Vol. 273, Issue 4, H2062-H2071, October 1997
MODELING IN PHYSIOLOGY
Modeling the myocardial dilution curve of a pure intravascular
indicator
J. S.
Lee,
J.
Karch,
A. R.
Jayaweera,
J. R.
Lindner,
L. P.
Lee,
D. M.
Skyba, and
S.
Kaul
Department of Biomedical Engineering and Cardiovascular
Division, University of Virginia School of Medicine,
Charlottesville, Virginia 22908
 |
ABSTRACT |
The dispersion
and dilution of contrast medium through the myocardial vasculature is
examined first with a serial model comprised of arterial, capillary,
and venous components in series to determine their time-concentration
curves (TCC) and the myocardial dilution curve (MDC). Analysis of
general characteristics shows that the first moment of the MDC,
adjusted for that of the aortic TCC and mean transit time (MTT) from
the aorta to the first intramyocardial artery, is one-half the MTT of
the myocardial vasculature and that the ratio of the area of the MDC
and aortic TCC is the fractional myocardial blood volume (MBV). The use
of known coronary vascular morphometry and a set of transport functions
indicates that the temporal change in MDC is primarily controlled by
the MTT. An analysis of several models with heterogeneous flow
distributions justifies the procedures to calculate MTT and MBV from
the measured MDC. Compared with previously described models, the
present model is more general and provides a physical basis for the
effects of flow dispersion and heterogeneity on the characteristics of the MDC.
myocardial blood flow; blood volume; mean transit time; intravascular indicator
| |
INTRODUCTION |
ULTRAFAST X-RAY computerized tomography
(CT), magnetic resonance imaging (MRI), and myocardial contrast
echocardiography (MCE) measure time-intensity curves for assessing the
passage of intravascular contrast medium (or indicator) injected into
the aorta through a region of interest (ROI) placed over the myocardium
(8, 11, 13, 19, 27, 33, 34). The concentration of the indicator in the
aorta, microvessels, or coronary sinus is based on the blood volume
within these vessels and is reflected in the time-concentration curve
(TCC). The TCC of microvessels in turn is dependent on the aortic TCC
and the dispersive and heterogeneous nature of the flow through the
myocardial vasculature. The ensuing time-intensity curve in the ROI,
expressed as indicator concentration per unit volume of myocardium
tissue, is recognized as the myocardium dilution curve (MDC). When the
indicator concentration curve is normalized to unit blood volume of the
vasculature, it is referred to as the vascular dilution curve (VDC).
For a pure intravascular indicator, VDC reflects the TCCs of all
vessels in the ROI. Several approximations (3, 8, 10, 28, 34) have been
used to calculate from the noninvasively measured MDC the mean transit
time (MTT) through the myocardial vasculature, regional myocardial
blood flow (MBF), and blood volume (MBV). To justify these
calculations, it is important to know whether the approximations
adequately simulate the distribution of an indicator in the ROI as it
flows through the myocardial vasculature. In this paper, we use a
recently described comprehensive morphometric model of the coronary
vasculature (16-18) to define a set of TCCs to account for the
dispersive nature of blood flow, to simulate the flow heterogeneity in
the myocardial vasculature, and to analyze the interrelation between
TCC and VDC. The equations derived from this morphometric and
heterogeneous model provide a practical procedure to assess perfusion
more specific to myocardium.
| |
ANALYSIS |
The myocardial vasculature in the ROI is modeled as eight vascular
components arranged in series, adapted from the detailed description of
the pig coronary vasculature by Kassab et al. (16-18). Each
component has its own TCC, such that the summation of the TCCs forms
the VDC. Based on the general characteristics of TCC, two equations are
derived using parameters of VDC to calculate MBV and the MTT for the
indicator to traverse the myocardial vasculature. We assume that the
flow is similar among microvessels within a single component and use
partitioned transport functions to construct the functional form of VDC
and to examine how the VDC is affected by changes in MBF and MBV.
Finally, we analyze the VDC of a vasculature formed by many serial
models in parallel to assess whether heterogeneous flow distribution in
the myocardial vasculature alters the calculated MBV and MTT.
Characteristics of serial model.
In the detailed morphometric description of the pig coronary
vasculature by Kassab and colleagues (16-18), the arterial tree is
composed of 11 orders of blood vessels with the parent vessel (right,
left anterior descending, or left circumflex artery) being of the 11th
order. The 10th-order artery branches into 18 intramyocardial arteries,
whereas the 1st-order arteries terminate into ~3 million arterial
capillaries. The venous system consists of 12 orders with the coronary
sinus being of the 12th order. The 11th-order vein has 16 intramyocardial veins, whereas there are ~5 million venous
capillaries connected to the 1st-order veins.
Because the myocardial vasculature does not include epicardial vessels,
the intramyocardial arteries are taken as the first component of the
serial vasculature and the intramyocardial veins as its last component.
