Measurement of rCBF by H2 Clearance:
Theoretical Analysis of Diffusion Effects
ROBERT A. PEARCE AND J. MILTON ADAMS, P H . D .
SUMMARY Although experimental evidence now indicates that diffusion of hydrogen can influence the
measured clearance curves from which local blood flow is inferred, its exact role has not yet been well
defined. For this reason we have developed a theoretical treatment of the effects of diffusion near a
boundary separating regions of inhomogeneous perfusion (e.g. the gray-white matter interface), and reexamined the appropriateness of the currently used bi-exponential model. Using our model, we confirmed
empirical estimates of important diffusion effects up to approximately 2 mm from an inhomogeneity, and
further refined the concept of spatial resolution. We also showed that fitting data to bi-exponential curves
may be incorrect and lead to inaccurate results. We conclude from these studies that diffusion does indeed
have an important effect on the clearance curves measured near an inhomogeneity.
Stroke, Vo! 13, No 3, 1982
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THE USE OF DISSOLVED MOLECULAR HYDROGEN gas as an inert diffusible marker for the measurement of cerebral blood flow has several major advantages over other techniques commonly used. '•2 Unlike
the radioactive gases, hydrogen is inexpensive, is easy
to obtain, and its concentration is relatively easy to
measure using a polarized microelectrode. One of its
distinguishing characteristics is that the small size of
the measuring electrode allows one to measure the
clearance of dissolved hydrogen gas from a much
smaller tissue volume3 than is possible using a scintillation counter to measure the clearance of radioactive
Xenon or Krypton.4 Its advantage over the microsphere technique is principally that repeated flow measurements under varying conditions are possible with
the gas clearance technique.
The hydrogen method has, however, suffered from a
significant limitation in the past; the high diffusivity of
hydrogen has prevented researchers from attaining the
fine spatial resolution hoped for. It is true that the
measurement can reflect the concentration of dissolved
gas in a very local volume depending on the size and
shape of the microelectrode. However, the concentration vs. time curve at any point is determined not only
by the washout due to local flow, but also by diffusive
exchange with nearby areas of differing concentration.
Perfusion rates which differ in adjacent compartments
(e.g. gray and white matter) can set up this concentration gradient and lead to exchange between the
compartments. When this happens, the assumption of
independent compartments made in deriving the multiexponential washout model (the one presently used in
interpreting clearance curves) is incorrect.
That diffusion of hydrogen in tissue is significant
has now been firmly established experimentally5-7 and
has led Halsey et al7 to conclude that unless diffusion is
taken into account quantitatively, the ultimate spatial
From Division of Biomedical Engineering, University of Virginia,
Charlottesville, Virginia 22908.
Supported by NIH GM07267 and HL25606-2.
Address for correspondence: Dr. J. M. Adams, Division of Biomedical Engineering, Box 377, Medical Center, University of Virginia,
Charlottesville, Virginia 22908.
Received June 2, 1981; revision accepted January 6, 1982.
resolution of the hydrogen method is about 2 mm. For
this reason we have developed a mathematical treatment of diffusion effects on the clearance of H2 gas
near a boundary separating regions of inhomogeneous
perfusion. Using our model we confirm theoretically
the empirical estimate of important diffusion effects up
to approximately 2 mm, and show that diffusion can
lead to some of the "bizarre" curves which have been
measured.7 We also show that the conventional biexponential analysis of clearance curves is incorrect,
and that it can lead to inaccurate estimates of local
blood flow.
Theory
When considering the problem of diffusion, it is
necessary to define a "model" situation close enough
to reality for the results to have meaning, yet simple
enough to treat in a tractable quantitative fashion. The
model which we propose is the following. Two semiinfinite tissues abut each other, their adjoining surfaces
defining a plane (see fig. 1). The perfusion rates in the
two tissues are different and unchanging but their input
(arterial) concentrations are identical. We want to find
the marker concentration, as a function of time and
distance from the boundary, in response to a known
input concentration change. The two cases in which we
will be interested are the responses to a step change in
the input concentration and a pulse in the input
concentration.
Three main assumptions are made to derive an equation for the marker concentration (see Appendix).
First, we assume that the microelectrode measurement
reflects the concentration of H2 from a volume of tissue
whose size is on the order of several capillaries and
surrounding tissue. Second, the tissue is assumed to be
in equilibrium with the blood as it leaves the tissue.
