Myocardial transit time of the echocardiographic contrast media

Ultrasound in Med. & Biol. Vol, 19, No. 8, pp. 635-648, 1993 0301-5629/93 $6.00 + .DO Printed in the USA © 1993 Pergamon Press Ltd.

OOriginal Contribution

MYOCARDIAL TRANSIT TIME OF THE ECHOCARDIOGRAPHIC CONTRAST MEDIA

V I C T O R M O R - A v I , * SOLANGE AKSELROD, ~" D A N I E L DAVID,*

LAUR ENC E KESELBRENER t and YVES B I T T O N t *Medical Physics Laboratory, School of Physics, Tel Aviv University, and

*Department of Cardiology, Meir General Hospital, Kfar Saba, Israel

(Received 12 October 1992; in final form 17 May 1993)

Abstract--The mean transit time of a tracer through a sample of tissue is a quantitative marker most closely related to regional tissue blood flow. Therefore, an accurate estimation of the mean time of transit of an ultrasonic tracer through a sample of myocardial tissue, obtained by contrast echocardiography, may provide a quantitative noninvasive estimate of myocardial perfusion. We hereby present an algorithm for the determination of the mean transit time by computerized analysis of a series of contrast-enhanced ecboeardiographie images. The algorithm comprises the evaluation of the echocardiographic impulse response function of a selected region of interest, using a deconvolution technique based on a fast Fourier transform and a frequency domain division of the videointensi- ties measured in the sample, by that measured in a predetermined reference region. An extensive computer simulation study was designed to facilitate the optimization of the steps of analysis. We present the results of the evaluation study performed in order to assess the accuracy of the procedure in compoter-simuiated ecboeardio- graphic images. Within a wide range of parameters chosen to define these functions, the analysis is shown to be essentially independent of the rise and decay times of the impulse response function of the tissue sample as well as of the simulated intensities. The effects of random noise introduced into the simulated intensity curves and of their variable width were investigated. The mean transit time was found to be accurately evaluated within about 10% of error for the variety of widths and noise levels permitted. The reconvolution error did not correlate with the accuracy of the evaluation of the mean transit time, indicating that the reconvolution error cannot be used as an estimate of the accuracy of the procedure. The numerical methods and the results of the computer study are discussed in detail. The approach is proposed to be used as part of a more general technique for the quantitative measurement of regional myocardial tissue blood flow.

Key Words: Indicator dilution, Mean transit time, Contrast echocardiography, Myocardial perfusion, Fast Fou- rier transform, Impulse response, Deconvolution.

I N T R O D U C T I O N

Indicator dilution principles, first introduced at the onset o f the 20th century, constitute the basis for different techniques for the measurement o f flow in various hydrodynamic systems (Gonzalez-Fernandez 1962; Meier and Zierler 1954; T rau t m an and New- bower 1984; Zieder 1958, 1962), including assess- men t of flow in the cardiovascular system (Marcus et al. 1987). Instantaneous injection of an indicator substance into the blood circulation and cont inuous moni tor ing of its concentrat ion at a certain point downstream allows the evaluation of the mean transit

Address correspondence to: Prof. S. Akselrod, Ph.D., Head of Medical Physics Laboratory, School of Physics, Tel Aviv Univer- sity, Tel Aviv 69978, Israel.

635

t ime of the indicator between the two sites (Gonzalez- Fernandez 1962; Meier and Zierler 1954; T rau tman and Newbower 1984; Zierler 1958, 1962). The ideal indicator substance must comprise small inert particles not affecting the fluid dynamics of the system. Therefore, the mean transit t ime of such indicator represents the t ime for the entire fluid volume to be replaced by the inflow of fresh fluid, and it is inversely proport ional to the flow rate. In case the total volume of fluid is known, and the mean transit t ime is found, a quantitative estimate of the flow can be obtained.