The remaining orders in the arterial and venous tree are merged into
five components under the criterion that their blood volumes are
sufficiently small for adequately calculating indicator mass in each
component. The capillaries represent the fourth component. The blood
volume Vi of each component
derived from Kassab and co-workers (16-18) is listed in Table
1. With 
signifying the definition of a
quantity, the total MBV in the myocardial vasculature, therefore, is
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(1)
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The MTT for the indicator to traverse from the aorta to a point midway
in the ith component is identified as
Ti. The location subscript x of
Tx, when
identified as A (aortic entrance to the coronary vasculature), a
(arterial entrance to the myocardial vasculature), v (venous entrance
to the coronary sinus), or cs (end of coronary sinus where it enters
the vena cava), pinpoints the location along the serial myocardial
vasculature (Fig. 1) to which the quantity
T is referred.
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Fig. 1.
Coronary vascular model formed by 5 components in series. Q, total
blood flow; Vmv, total blood
volume; a, entrance of intramyocardial artery; v, exit of
intramyocardial veins. The ith
component is composed of multiple identical microvessels arranged in
parallel with a blood volume of
Vi. Mean transit time (MTT) from
arterial entrance of model to midpoint of
ith component is identified as
Ti.
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Based on the indicator dilution theory, the MTT
(Ti) is the
volume of vessels situated between the aorta and the
ith location divided by the MBF (Q).
Accordingly, we
have
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(2)
|
Similarly, the MTT from the aorta to the entrance of the
first component is
Ta (=
Va/Q), to the end of the eighth
component is Tv
[= (Va + Vmv)/Q] and to the exit of
the coronary sinus is Tcs [=
(Va + Vmv + Vcs)/Q]. The values of
Ti for each
component are listed in Table 1. The MTT of the myocardial vasculature, Tmv, is
|
(3)
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Suppose an indicator is injected at
time
0 as an instantaneous pulse into the
aortic root and that there is no recirculation. The transport function
hx is the TCC at
location x normalized to have an area
(i.e., its integration from time
0 to 
, which is taken as a time
that concentration becomes negligible) of unity. The
Tx defined in
Eq. 2 has been shown by the indicator dilution theory to equal the first
moment of hx
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(4)
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Vascular dilution curve of serial model.
Normally, the injection of an indicator into the aorta does not
resemble an instantaneous pulse. If we assume the TCC of aortic blood
flowing into the coronary artery to be
CA(t),
then the indicator dilution theory for the serial model yields the TCC
at location x as
|
(5)
|
where
symbolizes the convolution integral of the two
functions.
The first moment of
Cx is identified
as tx. (Because
Cx is not the
TCC of a pulse aortic input, a lower case is used to highlight that
tx is not
Tx and does not
have the physical meaning of MTT defined as blood volume divided by
flow). For convolutions in the form of
Eq.
5, it can be shown that the area of
Cx is the product of the areas of
CA and
hx. It can also
be shown that the first moment of
Cx is the sum of
the first moments of
CA and
hx (23). The
area of hx is
unity, and its first moment is
Tx. The application of Eq.
5 to calculate the areas and first
moments to all locations along the myocardial vasculature yields the
following equalities
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(6)
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(7)
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Most components of the serial model either have a small volume or a
short vessel length. Thus we can neglect the concentration variations
within each component and calculate the mass of indicator in the
ith component to be
ViCi(t).
The VDC can now be identified as the summation of the indicator masses
in all components normalized by their blood volume, i.e.
|
(8)
|
Based on Eq.
6, the integration of the VDC from 0 to can be rewritten to
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(9)
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By using the equalities in
Eq.
7, the first moment of
Cmv(t)
can be expressed as
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(10)
|
The terms in braces after the substitution for
Ti 
Ta from
Eq. 2 will become
which is
Vmv/(2Q)
(Tmv/2).
(Some quantities in this derivation are underlined to show the
correspondence.) Thus
Tmv relates to
tmv by
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(11)
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This MTT is derived without assigning any specific form for TCC.
For the purpose of illustration, the functional form of Cmv is calculated with the following
CA and hcs (4)
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(12)
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![<IT>h</IT><SUB>cs</SUB>(<IT>t</IT>) = <IT>b</IT><SUP>2</SUP>(<IT>t</IT> − <IT>t</IT><SUB>ap</SUB>)<IT>u</IT>(<IT>t</IT> − <IT>t</IT><SUB>ap</SUB>) exp [−<IT>b</IT>(<IT>t</IT> − <IT>t</IT><SUB>ap</SUB>)]](/content/vol273/issue4/fulltext/H2062/img024.gif)
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(13)
|
where
u(t)
is a unit step function,
C0 a constant,
tap the
appearance time, and a and
b are rate constants. The first moment of the aortic TCC is 1/a. The functional form in
Eq. 13 is known as the gamma variate,
and its first moment is 2/b + tap, which is
Tcs. With
Fi as
Ti/Tcs,
we set hi as
L
1{L[hcs]}Fi
with L as the Laplace transform operator and
L1 the
inverse Laplace transform operator (26). Then the TCC at the
ith component
is
![× {&ggr;[2<IT>F</IT><SUB><IT>i</IT></SUB>, (<IT>b</IT> − <IT>a</IT>)<IT>t</IT>*]/[(1 − <IT>a</IT>/<IT>b</IT>)<SUP>2<IT>F</IT><SUB><IT>i</IT></SUB></SUP>&Ggr;(2<IT>F</IT><SUB><IT>I</IT></SUB>)]}](/content/vol273/issue4/fulltext/H2062/img026.gif)
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(14)
|
where t* is
t Fitap,
[ ] an incomplete gamma function, and
(k) the gamma factorial (1, 30).
