Journal of Neuroscience Methods, 44 (1992) 179-196
179
© 1992 Elsevier Science Publishers B.V. All rights reserved 0165-0270/92/$05.00 NSM 01410
Computer simulation of the blood-brain barrier: a model including two membranes, blood flow, facilitated and non-facilitated diffusion G e o r g e A. Oyler 1, R ober t . B. D uckrow 2 and Ri chard A. Hawkins 3 Departments of 1 Pharmacology and 2 Medicine, The Milton S. Hershey Medical Center, Hershey, PA 17033 (USA) and 3 Department of Physiology and Biophysics, University of Health Sciences / The Chicago Medical School, North Chicago, IL 60064 (USA) (Received 22 April 1992) (Revised version received 7 July 1992) (Accepted 22 July 1992)
Key words: Blood-brain barrier; Substrate transport; Cerebral nutrition; Mathematical modeling; Radioactive tracers; Essential nutrients A mathematical model of blood-brain barrier (BBB) transport was developed to assist in experimental design and data analysis. The model includes the luminal and antiluminal endothelial cell membranes, each with separate transport systems. Substrate movement between 3 compartments can be calculated: the capillary lumen, the endothelial cell cytoplasm, and the brain parenchyma. Blood flow, substrate concentration and competition in each compartment, concentration gradients along the capillary, and non-steady-state conditions are considered. The utility of the model is demonstrated by predicting: (1) complex concentration profiles along the length of the capillary lumen under different circumstances, (2) the permeability-surface area products along the capillary lumen, (3) the time course of events during brain-uptake index (BUI) experiments, (4) the accuracy of the BUI in measuring glucose transport over a range of endogenous glucose concentrations, (5) the influence of 2 membranes in series with different kinetic constants, and (6) a comparison of kinetic constants expected from high-flow infusion and BUI experiments.
Introduction
Brain capillaries consist of a cylindrical layer of endothelial ceils surrounded by an acellular basement membrane and an almost complete sheath of astrocyte processes. The endothelial cells are joined to one another by tight junctions (zonulae occludens) (Reese and Karnovsky, 1967; Brightman and Reese, 1969). This prevents the paracellular movement of hydrophilic molecules
Correspondence:
Richard A. Hawkins, Ph.D., U H S / T h e Chicago Medical School, 3333 Green Bay Road, North Chicago, IL 60064, USA. Tel.: (708) 578-3280; FAX: (708) 578-3265.
and forms a blood-brain barrier (BBB). While lipophilic molecules cross the endothelial cell by diffusion, hydrophilic molecules, which includes most essential nutrients, must be transported by a facilitated process across both membranes. Several carrier systems have been identified (Pardridge, 1983). They are stereospecific and do not require energy or Na ÷ gradients. Most kinetic analyses of the transport systems assumed steady-state conditions and treated the endotheldial cell barrier as a single membrane. These analyses provided useful knowledge, but further understanding of the BBB requires a more complete model of the endothelial cell barrier that includes the effect of transport across both endothelial cell membranes under non-steady-state conditions.
180
Pappenheimer and Setchell (1973) recognized the implications of a barrier consisting of 2 membranes in series separated by a cytoplasmic volume and a changing capillary concentration of tracer. They described a circumstance in which the Vm~x and K t for transport were identical in each membrane and represented the capillary concentration as an exponential decrease. Gjedde and Christensen (1984) developed a more comprehensive description of the kinetics of a 2membrane BBB. Their model was restricted to steady-state conditions and did not include the influence of endogenous substrates. Gjedde and Christensen described how 2 membranes with different Vm~X could affect the apparent kinetics of BBB transport. They concluded from an evaluation of the available experimental data that the glucose transport carrier was symmetrical, as had been assumed by Pappenheimer and Setchell. Cunningham et al. (1986b) derived more elaborate analytical solutions to equations describing the kinetics of a double-membrane BBB. They considered each membrane as having either simple irreversible Michaelis-Menten kinetics, assuming independent influx and efflux, or re-
versible Michaelis-Menten kinetics corresponding to stereospecific membrane pores. Their model also allowed for changes in tissue glucose content and back-diffusion of tracer during the experiments. We have developed a more dynamic (nonsteady state) model of the BBB than previous models. The model combines many of the features of previous double-membrane models of BBB transport with more complex features of capillary exchange developed for other organs (Sangren and Sheppard, 1953; Bassingthwaighte et al., 1970; Goresky et al., 1970). The model is illustrated by the application to several biological phenomena of current interest.
Model The model considers 3 compartments: the capillary lumen, the endothelial cell and the brain parenchyma (Fig. 1) between which substrate movement can be calculated. The compartments are separated from each other by the luminal and antiluminal endothelial cell membranes, which
Cerebral Tissue-compartment 3
; I
1
F,ow ...>
.-
Endothelial Cell-compartment 2
/
•
; I
/
,""
a
(
Antiluminal Membrane Luminal Membrane
Capillary" Lumen-oompar;ment 1 [
Element(I,1) \, Element(1,2) x,,.
,
~
j
Element(2,1)
i Element(3,1) !
I
,,i,
I !
i !
I
1 I
Fig. 1. Diagram of the capillary-tissue system. The capillary-tissue column is divided into 3 compartments by the luminal and antiluminal membranes of the endothelial cell. The capillary-tissue column is further divided along its length into volume elements for the purpose of numerical calculations (shown by broken lines). Facilitated transport across the luminal membrane is shown as arrows a and b. Transport across the antiluminal membrane is depicted as arrows c and d. Metabolism of substrate is indicated by e. The distance from the center of the capillary to the luminal membrane is indicated by r,. and the distance from the center of the capillary to the antiluminal membrane by r~.
181
have separate transport systems that are independent of each other. Kinetic parameters are assigned to each side of the luminal and antiluminal membranes. Blood flow, arterial substrate concentration, and the initial substrate concentrations in the other compartments are also assigned. To assess the effects of 2 endothelial cell membranes more accurately, the model calculates the exchange of material between the 3 compartments on a moment-to-moment basis. Thus the concentrations of substrate can be predicted in each of the 3 compartments and for any point along the length of the capillary at any time. While Michaelis-Menten kinetics were used to illustrate the utility of the model hereinafter it is not restricted to this particular mechanism: other kinetic descriptions could be assigned to each membrane as desired. A complete explanation of the model is given in the following section along with a glossary of terms. However, the reader who is less interested in details could go directly to the Results and discussion section without loss of continuity.