Solutions containing microscopic air bubbles, which cause echocardiographic contrast enhancement , were suggested as a possible indicator substance for the assessment of regional blood supply to myocardial tissue (Armstrong et al. 1982, 1983; Feinstein et al. 1986; Kaul et al. 1984; Maurer et al. 1984,

636 Ultrasound in Medicine and Biology Volume 19, Number 8, 1993

Meltzer et al. 1980; Roelandt 1982; Sakamaki et al. 1984; Tei et al. 1983, 1984). Assuming a quasilinear relation between the ultrasonic echo intensity and the concentration of the microbubbles, their mean transit time may be evaluated by analyzing the variations in the regional ultrasonic backscatter, induced by contrast injection. The aim of this study was to develop a detailed algorithm for the evaluation of the mean transit time based on the analysis of the time dependence of the regional echocardiographic signal following an echo-enhancing injection. A computer simulation model was designed to achieve optimization of the various steps of the analysis procedure and to assess its precision under different conditions of signal- to-noise ratio (SNR) and duration of contrast injection. This approach provides a relatively simple solution to an, as yet, unresolved problem, which may provide, with the use of a proper ultrasonic contrast medium, the basis for the noninvasive measurement of regional myocardial perfusion (Feinstein 1986; Meltzer 1985).

BASIC PRINCIPLES

According to the classical indicator dilution theory, each tracer particle introduced instantaneously into a multiple branching system, with single input and single output, leaves the system with a specific time delay. When a certain amount of indicator solution is injected instantaneously into the system input and its concentration Cout(t) is monitored contin- uously at the output, the mean time of transit may be determined using the following expression:

fo° tCu,,(t)dt ( t ) -

f l Cout(t)dt (l)

This approach is restricted to a system characterized by a single input and a single output. In a system with a single input and more than one output, tracer concentration as a function of time may vary significantly between the different outputs, due to varying pathways and varying degrees of mixing.

Since a sample of myocardial tissue cannot be regarded as a single output model, the simultaneous measurement of tracer concentration at all output points is required. Assuming a nearly linear relation between the ultrasonic echo intensity and the concentration of microbubbles in an appropriate range of concentrations, the requirement of multiple point measurement of tracer concentration is satisfied by the simultaneous measurement of the echo intensity

at all points of the sample tissue. Thus, considering mean echo intensity in the sample tissue, Is(t), rather than tracer concentration, Cout(t), we may noninva- sively obtain a quantitative estimate of the mean transit time of a particle through the sample tissue by:

fo ° tI~(t)dt ( t ) - (2 )

f i I,(t)dt

The function, Is(t), is obtained by averaging the intensities over the entire sample, which increases the SNR of the intensity curves, and thus improves the precision of the procedure.

All of the above assumes an instantaneous bolus injection at the system input. However, the coronary circulation does not fully comply with this assumption, since even a direct intracoronary injection of contrast medium does not deliver the entire amount of tracer as a short bolus into the region of interest. Since the sample is in part supplied by blood entering from the capillary bed of the adjacent regions, each having its specific distribution of transit times, the inflow function of tracer is subject to delay, widening and damping. It becomes more widened and damped, the more distant the sample is from the site of injection. This effect will be even more pronounced in case of a peripheral intravenous injection because microbubbles introduced intravenously must traverse the lungs before entering the left heart, obviously involv- ing different pathways and different arrival times.

The echo-intensity Is(t), measured in the tissue sample depends in turn on the configuration of the tracer inflow, so that a prolonged inflow will produce an essentially different, widened and damped Is(t). Therefore, the evaluation of the mean transit time according to eqn (2) would result in an overestima- tion of the mean transit time. This difficulty may, however, be resolved by using the following mathe- matical approach, which is well known in the linear systems theory.

We define Io(t) as the intensity measured at the port of entry to the tissue sample. Accordingly, Is(t) may be considered as a convolution of Io(t), with an appropriate response function Rs(t) characteristic for the specific sample and independent of the tracer inflow configuration:

Is(t) = Io(t)* Rs(t). (3)

In case of an instantaneous injection of indicator directly at the input point, Io(t) may be idealized by a Dirac 6-function, defined as follows:

Myocardial transit time of echo contrast • V. MoR-AVl et al. 637

- t o ) - = [ ~ ' f o r t = t o (4) 8(t t 0, for t # to

where to is the instant of injection. The definition of the 6-function also includes the constraint:

f >~(t - t o - t ' ) d t ' = 1. (5) oO

Assuming to = 0, the convolution integral in eqn (3) may be rewritten and evaluated:

Is(t) = Rs(t')6(t - t ' )dt '= Rs(t). o ~

(6)