The appearance time for
Ci is
Fitap,
and its first moment is
Ti + tA. In the case that a equals
b, the terms in braces in of
Eq.
14 are simplified to
(at*)2Fi/(2Fi + 1).
Heterogeneous model.
One basic assumption of the serial model is that the flow is similar in
microvessels within each component and the lengths of the microvessels
are also identical. This uniformity allows the use of one TCC to
characterize the transport of indicator from the aorta to the midpoint
of each component. In reality, the flows and hence TCCs may differ
among microvessels of the same order. Also, the branching orders of
vessels of similar size may be different. To account for such
heterogeneities, let us assume that the myocardial vasculature has
k branches arranged in parallel
(25). Being a serial model, its jth branch has a flow
of Qj and a blood volume of
Vj. The branches and flow subdivide from the
coronary artery (location designated as a) and rejoin of the coronary
sinus (location designated as v).
The diagrammatic sketch in Fig. 2
illustrates a simple heterogeneous model with two branches, each with
three components in series. Their cross-sectional areas and flows are
different. Two first components join to form one artery, and two third
components join to form one vein. Because Q,
Vmv,
CA(t),
and
Cv(t)
are quantities associated with the entire myocardial vasculature, it is
not necessary to differentiate them from the serial model. A subscript
j will be added to quantities
exclusive to the jth branch.
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Fig. 2.
Heterogeneous vascular model formed by 2 branches
(j = branch no.) in parallel, each in turn
being a serial model of 3 components. V and Q of each branch could be
different.
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Let the blood flow through the jth
branch be Qj (a fraction
q j of Q), its blood volume
Vj (a fraction vj of
Vmv), and the TCC at the exit of
the jth branch (location designated as
v)
Cv, j(t).
Because the indicator concentration in venous blood is the collection
of all
Cv,j but weighted by the flow fraction
qj,
Cv(t)
is
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(15)
|
The MTT from the aorta to the entrance of the intramyocardial artery is
Ta and that from
the aorta to the jth branch to
location v is Tv, j.
Thus the MTT for blood to flow through the jth branch
Tmv,j
is
Tv, j Ta. The
following relationship between the overall
Tmv and
individual
Tmv, j can be derived from the first moment of
Eq.
15 (25)
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(16)
|
The VDC of the jth branch
Cmv, j(t)
is represented in Eq.
5 after the insertion of subscript
j. The overall VDC is the sum of
Cmv, j(t)
weighted by the blood volume fractions
vj, in the
jth branch, i.e.
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(17)
|
The application of Eq.
9 to the
jth branch, the integration of
Eq.
17, and the sum of
vj being unity allow us to obtain
the following equalities
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(18)
|
where
j ranges from 1 to
k. The first moment of
Cmv(t)
computed from Eq.
17 can be expressed as
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(19)
|
For each branch,
tmv,j tA Ta = 0.5Tmv,j.
Using Eq.
16, we reexpress
Eq.
19 as
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(20)
|
Note that the term in braces is zero. For a parallel model
with identical branches, vj
equals qj,
Tmv,j equals Tmv, and
Eq.
20, therefore, reduces to
Eq.
11, which is that of a serial model.
Within the ROI, the heterogeneity may be characterized by small
differences between vj and qj and between
Tmv,j
and Tmv. Then the
term on the right-hand side of Eq.
20, two orders of magnitude smaller, is used as an accuracy assessment of the following approximation
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(21)
|
Because
Tmv is
Vmv/Q, we have
![Q/V<SUB>my</SUB> = (V<SUB>mv</SUB>/V<SUB>my</SUB>)/[2(<IT>t</IT><SUB>mv</SUB> − <IT>t</IT><SUB>A</SUB> − <IT>T</IT><SUB>a</SUB>)]](/content/vol273/issue4/fulltext/H2062/img037.gif)
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(22)
|
where
Vmy is myocardial tissue volume.
One can calculate from the following definition the second moment of
Cx as
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(23)
|
The
transport function of the coronary sinus, which is the gamma variate in
Eq.
13, has
2/b2 as its
second moment. It can be calculated that 74% of the indicator in a
bolus injected at time
0 passes through the coronary sinus within the period of
Tcs 
cs to
Tcs + cs. Because Gaussian normal
distributions also have such a characteristic, we refer to
cs here also as the standard
deviation. The area,
tx, and x of
Cx serve as
parameters to compare the functional characteristics of TCC.