Glossary A element(i, j)
Cross-sectional area of the capillary lumen. A physical volume-element in the capillary-tissue system where the position is designated by the indices i and j; i indicates the compartment ranging from 1 to 3 (1 = capillary, 2 = endothelial cell, 3 = brain parenchyma) and j indicates the position along the length of the capillary ranging from 1 to
Ux.
c(i, j) C *(i, j) Cart
Concentration of total substrate (labeled and unlabeled) in the element(i, j ) (/zmol/ml). Concentration of tracer present in the element(i, j ) ( ~ m o l / m l ) . Concentration of total substrate at the arterial entrance of the capillary. Cart and C(1, 1) will have the same value (/zmol/ml).
Concentration of labeled tracer substrate at the arterial entrance of the capillary (pmol/ml). Concentration of the total subCve. strate at the venous exit of the capillary ( ~ m o l / m l ) . Concentration of tracer at the veCv~ n o u s exit o f the c a p i l l a r y (pmol/ml). C(t) Mean concentration of substrate me,m in the capillary calculated by taking the arithmetic mean of the values of the concentration of substrate in each element of the capillary compartment at each iteration in time ( ~ m o l / m l ) . Mean concentration of substrate C'mea. in the capillary averaged over time (~mol/ml). D Diffusion constant (cmZ/min). dq(i, j ) / d t The rate of change in the quantity of total substrate in an element(i, j). Values of dq(i, j ) / d t are calculated from mass balances around each element. dq *(i, j ) / d t Rate of change in the quantity of tracer in an element(i, j). E Extraction of a substrate during a single pass through a capillary. By definition E = (Car t - Cven)Qr t. F Brain blood flow rate ( m l / m i n / g of brain). F~ Capillary blood flow ( m l / m i n / capillary). Michaelis-Menten constant assigned to the luminal endothelial cell membrane (/zmol/ml). Michaelis-Menten constant of the antiluminal membrane of the endothelial cell (/~mol/ml). K am pp Apparent Michaelis-Menten constant for net transport through both membranes of the BBB (~mol/ml). Kd Permeability constant for a nonsaturating component of BBB transport ( m l / m i n / g ) . Lc Mean length of a single capillary Car*
182
Uc
PA Q(i, j) Q *(i, j) rc
re
rt
t
tin fusion
T
Vm'ax V ax
in rat brain. A value of 410 /~m was used. This value was calculated by dividing a mean length of 820 m of capillary/g of rat cortex by N~, the number of capillaries/g (Craigie, 1921). Number of individual capillaries/ g of rat brain. A value of 2,000,000 was used. Number of elements into which the capillary was divided. Number of iterations in time during a modelled experiment. Permeability-surface area product ( m l / m i n / g ) . Total quantity of substrate in the element(i, j). Quantity of tracer in element(i, j). Mean capillary radius measured from the central axis of the caprilary to the luminal membrane of the endothelial cell. A value of 2.40 ~ m was used (Laursen and Diemer, 1980). Mean radius from the central axis of a rat brain capillary to the antiluminal membrane of the endothelial cell. A value of 2.60/~m was used. The thickness of the endothelial cell is equal to r e - r c = 0.20 ~zm. Radius of a tissue column taken to be the distance from the central axis of the capillary to the midpoint on a side of the hexagonal cross-section of the tissue column. A value of 19.8 # m was used. Time during experiment. Length of time tracer is entering the capillary (min). Time at end of experiment. Maximum velocity of transport across the luminal membrane of the BBB (t~mol/min/g). Maximum velocity of transport across the antiluminal membrane of the BBB ( # m o l / m i n / g ) .
Apparent maximum velocity of transport across the BBB as determined from the net velocity as a function of substrate concentration data predicted by the model (/.Lmol/min/g). The brain is considered to be assembled from a parallel series of hexagonal columns of tissue surrounding capillaries which pass through the long axis of the column. Thus, the tissue columns of the model are similar to Krogh cylinders (Krogh, 1919). The length of a tissue column is equal to the mean length of a brain capillary. Flow through the capillaries of the individual columns is considered to be concurrent. By assuming concurrent flow and homogeneity of capillary-tissue column dimensions, the concentrations of diffusible materials will be equivalent at the tissue column boundaries. Therefore, no net exchange of material will occur between tissue columns, and the behavior of the entire brain could be modeled by the response of a single capillary-tissue column. The assumption of a parallel system of capillaries is perhaps the simplest system to describe transport processes in the brain. Nevertheless, it should be recognized that the structure of capillaries in brain is considerably more complex. Please see the concluding comments. ~Zmapp ax
Compartments The tissue-capillary system is divided into 3 compartments separated by membranes. Compartment 1 contains the volume within the capillary lumen. The boundaries are at the arterial and venous ends of the capillary and the luminal membrane of the endothelial cell (see Fig. 1). The volume of compartment 1 is: vol 1 = L c ' ~ - ' r ~
(1)
where rc is the radius of a brain capillary and L~ is the capillary length. The parameters of this model are calculated on a basis of 1 g of brain. The volume of the capillary c o m p a r t m e n t / g of brain is: vol (1) = L~'w'r2c'Nc
(2)
where N c is the number of capillaries/g of rat brain.