Therefore, following an instantaneous injection, the response functions Rs(t) and Is(t ) are in fact identical functions. For this reason, the response function is often referred to as the "impulse response." Thus, the mean transit time of the indicator particles may be generally evaluated for an injection of any configuration by:

/ ,

_ Jo tRs(t)dt <t>

Rs(t)dt (7)

provided the response function R~(t) is known. Theoretically, if Io(t) and Is(t) are known or mea-

surable, R~(t) may be determined as follows. Applying a Fourier transform to both sides ofeqn (3) and using the convolution theorem, we obtain the following:

tissue, such as the aortic root or the left ventricle. Us- ing this approximation requires additional correction for the time of passage from the pool to the entrance to the sample:

to(t) ~- 1At + ~), ( l l )

where the time delay r can be estimated by measuring the time between contrast appearance in the pool and its appearance in the tissue sample.

NUMERICAL METHODS

Data simulation The computer study was performed using an

image processing system based on an Olivetti M380 personal computer with Data Translation frame grabber DT-2851 board (Marlboro, MA). The software was written in the Pascal language. The internal buffers of the frame grabber were used to perform all image processing operations, including the display of the images and the evaluation of the mean intensity within predetermined regions of interest.

Figure 1 presents an example of computer-generated short-axis images of the left ventricle at different phases of contrast enhancement. These 4 examples were selected out of 64 (=26) frames representing con- secutive heart beats. The inner circle represents the left ventricular cavity, whereas the surrounding area represents the myocardial tissue. Panel (a) shows the different intrinsic background precontrast intensities in the two regions; (b) corresponds to maximal echo

[~(~) = [o(,O)~s(O,), (8)

where the twiddle indicates the Fourier transform of each function in the frequency domain. The transfer function is therefore given by:

/~s(~o) ~ (9) = Io(~) '

and the impulse response is obtained by the inverse Fourier transform:

Rs(t) = FT-'IR~(o~)]. (10)

An approximate determination of the impulse response Rs(t) is possible, since Is(t) is easily obtainable from the sample tissue, and Io(t) may be replaced (as a first approximation) by the intensity of the reflection from a homogeneous blood pool upstream the sample

Fig. 1. Computer-generated echocardiographic images dur- ing contrast enhancement.


contrast enhancement of the ventricular cavity; in (c) the myocardial tissue shows mild contrast enhancement, as compared with (a) and (b); and (d) is obtained following total washout of the contrast solution and is practically identical to (a).

The videointensity in the ventricular cavity as a function of time was determined according to the following function:

Ip(t) = A. t". e -~', (12)

where A, a, u and I are variable parameters. The power term assures intensity buildup, and the expo- nential term is responsible for its subsequent decay.

The time sequence of the videointensity in the myocardium was determined by the convolution of lp(t) with the following response function:

R~(t) = B . t ~. e -b¢, (13)

where B, b, v and k are variable parameters. Positive time shifting by z, the minimal time of

passage from the blood pool to the sample, is performed, thus providing the echo-intensity time sequence curve in the myocardial tissue:

I~(t + r) = Ip(t) ,R~(t) . (14)

To imitate echocardiographic images, the intensity set to each pixel was computed by adding 25% random noise to the values of Ip(t) and ls(t). An example of the mean intensity in the two regions as a function of time is presented in Fig. 2(A). The above intrinsic noise is essentially eliminated by averaging over each region, which results in almost smooth curves.

Additional random uncorrelated noise of different levels is added to each of these curves following the convolution in eqn (14), in two different ways. First, in protocol A, the noise was added to the simulated signal so that the maximal amplitude of noise at a certain instant was allowed to be proportional to the value of the signal at that specific instant. Then, in protocol B, we added noise to the same curves so that the maximal amplitude of noise at a certain instance was allowed to be proportional to the peak amplitude of the myocardial signal (ls).