Myocardial blood volume and calculation procedure.
The MDC
Cmy(t)
is an indicator concentration curve normalized by the myocardial tissue
volume (Vmy). Accordingly, the
mass of the indicator in the myocardium is
VmyCmy(t).
Based on the definition of VDC, the mass of indicator in the myocardial
vasculature is
VmvCmv(t).
With no indicator destruction or leakage from the myocardial
vasculature, the equality of these two masses at any given time yields
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(24)
|
By using the equalities in Eq.
18 of the heterogeneous model, the
integration of Eq.
24 can be rewritten as
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(25)
|
If
we assume that the signals
Cmy and
CA could be
derived from the intensities of contrast medium being assessed by CT,
MRI, or MCE from the myocardium and aorta, then
Eq.
25 allows us to calculate the
fractional blood volume in the myocardial tissue Vmv/Vmy.
Similar substitutions in Eq.
10 lead to the determination of
tmv and
tA. The velocity
distribution in a Poiseuille flow has an average velocity of one-half
the maximum velocity. The length of the tube divided by the maximum
velocity is the appearance time, and the length divided by the average
velocity is the MTT. As a result, the MTT of the TCC of a Poiseuille
flow is twice its appearance time. If we consider that the blood flow
from the aortic ROI to the intramyocardial artery
(Ta) has the
Poiseuille velocity distribution, we can then estimate
Ta to be twice
the time taken between the appearance of indicator from the aorta to
the intramyocardial artery, i.e., twice the time between the appearance
of signals CA and
Cmy. The
substitution of
tmv,
tA, Ta, and
Vmv/Vmy
so assessed into Eq.
22 yields MBF per unit myocardial tissue volume Q/Vmy.
| |
RESULTS |
Figure 3 illustrates the TCCs at the aorta,
the second and fourth components of the serial microvascular model, and
the exit of the coronary sinus calculated from
Eq.
14 with a
Tcs of 8 s, tap of 4 s,
a of 0.5 s1, and the values listed
in Table 1. These TCCs depict the progressive dispersion of an
intravascular indicator that remains solely within the myocardial
vasculature. The closer the site of TCC measurement is to
the venous exit, the longer the appearance time, the smaller the peak
concentration, and the larger the second moment (or the width of the
TCC). These features are comparable to the TCC used by Dawson and
colleagues (7) for the pulmonary microvascular network.
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Fig. 3.
Family of time-concentration curves (TCC) along myocardial serial model
specified in Table 1.
CA(t),
TCC at entrance of coronary artery;
C2(t),
TCC at midpoint of 2nd component;
C4(t),
TCC at midpoint of 4th component (capillaries);
Ccs(t),
TCC at exit of coronary sinus. Note delay in indicator appearance of
C2(t),
C4(t),
and
Ccs(t)
from
CA(t),
decrease in peak concentrations, and broadening of TCCs as indicator is
transported by dispersive flow from coronary arteries to coronary
sinus. Cx, TCC
at location x;
C0, CA(t)
at t = 0.
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The VDC for this serial model is depicted in Fig.
4 together with the TCCs of the aorta and
coronary sinus. The short appearance time of the VDC is a consequence
of the TCC in the first component, and the jaggedness of VDC results
from the use of only eight components to simulate the myocardial
vasculature. Although the first moment of the VDC (5.65 s) is smaller
than that of the TCC of the coronary sinus (10 s), their standard
deviations (VDC, 3.17 s; TCC, 3.46 s) are comparable. The
characteristics of the VDC in relation to the aortic TCC are similar to
those previously reported (34).
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Fig. 4.
Aortic TCC (CA,
input), vascular dilution curve (VDC,
Cmv), and TCC
at coronary sinus
(Ccs, output).
CA is derived
from Eq.
12 with a rate constant
a of 0.5 s1, and
Cmv and
Ccs are derived
from Eqs.
8 and
13 for serial model described in Table
1. VDC has a very broad distribution, a characteristic of summing TCCs
of components situated within input and output.
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One can apply this myocardial model to examine how various changes in
flow and volume alter the form of VDC. The case presented in Fig. 4
serves as control and is reproduced as the solid curve in Fig.
5. The second case has a flow of 2Q with no
change in volume. The
Tcs is shortened
to 4 s, and the new VDC is the curve with the dotted line in the same
figure. The third case has the VDC and venous TCC represented by curves
with the dashed lines, which correspond to a flow of 0.5Q and a
Tcs of 16 s.
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Fig. 5.
VDC (Cmv,
A) and TCC
(Ccs,
B) at different flow settings.
Curves with solid lines are calculated from
Eq. 8 with flow and volumes given in Table 1, curves with dotted lines are
derived from a flow of 2Q, and curves with dashed lines are derived
from flow of 0.5Q. Further calculations show that curves with dotted
lines are VDC and TCC during vasodilation with a volume increase to
2Vmv and a flow increase to 4Q,
whereas those with dashed lines are VDC and TCC during vasoconstriction
with a volume decrease to 0.5Vmv
and a flow decrease to 0.25Q.
