data indicated that the slopes of the tissue concentration profiles increased with time as shown in Fig. 2.
TABLE 3. Apparent tissue dz&sion coeficients (I)) and capillary exchange t&s found af various limes ~-~~----~__l__l-----~ -LLLL---Approximate --__1_1~
Material 2 I
-------..
CreatinineD tll:!
n Sodium
Kmnitol
--
D 2.5 + 0.3 t1/2 38 LIZ 5 n 4
35
--- --
5.5
;;;y;
I_
1.8 It 0.1 1.9zto.22*0+ 29 It 3 41 * 2 6 4 4.1 86 1
b
5
“--_~-.-.-l--l”
2,2 & 0.1 25 zt 1 3
Times,
4
D 6,3ztO.5 h/z 56 xt 3 n 4
I
Perfusion
3
h
117~ 4
0.21.9*0.1 91 dz 3 2 15
0.8
5.1
18
175 dz37 3
+
0.85.6
zt 0.4
12
1.8
-.-
1
Diffusion coefficient V alues are means (X l@ cm2/s) Capillary exchange tl/2 values are means (min) % SE.
+
SE.
From such plots apparent tissue diffusion coefficients were calculated. The results are presented in Table 3. The constancy or near constancy of each particular material’s Dt with time indicates that the tissue concentration profiles continued to change as a function of time and that no tissue steady state was achieved for them within the perfusion periods used in the present work.
A Exp.
J33: time= D=2.2
+ Exp. 716:
,005: ,005 .OOZ.002 .oor .OOll
240min x IO-‘cm’kec
time= 129 min D= 2.2 x IO-’ cm’kec
\
\
r F ’
’ .08
’
’ .I6
DISTANCE
(Xl
’ FROM
’ .24
’
’ -32
’
VENTRICULAR
’ .40
’
SURFACE
’ .48
’
’
IN CM
2. Inverse complimentary error function graph of [14C]creatinine tissue concentration profiles in dog caudate nucleus after 2 different periods of perfusion. Lines are regression analysis fits to separate experiments : ex@riment 133-broken lines, experiment 716-continuous line. FIG.
2. Apparent capillary mchange half-times various perfusion times
(min) found at
TABLE
Approximate
Molecuk. 1
3HzQ
2
Time,
are means
IIZ SE:.
19 (n=
Average of all Times
4
1.4
17 * 3 (n = 4) 16 zt 3 (n=3)
[W]ethylene glycol
h
3
1.6 & 0.4 (n = 3)
[“C]urea
Values
Perfusion
1)
(n 14 (n 17 (n
It = zk = * =
0.2 3) 2 5) 2 3)
1.5 (n 15 (n 17 (n
=t = * = & =
0.2 6) 2 9) 1 7)
After completion of the preliminary group of experiments in this study, the need for some objective or relatively objective way to evaluate the acceptability of the data for a given tissue series became apparent. The criterion chosen for this purpose was the form of the inverse complimentary error function plot of the extracellular marker’s data+ If the points yielded a line which either bent up or down, the results with both the extracellular and the test marker were not included in the reportable data; if, on the other hand, the extracellular material’s points gave a straight line, the data were considered acceptable and included. Initially, about 50 % of our data were eliminated by this test. In our latest set of experiments, a group of 20 which were run in an identical manner over a 2-mo period, 50 out of the 60 tissue series gave linear erfc-l graphs of the extracellular marker data. The experimental points of the companion test materials in this group all yielded either good erfc-1 or semilog plots. For the graphs of the test material’s results in the remaining 10 tissue series, approximately onehalf produced straight-line erfc-l or semilog plots; whereas the other half did not. The application of this criterion, therefore, did not eliminate a very large portion of our results (s of this series) once the experimental technique was perfected and may have, if anything, excluded some “good” tissue data. Although in the previous paragraph we have emphasized the use of the graphs as a means of analyzing the transport mechanism involved in the exchange of material between blood, brain, and CSF during a ventriculocisternal perfusion, such plots in themselves are not clear proof that the material is moving through the tissue ECF solely by diffusion (an erfc-l plot) or that a tissue steady state has been reached (a semilog plot). Because of the scatter which
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882
C. S. PATLAK
is found in the experimental results and the ability of the computer program to force a fit, the same group of data points often yielded fairly good straight lines on both types of graphs and two sets of transport numbers could be calculated. More definitive evidence of the mechanism or mechanisms involved emerges from an examination of the apparent tissue diffusion coefficients at various times (the Dt’s should be constant if diffusion through the ECF is the principal transport process) or the capillary exchange halftimes determined at different times (the 11,2’s should be approximately equal for two different perfusion periods if a true tissue steady state has been reached). Perfusions of varying lengths of time were performed and reported (Tables 1, 2, and 3) in an attempt to more clearly establish and quantitate the transport phenomena involved. The apparent diffusion coefficients and the distribution spaces of sucrose in the caudate nucleus were constant with time. The indicated independence of sucrose’s Dt and distribution space from time is consistent with the assumptions that: I) little or no bulk flow of brain ECF occurs within the caudate nucleus of the dog; 2) extracellular materials move through the ECF of this brain region primarily by diffusion; 3) sucrose remains extracellular in its distribution within the caudate nucleus for up to 5 h, and 4) sucrose exchange between ECF and blood in this portion of the CNS is small. The average (1-5 h) Dt reported in Table l is roughly equal to 45 % of sucrose’s diffusion coefficient in water at 37°C (D, = 7 X 1W6 cm2/s; ref. 17 and this study). A similar reduction in the tissue digusitivity of two other extracellular markers, inulin (9) and EDTA (this study; mean Dt it SE = 3.3 & 0.3 X lo+ cm2/s, N = 7;D, = 6.8 X 10e6 cm2/s) have been found for dog caudate nucleus. This lowering of the apparent D probably is the result of the tortuosity of the extracellular space as has been previously suggested by Levin et al. (17), and not of the restriction to free diffusion imposed by “tight” extracellular channels. The tortuosity factor (X), a measure of the increase in the path length for diffusion, is given by x=dmt.U’gh sin t is equation and a D,/Dt ratio of 2.2 (7.0/3.1), the calculated X for sucrose in the dog caudate nucleus is 1.5. Estimates of X from the previously mentioned inulin (9) and EDTA (this study) data yield values of x = 1.5 and 1.45, respectively, for these two extracellulartype molecules, indicating that the microscopic increase in the extracellular diffusion path in this CNS region is around 50 %. For the calculations of the /cp’s by equation 8, either the D or the x term must be adjusted for the path length. In this study the appropriate correction was applied to the distance factor, x. Equation I indicates that the tissue concentration profile at any time is a function, in part, of the materials’ exchangeability across the capillary complex. To successfully evaluate the transfer constant, k,, during the transient period, all of the other terms in equation I, notably the cellular exchange and/or binding, must also be measured. As the number of constants which are to be determined increases, the number of experiments which must be performed to evaluate them also increases. The tissue steady state provides a way of avoiding this problem in the measurement of k, for materials of moderate-to-high exchangeability. To properly ascertain that a constant tissue con-
AND
J. De FENSTERMACHER
centration profile has been reached and that a capillary transfer coefficient can be measured, the similarity of the slopes and the calculated f 1,~‘s for two relatively divergent times must be clearly demonstrated. This is the essential point in the results presented in Table 2. Computer modeling of quution I indicates that the capillary transfer coefficients which are determined before a true tissue steady state has been reached will be erroneously high, and the related capillary exchange times will be too low. In agreement with this, the results presented in Table 3 show steadily increasing f 1,2’s, which result from changing tissue concentration profiles, for creatinine, sodium, and mannitol. In order to quantitate capillary exchange phenomena, various assumptions are made to take care of unmeasurable or unknown components in the system. (For a comprehensive review of the standard assumptions, see ref* 16.) In the present work the assumption that a steady-state concentration profile exists within the capillary from the arterial to the venous end was made. This assumption would appear to be satisfied in our system since two factors-a steady state and low (l-4 times background) concentrations of test material in the arterial blood-would combine to give constant exchange rates along the capillary and an intracapillary steady state. Such an assumption is a significant problem for the tissue clearance technique (e.g., 6) and would be a problem for single-injection experiments (e.g., 3, 20) i f si g ni fi cant backflux of material from tissue fluid to blood occurs during their respective experimental periods. Radial mixing of the tracer material within the capillaries, another assumption made in this study, is likely to be complete, since the capillary diameter is small, the diffusion coefficients are relatively high (D > 1 .O X lo-” cm2/s), and the blood cells serve as plasma stirrers. In connection with this, Taylor diffusion, which often complicates the single-injection technique by causing the intravascular separation of test and reference substance (15, 28), does not affect the determination of the transfer constant in this study. A uniform concentration of test material in the extracellular fluid around a capillary was also assumed. In the steady-state condition of our system, there is a concentration gradient in the ECF in the x direction; however, since the capillaries are randomly oriented, the diffusion coefficient is of reasonable magnitude (1 .O X 1Om5 cm2/s), and the steady-state gradient is not too steep (exchange 1112> 1 min), the change in ECF concentration across a capillary-tissue unit would not be appreciable for most capillaries. The two terms used in equation 8 to calculate the capillary exchange constant are the diffusion coefficient and the distance from the ventricular surface to a point in the tissue. The D that was used in this study to obtain the values of kpi was the D for the compound in water at 37°C; however, the appropriate value of the distance, x, was harder to ascertain. Since molecules such as urea and ethylene glycol would be expected to move into deeper tissue sites primarily by diffusion through the extracellular space (ECS), the assumption was made that the x should be corrected for the tortuosity of this space and multiplied by 1.5. On the other hand, water enters most cells very rapidly (t I,2 of exchange in human red cells = 4 ms; ref. 23). The movement of this molecule into the caudate nucleus prob-
