Pitfalls in Digital Computation of the Impulse Response
of Vascular Beds from Indicator-Dilution Curves
By John Gamel, W i l l i a m F. Rousseau, Charles R. Katholi, and Emmanuel Mesel
ABSTRACT
Many methods are available for digital computation of the impulse response from
indicator-dilution measurements representing input and output signals. In all instances,
the only criterion for validity of the computation is comparison of the reconvolution of
the computed impulse response and the input with the actual output. In this paper, a
model mathematical system was constructed with a known impulse response; noise and
time variation could be introduced independently or simultaneously in the input and
the output data. Six methods for digital computation of the impulse response were
applied to data from this system and to actual dye-dilution data. Precision of reconvolution did not assure that the computed response would resemble the actual response of
the system. Some numerical considerations also significantly affected the digital computation of a valid response.
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
KEY WORDS
reconvolution
transfer function
mathematical models
• Zierler (1) originally described the significance
of the impulse response of vascular beds and its
relationship to indicator-dilution measurements,
and this function has since been used for a variety
of purposes. Scheel et al. (2) used a relationship
identical to that in Eq. 1 (see below) to correct for
catheter distortion of sampled dye concentrations.
Coulam et al. -(3) examined changes in the impulse
response of various in vivo vascular beds resulting
from injections of angiotensin, acetylcholine, and
adenosine triphosphate. Lange et al. (4) examined
the relationship between the arterial circulatory
transfer function and respiratory control. Moreover,
Bassingthwaighte et al. (5), Knoop et al. (6), and
Greenleaf et al. (7) have attempted to use the
nature of the calculated impulse response to draw
conclusions about the basic physiology of vascular
beds.
A number of digital methods have been applied
to the computation of the impulse response (2-8). In
spite of the number of experimental results reported,
there is little evidence that experimenters are fully
aware of the pitfalls involved in the calculation of
the impulse response. Therefore, in this paper,
many of these pitfalls are examined in detail and
discussed.
From the Department of Information Sciences, University
of Alabama in Birmingham, Birmingham, Alabama 35294.
This work was supported by U. S. Public Health Service
Grants FR 00145, FR 003il, and HE 11996 from the
National Heart and Lung Institute.
Received April 12, 1971. Accepted for publication
February 20, 1973.
516
deconvolution
Riemann sum
Methods
A number of analytical approaches with their
corresponding assumptions have been considered for
the computation of the impulse response. In the
approach considered in this paper, the indicator was
injected at a point well upstream from the system under
study. The concentration of indicator was sampled at
two points: one just upstream from the region of study
and one just downstream. Assuming linearity and time
invariance, the measurements were related to the
impulse response by the convolution integral,
o(t) = i(t) * h(t) = I i(s)h(t - s)ds,
Jo
(1)
where i(t) is the upstream (input) signal, o(t) is the
downstream (output) signal, and h(t) is an impulse
response. This formulation is still applicable when
indicator recirculation is present within the time
interval of sampling.
In application, this method involves the replacement
of Eq. 1 by a discrete analogue. In general, one of the
various formulas for numerical quadrature, e.g., the
simple Riemann sum or the trapezoidal rule, is applied
to get a discrete approximation to Eq. 1. Such a
procedure leads to a set of linear equations (in
triangular form), which can easily be solved to yield
approximate values of h(t) at a collection of equally
spaced points, t0, tu , . ., tN. Although this technique
leads easily to a unique set of values for h at the points
tt, unavoidable errors in the measured values of i(t)
and o{t) could result in physically unreasonable values
for h(t), e.g., negative values or rapid oscillatory
behavior (Fig. 2). Clearly a method that provides a
smoothing or a filtering of the calculated value of h is
required, and naturally the resulting h will no longer
satisfv the discrete analogue of Eq. 1 exactly.
Whenever experimental data are used, there is no
method for establishing a resemblance between the
calculated function and the real function, since the real
Circulation Research, Vol. XXXII, April 1973
COMPUTATION OF IMPULSE RESPONSE
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
function is, of course, unknown. Furthermore, any
investigator must closely examine the validity of the
assumption of time invariance, since vascular systems
are generally pulsatile and, therefore, vary significantly
in flow and volume. Thus, the results of any method
that has been applied only to experimental data are of
questionable validity.