Third, we assume that the diffusion of H2 through
tissue follows Fick's law, using a "facilitated diffusivity" to account for the effect of micro-flow on the
movement of H2 in the tissue. Application of the conservation of mass principle leads to a differential equation which states that the rate of change of concentration consists of two components, one due to diffusion
and one due to blood washout. We assume that initially
the tissue is at a uniform concentration, and that far
348
STROKE
SIDE
B
SIDE A
VOL 13, No
3, MAY-JUNE
1982
DJDeA and qBlqA. This also shows that diffusion effects should extend the same order of magnitude in
distance as the characteristic length \lDJq. These results are useful in knowing what cases we need to look
at in our simulation, and what the approximate length
scales are.
Results
Response to a Step Change
FIGURE 1. Geometry assumed. The two sides are assumed to
diffr in bloodflow,q and effective diffusivity, De.
Downloaded from http://stroke.ahajournals.org/ by guest on June 17, 2017
enough from the boundary, diffusion effects are negligible. This yields the following equation (1):
dc
Bt
= £>
dx2
qc
slope (j-) at the edge (x = ± L) as a percentage of the
which is applied to each side of the boundary. De is the
effective diffusivity, q is blood flow rate, c is hydrogen
concentration and t and x are time and distance from
the boundary (see table 1).
Solution of the Equations
We have solved equation 1 using a finite difference
technique8 modified to accommodate the change in
diffusivity across the boundary.
Dimensional Analysis of the Problem
It is often useful to rewrite a problem or set of
equations in nondimensional or "normalized" terms;
insight can be gained into which variables or combinations of variables are important and how they influence
the process under investigation.9 If we perform the
change of variables,
*A =x/ V
DJqA
c = clc0
where A and B denote the two sides of the boundary,
we are left with a modified form of equation 1:
Side A(x&
0):
JLS + dc + c = 0
dx2.
31A
Side B(x^O)
-(DJDJ
+ dc/d1A + (qB/qA)c = 0
Using the methods outlined, we performed simulations of the tissue response to a step change in the input
(arterial) concentration. The results from one such
simulation are shown in figure 2, where concentration
is plotted in three dimensions as a function of time and
distance from the center boundary.
The variables, time, distance, and concentration,
are shown in nondimensional units. The two important
parameters, qA/qB (ratio of perfusion rates) and DJDeB
(ratio of effective diffusivities), are also listed. We
examined convergence and precision of the numerical
method and found them adequate, and we estimated
that conservative bounds on the solution are c ± 0.01.
Also computed and shown is the value for the largest
(2A)
d2c
^r
(2B)
Equation 2 shows that the desired solution c as a
function of f and x depends on only two parameters,
largest slope on the graph. This serves as a check on
the assumption that far enough away from the boundary diffusion is negligible.
One of the questions we wish to address is how far
away from the inhomogeneity is its effect felt. A measure of this distance at any time is given by the point in
space at which the concentration differs from the simple exponential decline (that is, the concentration at the
TABLE 1 Definition of Symbols
A,B indices for each side of the boundary
C
concentration of species (M oles/ cm3)
Cr final concentration (as t—°°) (Moles/cm3)
C-Cy(Moles/cm 3 )
initial concentration (t < 0) (Moles/cm3)
c/c0 (dimensionless)
molecular diffusivity (cm2/sec)
De= D + DF, effective diffusity (cm2/sec)
DF
facilitated diffusivity coefficient (cm2/sec)
JD
diffusive flux due to molecular diffusion
(Moles/cm2 • sec)
flux due to facilitated diffusion (Moles/cm2 • sec)
JF
L
distance far enough away from x = 0 for diffusion to
have negligible effect (cm)
blood flow rate (cm3 blood/ sec • cm3 (tissue)
4
t
time (sec)
tqA (dimensionless)
~t.
tqB (dimensionless)
distance from boundary (cm)
y^lslDeAjqA (dimensionless)
x
/ V^efl/^s (dimensionless)
ANALYSIS OF H2 CLEARANCE AND DIFFUSION/P<?ar«? and Adams
349
= 0.64
FIGURE 2. Plot of nondimensional concentration versus nondimensional time and distance. The boundary is atx = 0. The maximum
slope on the edge, 3c/dx at x = ±L, was
1.97% of the maximum slope on the plot.
3.58
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AT
ANY X ON PLOT
edge) by a fraction " e " of the difference in concentrations between the two edges. Expressed mathematically, it is that point where [c(L,t) — c(x,t)]/[c(L,t) -
c(-L,t)]
= s.