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BLOOD-BRAIN
TRANSFER
883
COEFFICIENTS
ably occurs at fairly similar rates through both extracellular and intracellular pathways; therefore, the correct value of X lies somewhere between 1.0 and 1.5. For the water results presented in Table 2, the tortuosity factor was assumed to be 1. If in fact the x should be fully corrected for the tortuosity of the ECS (X = 1.5) in the calculation of tritiated water’s k,i (a situation which seems highly unlikely), the tl/ 2 reported in Table 2 would be 2.5 times larger. For metabolically inert substances, which were used in this study, it is likely that the transfer constants are equal and hence the relationship between the transfer constant, k,, and the usual permeability-surface area product, PS, may be found from equation 16. The transfer constant is dependent not only on PS but also F and V,. For this study F is the flow rate of that particular volume of the capillary blood in which the test material distributes during its initial passage through the capillary. Based on Freygang and Sokoloff’s (11) measurement of blood flow to the caudate nucleus in unanesthetized cats, the 1ikeIy values of F range from 0.55 (the plasma water flow rate) to 1.0 ml/g-min (the blood water flow rate). The volume term, V,, refers to the tissue volume or space from which the material exchanges directly with the intracapillary fluid. In most cases V, equals the ECS; however, for substances which move in and out of some or all of the brain cells extremely rapidly, the functional exchanging space would be larger and approach the tissue water space in size. For the caudate nucleus the likely range of the V./F ratio extends from (X3-0.8 min. According to equation 16 and the related discussion, k, equals PS/V, if k,,V,/F < 0.1. Using values of V./F = 0.3 and 0.8, the limits Of ~I/Z which satisfies this condition are 2.1 and 5.6 min, respectively. For either V./F ratio, a much larger tl/ 2 (much smaller kp) was found with urea and ethylene glycol; therefore, for these two compounds, the approximation of PS = k,V, is valid and yields PS g 1.5 X Mm4 cm3/g-s. On the other hand, the t l/2 of [3H]water is below these limits and equation 16 must be used to calculate PS. Taking the water t I/ 2 from Table 2 (E 1.5 min) and assuming that V, = 0.8 cm3/g and F = LO cm3/g-min, the PS product for water is about 0.46 cm3/g-min (7.7 X 1W3 cm3/g-s) . A comparison of the results in this study to other works
on blood-brain exchange can be made for several of the substances. By using the working equation given by Crone (3) for the calculation of P and our equation 16, the following relationship between the fractional loss or extraction (E) as determined in single-injection experiments and k, emerges: E = k&/F. For urea, Crone reported an E of 0.11 in dogs (3) ; in the present study a value of E around one-fifth that of Crone’s was determined. Subsequent work by Yudilevich and DeRose (29) suggested that the E measured by Crone for urea was probably artifactual and mainly caused by Taylor diffusion. Oldendorf’s work (19) also indicates that the permeability of the rat blood-brain barrier to urea is very low (BUI < 2 o/I,; E _< 0.02). Bering (1) reported extremely large exchange rates of water between blood and brain in dogs; using compartmental analysis, gray matter appearance half-times around 12 s were calculated from the data. The sharp fall in the vascular ID20 levels during the first several minutes of the experimental period may account, in part, for the higher permeability figures obtained by Bering for water. Raichle et al. (25) and Eichling et al. (6) have recently reported for rhesus monkeys that water does not freely equilibrate between blood and brain tissue at normal rates of cerebral blood flow and that the uptake of labeled water by the brain is partly limited by the permeability of the bloodbrain barrier. Eichling et al.% estimate of brain capillary PS for H2150 in the monkey (0.019 cm3/g-s) is in fair agreement with the one we calculated for dog caudate nucleus. The relatively low permeability of the brain capillary complex to mannitol and sodium found by ventriculocisternal perfusion is supported by the study of Yudilevich and DeRose (29) with the indicator dilution or single-injection technique in dogs. We are grateful to Ernest S. Owens and Roosevelt Hyman for technical assistance, Qrs. Victor A. Levin and David P, Rall for aid in developing the technique, and Ms. Mary Tyler for secretarial support. Preliminary reports of portions of this work have been presented to the Fifty-fourth Annual Meeting of the Federation of American Societies for Experimental Biology, Atlantic City, N. J., April 1970 and the Sixteenth Annual Meeting of the Biophysical Society, Toronto, Canada, February 1972. Received
for publication
5 November
1974.
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259-277, 1957. AND R. KAPLAN. Diffusion of non-electrolytes brain tissue. Brain Res. 17 : 407-4 16, 1970. 25. RAICHLE, M,, J* EICHLING, AND R. GRUBB. Brain permeability of water. Arch. Neural. 30 : 319-321, 1974. 26. RALL, D., W. OPPELT, AND C. PATLAK. Extracellular space
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