To study these issues in a precise, controlled system,
we constructed mathematical systems that analytically
determined the output from known functions representing the input and the impulse response. The systems are
shown in Appendix 1. In these systems, variables such
as noise and time variation could be controlled
precisely. The various parameters had no physiological
significance; however, the impulse responses were
precisely known and thus at least some of the hidden
analytical difficulties which could be encountered in the
computation of the impulse response of real systems
were detected.
In the first section of Results, various types of input
and output data were created and processed by a
number of programs designed to calculate the impulse
response. The effectiveness of each program was iudged
by several criteria, the most important being the
deviation of the calculated impulse response from the
actual impulse response of the system. In the second
section of Results, each of the programs was applied to
real dve-dilution data. In this case, no criterion existed
as to the true impulse response of the system. However,
it was informative to compare the responses calculated
by the various programs.
517
FIGURE 1
Curves for noise-free data; i = input function, o = output
function, and h = deconvoluted impulse response.
reconstruction of output alone was not sufficient to
prove that an accurate impulse response had been
obtained for the system under study.
NOTATION
Standard mathematical notation was adopted. The
subscript s was added to distinguish the discretized
functions; thus, for example, os(n) = o(nAt). The
operation of discrete convolution, i.e., application of a
numerical quadrature method, was denoted by *..
Finally, a computed impulse response was denoted by
hc to distinguish it from the theoretical impulse
response, h.
DIRECT COMPUTATION OF IMPULSE RESPONSE
When Eq. 1 was replaced by a simple Riemann sum,
hc could be easily calculated. Figure 1 illustrates the
results of this computation for the case of a noise-free,
time-invariant system, using analytically produced test
data. The computed results were so: close to the
analytical results that they could not be distinguished in
the
figure.
:
Figure 2 illustrates the extreme sensitivity of the
direct computation to data errors or noise. The
computation was performed with noise levels of 1:100
(Fig. 2C) and 1:1000 (Fig. 2D). The impulse
responses from the deconvolution by the direct
computation were ragged and assumed significantly
negative values. The result plotted in Figure 2D was
actually worse than is indicated, because the.computation was terminated by exponent overflow in the
floating point numbers of the computer. We forced the
uncomputed impulse response to zero to obtain the
reconvolution. in Figure 2F.
The accurate reconstructions of the output by
reconvolution illustrated in Figure 2E and F show that
Circulation Research, Vol. XXXII, April
1973
FIGURE 2
Effects of noise on deconvolution by direct method. A: True
impulse response. B: Deconvolution with no noise. C: Deconvolution with noise present, 1:100 (1/2 scale). D: Deconvolution with noise present, 1:1000 (1/4 scale). E: Output
(line) and reconvolution of C with input (symbols). F: Output
(line) and reconvolution of D with input (symbols).
518
GAMEL, ROUSSEAU, KATHOLI, MESEL
OTHER METHODS
Most methods for computing hc achieved a smooth,
nonnegative answer by constraining the computed
impulse response to behave in a certain fashion. Since
the constrained function usually did not exactly satisfy
the discrete analogue of Eq. 1, the criterion for a
satisfactory hc was that it "satisfy" the approximation
within certain limits, or more precisely that
ll* e **.-°.||<a||°.ll
(2)
for some specified a, o, and norm (denoted by
1
II ID-
PARTICULAR METHODS CONSIDERED
We programed six methods of computing impulse
responses from discretely sampled input and output
data. In each case, the convolution integral (Eq. 1)
was approximated by a simple Riemann sum,
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
os(n) = At ^ i/i)hc(n - j).
i=
(3)
i
Brief descriptions of the various programs follow.
Linefit Program.—Linefit constrained hc to be
continuous and to take the form of a series of
consecutive, connected straight-line segments, where
each segment covered a fixed number of data points.
The slopes of the various segments were selected to
minimize the Euclidean norm of the absolute reconvolution error. This procedure led to a set of simultaneous
linear equations equal in number to the number of
segments used.
Segfit Program.—Segfit constrained hc to be continuous and to take the form of a series of consecutive,
second-order polynomial segments whose slopes were
equal at the points of intersection. The coefficient of the
polynomials was again selected on the basis of
minimizing the Euclidean norm of the reconvolution
error. Finding hc was thus reduced to solving a set of
simultaneous linear equations with dimension equal to
the number of segments.
Polyfit Program.—Polyfit constrained hc to take the
form of one polynomial over the entire interval of
computation. The order of the polynomial could be
selected by the program user. The coefficients were once
again selected on the basis of minimizing the Euclidean
norm of the reconvolution error. A set of simultaneous
linear equations resulted with dimension equal to the
degree of the polynomial.