The addition of noise creates data which may be expected to be similar to the experimentally obtained data. Since noise in real echo contrast curves is probably of multiple sources, it is difficult to determine to what extent it is correlated with the signal. Therefore, these two protocols may represent the two extreme cases, A l t h e correlated random noise, and B--the uncorrelated random noise. In reality, the noise must

1.0

~0.5

0.0

A

• b c time d

1.0 B

0 . 0 ~

C 1.0

I 0 . 5

Fig. 2. Time dependence of the mean intensities: (A) as obtained from a series of computer-generated images (the arrows point at the phases of the images in Fig. 1); (B) after adding random noise (up to 40% of signal amplitude for each point); and (C) after adding random noise [up to 40% of the peak value of l~(t)]. All signals are shown after intrin-

sic background intensities are added.

be a combination of the two. It is therefore important to study the ability of the proposed algorithm to treat each kind of noise. Two examples of simulated noisy curves produced by protocols A and B are presented in Figs. 2(B) and 2(C). These curves were subjected to further analysis for evaluation of the mean transit time.

Myocardial transit time of echo contrast • V. MoR-Avl et aL 639

Data analysis The analysis procedure comprises the following

steps: 1. Intrinsic background values in the tissue sample

and in the homogeneous pool are evaluated by averaging the intensities prior to contrast enhancement [phase a in Fig. 2(A)].

2. Each background value is then subtracted from the corresponding function, resulting in pure indicator-produced intensities.

3. The instants of contrast appearance in each region are marked, and the videointensity curves are shifted to zero time accordingly. The determination of the time of contrast appearance in the sample tissue is not always easy and clear-cut due to noise. A small error (one or two heart beats to either side) in the identification of the contrast appearance in tissue does not affect much the evaluation of the mean transit time. However, larger errors may result in inaccurate analysis.

4. In order to minimize a possible artifact produced by the fast Fourier transform applied to functions with uneven edges, the time series Is[n] and Ip[n] are multiplied by the function:

1.0

t/) 0.5

O , 0 , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , 0 32 64

n

Fig. 3. A plot of the function S(t) defined by eqn (15) where nf= 32. The power p is directly related to the decrease rate of the function so that a large value ofp corresponds to a steep

decrease around nf

.

1 8. S[nl = 1 (15)

1

where ny, the right edge point, is marked man- ually and p is set to 30. The meaning of the variable parameters nland p is depicted in Fig. 3. This stage of the analysis and the preceding background subtraction result in equalized zero edges. This equalization of edges was found to significantly reduce the variability of the results. In cases in which the function Is(t) decays to a level different from the initial background, equalization may be useful in eliminating artifacts produced by uneven edges due to random noise. Gradual smoothing of the edge achieved by using a sigmoid function (eqn 15) was found to be pref- erable to the assignment of zero values to the edge points.

5. The curves are subjected to the following smoothing procedure:

Is[nl = ¼Is[n - 11 + ½I~[nl + ¼1,[n + 11, (16)

Ip[n] = ¼IpIn - 11 + lip[n] + ¼Ip[n + 11. (17)

6. The time series/~[n] and In[n ) are both divided by the maximal value of In[n], which results in val-

ues in the range [0; 1]. This normalization of data places all intensity values between 0 and 1, which is essential for the success of the subsequent steps. The series are transformed from the time domain to the frequency domain using the fast Fourier transform. The complex values of the estimated transfer function are computed according to eqn (9) as following:

T~[m] • I / 'p[m] I + (18)

This expression uses a simplified one-dimensional version of the Wiener filter, which is one of the most widely used deconvolution techniques in practice in two-dimensional image processing (Bates and McDonnell 1986; Castelman 1979). It has been introduced here in order to minimize the artifact of"spectral contamination" of the response in the frequency domain due to the division by small, nearly zero values of~[m], mainly at high frequencies. The effects of different values of ~b were studied and the optimum was found to be 0.02 for functions normalized as specified in step 5 above.

The value of ~k must fit the specific data analyzed. For instance, excessive ~b values produce a flat and damped response function, as if subjected to a low-pass filter. On the other hand, low ~k values reduce the effect of the Wiener filter. In this case, fast and high-power oscillations, some- times referred to as spectral contamination (Bates and McDonnell 1986; Castelman 1979), appear in the transfer function.


The effect of such spectral contamination on the impulse response is dual and directly related to the symmetry of the Fourier transform regard- ing the time- and frequency domains. First, fast and irregular oscillations in the frequency domain result in building an erroneously increasing tail in the response function in the time domain, in analogy to the large high frequency spectral components characterizing fast oscillating function of time. Second, the contamination is most pronounced at the higher frequencies, where both the numerator and the denominator have near-zero values and the result of the division at each point may be unexpected and certainly have no relation to the true transfer function. Thus, irregular oscillations are produced in the response function in the time domain (Gamel et al. 1973). An effective choice of ~k is therefore cru- cial. Its value depends upon the typical amplitude of Ip[n] in the frequency domain (eqn 18). The above mentioned normalization of data to a defined range permits using a constant value of ~, rather than adjusting it for different inputs.