183 would be considered. W i t h such g e o m e t r y the v o l u m e of the third c o m p a r t m e n t / g of tissue is:
T h e second c o m p a r t m e n t is the region bet w e e n the l u m i n a l a n d a n t i l u m i n a l m e m b r a n e s of the e n d o t h e l i a l cell. Its v o l u m e is: vol (2) = L c ' T r " ( r ~ - r 2) "N c
vol(3)=[2(3)'/Z.rZ-~.rZ~].Lc.Nc
(3)
w h e r e r t is the m e a n intercapillary radius in b r a i n tissue.
w h e r e r e is the distance from the long axis of the capillary to the a n t i l u m i n a l m e m b r a n e . T h e third c o m p a r t m e n t ( p a r e n c h y m a l region) extends from the a n t i l u m i n a l m e m b r a n e of the e n d o t h e l i a l cell to the intercapillary b o u n d a r y . A hexagonal o u t e r cross-section was chosen so that w h e n the tissue c o l u m n s were a s s e m b l e d all space
Radial diffusion If substrate removal were fast in r e l a t i o n to diffusion, it would be possible to set u p diffusion gradients. T h e s e g r a d i e n t s would r u n from the c e n t e r axis of the capillaries, the site of the
VeRou8 Exit J:3
Arterial Entrance J=l
C(1,1)
(4)
1
2
C( I ,J),
C( 1
Capillary I = 1 4 ib
m
5
Endothelial Cell i : 2
C(2,1)
7
4_
m
Luminal Membrane
0
8
---- C ( 2 , J )
---- C ( 2 . 3 )
I I
~_Antlluminal , Membrane
I
lO Cerebral Tissue i = 3
11 C(3,1)
12 C(3,j)
C(3,3)
Fig. 2. The capillary-tissue column presented diagrammatically as a matrix of volume elements. The capillary-tissue system can be represented abstractly as elements in a matrix where each element corresponds to a physical volume. The index i of the matrix determines the compartment of the element and the index j denotes the position along the capillary. The concentration in an element can be determined by mass balances around the element. Mass-transfer vectors 1 and 2 represent flow of substrate into and out of element(i, j). Vectors 3 and 4 represent diffusion into and out of element(i, j). Vectors 5 and 6 represent facilitated transport across the luminal membrane. Vectors 7 and 8 represent diffusion into and out of element (2, j). Vectors 9 and 10 represent facilitated transport across the antiluminal membrane with potentially different kinetic constants (e.g., K m and Vm,x). Vectors 11 and 12 represent diffusion into and out of element (3, j). The capillary was divided into 3 divisions along the length only for the purpose of illustration; in simulated experiments Nx was larger.
184
highest potential concentration, to the luminal membrane, the site of substrate removal. These gradients would, therefore, create what is known as 'unstirred layers' (Barry and Diamond, 1984). The effect of unstirred layers increases with the distance from the center axis to the luminal membrane. However, the distance in cerebral capillaries is small (2-3 p.m) in relation to diffusion and the radial concentration gradients were calculated to be inconsequential even for the most rapid facilitated transport system known (Barry and Diamond, 1984). The possibility of radial gradients were, therefore, ignored. Others have demonstrated that even where radial diffusion distances are much larger (e.g., hepatic sinusoids) the radial concentration gradients are negligible (Bass et al., 1976).
Diffusion barriers The only barriers to transport considered were the luminal and antiluminal endothelial cell membranes. Once past the antiluminal membrane a molecule was assumed to have entered a homogeneous tissue compartment where gradients were neither created by substrate consumption nor production. Brain parenchyma equilibrates relatively rapidly with the extracellular fluid of the brain (Lund-Andersen and Kjeldsen, 1977), and diffusion through the extracellular spaces in the intact brain is rapid compared with BBB (luminal and antiluminal membranes) transport (Diemer, 1968).
Matrices A variation of the finite difference analysis was applied to allow a numerical solution of the equations. After dividing the capillary-tissue system into 3 compartments, it was further divided into N x divisions along its length. All calculations were performed on elements of 3 by N x matrices where each matrix element corresponds to a physical volume element of the capillary-tissue system (see Fig. 1). The ith dimension of the matrix, varying from 1 to 3, represents the capillary, endothelial and tissue compartments, respectively. The jth index, ranging from 1 to N x represents the axial position of an element along the length of the capillary-tissue system.
Vectors: unlabeled substrate The exchange of material between individual elements was determined by mass balances around each element. The mass balances are represented schematically in Fig. 2 where net flux is represented as mass-transfer vectors. The mass balances were calculated over time in small increments to model substrate movement. Substrates enter the capillary-tissue system through flow into the capillary. Substrates are lost from the system through flow out of the capillary and consumption within the tissue compartment. Mass-transfer vector 1 represents the flow of substrate from the ( j - 1) element in the capillary compartment (i = 1) into the jth element of compartment 1 and is described by the equation: vector 1 d q ( l , j ) / d t = F c . C ( a , j -
1)
(5)
where Fc is the rate of plasma flow/capillary and C(1, j - 1 ) is the concentration of substrate in the ( j - 1) element of the capillary compartment. Fc can be determined by dividing the rate of plasma flow/g of brain, F, by the number of capillaries/g, N c. (The model was applied to situations in which erythrocytes were not present. Consequently the effect of erythrocytes was not considered.) Vector 1, flow into element(l, j) is accompanied by vector 2, the flow of substrate from element(l, j) into element(l, j + 1). This second vector may be expressed as: vector 2 dq(1, j ) / d t = - F ~ ' C ( 1 , j)
(6)
Substrate can also enter and leave element(l, j) by diffusion along concentration gradients. Axial diffusion within the lumen from element(l, j - I) into element(l, j) is described by Fick's law: vector 3 dq( 1, j ) / d t
=D.A/AX[C(1,
j-1)-C(1,
j)]
(7)
where A is the cross-sectional diameter of the capillary lumen, AX is the length of 1 axial division and D is the diffusion constant. The length of an axial division is equal to the total length of the capillary divided by the number of axial divisions.