We call this point xe and it can be thought of as the
leading edge of diffusive effects. To transform this
non-dimensional quantity into an actual distance, representative values for q and De were assigned to each
side, and the resulting values for xe plotted as a function of time in figure 3. The faster the influence of
diffusion spreads, the faster we will see xe rising. If,
however, diffusion is unimportant, we will see a line
following xe = 0. Note how the effect spreads with
time, making an "envelope" inside of which diffusive
effects contribute significantly to the washout of hydrogen. (Just how significantly depends on the choice
of £.)
Perhaps a more pertinent question from an experimental point of view is what blood perfusion rates we
would infer by using conventional data analysis if we
measured with a probe the concentration in the tissue at
a point where the effects of the inhomogeneity are
important. We answer that by fitting our simulated
data to exponential and bi-exponential curves for various points along the x-axis, the results of the curve
fitting giving "measured" flow rates. We may compare these "measured" flow rates with the known
perfusion rates to see how well the single or double
compartment model4 predicts the true flow.
Such an analysis was carried out for two points, one
within the xoos envelope where the other side's influence is felt, and one outside the envelope (table 2).
Note the errors in predicted flow rates for these different points. The residuals (measured-predicted) from
one plot are shown in figure 4. The distinctive pattern,
as opposed to a random scatter, indicates an unacceptable fit to the data.
A final question we wish to ask is: how will changing the physical parameters q and De change the shape
of our solution? Specifically, how will they influence
how far away from the inhomogeneity its effect will be
felt? In order to answer that question, simulations were
performed for differing ratios of q and De. The results
are shown in figure 5, where xAo w is plotted as a function of fA. Notice not only that the effect spreads out
with time, but also that the shape of the solution is
much more sensitive to the parameter qBlqA than to
DJDeA.
Response to a Pulse
In addition to step changes in the arterial concentration of hydrogen, experiments are often performed
where the blood concentration approximates a pulse.
This typifies bolus injections of marker or the breathing of a gas for a short time. We ran simulations of the
pulse response in order to compare our theoretical re-
SIDE A
«=0.I0
— — -«=0.05
— ~__
TIME (min)
SIDE B
QB!
120
D e B = nxio" 5
FIGURE 3. Curves for the point, xE, at which the concentration begins to show the effect of diffusion (defined in text). The
values of q (cm3 blood/cm3 tissue • sec) and De (cm21sec) are
"worst case" values.
350
STROKE
VOL 13, No 3, MAY-JUNE
1982
TABLE 2 Results of Fitting Exponentials
Including points inside
envelope of diffusion effects
c(r)=P,ep2' + P3ep4'
Excluding points inside
envelope of diffusion effects
c(F) = PleFi'
CASE A: Bi-exponential fit of entire
curve, including those points
inside of diffusion envelope
(m = 20)
c measured at: xB = -\A3
"true flows"
qA = 2 5
qB=\2Q
Parameters
P, = 0.286
estimated
P2 = -0.40
P3= 0.711
P4 = -1.24
"measured
qA = 48 (92% error)
flows"
qB= 149 (24% error)
CASE B: Exponential fit of same location as in Case A, but including only points outside of
diffusion envelope (m = 3)
x B = -1.13
qA- 25
qB =120
P,=
0.998
P2 = -0.959
qB = 115 (4.2% error)
CASE C: Exponential fit for location
far from boundary, where envelope never reaches (m = 20)
xB = -3.0
qA = 25
9s =120
P, = 0.996
P2 = -0.994
qB= 119.3 (0.6% error)
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ml blood
* Units: -———
—
100 ml tissue • min
suits with those obtained experimentally. A typical
result is shown three dimensionally in figure 6 (The A
and B sides have been exchanged for visual clarity).
The curves at three points along the x axis are also
plotted in figure 7 next to curves measured by Halsey
et al.7 It is apparent that the complex curves measured
experimentally can be reproduced by our simulations.