Lagnorm Program.—Lagnorm required h0 to take
the form of a lagged-normal density curve. A laggednormal density curve was produced by convolving a
normal density function with an exponential decay. Fit
was obta'ned by trial-and-error adjustment of parameters with best values judged by the Euclidean norm of
the reconvolution error. Our program was a modification of the procedure suggested by Bassingthwaighte
(5).
]
A norm is a generalized distance. Normal distance is
obtained with the Euclidean norm computed by the square
root of the sum of the squares of vector elements. We also
used a sum of the absolute-values norm (Appendix 2). In
the appendix, the factor of 100 is to convert cc to percent.
Point-by-Point Program.—Point-by-point did not
force hc to take any preconceived mathematical form
except that hc was not allowed to go negative. The
program used an iterative technique that adjusted the
points of hc one at a time, starting with hc(0), to
minimize the Euclidean norm of the reconvolution error
on a selected number of data points. Only positive or
zero values were selected. The cycle through the points
of hc could be repeated for as many iterations as
desired. Starting the initial iteration with hc = 0 was
satisfactory. Solution of simultaneous equations was not
required.
Fourier Program.—Fourier used the fast Fourier
transform algorithm. From Fourier transform theory,
convolution in the "time" domain was multiplication in
the transformed ("frequency") domain. Therefore,
deconvolution was simply division. The Fourier program transformed the data to the frequency domain,
filtered it by dropping the high-frequency terms, and
computed the transform of h,. by division. The
transform of hc was then converted back to the time
domain by the inverse fast Fourier transform (which is
the fast Fourier transform with a sign change). Because
of the lost high-frequency terms, the results of the
Fourier program showed only one-fourth as many data
points.
CONSIDERATIONS OF DATA STARTING POINT
A significant problem encountered in computing the
impulse response from experimental data is the starting
point of the input and the output signal. Clearly the
first point of the input signal is the first point
significantly greater than the background noise. The
output data remains zero from the beginning of the
input signal until the minimum transit time has elapsed.
Since the minimum transit time for real systems is not
known, the beginning of the output signal must be
determined. This determination is considerably more
difficult than it first appears. The reason for this
difficulty can be seen in Figure 1. Even though the i
and h curves rose quite steeply, the o curve rose slowly,
being the convolution of i with h, so the first few data
points were only a small fraction of the maximum
signal. A similar condition existed with experimental
data, so that the first few data points of the actual
output signal were not significantly greater than the
noise level. This phenomenon creates a serious problem,
since an erroneous extension of the minimum transit
time produces an invalid impulse response. However,
an underestimation of the minimum transit time also
introduces serious error. Therefore, hr is not constrained to be zero throughout the minimum transit
time: it assumes small but significant values during this
interval, especially just prior to the appearance of the
output signal. This phenomenon blunts the rise of the
calculated response and creates a corresponding error in
the later points of hc (Fig. 3).
Furthermore, it is inefficient to include in the
computation those data points of the output that are
zero or not significantly greater than the background
noise. The simplest approach to avoid this inclusion is
to transpose the output curve toward the origin a
distance equal to the minimum transit time so that the
first nonzero output point is adjacent to the origin. After
Circulation Research, Vol. XXXII, April 1973
519
COMPUTATION OF IMPULSE RESPONSE
H. LINEFIT
C. POLTFIT
^u..^^r^-.
E. POINT-BY-POINT
B. SEGFIT
(V
0 . LRSNORH
F.
FOURIER
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
FIGURE 3
Impulse responses computed by the six programs for analytical data with no noise and no time variation. The impulse
response was unconstrained during the minimum transit time.
Symbols are the true impulse response.
the impulse response is calculated, it is then transposed
away from the origin a distance equal to the minimum
transit time. This transposition is equivalent to
constraining hc to be identically zero until the
minimum transit time has elapsed, and it is easy to
program.
Our most successful method for determining minimum transit time was the point-by-point program. As is
shown in Figure 3, this program was not significantly
disrupted by removal of the constraint that hc — 0 until
the minimum transit time has elapsed. Thus, it could be
used to determine the minimum transit time for real
data, even if another method was used to calculate the
impulse response. We obtained the much better fit
shown in Figure 4A by this method (compare with Fig.
3).
PROBLEMS IN DISCRETE APPROXIMATION
Any numerical integration formula has associated
with it an inherent truncation (discretization) error.