aX = Lc/N X
(8)
185 Substituting the appropriate values: vector 3 dq(1,
j)/dt
=O" [~r'r2l[Nx/Lc] × [C(1, j vector 4 d q ( 1 ,
1) - C(1, j ) ]
(9)
j)/dt
= -D.[Tr.r2][Nx/L~] × [C(1, j ) - C(1, j + 1)]
(10)
Vectors 3 and 4 represent the transfer of mass by axial diffusion in and out of element(l, j), respectively. Substrates move out of the capillary luminal compartment across the luminal membrane by facilitated transport systems. The transport systems in each membrane were considered to transport substrate with the same kinetics in either direction. Each membrane was assigned kinetic constants that were independent of the other membrane. Transport processes across both membranes were assigned 2 components: a saturable component obeying Michaelis-Menten kinetics and a non-saturable component. Thus:
dq/dt=Vmax. C / ( K m + C ) + K d . C
(ll)
The kinetics constants Vma* and K d were scaled to the area of luminal membrane of an individual element. The Michaelis-Menten constant K m is a property of the transport system and did not require scaling. Mass-transport vector 5 represents the passage of substrate across the luminal membrane from element(l, j) into element(2, j ) in the endothelial compartment. vector 5 d q ( 1 ,
j)/dt
= --[Vmlax" C(1,
j)/(Klm + C(1, j ) )
+ K d "C(1, j ) ]
(12)
Vector 6 represents transport from the endothelial cell compartment element(2, j) into the capillary element(l, j): vector 6 d q ( 1 ,
j)/dt
1 = V~a x "C(2,
j)/(Klm + C(2, j ) )
+ K d " C(2, j )
The mass balance for all elements(l, j) was calculated by summing the rate of change described by vectors 1-6. Similar calculations were made for all the elements of the endothelial cell (compartment 2) and brain parenchymal tissue (compartment 3). A mass balance on element(2, j) involved the summing of vectors 5-10 with respect to element(2, j). Vectors 7 and 8 express axial diffusion in the endothelial compartment, and vectors 9 and 10 represent transport of substrate across the antiluminal membrane. The equations for vectors 7 and 8 are identical to the axial diffusion equations for vectors 3 and 4, except that the endothelial cross-sectional area and diffusion constant were substituted. The form of vectors 9 and 10 was the same as for luminal membrane transport, except that the Vaax and K a values of the antiluminal membrane were used.
j)/dt = - [Vaax "C(2, j ) / ( K a + C(2, j ) )
vector 9 dq(2,
+ K d "C(2, j)] vector 10 d q ( 2 ,
j)/dt
= V~ax . C ( 3 , j)//(g~n q- C(3, j ) ) + K o • C(3, j )
(15)
Axial diffusion (diffusion parallel to the long axis) in the tissue compartment may be considered by 2 additional vectors 11 and 12. The form of the equations for vectors 11 and 12 is the same as those for vectors 3 and 4 but with the diffusion constant and cross-sectional area of the tissue compartment substituted. After calculating the rate of change of substrate in each element (dq(i, j)/dt) the absolute change (AQ(i, j)) was calculated per increment of time At by:
AQ(i, j ) = dq(i, j ) / d t . At
(16)
The total quantity of substrate in element(i, j) after each At was calculated by adding the change in quantity to the previous amount of substrate in the element: Qnew(i, j ) = Qola(i,
(13)
(14)
j) +AQ(i, j)
(17)
A new set of substrate concentration values in
186
each element was found by dividing the quantity of substrate by the volume of that element. These concentration values were used in the mass-balance equations to determine another set of dq(i, j ) / d t values. Thus, the program proceeded through time by iteratively calculating dq(i, j ) / d t for each element(i, j) based on the existing concentrations. The time increment was set according to the volume of an individual capillary element divided by the capillary flow rate.
At = v o l ( 1 ) / ( F c .Ux)
(18)
Thus At represented the time required for flow to fill 1 capillary element. Time at any point was calculated by summing the time increments. The initial conditions of the numerical solutions were established by assigning values to the concentration array (thereby setting the initial condition values for the quantity array as well). The concentration in the first element at the arterial end of the capillary compartment, C(1, 1), was set equal to the concentration of substrate in the injection or infusion (an assigned value in the model). The remaining elements of the capillary compartment and all endothelial and parenchymal tissue elements were assigned initial concentrations of zero or a value corresponding to the endogenous concentration of substrate. The boundary condition applied to the solution was established by assigning values to the concentration C(1, 1) at the start of each iteration in time.
in which competition between labeled tracer and unlabeled substrate was taken into account. The flux of labeled substrate across a membrane is calculated by multiplying the rate of transport of total substrate by the fraction of total substrate that is labeled. Thus transport from an element(i, j) in the capillary compartment to an element(2, j) in the endothelial compartment (vector 5 in Fig. 2) would be described by: dq *(1, j ) / d t = -[(Vmla,," C(1, j ) ) / ( K l, + C(1, j ) ) + K d . C ( 1 , j)]C *(1, j ) / C ( 1 , j)
(19)
where the quantity C *(1, j)/C(1, j) represents the quantity of total substrate that is labeled. Similar equations can be written for the remaining vectors 6, 9 and 10 in Fig. 2. A different set of initial and boundary conditions were applied for tracer. Because tracer cannot be endogenous to the endothelial and tissue compartments, the tracer concentrations in those compartments were initially set to zero. The concentration of tracer at the entrance of the capillary C *(1, 1) was initially set equal to the concentration of labeled tracer in the fluid entering the capillary. The boundary condition value of C *(1, 1) was set equal to the concentration of tracer in fluid entering the capillary as long as it continued to enter. When tracer is no longer entering the capillary, C *(1, 1) was set to zero.
Vectors: tracers In experimental techniques measuring transport kinetics, a radiolabeled tracer is used in conjunction with varying concentrations of unlabeled substrates. To model such experiments, the movement and distribution of labeled tracer substrate was considered separately from the movement of total substrate. Separate arrays were established for the concentration, rate of change of quantity, and quantity of tracer substrate in each element of the capillary-tissue system. These arrays were designated C *(i, j), dq *(i, j)/dt, and Q *(i, j) respectively. The mass balances applied to tracer substrate were essentially the same as those used for total substrate with the exception of the membrane transport equations
Calculation of net extraction The calculations were stopped when the elapsed time was equal to the experimental time, and the concentration and quantity arrays from the final iteration were retained. Extraction from the capillary was calculated by summing the quantities of tracer in each element to determine the total quantity of tracer in the system at the end of the experiment and dividing by the total quantity of label that had entered by flow during the experiment. Thus:
E=
s: Q * / F .
C.r*t ( t ) d t
(20)
When the value of Car* is constant over one
187
infusion period and is zero when the infusion ends, Eqn. 20 can be simplified to:
E
=
Q */F.