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During microvascular vasodilation, changes in blood volume are coupled
with changes in flow. Suppose that MBF is increased to 4Q and MBV is
increased to 2Vmv.
The Tcs of this
case is shortened to 4 s. Also assume that the blood volume change of
each component is proportional to the total MBV change, i.e., they have
the Fi values
listed in Table 1. Then the VDC is found to be similar to the curve
with the dotted line in Fig. 5, which has the same 4-s
Tcs as well.
Conversely, if vasoconstriction results in a decrease in MBV to
0.5Vmv and flow to 0.25Q, then
Tcs is lengthened to 16 s. This time we find that the VDC is identical to the curve with
the dashed line in Fig. 5, whose
Tcs is 16 s.
Specific vasoactive agents could produce a doubling of flow by dilation
of small arteries and arterioles (e.g., a doubling of volumes of 2nd
and 3rd components of serial model in Table 1). These internal volume
adjustments lead to a new set of
Fi values and a
new VDC, which is depicted in Fig. 6 as the
curve with a solid line. The
Tcs of this
vasodilation is 4.45 s. The curve with the dashed line in Fig.
6A is the VDC with a flow increase that results in a change of
Tcs to 4.45 s.
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Fig. 6.
Changes in VDC when volume distributions among various components are
altered. A: curve with solid line is
VDC for which small arteries and arterioles double their blood volumes
from those listed in Table 1 and total flow also doubles (an increase
of Tcs to
4.45 s). Curve with dashed line is calculated with no volume
changes from Table 1 and an increased flow to yield same
Tcs.
B: changes with a doubling in
capillary blood volume and total flow. Curve with dashed line has same
MTT and no volume changes. Minimal differences shown in Fig. 5 and here
indicate that value of
Tcs, not flow or
volume individually, is primary factor in determining form of
Cmv.
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As another example, the flow may be doubled because of a doubling in
the capillary blood volume caused by recruitment.
Tcs now increases
to 5.26 s, and VDC is represented by the solid curve in Fig.
6B. The curve with the dashed line in
the same figure shows the VDC with a flow decrease that is proportional
to the increase in
Tcs. A comparison
of these case studies over these wide blood volume changes indicates
that the functional form of VDC is mainly controlled by the value of
Tcs. Because the
VDC is composed of the convolutions of the aortic TCC and transport functions, the functional form of VDC is also affected by the form
representing the aortic TCC.
Equation 16 relates the actual MTT of
a heterogeneous model to the MTT values of its branches. Without the
latter information, Eq.
21 becomes a more practical procedure
to calculate the MTT. Can Eq.
21 provide a reasonable estimate of
the actual MTT? To address this question, four cases of heterogeneity
are considered under the condition that the blood volume of and blood
flow to the entire vasculature remain unchanged.
The first case is composed of two branches with equal flow but
different MTT (7 and 9 s, respectively). These MTT values are chosen so
that their flow-weighted average remains 8 s. Because the product of
MTT and flow of an individual branch is its volume, we have the
corresponding blood volumes of the branches as 0.4375 Vmv and 0.5625 Vmv, which add up to the same
total blood volume. The substitution of these fractional values into
Eq.
20 yields an approximate
Tmv that is
longer than the actual
Tmv by 0.125 s
(or 1.6% of
Tmv).
The second case simulates ischemia in one branch. Its flow is lowered
to only 20% of the total blood flow, and its MTT is lengthened to 15 s. This ischemic branch has a blood volume of 0.375Vmv. For the normal branch,
fractional flow is 80%, MTT 6.25 s, and fractional volume 0.625. The
error in estimating the actual MTT with
Eq.
21 is 19%. This large error can be
avoided by using ROI for the ischemic and normal regions within a
vascular bed.
The third case has three branches in parallel with flows of 0.25Q,
0.5Q, and 0.25Q, respectively. If we choose their MTT values as 6, 8, and 10 s respectively, the overall MTT is 8 s. The corresponding volumes of the branches are
0.1875Vmv,
0.5Vmv, and
0.3125Vmv. The predicted
Tmv for this
microvascular system based on Eq.
20 is longer than the actual one by
0.25 s, or 3.1%.
Finally, consider a heterogeneous vascular tree formed by 11 branches
in parallel. In this fourth case, the distribution of flows among the
branches classified according to
Tmv is depicted in Fig.
7A. The
only constraint in constructing this distribution is that the total
blood volume is Vmv. The
substitution of these distributions in
Eq.
20 yields a difference between the
estimated and actual
Tmv to be 0.2 s,
or 2.5% of Tmv.
The VDC and TCC of the coronary sinus for this system are plotted in
Fig. 8 and indicate that the wide flow
heterogeneity in Fig. 7 changes the VDC and TCC at the coronary sinus
only minimally from those of a serial model with uniform flow
distribution and same overall MTT of 8 s (curves with a dashed line).