Such errors are measured in terms of the distance
between the sample points. In the case of Eq. 3, the
error is on the order of At.2 Clearly then the sample
rate should be chosen so as to make At significantly
smaller than the noise in the data.
Careful choice of the particular integration method to
be used is also quite important. For example, use of the
trapezoidal rule (approximation on each subinterval by
2
In precise mathematical terms, one says that the error is
O[(At)]by which is meant that |error| <M(At) for
some constant M that does not depend on At.
Circulation Research, Vol. XXXII, April 1973
D. LflGNORM
E.
POINT-BT-POINT
FIGURE 4
A: Impulse responses computed by the six programs for
analytical data with no noise and no time variation. The
impulse response was constrained to zero during the minimum transit time. Symbols are the true impulse response. B:
Reconvolution of impulse responses shown in A. Symbols
are the true output.
trapezoids rather than rectangles), which has a
truncation error on the order of A*2, was found to yield
better results in the direct computation without
noise.
Both methods, however, suffer from the same
inherent sensitivity to noise. In each case, the
coefficient matrix of the system of equations to be
solved is numerically ill conditioned. Thus any method
GAM EL, ROUSSEAU, KATHOLI, MESEL
520
which does not involve some form of smoothing is
doomed to failure.
Results
APPLICATION OF THE PROGRAMS TO ANALYTICAL SYSTEMS
Figures 4 and 5 illustrate the application of the
programs to the analytical' systems in Appendix 1.
Table 2 compares the performance of each program
by the measures in Table 1. All of the computed
impulse responses satisfied the criterion of reconvolution much better than they satisfied comparison
with the actual response. In some instances, the
computed response bore little resemblance to the
actual response, even though it reconvolved with
very little error.
The addition of time variation to the data
perturbed the computation of the impulse response
less than might be expected. In most programs, the
filtering process was able to smooth out the
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
C. POLYF"IT
. POINT-BY-POINT
/ \ / \ ,
E.
POINT-BY-POINT
E.
POINT-BY-POINT
FIGURE 5
A: Impulse responses computed by the six programs for
analytical data with time variation and noise. Symbols are
the true impulse response. B: Reconvolution of impulse responses shown in A. Symbols are the true output.
FIGURE 6
A: Impulse responses computed by the six programs for
actual dye-dilution data. B: Reconvolution of impulse responses shown in B. Symbols are the true output.
Circulation Research, Vol. XXXII, April 1973
521
COMPUTATION OF IMPULSE RESPONSE
TABLE 1
Comparison of Impulse Response Computed by Direct Method with Actual Impulse Response
N
1
2
3
4
5
6
7
8
9
10
h(N)
hc(N)
N
h( N )
hc(N)
N
h(N)
MN)
0.1915
0.3000
0.3523
0.3678
0.3600
0.3382
0.3090
0.2765
0.2435
0.2119
0.0198
0.1926
0.2991
0.3530
0.3672
0.3605
0.3378
0.3093
0.2762
0.2438
11
12
13
14
15
16
17
18
19
20
.0.1825
0.1559
0.1322
0.1115
0.0935
0.0781
0.0650
0.0539
0.0445
0.0367
0.2117
0.1827
0.1557
0.1324
0.1114
0.0936
0.0780
0.0651
0.0538
0.0446
21
22
23
24
25
26
27
28
29
30
0.0302
0.0248
0.0203
0.0166
0.0135
0.0110
0.0089
0.0073
0.0059
0.0048
0.0367
0.0302
0.0247
0.0203
0.0165
0.0135
0.0110
0.0090
0.0072
0.0059
N = sample number, h = value of the actual impulse response at sample N as computed from the equation
in Appendix 1 for the time-invariant, noise-free system, and hc = impulse response computed using Eq. 3
on noise-free, time-invariant data computed from the functions in the first section of Appendix 1.
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
a comparison of the performance of the individual
programs.
Since the actual response of the system was not
known, the validity of the computed responses
could not be tested. All programs performed well
by the criterion of reconvolution. Those programs
that smoothed the impulse response by forcing it to
take the form of a preconceived function remained
smooth with both noisy analytical and real data;
those programs that computed each point of the
response independently (point-by-point and Fourier ) became irregular with real data.
relatively rapid oscillations to produce a response
that represented the time-invariant response of the
system about as well as with time-invariant data.