C a r *" t tinfusion
(20.1)
The extraction can be used to determine the apparent average velocity of transport through BBB as: average velocity = ( E . F . Car* ) = Q
*/tinfusion (21)
The average capillary concentration at each iteration was calculated by summing the C(1, j) values from j = 1 to N x and dividing by N x.
j=Ux C
179
© 1992 Elsevier Science Publishers B.V. All rights reserved 0165-0270/92/$05.00 NSM 01410
Computer simulation of the blood-brain barrier: a model including two membranes, blood flow, facilitated and non-facilitated diffusion G e o r g e A. Oyler 1, R ober t . B. D uckrow 2 and Ri chard A. Hawkins 3 Departments of 1 Pharmacology and 2 Medicine, The Milton S. Hershey Medical Center, Hershey, PA 17033 (USA) and 3 Department of Physiology and Biophysics, University of Health Sciences / The Chicago Medical School, North Chicago, IL 60064 (USA) (Received 22 April 1992) (Revised version received 7 July 1992) (Accepted 22 July 1992)
Key words: Blood-brain barrier; Substrate transport; Cerebral nutrition; Mathematical modeling; Radioactive tracers; Essential nutrients A mathematical model of blood-brain barrier (BBB) transport was developed to assist in experimental design and data analysis. The model includes the luminal and antiluminal endothelial cell membranes, each with separate transport systems. Substrate movement between 3 compartments can be calculated: the capillary lumen, the endothelial cell cytoplasm, and the brain parenchyma. Blood flow, substrate concentration and competition in each compartment, concentration gradients along the capillary, and non-steady-state conditions are considered. The utility of the model is demonstrated by predicting: (1) complex concentration profiles along the length of the capillary lumen under different circumstances, (2) the permeability-surface area products along the capillary lumen, (3) the time course of events during brain-uptake index (BUI) experiments, (4) the accuracy of the BUI in measuring glucose transport over a range of endogenous glucose concentrations, (5) the influence of 2 membranes in series with different kinetic constants, and (6) a comparison of kinetic constants expected from high-flow infusion and BUI experiments.
Introduction
Brain capillaries consist of a cylindrical layer of endothelial ceils surrounded by an acellular basement membrane and an almost complete sheath of astrocyte processes. The endothelial cells are joined to one another by tight junctions (zonulae occludens) (Reese and Karnovsky, 1967; Brightman and Reese, 1969). This prevents the paracellular movement of hydrophilic molecules
Correspondence:
Richard A. Hawkins, Ph.D., U H S / T h e Chicago Medical School, 3333 Green Bay Road, North Chicago, IL 60064, USA. Tel.: (708) 578-3280; FAX: (708) 578-3265.
and forms a blood-brain barrier (BBB). While lipophilic molecules cross the endothelial cell by diffusion, hydrophilic molecules, which includes most essential nutrients, must be transported by a facilitated process across both membranes. Several carrier systems have been identified (Pardridge, 1983). They are stereospecific and do not require energy or Na ÷ gradients. Most kinetic analyses of the transport systems assumed steady-state conditions and treated the endotheldial cell barrier as a single membrane. These analyses provided useful knowledge, but further understanding of the BBB requires a more complete model of the endothelial cell barrier that includes the effect of transport across both endothelial cell membranes under non-steady-state conditions.
180
Pappenheimer and Setchell (1973) recognized the implications of a barrier consisting of 2 membranes in series separated by a cytoplasmic volume and a changing capillary concentration of tracer. They described a circumstance in which the Vm~x and K t for transport were identical in each membrane and represented the capillary concentration as an exponential decrease. Gjedde and Christensen (1984) developed a more comprehensive description of the kinetics of a 2membrane BBB. Their model was restricted to steady-state conditions and did not include the influence of endogenous substrates. Gjedde and Christensen described how 2 membranes with different Vm~X could affect the apparent kinetics of BBB transport. They concluded from an evaluation of the available experimental data that the glucose transport carrier was symmetrical, as had been assumed by Pappenheimer and Setchell. Cunningham et al. (1986b) derived more elaborate analytical solutions to equations describing the kinetics of a double-membrane BBB. They considered each membrane as having either simple irreversible Michaelis-Menten kinetics, assuming independent influx and efflux, or re-
versible Michaelis-Menten kinetics corresponding to stereospecific membrane pores. Their model also allowed for changes in tissue glucose content and back-diffusion of tracer during the experiments. We have developed a more dynamic (nonsteady state) model of the BBB than previous models. The model combines many of the features of previous double-membrane models of BBB transport with more complex features of capillary exchange developed for other organs (Sangren and Sheppard, 1953; Bassingthwaighte et al., 1970; Goresky et al., 1970). The model is illustrated by the application to several biological phenomena of current interest.
Model The model considers 3 compartments: the capillary lumen, the endothelial cell and the brain parenchyma (Fig. 1) between which substrate movement can be calculated. The compartments are separated from each other by the luminal and antiluminal endothelial cell membranes, which
Cerebral Tissue-compartment 3
; I
1
F,ow ...>
.-
Endothelial Cell-compartment 2
/
•
; I
/
,""
a
(
Antiluminal Membrane Luminal Membrane
Capillary" Lumen-oompar;ment 1 [
Element(I,1) \, Element(1,2) x,,.
,
~
j
Element(2,1)
i Element(3,1) !
I
,,i,
I !
i !
I
1 I
Fig. 1. Diagram of the capillary-tissue system. The capillary-tissue column is divided into 3 compartments by the luminal and antiluminal membranes of the endothelial cell. The capillary-tissue column is further divided along its length into volume elements for the purpose of numerical calculations (shown by broken lines). Facilitated transport across the luminal membrane is shown as arrows a and b. Transport across the antiluminal membrane is depicted as arrows c and d. Metabolism of substrate is indicated by e. The distance from the center of the capillary to the luminal membrane is indicated by r,. and the distance from the center of the capillary to the antiluminal membrane by r~.
181
have separate transport systems that are independent of each other. Kinetic parameters are assigned to each side of the luminal and antiluminal membranes. Blood flow, arterial substrate concentration, and the initial substrate concentrations in the other compartments are also assigned. To assess the effects of 2 endothelial cell membranes more accurately, the model calculates the exchange of material between the 3 compartments on a moment-to-moment basis. Thus the concentrations of substrate can be predicted in each of the 3 compartments and for any point along the length of the capillary at any time. While Michaelis-Menten kinetics were used to illustrate the utility of the model hereinafter it is not restricted to this particular mechanism: other kinetic descriptions could be assigned to each membrane as desired. A complete explanation of the model is given in the following section along with a glossary of terms. However, the reader who is less interested in details could go directly to the Results and discussion section without loss of continuity.