Alternatively, the algorithm could be made adaptive to the observed temporal noise spectrum. On one hand, this would probably improve the performance of the algorithm under different noise conditions. However, on the other hand, obtaining a reliable picture of the noise spectrum would require acquisition and processing of an additional series of 64 baseline images without contrast enhancement, which would double the assessment time. We chose to base our approach on a less flexible algorithm of Fourier deconvolution, which would provide promptly a satisfac- tory estimate of the response function.

9. The transfer function is subjected to a smoothing procedure similar to that applied to the original functions in step 5 above:

ER,[m] = ¼El~s[m - 1]

+ ½E/~,tml + ¼El~s[m + 11. (19)

The aim of this smoothing is to eliminate residual oscillations in the frequency domain, which may still produce a "tail" in the evaluated response after inverse Fourier transform.

10. The following frequency domain low-pass filter is applied to the transfer function:

11.

This filter is of the form of the function S[n] defined in eqn (15) and its action as a low-pass filter may be easily understood by reexamining Fig. 3. It is important to remember that the discrete Fourier transform is wrapped around, and the filter should be symmetrical about the true zero frequency. Thus, in our case, eqn (20) is used to build the filter for m in range [ 1; 64], and the rest of the filter is created by the mirror image of this range. The parameter my, the cutofffrequency, is related to the full width at half maximum (FWHM) of Iu(t) according to the following em- pirical formula:

72 me= I1 + FWH------M' (21)

12.

1 Filter[m] = I (20)

1 + 13.

and p was set to 20. The aim of this step is to remove possible

residual oscillations from the response curve. The cutoff point of the filter might be determined according to the width of the transfer function. However, the width is difficult to estimate accurately due to the rapid oscillations in the transfer function, and it was found to be largely dependent on the specific shape of oscillations. Alterna- tively, the cutoff frequency should correspond to a point where ip(O~) becomes so small that putting its values in the denominator (eqn 18) results in spectral contamination of the transfer function. The location of this point depends on the width of [p(~o), which is inversely proportional to FWHM. Hence the choice of the relation in eqn (21). The coefficients were optimized empiri- cally, and their values are probably dependent on the width of the sample window, which was 64 frames in this study. Choosing a different window would require the reevaluation of these parameters. The evaluated transfer function is subjected to an inverse fast Fourier transform, resulting in the time domain impulse response, according to eqn (10). The evaluated impulse response ERs[n] is multiplied by S[n] defined by eqn (15), where p is set to 20 and nf = CT, which is the cutofftime, defined as the time where ERs goes down below 0.01. The last points corresponding to this tail are set to zero by this multiplication, so as to provide in the time domain the final correction to the artifact treated by step 9 above. Finally, the mean transit time is evaluated according to eqn (7).

Myocardial transit time of echo contrast • V. MOR-AVl et al. 641

Evaluation o f the analysis The following parameters were defined in order

to provide a quantitative estimate of the accuracy of the above analysis of the simulated data: • The response error is computed according to the

following definition:

64 F,.=1 IERs[n] - Rs[n]l Response Error = (22)

~6~, Rs[n ]

• The function lp(t) is convolved with the evaluated impulse response resulting in a reconvolution, Rec(t):

Rec(t) = I ,( t ) , ERs(t). (23)

which in case of a precise evaluation of the response should be identical to the intensity Is(t).

• The reconvolution is compared with Is(t) by evaluating the following reconvolution error:

Reconvolution error

Z 64, I Rec[nl - IJn]l (24)

The response error and reconvolution error (eqns 22 and 23) could also be differently defined. The use of square differences (Gamel et al. 1973), for example, would indeed change the typical ranges of both errors. However, it does not change significantly the functional dependence of these errors upon the noise level and the width of the curves.

For each combination of parameters in each protocol, A and B, the relation between the two errors defined by eqns (22) and (24) was studied by evaluating the Pearson product moment correlation coefficient.