The VDC of the serial model has a standard deviation of 3.5 s. The
heterogeneous flow distribution characterized in Fig. 7 increases the
standard deviation of the VDC to 3.7 s. One can also see from Fig. 8
that heterogeneity evens out the jaggedness of the VDC depicted in Fig.
4.
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Fig. 7.
Distribution of fractional blood flow
(qj;
A) and blood volume
(vj;
B) fractions for a myocardial
vascular model with heterogeneous flows. Model consists of 11 branches
in parallel. Height of each block represents fraction of total flow
passing through branch with MTT
(Tmv,j)
specified on axis.
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Fig. 8.
VDC (Cmv) and
TCC (Ccs) at
coronary sinus of 11-branch heterogeneous model described in Fig. 7.
Tcs of entire
vasculature is 8 s. Curves with dashed lines are from serial model
having a Tcs of 8 s.
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DISCUSSION |
Generalization of model.
Unlike previous descriptions of mathematical models for the estimation
of MBF and MBV that are limited to very specific settings and are based
on specific assumptions, the present model is more general. It shows
that the first moment of VDC, adjusted for that of the aortic TCC and
MTT from the aorta to the first intramyocardial artery, is one-half of
the MTT of the myocardial vasculature and that the ratio of the area of
the MDC and aortic TCC is the fractional MBV. The use of recently
described, detailed coronary vascular morphometry and a set of
transport functions to characterize the indicator dispersion along the
myocardial vasculature indicates that the temporal change in VDC is
primarily controlled by the MTT.
We have used 8 vascular components in our model rather than the 22 generations of the entire coronary tree. Because of the manner in which
Eqs. 16 and 22 are derived, the same equations
could still be deduced if the serial vasculature were divided into more components. In determining the effect of heterogeneities in regional flow and microvascular anatomy, several branch models are used. The
small difference between the MTT estimated for a 2-branch model and an
11-branch model from the actual MTT indicates that Eq.
21 will yield an adequate estimation
of MTT for a vasculature with more branches on the distributions of
flow and volume. The primary effect of these simplifications (merging
generations into components or taking a finite number of branches) is
on the form of the VDC. When we expand the 8-component model to one
with 16 components, the only modification is to smooth out the initial portion of the VDC with jaggedness. For the 2-branch model, the VDC
exhibits two peaks, which are smoothed out when the 11-branch model is
used.
The gamma variate function has been shown to be a good analytic
representation for the venous TCC. The partition scheme to obtain
Ci(t)
from
Cv(t)
accurately reflects the distribution of MTT along the myocardial
vasculature. It also gives a specific distribution for the standard
deviations of TCC. The VDC is a sum of TCCs. Thus the redistribution of
standard deviation among the TCCs, because of the nature of flow
(laminar or turbulent) or the anatomy of the vasculature while
Tcs and
cs are kept constant, may not
change the functional forms of VDC significantly from those shown in
Fig. 8.
The equalities in Eqs. 6,
18, and
25 for calculating the areas of TCC
and MBV are derived for a complete vascular network with a blood flow
of Q. Because the approach selected to account for flow heterogeneity
can be readily expanded to cover actual vascular flows,
Eq.
25 is valid whether or not the
vasculature within the ROI has homogeneous flow among the branches.
When a thin section of myocardium is selected for placement of the ROI, the myocardial vasculature will have some vessels directing flow out
and others bringing flow into it. The equalities may still be
applicable to the partial myocardial vasculature in the thin section,
because the blood flow and blood volume of the section as calculated by
Eqs. 22 and 25 with CT correlated well with those measured by the radiolabeled microspheres and direct morphometry (34).
The correlations reported in that study have a slope close to unity and
a positive intercept that may result from setting their
Ta to zero in
their calculation of MBF (34).
Physical nature of vascular flow.
The serial model presented here is comparable to the one-dimensional
dispersion analysis carried out by Harris and Newman (12). The approach
of expanding the serial model into a parallel model provides the basis
to fully simulate heterogeneous branching flows while accounting for
the flow dispersion within the branch. Vascular flow has been modeled
as one through a set of parallel branches (22, 25). The analysis of
Knopp et al. (22) aims to establish the fit to the measured venous TCC,
whereas our analysis and that of Lee (25) aim to describe the physical
nature of the VDC.
The TCC at the exit of a well-mixed chamber is characterized by
Eq.
12. The transport function of a plug
flow is a delta function with a time delay. The convolution of TCCs of
two well-mixed chambers and a plug flow produces a gamma variate with a
delay in the appearance of the indicator. Because the gamma variate
fits well with the TCC of many organs (4, 6) and the VDC obtained by
MCE (14, 19), it has been suggested that the indicator transport in a vasculature be compartmentalized as a chambers-plug-flow series. Such a
modeling of the central circulation is physically appropriate, because
a beating heart mixes the blood in the heart chamber like that of a
well-mixed chamber. In contrast, flow through a complex vascular
network is very different from such a compartmentalized model. Thus a
good fit of the gamma variate to the venous TCC of an organ does not
necessarily trivialize the disparate nature between vascular flow and
compartmental model.