Glossary A element(i, j)
Cross-sectional area of the capillary lumen. A physical volume-element in the capillary-tissue system where the position is designated by the indices i and j; i indicates the compartment ranging from 1 to 3 (1 = capillary, 2 = endothelial cell, 3 = brain parenchyma) and j indicates the position along the length of the capillary ranging from 1 to
Ux.
c(i, j) C *(i, j) Cart
Concentration of total substrate (labeled and unlabeled) in the element(i, j ) (/zmol/ml). Concentration of tracer present in the element(i, j ) ( ~ m o l / m l ) . Concentration of total substrate at the arterial entrance of the capillary. Cart and C(1, 1) will have the same value (/zmol/ml).
Concentration of labeled tracer substrate at the arterial entrance of the capillary (pmol/ml). Concentration of the total subCve. strate at the venous exit of the capillary ( ~ m o l / m l ) . Concentration of tracer at the veCv~ n o u s exit o f the c a p i l l a r y (pmol/ml). C(t) Mean concentration of substrate me,m in the capillary calculated by taking the arithmetic mean of the values of the concentration of substrate in each element of the capillary compartment at each iteration in time ( ~ m o l / m l ) . Mean concentration of substrate C'mea. in the capillary averaged over time (~mol/ml). D Diffusion constant (cmZ/min). dq(i, j ) / d t The rate of change in the quantity of total substrate in an element(i, j). Values of dq(i, j ) / d t are calculated from mass balances around each element. dq *(i, j ) / d t Rate of change in the quantity of tracer in an element(i, j). E Extraction of a substrate during a single pass through a capillary. By definition E = (Car t - Cven)Qr t. F Brain blood flow rate ( m l / m i n / g of brain). F~ Capillary blood flow ( m l / m i n / capillary). Michaelis-Menten constant assigned to the luminal endothelial cell membrane (/zmol/ml). Michaelis-Menten constant of the antiluminal membrane of the endothelial cell (/~mol/ml). K am pp Apparent Michaelis-Menten constant for net transport through both membranes of the BBB (~mol/ml). Kd Permeability constant for a nonsaturating component of BBB transport ( m l / m i n / g ) . Lc Mean length of a single capillary Car*
182
Uc
PA Q(i, j) Q *(i, j) rc
re
rt
t
tin fusion
T
Vm'ax V ax
in rat brain. A value of 410 /~m was used. This value was calculated by dividing a mean length of 820 m of capillary/g of rat cortex by N~, the number of capillaries/g (Craigie, 1921). Number of individual capillaries/ g of rat brain. A value of 2,000,000 was used. Number of elements into which the capillary was divided. Number of iterations in time during a modelled experiment. Permeability-surface area product ( m l / m i n / g ) . Total quantity of substrate in the element(i, j). Quantity of tracer in element(i, j). Mean capillary radius measured from the central axis of the caprilary to the luminal membrane of the endothelial cell. A value of 2.40 ~ m was used (Laursen and Diemer, 1980). Mean radius from the central axis of a rat brain capillary to the antiluminal membrane of the endothelial cell. A value of 2.60/~m was used. The thickness of the endothelial cell is equal to r e - r c = 0.20 ~zm. Radius of a tissue column taken to be the distance from the central axis of the capillary to the midpoint on a side of the hexagonal cross-section of the tissue column. A value of 19.8 # m was used. Time during experiment. Length of time tracer is entering the capillary (min). Time at end of experiment. Maximum velocity of transport across the luminal membrane of the BBB (t~mol/min/g). Maximum velocity of transport across the antiluminal membrane of the BBB ( # m o l / m i n / g ) .
Apparent maximum velocity of transport across the BBB as determined from the net velocity as a function of substrate concentration data predicted by the model (/.Lmol/min/g). The brain is considered to be assembled from a parallel series of hexagonal columns of tissue surrounding capillaries which pass through the long axis of the column. Thus, the tissue columns of the model are similar to Krogh cylinders (Krogh, 1919). The length of a tissue column is equal to the mean length of a brain capillary. Flow through the capillaries of the individual columns is considered to be concurrent. By assuming concurrent flow and homogeneity of capillary-tissue column dimensions, the concentrations of diffusible materials will be equivalent at the tissue column boundaries. Therefore, no net exchange of material will occur between tissue columns, and the behavior of the entire brain could be modeled by the response of a single capillary-tissue column. The assumption of a parallel system of capillaries is perhaps the simplest system to describe transport processes in the brain. Nevertheless, it should be recognized that the structure of capillaries in brain is considerably more complex. Please see the concluding comments. ~Zmapp ax
Compartments The tissue-capillary system is divided into 3 compartments separated by membranes. Compartment 1 contains the volume within the capillary lumen. The boundaries are at the arterial and venous ends of the capillary and the luminal membrane of the endothelial cell (see Fig. 1). The volume of compartment 1 is: vol 1 = L c ' ~ - ' r ~
(1)
where rc is the radius of a brain capillary and L~ is the capillary length. The parameters of this model are calculated on a basis of 1 g of brain. The volume of the capillary c o m p a r t m e n t / g of brain is: vol (1) = L~'w'r2c'Nc
(2)
where N c is the number of capillaries/g of rat brain.