Discussion
We have introduced the concept of an envelope of
important diffusive contributions to washout. We have
further shown that if analysis is excluded from points
within that envelope a mono-exponential curve can
accurately predict blood flow rates, but if analysis penetrates the envelope, bi-exponential analysis is incorrect and can lead to gross inaccuracies in predicted
flow rates. We have also shown that simulation of a
bolus injection leads to curves qualitatively similar to
experimental results of Halsey et al.7
^
-0.4 +
0
-i
2.0
1
4.0
TIME
-l
6.0
1
8.0
(tB)
FIGURE 4. Plot of the residuals (measured — predicted) for
fitting a bi-exponential to c vs. I data which penetrate the
diffusion effect envelope (case A, table 2). If the fit were acceptable the residuals should have a random scatter about zero.
Methods
Although we assumed a planar interface between
compartments, the model will be acceptable for a
curved interface if the radius of curvature is much
greater than the characteristic length ^jDJq. This is
reasonable for some regions of cortex/white matter
interface and for the border of the large deep nuclei.
The use of "effective diffusivity" requires only that
the transfer of hydrogen be a linear function of its
concentration gradient; this has not been experimentally verified.
A further complication in the interpretation of experimental measurement of hydrogen washout which
we have not considered here regards the effect of the
microelectrode on the measurement itself. The interpretation of some of our results rests on the assumption
that the measurement itself does not significantly affect
the clearance of hydrogen from tissue, and that the
sphere of influence of the measuring electrode is the
size of several well-stirred mini-compartments. While
we have not found an adequate treatment of this question for hydrogen, the analogy with oxygen measurement by the same electrode leads us to conclude that
these conditions may be met by the small electrodes
being used currently.3
Results
What then does our model tell us? First, let us see
how closely we may approach an inhomogeneity before the measurement becomes distorted; this is how
we define the spatial resolution. Because the solution
is stated in non-dimensional terms, the only two parameters which influence the SHAPE of the solution
are the ratios qBlqA (and DeB/DeA). The exact values for
q and De for each side determine the SCALE of the
solution.
For the scales shown infigure3, values for q and De
were chosen to model the cortex-white matter interface. The diffusion coefficients are rough estimates
based on diffusivity of H2 in water (6 x 10-5 cm2/sec),
modified by analogy with what Adams et al.10 found to
ANALYSIS OF H2 CLEARANCE AND DIFFUSION/Pearce and Adams
-. b
2.0
351
DeB / D e A
1.3
De B /De A = i-zo
0.0
1 —
i.o
2.0
—I
3.0
t
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FIGURE 5. Plot of the diffusion effect envelope, xcA (x/VDeA/qAj vs. fA (tqA) for different values ofq and De. (a)
Effect of different ratios of qB/qA for DeB/DeA = 1.2. (b) Effect of different ratios of DcB/DcA for qB/qA = 5.0.
be the case for thermal conductivities in perfused tissue. Figures 3 and 5 show that the diffusion effect
spreads with time. This means that the "resolution"
depends upon how much of the clearance curve is used
to infer blood flow. As figure 3 shows, five minutes
after the step change the diffusion effect has spread to
about 3.0 mm in the slow (white matter) side, and in
the fast compartment (gray matter) has spread to about
1.4 mm. This may be compared to an estimate by
Halsey et al.7 based on experiments and a review of the
literature, that the resolution of the hydrogen method is
about 2 mm.
Since we do not know with certainty how the diffusivity is influenced by the perfusion rate, it is important to see how sensitive the shape of our solution is to
different ratios of diffusivity, as well as different ratios
of flow rates. Figure 5 shows that the xe envelope (at
least) is relatively insensitive to DeBIDeA compared to
qBlqA. This indicates that the parameters of interest,
i.e., theflowrates, are the primary determinants of the
solution shape.
The fact that diffusional effects spread out in time
explains why measurements of the initial slope of a
clearance curve are more successful in predicting local
flow rate than are conventional analyses when the measuring electrode lies near two tissue compartments."
As long as the analysis is performed outside of the xe
envelope, the clearance curve can be expected to be
mono-exponential. A knowledge of the envelope dimensions is useful in knowing how much of the clear-
DeB / D e A = -67
0.00
0.00
-3.08
FIGURE 6. Response to an increase and then decrease in arterial concentration (pulse). The A and B sides have
been exchanged from figure 2 for visual clarity.