APPLICATION OF THE PROGRAMS
TO REAL DYE-DILUTION DATA
Real-time dye-dilution curves were obtained by
injecting a bolus of Cardio-Green indicator into the
inferior vena cava of an adult dog and then by
simultaneously sampling with paired catheters in
the pulmonary artery and the left ventricle. The
density of dye (mg/liter) withdrawn from the
pulmonary artery plotted against time represented
the input function, while that from the left ventricle
represented the output function. Since the impulse
response of the catheters canceled out, the impulse
response obtained from these input and output data
presumably was that of the lungs.
Figure 6 illustrates the application of the
programs to the experimental data. Table 2 contains
Discussion
The programs presented in this paper all
produced a credible impulse response from the
varying types of input and output data, although
the response produced by point-by-point and
Fourier analysis became jagged when there was
TABLE 2
Evaluation of Programs
Type of data
Analytical, noise-free,
time-invariant
Analytical, noise-added,
time-invariant
Analytical, noise-free,
time-vary ing
Analytical, noise-added,
time-varying
Real dye-dilution data
No. FORTRAN statements
Criterion*
IE
RE
IE
RE
IE
RE
IE
RE
RE
Linefit
12.6
0.04
14.29
0.42
12.6
Segfit
13.0
0.21
12.1
0.50
11.8
Program
Lagnorm
17.7
22.1
2.6
61
7.4
3.7
22.3
17.6
12.6
3.7
0.9
1.0
2.8
1.1
28.9
12.7
11.2
12.6
0.4
1.3
75
1.0
1.8
2.6
80
1.0
3.8
40
1.43
97
IE = impulse response error, and RE = reconvolution error.
*See Appendix 2.
Circulation Research, Vol. XXXII, April 1973
Polyflt
Point-by-point
11.1
0.07
11.2
1.8
9.8
1.8
9.8
1.8
1.6
30
Fourier
40.27
0.90
82.4
2.0
41.5
1.1
105
1.1
2.3
100
GAMEL, ROUSSEAU, KATHOLI, MESEL
522
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
noise in the data. By smoothing the computed
response, the programs yielded an answer that
satisfied Eq. 3 within small limits and behaved itself
reasonably well. However, the smoothing can
distort the computed response, causing it to deviate
from the actual response of the system examined
and the application of the calculated response.
With the model mathematical system used,
precision of reconvolution did not assure that the
computed response would closely represent the
actual response of the system. This finding does not
imply that the same difficulty will necessarily arise
with every system, but it does show that such a
possibility exists. In real systems, where noise and
time variation exist in unknown degrees, the
disparity between precision of reconvolution and
precision of the calculated response might be
significantly greater than it is with the mathematical
model used in this paper. Thus, researchers who
utilize the digital computation of the impulse response should rely on criteria other than precision
of reconvolution to validate their method. A more
valid criterion is testing of the digital computation
method with data from a mathematical system such
as that used in this paper.
Since the computed impulse response is so
sensitive to the method of computation, the
researcher should thoroughly document the method
of computation along with his results. The use of
French curves or analytical functions to smooth the
experimental data might be justified in some
instances. However, the use of these techniques
should be acknowledged, and some attempt should
be made to confirm their validity.
Appendix 1
THE MATHEMATICAL SYSTEM USED TO TEST THE PROGRAMS
i(t) — Input signal.
h(t) = Impulse response.
o(t) = Output signal.
t
= Time = n\t.
r = A random number uniformly distributed between
—R and R.
(0.01 for noise ratio of 1:100.
R =
(0.001 for noise ratio of 1:1000.
hv(t, s) = Time varying impulse response,
impulse at time s.
At
= 0.1.
TIME-VARYING, NOISE-FREE SYSTEM
hjt,s) = \(t-
s)b(t)
( 0, t < s. L
TIME-INVARIANT
J
c(t)
NOISE-FREE SYSTEM
i(t) = Per<*.
h(t)= ter*.
o(t) = I
(t -s)c(t) + b(t) \e-(t-»*<t>, t > s.
b(t).
= 1+
= -
ff
V sin(Tt)
i
cos(Tt) - sin(Tt)
]•
T = 2TT to yield a frequency variation of 1 Hz.
V is chosen so that
h(t -s)- hv(t, s)
Max t, s
= 0.1.
i(s)h(t - s)ds
Ht-s)
Jo
where a — 2 to approximate dye-dilution data.
Therefore a graph of hv(t, s) for any s resembles h(t).