183 would be considered. W i t h such g e o m e t r y the v o l u m e of the third c o m p a r t m e n t / g of tissue is:
T h e second c o m p a r t m e n t is the region bet w e e n the l u m i n a l a n d a n t i l u m i n a l m e m b r a n e s of the e n d o t h e l i a l cell. Its v o l u m e is: vol (2) = L c ' T r " ( r ~ - r 2) "N c
vol(3)=[2(3)'/Z.rZ-~.rZ~].Lc.Nc
(3)
w h e r e r t is the m e a n intercapillary radius in b r a i n tissue.
w h e r e r e is the distance from the long axis of the capillary to the a n t i l u m i n a l m e m b r a n e . T h e third c o m p a r t m e n t ( p a r e n c h y m a l region) extends from the a n t i l u m i n a l m e m b r a n e of the e n d o t h e l i a l cell to the intercapillary b o u n d a r y . A hexagonal o u t e r cross-section was chosen so that w h e n the tissue c o l u m n s were a s s e m b l e d all space
Radial diffusion If substrate removal were fast in r e l a t i o n to diffusion, it would be possible to set u p diffusion gradients. T h e s e g r a d i e n t s would r u n from the c e n t e r axis of the capillaries, the site of the
VeRou8 Exit J:3
Arterial Entrance J=l
C(1,1)
(4)
1
2
C( I ,J),
C( 1
Capillary I = 1 4 ib
m
5
Endothelial Cell i : 2
C(2,1)
7
4_
m
Luminal Membrane
0
8
---- C ( 2 , J )
---- C ( 2 . 3 )
I I
~_Antlluminal , Membrane
I
lO Cerebral Tissue i = 3
11 C(3,1)
12 C(3,j)
C(3,3)
Fig. 2. The capillary-tissue column presented diagrammatically as a matrix of volume elements. The capillary-tissue system can be represented abstractly as elements in a matrix where each element corresponds to a physical volume. The index i of the matrix determines the compartment of the element and the index j denotes the position along the capillary. The concentration in an element can be determined by mass balances around the element. Mass-transfer vectors 1 and 2 represent flow of substrate into and out of element(i, j). Vectors 3 and 4 represent diffusion into and out of element(i, j). Vectors 5 and 6 represent facilitated transport across the luminal membrane. Vectors 7 and 8 represent diffusion into and out of element (2, j). Vectors 9 and 10 represent facilitated transport across the antiluminal membrane with potentially different kinetic constants (e.g., K m and Vm,x). Vectors 11 and 12 represent diffusion into and out of element (3, j). The capillary was divided into 3 divisions along the length only for the purpose of illustration; in simulated experiments Nx was larger.
184
highest potential concentration, to the luminal membrane, the site of substrate removal. These gradients would, therefore, create what is known as 'unstirred layers' (Barry and Diamond, 1984). The effect of unstirred layers increases with the distance from the center axis to the luminal membrane. However, the distance in cerebral capillaries is small (2-3 p.m) in relation to diffusion and the radial concentration gradients were calculated to be inconsequential even for the most rapid facilitated transport system known (Barry and Diamond, 1984). The possibility of radial gradients were, therefore, ignored. Others have demonstrated that even where radial diffusion distances are much larger (e.g., hepatic sinusoids) the radial concentration gradients are negligible (Bass et al., 1976).
Diffusion barriers The only barriers to transport considered were the luminal and antiluminal endothelial cell membranes. Once past the antiluminal membrane a molecule was assumed to have entered a homogeneous tissue compartment where gradients were neither created by substrate consumption nor production. Brain parenchyma equilibrates relatively rapidly with the extracellular fluid of the brain (Lund-Andersen and Kjeldsen, 1977), and diffusion through the extracellular spaces in the intact brain is rapid compared with BBB (luminal and antiluminal membranes) transport (Diemer, 1968).
Matrices A variation of the finite difference analysis was applied to allow a numerical solution of the equations. After dividing the capillary-tissue system into 3 compartments, it was further divided into N x divisions along its length. All calculations were performed on elements of 3 by N x matrices where each matrix element corresponds to a physical volume element of the capillary-tissue system (see Fig. 1). The ith dimension of the matrix, varying from 1 to 3, represents the capillary, endothelial and tissue compartments, respectively. The jth index, ranging from 1 to N x represents the axial position of an element along the length of the capillary-tissue system.
Vectors: unlabeled substrate The exchange of material between individual elements was determined by mass balances around each element. The mass balances are represented schematically in Fig. 2 where net flux is represented as mass-transfer vectors. The mass balances were calculated over time in small increments to model substrate movement. Substrates enter the capillary-tissue system through flow into the capillary. Substrates are lost from the system through flow out of the capillary and consumption within the tissue compartment. Mass-transfer vector 1 represents the flow of substrate from the ( j - 1) element in the capillary compartment (i = 1) into the jth element of compartment 1 and is described by the equation: vector 1 d q ( l , j ) / d t = F c . C ( a , j -
1)
(5)
where Fc is the rate of plasma flow/capillary and C(1, j - 1 ) is the concentration of substrate in the ( j - 1) element of the capillary compartment. Fc can be determined by dividing the rate of plasma flow/g of brain, F, by the number of capillaries/g, N c. (The model was applied to situations in which erythrocytes were not present. Consequently the effect of erythrocytes was not considered.) Vector 1, flow into element(l, j) is accompanied by vector 2, the flow of substrate from element(l, j) into element(l, j + 1). This second vector may be expressed as: vector 2 dq(1, j ) / d t = - F ~ ' C ( 1 , j)
(6)
Substrate can also enter and leave element(l, j) by diffusion along concentration gradients. Axial diffusion within the lumen from element(l, j - I) into element(l, j) is described by Fick's law: vector 3 dq( 1, j ) / d t
=D.A/AX[C(1,
j-1)-C(1,
j)]
(7)
where A is the cross-sectional diameter of the capillary lumen, AX is the length of 1 axial division and D is the diffusion constant. The length of an axial division is equal to the total length of the capillary divided by the number of axial divisions.