RESULTS Two sets of parameters v, and k defined in eqn

(13) were used (Table 1) to produce different rise and

0.014

g- O , Response II

• 0.007 e,- o ffl

0 . 0 0 0 , ~ 0 32 64

time Fig. 4. Two curves representing the response function R,(t) produced by each set of parameters in Table 1. All parameters were selected so as to produce two curves of approxi-

mately equal peak amplitude and width.

decay rates in the response function Rs(t ). Figure 4 shows two curves representing the response function Rs(t) produced by each set of parameters. Curve I is characterized by a sharper upstroke and a more gradual decline, as compared with curve II. All parameters were selected so as to produce two curves of approxi- mately equal peak amplitude and width (Fig. 4).

For each response function, three sets of parameters u, and I defined in eqn (12) were used (Table 2) to produce different rise and decay rates in the function Ip(t). Figure 5 shows three curves representing the function Ip(t) produced by each set of parameters.

The reliability of the determination of the mean transit time was examined under different conditions of contrast inflow duration and random uncorrelated noise. This was achieved by randomizing the parameter a within a range shown in Table 1, so as to produce curves decaying to the background level within 64 frames, and introducing random variations into the noise level added independently to each function, In(t) and Is(t), between 0% and 50% of the amplitude of the signal. The analysis was carried out separately for curves obtained with protocol A and protocol B of noise addition. For each protocol, the analysis was performed 600 times for each set of parameters with

Table 1. Two sets o f parameters de te rmin ing the response funct ion Rs(t), defined by eqn (13).

Set I Set II

rn 1.0 2.0 k 1.0 2.0 b 0.30 0.03

Table 2. Three sets of parameters determining the function Ip(t), defined by eqn (12).

S e t l S e t l l ~ t l I l

n 2.0 0.5 0.5 l 1.0 3.0 2.0 a 0.15-0.60 0.0001-0.0040 0.002~.050

642 Ultrasound in Medicine and Biology Volume 19, Number 8, 1993 10 . Ill

' 0 . 5

0.0 ~ 0 3 2 6 4

n

Fig. 5. Curves representing the function Ip(t) produced by each of the three sets of parameters in Table 2. Parameter a was chosen for each case so as to produce three curves of

equal FWHM.

each response curve. The evaluated mean transit time normalized by its true value results in 1.00 in case of a completely successful analysis, and values smaller or greater than 1.00 in cases (t) are underestimated or overestimated correspondingly.

For each protocol, a histogram was obtained for each combination of parameter sets, without separat- ing the results according to variations in noise level and width of the curves (Fig. 6), It can be seen that in the majority of runs the data are analyzed within about 10% error, for all noise levels and curve widths considered.

For each protocol (A and B), the same data were used to evaluate (t) as a function of the noise level, independently of FWHM [Figs. 7(A) and 8(A)] and as a function of FWHM, independently of noise [Figs. 7(B) and 8(B)]. In protocol A, the analysis is essentially independent of the noise level, as one can con- elude from the dense duster of dots surrounding the horizontal line corresponding to the constant level ( t) -~ 1 [Fig. 7(A)]. In protocol B, the distribution of results becomes wider as the noise level increases [Fig. 8(A)], indicating that the analysis becomes less precise.

The response error does not show a clear dependence on the noise level. Low-noise signals result in about 40% error in the evaluated response function in both protocols, A and B [Figs. 7(C) and 8(C)]. As the noise level increases, the distribution of the response error becomes wider. The distribution of the response error as a function of FWHM values does not show a clear-cut trend in any of the two protocols [Figs. 7(D) and 8(D)].

The reconvolution error seems to be only mildly

dependent on the noise level in protocol A [Fig. 7(E)], and the function Is(t ) is reconstructed within similar error limits for all noise levels considered. In protocol B, the reconvolution error shows a trend of protocol A [Fig. 7(E)], and the function Is(t) is reconstructed within similar error limits for all noise levels considered. In protocol B, the reconvolution error shows a trend of increase with increasing noise [Fig. 8(E)]. The reconvolution error shows in both protocols a wide distribution without clear width-dependence [Figs. 7(F) and 8(F)].