STROKE
352
a)
96n
VOL 13, No 3, MAY-JUNE
1982
XA»4.0
48to
T—i—i—i—i—i
2
3
r
4
XA=-0.92
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XA=-I.53
X
0
TIME (min)
i i I I i i i I I I
3
6
9
rn-T
12 15
TIME (min)
FIGURE 7. Comparison of model prediction (on left) and curves redrawn from Halsey et al.7 on right, (a) From
fig. 1 in Halsey et al., from rabbit cortex, and model at xA = 4.0 infigure6. (b) Also redrawn from their fig. 1 but
in subcortical white matter, and model at xA = 0.92. (c) Fromfigure2 in Halsey etal. " . . . apparently in contact
with fast flow compartment but also receiving diffusion input. . ." and model at xA = - 1 . 5 3 .
ance curve may be used to fit data to an exponential
curve. However, errors arising from other sources
such as measurement noise will be minimized by using
as much of the clearance curve as possible, and not just
the initial slope.
Flow Rates by Exponential Analysis
We can see what perfusion rates would be inferred
using a probe to measure clearance rates from a point
in our "tissue" if the data were analyzed by conventional techniques. Table 2 shows the results using a
nonlinear least squares fit. If the probe measures clearance outside the xe envelope (at xB — —3.0), the data
fit to a mono-exponential curve predict the "true"
flow with an error of 0.6%. However, if measurements
are made within the xe envelope (at xB = — 1.13) and
the data fit to a bi-exponential curve, the errors in
predicted flow rates are 92% and 24% for the white and
gray compartments respectively. Furthermore, the plot
of residuals (fig. 4) shows that assuming that the data
fit a bi-exponential curve is in fact not correct.
If we fit the points at xB - -1.13 which lie outside
the x0 os envelope (taking the initial slope in essence)
we find that according to our exponential fit they predict a blood flow within 4% of the true flow. Here, the
initial portion of the washout curve is a good indicator
of perfusion rate if only points outside the x0 os envelope are included. This finding is in accord with experimental results of Rowan."
To understand why a multi-exponential clearance
equation may be valid when measuring the clearance
of radioactive Xenon or Krypton, but not hydrogen, it
is necessary to understand the actual measurement being made, as well as the underlying clearance phenomenon. Whether the marker being measured is the highly diffusible hydrogen molecule or the much less
diffusible Xenon atom, some finite amount of intercompartmental diffusion will occur. The necessity of
accounting for that diffusion depends on the nature of
the measurement. For the case of hydrogen, the measuring electrode may sense the hydrogen concentration
within only a few tens of microns from the tip, depending on the specific design of the electrode. As we have
shown, the clearance curve for such a mini-compartment is not necessarily exponential if the measurement
is made at a point where diffusional flux is significant.
By contrast, a scintillation counter averages the concentration from many such mini-compartments. If the
measurement extends over a broad area, including regions far from inhomogeneity, the many mini-corn-
ANALYSIS OF H 2 CLEARANCE AND DIFFUSION/Pearce and Adams
partments where diffusion is not important will
' 'drown out'' the effects near the boundary, and simple
exponential or multi-exponential clearance curves will
be measured. In addition, the lower diffusivity of Xenon will make the effects of the inhomogeneity less
pronounced. It is the fundamental difference in the
nature of the measurements, however, which leads to
the bi-exponential clearance of radioactive gasses but
not hydrogen. Indeed, a multi-exponential clearance
curve should never be expected when measuring hydrogen clearance from such a mini-volume.
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Response to a Pulse
We now turn briefly to examine the response to a
pulse input. Although we will not attempt to glean any
quantitative information from this simulation, it is presented because, from a theoretical viewpoint, the results help to explain the findings of others. Graphs of
three points from the response are shown in figure 7,
compared to several of the "bizarre" findings of Halsey et al.7 They guessed that the distorted curves they
measured must be due to diffusion, and we show here
that this interpretation is consistent with our simulation. However, in figure 7C these results are not exactly analogous since this particular curve was for a point
in our slow compartment and assumed equilibrium of
H2 before the arterial bolus and an adequately small
electrode. It is apparent from this simulation that diffusion of hydrogen does play a significant role in the time
response of the tissue to an input pulse.
We did not simulate the method of Stosseck and
Lubbers12 where hydrogen is electrochemically generated in the tissue, and we are unable to evaluate whether it may improve the resolution of the H2 washout
technique.
Experimental Implications
For application, the main points of this paper are: (1)
to insure that inhomogeneities never influence the
clearance curve, measurements should be made 2-4
mm away from a boundary; (2) analysis of measurements closer than 2-4 mm should include only the
initial (mono-exponential) portion of the washout
curve; (3) bi-exponential curves are probably never
adequate to predict blood flow when points inside the
diffusion envelope are included; (4) a bolus injection of
marker may not improve the resolution of the H2 clearance technique.