aX = Lc/N X
(8)
185 Substituting the appropriate values: vector 3 dq(1,
j)/dt
=O" [~r'r2l[Nx/Lc] × [C(1, j vector 4 d q ( 1 ,
1) - C(1, j ) ]
(9)
j)/dt
= -D.[Tr.r2][Nx/L~] × [C(1, j ) - C(1, j + 1)]
(10)
Vectors 3 and 4 represent the transfer of mass by axial diffusion in and out of element(l, j), respectively. Substrates move out of the capillary luminal compartment across the luminal membrane by facilitated transport systems. The transport systems in each membrane were considered to transport substrate with the same kinetics in either direction. Each membrane was assigned kinetic constants that were independent of the other membrane. Transport processes across both membranes were assigned 2 components: a saturable component obeying Michaelis-Menten kinetics and a non-saturable component. Thus:
dq/dt=Vmax. C / ( K m + C ) + K d . C
(ll)
The kinetics constants Vma* and K d were scaled to the area of luminal membrane of an individual element. The Michaelis-Menten constant K m is a property of the transport system and did not require scaling. Mass-transport vector 5 represents the passage of substrate across the luminal membrane from element(l, j) into element(2, j ) in the endothelial compartment. vector 5 d q ( 1 ,
j)/dt
= --[Vmlax" C(1,
j)/(Klm + C(1, j ) )
+ K d "C(1, j ) ]
(12)
Vector 6 represents transport from the endothelial cell compartment element(2, j) into the capillary element(l, j): vector 6 d q ( 1 ,
j)/dt
1 = V~a x "C(2,
j)/(Klm + C(2, j ) )
+ K d " C(2, j )
The mass balance for all elements(l, j) was calculated by summing the rate of change described by vectors 1-6. Similar calculations were made for all the elements of the endothelial cell (compartment 2) and brain parenchymal tissue (compartment 3). A mass balance on element(2, j) involved the summing of vectors 5-10 with respect to element(2, j). Vectors 7 and 8 express axial diffusion in the endothelial compartment, and vectors 9 and 10 represent transport of substrate across the antiluminal membrane. The equations for vectors 7 and 8 are identical to the axial diffusion equations for vectors 3 and 4, except that the endothelial cross-sectional area and diffusion constant were substituted. The form of vectors 9 and 10 was the same as for luminal membrane transport, except that the Vaax and K a values of the antiluminal membrane were used.
j)/dt = - [Vaax "C(2, j ) / ( K a + C(2, j ) )
vector 9 dq(2,
+ K d "C(2, j)] vector 10 d q ( 2 ,
j)/dt
= V~ax . C ( 3 , j)//(g~n q- C(3, j ) ) + K o • C(3, j )
(15)
Axial diffusion (diffusion parallel to the long axis) in the tissue compartment may be considered by 2 additional vectors 11 and 12. The form of the equations for vectors 11 and 12 is the same as those for vectors 3 and 4 but with the diffusion constant and cross-sectional area of the tissue compartment substituted. After calculating the rate of change of substrate in each element (dq(i, j)/dt) the absolute change (AQ(i, j)) was calculated per increment of time At by:
AQ(i, j ) = dq(i, j ) / d t . At
(16)
The total quantity of substrate in element(i, j) after each At was calculated by adding the change in quantity to the previous amount of substrate in the element: Qnew(i, j ) = Qola(i,
(13)
(14)
j) +AQ(i, j)
(17)
A new set of substrate concentration values in
186
each element was found by dividing the quantity of substrate by the volume of that element. These concentration values were used in the mass-balance equations to determine another set of dq(i, j ) / d t values. Thus, the program proceeded through time by iteratively calculating dq(i, j ) / d t for each element(i, j) based on the existing concentrations. The time increment was set according to the volume of an individual capillary element divided by the capillary flow rate.
At = v o l ( 1 ) / ( F c .Ux)
(18)
Thus At represented the time required for flow to fill 1 capillary element. Time at any point was calculated by summing the time increments. The initial conditions of the numerical solutions were established by assigning values to the concentration array (thereby setting the initial condition values for the quantity array as well). The concentration in the first element at the arterial end of the capillary compartment, C(1, 1), was set equal to the concentration of substrate in the injection or infusion (an assigned value in the model). The remaining elements of the capillary compartment and all endothelial and parenchymal tissue elements were assigned initial concentrations of zero or a value corresponding to the endogenous concentration of substrate. The boundary condition applied to the solution was established by assigning values to the concentration C(1, 1) at the start of each iteration in time.
in which competition between labeled tracer and unlabeled substrate was taken into account. The flux of labeled substrate across a membrane is calculated by multiplying the rate of transport of total substrate by the fraction of total substrate that is labeled. Thus transport from an element(i, j) in the capillary compartment to an element(2, j) in the endothelial compartment (vector 5 in Fig. 2) would be described by: dq *(1, j ) / d t = -[(Vmla,," C(1, j ) ) / ( K l, + C(1, j ) ) + K d . C ( 1 , j)]C *(1, j ) / C ( 1 , j)
(19)
where the quantity C *(1, j)/C(1, j) represents the quantity of total substrate that is labeled. Similar equations can be written for the remaining vectors 6, 9 and 10 in Fig. 2. A different set of initial and boundary conditions were applied for tracer. Because tracer cannot be endogenous to the endothelial and tissue compartments, the tracer concentrations in those compartments were initially set to zero. The concentration of tracer at the entrance of the capillary C *(1, 1) was initially set equal to the concentration of labeled tracer in the fluid entering the capillary. The boundary condition value of C *(1, 1) was set equal to the concentration of tracer in fluid entering the capillary as long as it continued to enter. When tracer is no longer entering the capillary, C *(1, 1) was set to zero.
Vectors: tracers In experimental techniques measuring transport kinetics, a radiolabeled tracer is used in conjunction with varying concentrations of unlabeled substrates. To model such experiments, the movement and distribution of labeled tracer substrate was considered separately from the movement of total substrate. Separate arrays were established for the concentration, rate of change of quantity, and quantity of tracer substrate in each element of the capillary-tissue system. These arrays were designated C *(i, j), dq *(i, j)/dt, and Q *(i, j) respectively. The mass balances applied to tracer substrate were essentially the same as those used for total substrate with the exception of the membrane transport equations
Calculation of net extraction The calculations were stopped when the elapsed time was equal to the experimental time, and the concentration and quantity arrays from the final iteration were retained. Extraction from the capillary was calculated by summing the quantities of tracer in each element to determine the total quantity of tracer in the system at the end of the experiment and dividing by the total quantity of label that had entered by flow during the experiment. Thus:
E=
s: Q * / F .
C.r*t ( t ) d t
(20)
When the value of Car* is constant over one
187
infusion period and is zero when the infusion ends, Eqn. 20 can be simplified to:
E
=
Q */F.
C a r *" t tinfusion
(20.1)
The extraction can be used to determine the apparent average velocity of transport through BBB as: average velocity = ( E . F . Car* ) = Q
*/tinfusion (21)
The average capillary concentration at each iteration was calculated by summing the C(1, j) values from j = 1 to N x and dividing by N x.
j=Ux C