Using a constant value of the parameter a (a = 0.0006), which determines FWHM (eqn 12), the noise dependence of the mean transit time and the response and reconvolution errors can be studied for nearly constant FWHM vaue of l0 (Fig. 9). Minor variations in FWHM (_+ l) were produced by the noise added to the curves. Each point in Fig. 9 was obtained by averaging the results of 25 separate runs, in order to minimize the effect of the random character of the noise of different levels. The data in Figs. 9(A) and 9(D) further confirm our observations presented in Figs. 7(A) and 8(A), and clearly demonstrates that using the above algorithm, (t) can be evaluated, nearly without dependence on the noise superimposed on the curves. Panels (B) and (E) show the correlation between the noise level and the response error. Panels (C) and (F) confirm the mild increase with noise in reconvolution error in protocol A, and a more pronounced increase with noise in protocol B.

The use of noiseless functions (noise = 0) of different widths reveals a quite steady evaluation of the mean transit time for all widths considered [Fig.

60

50

40 2 15

30

E == 20

10

O ~ 0,6 0.8 1.0 1.2 1.4 1.6 1.80

normalized mean transit time

Fig. 6. A histogram of the evaluated mean transit time normalized by its true value, obtained from 600 computer runs using response function I and parameters in set II, for protocol B. The histogram shows a peak around 1.00, suggesting that mean transit time is successfully evaluated in most

c a s e s .

Myocardial transit time of echo contrast • V. MOR-AVI et al. 643

1.0

I 1.4 %

• g ,uml an

1.0

"• 1.,

1.0

Q( 5

B • •

,,,.ii;i.: -.:: 5 10 15 20

no i se FWHM

1.2 t.2

t ~, i °"

G ~ Q1 ~2 ~3 G4 ~6 Q0

D

m . ' - i "

i11111;;;. :llnlnn!'. ,=,n

0.4

E .

n2 • ~ , , , , p / .

c~ qn

(14

¢13

o.~, , : : • n m . , , . l u . " -

0.1 e m ; l l

nO o 5 10 15 20 26

Fig. 7. Distribution of results obtained from 600 computer runs based on response curve I and parameter set II, in protocol A: the computed values of normalized mean transit time as a function of(A) noise level, and (B) FWHM; response error as a function of(C) noise level, and (D) FWHM; reconvolution error as a function of(E) noise level,

and (F) FWHM.

10(A)]. The response is deconvolved with similar error for narrow and wide curves, and shows only a slight decrease with increasing width [Fig. 10(B)]. The reconvolution errors are similar (= 10%) for all widths [Fig. 10(C)].

The results presented in Figs. 6 to 10 were all obtained using the response function I (Table 1), and the parameter set II of Ip(t) (Table 2). The results obtained for parameter sets I and III, as well as those obtained for all three parameter sets I, II and III for response curve II, displayed similar trends, and are therefore not presented. They were characterized by only minor differences, such as shifting up or down within the limits of a few percents, and slight differences in the variability of the results.

The response error and the reconvolution error, obtained in both protocols, A and B, for the above combination of parameter sets, were found to be un-

correlated. Correlation coefficients obtained for all combinations of parameters in both protocols range between 0.30 and 0.55.

DISCUSSION

A reliable noninvasive method for measuring regional tissue blood flow may provide the means for early detection and thus lessen the threat of coronary heart disease. Considerable efforts are made to use contrast echocardiography in order to quantitate regional myocardial perfusion by measuring different parameters related to perfusion (Armstrong 1986; Feinstein 1986; Feinstein and Shah 1986; Martin 1989; Meltzer 1985). The correlation between tissue blood flow and other measurable parameters, such as the peak gray level intensity, the time to peak contrast, the maximal slope and the area under the con-


t 8 1.0

• . . ~ B - " °

)"" " I ' " ! ' ' "

• . - . - . -

_ ~ . . ,,.. m-.mm.m . 1.0 ,m,m:.m....;-., l l l l ! ! l : : ' " u o

°~o n , a 2 (13 (14 (16 ~ 5 ~

~ w l . m 4 noise

,-.o m . . •

i ) """'""-'" | | | |h . . - : . . . (1 . i ! ! l l l i . - . . ; . , . .