Conclusions
Our initial attempts to take hydrogen diffusion into
account quantitatively have proven quite satisfactory
and have revealed several important concepts. First,
we have shown that in certain cases conventional exponential data analysis can lead to significant errors in the
inference of regional perfusion rates, and in which
situations this can be expected. We have shown how
diffusion effects spread with time, and how this gives
rise to the concept of an "envelope" describing the
leading edge of that spread. We have shown that when
using exponential curve fitting, data excluded from the
353
inside of this envelope yield better estimates of blood
flow rates than do data which penetrate the envelope.
This concept of a time-space envelope of significant
diffusional effects is a refinement of the spatial resolution concept. In the absence of absolute values for De
we can make estimates of the "spatial resolution" of
the hydrogen method which confirm others'
statements.
We are confident that the shape of our solutions is
essentially correct. However, the scale depends absolutely on the value used for effective H2 diffusivity in
perfused tissue; it is therefore worthwhile, even essential, to measure that value. When found as a function
of flow rate, it will simplify the model and leave blood
flow rates as the only unknown parameter, as befits a
good model for blood flow measurement. When found
as a function of concentration gradient, it will confirm
our hypothesis that Fick's law applies, or it will indicate the correct law to use in the equation.
Despite the limitations imparted by our assumptions
about the geometry of the problem, we have found
useful results. In practice, the boundary conditions for
the solution may not be the simple exponential declines
which we have used, but because they may be specified, a series of measurements may indicate more appropriate boundary conditions to be used in a specific
experimental situation. If this quantitative account of
diffusion is ever to find experimental utility, a more
detailed approach to the geometry of the problem will
be necessary.
Appendix
Theory and Methods
An important consideration regards the nature of flow
"through" tissues. Our treatment of this question depends heavily on the length scale of changes important to
us. The lengths we are interested in are distances from the
boundary where diffusional effects cause "average tissue
concentration" gradients. As we have shown, these
lengths are on the order of millimeters. On this scale the
exact nature of flow through capillaries, and gas exchange between capillary blood and adjacent tissue, can
be modeled sufficiently by treating each point in space as
a well-stirred "mini-compartment" in which concentrations are everywhere the same, and equilibration between
blood and tissue has occurred.I3 A differential mass balance14 leads us to treat perfusion of the tissue as a
distributed source or sink.3 One important precaution
that must be recognized is that in treating perfusion in
this manner, a "point" concentration refers to the effective concentration of a well-stirred "minicompartment."
A further assumption must be made concerning the
diffusion of H, gas in tissue. Fick's law of diffusion,
which states that the diffusive flux of a dissolved substance is directly proportional to its concentration graa. When transforming the conservation of mass from a finite balance to
a differential equation, the volume of the infinitesimal element must
approach the size of a capillary and its adjacent tissue for the limit
application to be consistent with our assumptions.
354
STROKE
dient (JD = -D-j-)
is undoubtedly true for a dilute
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solution of dissolved H2. However, Fick's equation
assumes that a concentration gradient in a homogeneous isotropic solution provides the only mechanism
for H2 movement. In the tissue the highly diffusible H2
molecule can diffuse from the tissue into a blood vessel, be carried along a distance, and diffuse back out.
This mechanism, as well as that of simple diffusion,
may allow H2 to move along a concentration gradient,
provided that there is no inherent direction associated
with the capillary flow. For this reason we propose a
"facilitated diffusion" concept, and assume that this
facilitated diffusion also follows the linear form of
dc
Fick's law (JF = —DF-j-), where DF is the "facilitated' ' diffusivity (analogous to the heat flow equation). I5
Given these assumptions, a pair of differential equations (one on each side of the boundary) describes the
response of the system to a step change in the arterial
concentration:
side A(x^ 0): -D,A d2cldx2 + dc/dt - qAc = 0
side B(x s= 0): - DeR d2c/dx2 + dc/dt - qBc = 0
(3A)
(3B)
(the substitution was made c = C — Cf symbols defined in table 1.)
Two second order equations in distance and first
order in time require four boundary conditions as well
as initial conditions. For the initial conditions we
specify some known concentration as a function of x.
For two of the boundary conditions we can specify any
functions of time at known distances from the boundary, one on each side. If we choose the distances to be
large enough that diffusion effects are minimal, the
concentration will be a function of time only; that
function is found by integrating Equation 1 with
d2c/dx2 = 0 to give c = c0e~*. (The familiar exponential washout where diffusion effects are unimportant.)