The transport function of Poiseuille flow (a tube flow with a parabolic
velocity profile) has a concentration jump at the appearance time and
then a decay in the form of
1/t3. Take the
volume contained in a well-mixed chamber or plug flow as one-half of
the Poiseuille flow. Although the gamma variate of this
chamber-plug-flow model shows an indicator appearance and a decay
portion qualitatively similar to that of the Poiseuille flow, their
physical nature to transport the indicator is different. Because of the
minute diameter of the capillary, the rapid diffusion of indicator in
the radial direction makes its transport along the capillary like that
of a plug flow with minimal longitudinal dispersion. However, indicator
transport across the pulmonary capillary network is significantly
dispersed because of flow heterogeneity among the capillaries (2).
Simulation of blood flow in the myocardial vasculature as a
chambers-plug-flow model may delineate the connection between TCC and
dispersive indicator transport produced by nonuniformities in
velocities and path lengths within a vascular network. Consequently,
the use of an analytic function to represent a TCC should be viewed
only as a mathematical means for solving convoluted integrals or
partitioning TCC.
Comparison with previous studies.
Many procedures have been developed to calculate from the VDC measured
by CT, MRI, and MCE for MBV, MBF, and MTT. For a comparison with our
model, these approaches could be divided into three categories. For the
sake of simplicity, we will deal with all TCCs and VDC that have been
deconvoluted and a myocardial vasculature that is simplified to the
case where Ta and
tA are zero. By
assuming that the standard deviation is proportional to the MTT with a proportional constant 
, the equation obtained by Gobbel et al. (10)
through a center of gravity analysis could be generalized to
](/content/vol273/issue4/fulltext/H2062/img041.gif)
|
(26)
|
Depending
on the vascular system (4, 7), the value of ranges between 0 and 1. For the current analysis with the myocardial vasculature as the ROI
(i.e., tROI = tmv),
Eq. 21 indicates that the value for
is 0.
One category of studies operates under the assumption that the VDC is a
selected combination of TCCs. In several studies (3, 10, 31), the
indicator concentration in the vasculature within the ROI is assumed to
be constant or well mixed. With this assumption, the VDC becomes the
venous TCC and the first moment of VDC equals the MTT, i.e., = 1. We note from Fig. 8 that the form of VDC looks quite similar to the
venous TCC; however, the TCC has a much longer appearance time than the
VDC. In treating the VDC as a multiple-point measurement, the analysis
done by Mor-Avi et al. (28) also yields an of unity.
Equation 26 with a unity and an
assessed MBV were used to calculate the MBF (38).
Wang et al. (34) approximated the MDC as
faCa(t) + fvCv(t)
(where fa and
fv are the fractional blood volume
in the arteries and veins in the ROI) and provided several
approximations (Eqs. 5 and 6 in the appendix of their paper) to
relate
Ca(t),
CA(t), Cmy(t),
and Cv(t).
Although the approximations relating the areas of these TCCs are
different from Eqs. 6 and 9, they also deduced Eq. 25 relating MBV with the area of
MDC. Despite the graphic nature in the determination of
Tmv, it can be
shown mathematically that these approximations lead to
Eq.
26 with as 0. The procedures so
derived have been used to assess the coronary microvascular site of
autoregulation and the relation between MBV and MBF (27, 39).
The reasons why Wang et al. (34) derived a value for similar to
ours may be serendipitous and empirical. If
Ca(t)
and Cv(t)
are selected from the family of TCCs (see Fig. 3) to construct the VDC
and MDC, they will exhibit two peak concentrations, because of
different peak times in
Ca(t)
and
Cv(t),
instead of being smooth as in Fig. 4. As tabulated in Table 1, the
fractional blood volume in the capillaries (45% of MBV) is larger than
that in the three arterial components (26%) or the four venous
components (29%). Furthermore, a myocardial vasculature composed of
arterial and venous compartments may be too simplistic to assess the
dispersive and heterogeneous factors relating the TCCs and VDC.
In the second category of studies, the MDC and the blood flows through
several coronary arteries were measured (8, 31, 33), from which rate
constants such as
1/tmv,
1/tap,
a, or
b in Eqs.
12 and 13 were
determined. These rate constants have a linear correlation with the
measured MBF, which is expected using a compartmental analysis. If the
inverse of the rate constant is taken as the MTT, then a blood volume
can be calculated from the measurements (38). Our modeling of VDC
provides a better physical understanding of the correlation between
tmv and flow. It
also yields a procedure to compute quantitatively MBV and MBF from the
measured MDC and aortic TCC.