TIME-INVARIANT SYSTEM WITH NOISE
o(t)= I* i(sMt,s)As
Jo
h(t) = te-K
o(t)=
(1+r)
T
t2
| _ ( l - f l )
4t
2
It
6
( l Circulation Research, Vol. XXXII, April 1973
523
COMPUTATION OF IMPULSE RESPONSE
4b2
126c
2
24bc
T "
6b2
2bct2
(b-a)
e-at _
+
(b - a)*
(b - a?
t+
TIME-VARYING SYSTEM WITH NOISE
24bc
6b
5 + •
The input and output functions of the time-varying
system are multiplied by random factors exactly as they
are for the time-invariant system with noise. A noise
ratio of 1:100 is used.
2
(b - a) ' (b-af
Appendix 2
CRITERIA FOR EVALUATION OF THE COMPUTED IMPULSE RESPONSE
RECONVOLUTION ERROR
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
100
lOOHo,
where ||
-i.*h.
i =o
|| denotes an absolute value norm.
Pulmonary to arterial circulatory transfer function:
Importance in respiratory control. J Appl Physiol
21:1281-1291, 1966.
IMPULSE RESPONSE ERROR
, - K\\
ir-811
5.
At.
ioo
n
E.H.: Application of the lagged normal density curve
as a model for arterial dilution curves. Circ Res
18:398-415, 1966.
_o
References
1. ZIERLER, K.L.: Theoretical basis of indicator-dilution
methods for measuring flow and volume. Circ Res
10:393-407, 1962.
6.
2.
7.
SCHEEL, K.W., LANGILL, A.W., AND MlLHORN, H.T.,
COULAM,
CM.,
WARNER,
H.R.,
WOOD,
E.H.,
AND
BASSINGTHWAICHTE, J.B.: Transfer function analysis
of coronary and renal circulation calculated from
upstream and downstream indicator dilution curves.
Circ Res 19:870-890, 1966.
4.
LANGE,
R.L.,
HORGAN,
J.D.,
BOTTICELLI,
TSACABIS, T., CARLISLE, R.P., AND KUIDA,
Circulation Research, Vol. XXXII, April 1973
J.T.,
H.:
KNOPP, T.J., GREENLEAF, J.F., AND BASSINGTHWAICHTE,
J.B.: Effect of flow on transpulmonary circulatory
transport function (abstr.). Proc Annu Conf Eng Med
Biol 10:16.4, 1968.
JR.: Correction of catheter distortion effects on mean
transit time: Dye-dilution method. J Appl Physiol
21:1637-1641, 1966.
3.
BASSINGTHWAICHTE, J.B., ACKERMAN, F.H., AND WOOD,
GREENLEAF,
J.F.,
COULAM,
T.J.,
AND
BASSING-
THWAIGHTE, J.B.: Identification of parallel pathway
systems (abstr.). Proc Annu Conf Eng Med Biol 10:
50.6, 1968.
8.
MASERI, A., CALDINI, P., PERMUTT, S., AND ZIERLER,
K.L.: Frequency function of transit times through
dog pulmonary circulation. Circ Res 26:527-543,
1971.
9. BERGLAND, G.D.: Guided tour of the fast Fourier
transform. IEEE Spectrum 6:41-52, 1969.
Pitfalls in Digital Computation of the Impulse Response of Vascular Beds from
Indicator-Dilution Curves
JOHN GAMEL, WILLIAM F. ROUSSEAU, CHARLES R. KATHOLI and EMMANUEL
MESEL
Downloaded from http://circres.ahajournals.org/ by guest on June 15, 2017
Circ Res. 1973;32:516-523
doi: 10.1161/01.RES.32.4.516
Circulation Research is published by the American Heart Association, 7272 Greenville Avenue, Dallas, TX 75231
Copyright © 1973 American Heart Association, Inc. All rights reserved.
Print ISSN: 0009-7330. Online ISSN: 1524-4571
The online version of this article, along with updated information and services, is located on the
World Wide Web at:
http://circres.ahajournals.org/content/32/4/516
Permissions: Requests for permissions to reproduce figures, tables, or portions of articles originally published in
Circulation Research can be obtained via RightsLink, a service of the Copyright Clearance Center, not the
Editorial Office. Once the online version of the published article for which permission is being requested is
located, click Request Permissions in the middle column of the Web page under Services. Further information
about this process is available in the Permissions and Rights Question and Answer document.
Reprints: Information about reprints can be found online at:
http://www.lww.com/reprints
Subscriptions: Information about subscribing to Circulation Research is online at:
http://circres.ahajournals.org//subscriptions/