The two remaining boundary conditions are matching conditions at the inhomogeneity. We specify that
the concentration at x = 0 is the same whether approached from side A or side B, and we further specify
that at x = 0 mass is conserved, which implies that the
fluxes into or out of the boundary from each side are of
equal magnitude and opposite sign. To summarize:
Initial conditions: at t = 0, c = ca (x) = a constant if the
tissue is at a steady state concentration
(3C)
Boundary conditions:
atjc= +L, c = c0e-iA'
atx=-L,
(3D)
t
c = c0e- >B<
lim c = lim c
x-^0+
x-^0
lim [-DeA dc_] =
lim [~DtB
x—>0+
dx
x—>0
(3£)
(3F)
Sc_j
dx
VOL
13,
No
3, M A Y - J U N E
1982
represent physical reality. Possible sources of error
stem from two categories — the assumptions leading
to the differential equations and the approximate nature
of the numerical solution. We consider each in turn.
Specifying the geometry as we did simplifies the
problem to one dimension in space. Although this assumption will not apply to all areas of the brain, it is
acceptable when the radius of curvature of a surface is
much larger than the characteristic length 'VDJq. Examples where this may apply includes areas of interface between cortex and underlying white matter and
also of the large deep nuclei.
Our treatment of hydrogen washout from tissue as a
distributed sink is an assumption that the size of a wellstirred mini-compartment is small compared to the distances involved in the diffusive effects under consideration here, as explained in the theory section. Our
simulation showed that for hydrogen, tissue concentration gradients occur over 1-3 mm, and this agrees with
experimental evidence.7 When this is compared to the
size of a well-stirred mini-compartment composed of a
capillary and its surrounding tissue, the assumption
seems justified.
Similar magnitude scaling considerations lead us to
treat the difference in perfusion rates between adjacent
tissues as an abrupt change. This is justified if the rate
of change of perfusion rate is much steeper than the
concentration gradient due to diffusion.
We assumed that blood flow was homogeneous on
each side of the boundary. In fact, blood flow varies
from capillary to capillary. Our model approximates
the situation where the difference from capillary to
capillary is several times less than the difference between the sides. This means that electrodes must be
large enough to not measure intercapillary gradients.
The final assumption, that we may acceptably model the diffusion effects in tissue by using the "effective
diffusivity'' approach, requires only that the transfer of
hydrogen be a linear function of its concentration grade
dient, —, that is, it must follow Fick's law. Neither
dx
have we, nor to our knowledge has anyone else, measured this. There are indications in the literature that it
may not be a bad assumption. Thermal dilution techniques used for the measurement of perfusion rates in
brain have relied upon the analogous "effective conductivity" and found it acceptable.10 On this basis we
made the assumption, and, further used the same ratio
of effective diffusivity to molecular diffusivity as Adams et al. did for thermal conductivity.
The numerical techniques used to arrive at a solution
to the differential equations are by nature inexact,
though the approximation can be good. By choosing a
Crank-Nicholson finite difference pointb, the analogs
written were 2nd order correct, which Ramirez8 sug-
(3G)
Discussion of Assumptions and Method
Before accepting the results of our mathematical
model we must consider how closely the simulations
b. Because the diffusivity changes abruptly across the boundary, so too
will the value of dc/dx, as Equation 3G shows. To insure the necessary continuity of all derivatives the finite difference analogs must be
written for point (i + 'h, m) in terms of (i + 'A, m + l)and(i + '/2,
m + 2) for side A; vice-versa for side B. A set of 21 + 1 equations
may then be written and solved, where / = LI step size.
ANALYSIS O F H 2 CLEARANCE AND D I F F U S I O N / / W c e and Adams
gests is acceptable for the equations being approximated. To insure the precision of the finite difference
solution, the step size was successively halved until the
solution converged. Our criterion for convergence was
that at all points the combined change produced by two
consecutive halvings was less than 0.5% of the original
input concentration, and that the change itself was
becoming smaller with each halving. With small
enough step sizes we could solve the set of equations to
within (conservative bounds of) 1% of the pre-washout
equilibrium concentration. For these reasons we have
confidence that the solution obtained accurately reflects the solution to the differential equations.
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Stroke. 1982;13:347-355
doi: 10.1161/01.STR.13.3.347
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