The third category of studies is based on residual analysis. The work
of Clough et al. (6) exemplifies this approach, which is conceptually
similar to ours. Flow dispersion and flow heterogeneity (2-branch
model) are considered in their analysis. Using the morphometry of
pulmonary circulation for the calculation, they examined how the ROI
kinematics and input dispersion affect the use of the height-to-area
ratio of the mass of indicator resident within the ROI (similar to the
MDC) as the MTT. In this study, we analyze the relation between VDC and
TCC to obtain Eq.
21 for estimating Tmv from
tmv.
Limitations of Model.
This model assumes that the indicator remains entirely within the
intravascular space and that the intensity measurement from a ROI can
be converted to the indicator concentration. Of the three imaging
techniques mentioned, this assumption of intravascular indicator is
strictly true only for MCE. Sonicated albumin microbubbles remain
entirely within the intravascular space, and their rheology is similar
to that of red blood cells both in microcirculatory preparations (21)
and in the beating heart (14, 24). Contrast agents used for CT leave
the intravascular space to briefly enter the interstitial space, thus
significantly lengthening the calculated MTT (5). Mathematical
manipulations have been proposed to correct for this effect (39). In
the case of MRI, paramagnetic agents enter the interstitial space and
this effect is prolonged if there is microvascular damage (24, 37).
Newer agents are currently being studied whose extravascular effect is
minimal (36).
Despite being true intravascular indicators, however, microbubbles used
for MCE have their own limitations. The relation between measured video
intensity signal and actual microbubble concentration is nonlinear for
two reasons, microbubble shielding and logarithmic compression of the
signal (33). Low doses of microbubbles are used to circumvent the first
problem (33). The second problem is currently being addressed by
developing procedures to measure acoustic intensity before applying
compression and postprocessing (15).
Another potential issue relates to microbubble destruction by
ultrasound (35). The concentration of microbubbles used for aortic
injection, however, is high enough not to be significantly affected by
this phenomenon (35, 9). Reducing acoustic power is also helpful in
this regard. Sonicated albumin microbubbles behave like red blood cells
only when the endothelium is functionally normal. They adhere to
abnormal damaged endothelium, prolonging the MTT (20, 29). Finally,
newer microbubbles have a small proportion of larger bubbles that can
become lodged within the microcirculation. When injected intravenously,
these bubbles are trapped by the pulmonary circulation and do not enter
the myocardium (32). When injected into the arterial system, however,
they can be trapped in the myocardial microvasculature, producing a long first moment or MTT (32).
In summary, analysis of indicator transport through the myocardial
vasculature with flow heterogeneity indicates that the areas, first
moments, and a timing comparison of time-concentration curves of
contrast agents in aortic blood and myocardium have the potential to
provide accurate estimations of MBV, MTT, and MBF.
| |
ACKNOWLEDGEMENTS |
This work was supported in part by National Heart, Lung, and Blood
Institute Grants R01-HL-40839 and R01-HL-48890. J. R. Lindner was the
recipient of a Fellowship Training Grant from the Virginia affiliate of
the American Heart Association, Glen Allen, VA. D. M. Skyba is the
recipient of NHLBI Postdoctoral Training Grant F32-HL-095410. S. Kaul
is an Established Investigator of the American Heart Association,
Dallas, TX.
| |
FOOTNOTES |
Address for reprint requests: J. S. Lee, Dept. of Biomedical
Engineering, Box 377, Medical Center, Univ. of Virginia,
Charlottesville, VA 22908.
Received 10 April 1996; accepted in final form 11 July 1997.
| |
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AJP Heart Circ Physiol 273(4):H2062-H2071
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Top
Abstract
Introduction
![· · · + V<SUB>8</SUB>(2V<SUB>1</SUB> + 2V<SUB>2</SUB> + · · · + 2V<SUB>7</SUB> + V<SUB>8</SUB>)]/(2QV<SUB>mv</SUB>)](/content/vol273/issue4/fulltext/H2062/img017.gif)
![= [V<SUB>1</SUB>V<SUB>1</SUB> + V<SUB>2</SUB>(V<SUB>1</SUB> + V<SUB>1</SUB> + V<SUB>2</SUB>) + V<SUB>3</SUB>(V<SUB>1</SUB> + V<SUB>2</SUB> + V<SUB>1</SUB> + V<SUB>2</SUB> + V<SUB>3</SUB>) + · · ·]/(2QV<SUB>mv</SUB>)](/content/vol273/issue4/fulltext/H2062/img018.gif)
![+ · · · + V<SUB>8</SUB>(V<SUB>1</SUB> + V<SUB>2</SUB> + V<SUB>3</SUB> + · · · + V<SUB>8</SUB>)]/(2QV<SUB>mv</SUB>) = <FENCE><LIM><OP>∑</OP><LL><IT>i</IT>=1</LL><UL>8</UL></LIM> V<SUB><IT>i</IT></SUB><FENCE><LIM><OP>∑</OP><LL><IT>n</IT>=1</LL><UL>8</UL></LIM> V<SUB><IT>n</IT></SUB></FENCE></FENCE>/(2QV<SUB>mv</SUB>)](/content/vol273/issue4/fulltext/H2062/img020.gif)