(1~0 (1, (12 (12 (1, (16 0% 6 ~o 16 ~o noise FWHM

0.4 04

(13 (13 •

f l i F " " ~ 2 • o (1~, o l | •

i ' . ' - ' . ' - ; • =:" = l

01 Ulllmll' '""

noise FWl.m4

Fig . 8 . Same data as in Fig . 7 , obtained for protocol B.

215

25

26

trast buildup and washout curves, and a variety of other parameters, have been studied using intracoronary injection of contrast media (Kaul et al. 1989, 1991; Kemper et al. 1985; Monaghan et al. 1988; Reisner et al. 1989; Ten Cate et al. 1984a; Vanderberg et al. 1989). The reproducibility of contrast echocardiographic measurements has recently been demon- strated (Shapiro et al. 1990).

The mean time of transit of the contrast solution through a selected sample of myocardial tissue is theoretically directly related to the regional blood flow (Meier and Zierler 1954; Zierler 1962). It may be measured by introducing a short intracoronary bolus of contrast-enhancing solution, and creating curves representing the regional intensity versus time (Ong et al. 1984). However, this measurement presents a considerable difficulty, when the tracer is injected intravenously. In this case, tracer inflow into the myocardial tissue sample does not comply with the basic requirement of indicator dilution theory for an instantaneous impulse input.

Present efforts in the contrast echocardiographic research are concentrated on the development of tracer solutions capable of traversing the lungs, thus permitting noninvasive echocardiographic a s s e s s -

m e n t of myocardial blood supply (Keller et al. 1987; Shapiro et al. 1989; Smith et al. 1989; Ten Cate et al. 1984b). Therefore, we felt that it was necessary to design an algorithm for the evaluation of the mean transit time, applicable to data obtained using any shape of injection function, including peripheral intravenous injection of contrast-enhancing media.

Impulse response analysis (Gamel et al. 1973) was recently applied to digital angiographic data, where the response was found by an iterative procedure based on fitting a two-compartment lagged normal density function to indicator dispersion in a flow- ing arterial system (Eigler et al. 1989, 1991). The analysis was shown to be useful in the assessment of coronary flow and coronary flow reserve (Eigler et al. 1989, 1991). Based on the assumption that myocardial passage of contrast material behaves as a linear

Myocardial transit time of echo contrast • V. MoR-Aw et al. 645

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Fig. 9. Data similar to those in Fig. 7, showing noise dependence o f the different parameters for functions o f constant width ( F W H M = 10 _+ 1). Each point is an average of 25 separate runs (with standard deviations graphed), performed in order to min imize the effect of the random character o f the noise. Panels (A) to (C) present

the results o f protocol A, and panels (D) to (F ) - -p ro toco l B.

system that can be described by its impulse response function (Eigler et al. 1989), we approached the computation of the echocardiographic impulse response of a region of interest in contrast-enhanced echocardiographic images, by direct deconvolution analysis.

The approach used in this study is based on sev- eral additional assumptions, the validity of which should be discussed. First, the question of linear relation between the concentration of bubbles and videointensity is fundamental to the application of the linear systems theory. This is probably the most disputable issue in the quantitative contrast echocardiography, and its different aspects are discussed in detail in our paper dealing with the evaluation of regional myocardial blood flow (Mor-Avi et al. 1993).

Also, the classical definition of the mean transit time (eqn 1) is based on a tracer concentration measurement at a point where the fluid leaves the system. This definition is replaced by an alternative one, suit- able for contrast echocardiography (eqn 2) and based

on measuring videointensity averaged over the entire region of interest. This change must result in an un- derestimation of the mean transit time, since it mea- sures the time of transit from the system input to some inner portion of the sample represented by the mean intensity. This point should be carefully considered while attempting to analyze experimental echocardiographic data.

The validity of approximating the reference intensity at the port of entry to the sample of myocardial tissue by the intensity measured in the left ventricle or the aortic root should also be questioned. Correcting the approximation for the minimal time of passage r from the left ventricle to the sample implies that the passage from the left ventricle involves a pure time shift of the reference function. This approach disre- gards the fact that the sample is in part supplied via coronary arteries entering the sample, and in part through the capillary network continuing from the adjacent regions. Thus, additional mixing of tracer in


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Fig. I0. Data similar to those in Fig. 7, obtained for noiseless functions of variable width: panels (A) to (C)--protocol A, (D) to (F)--protocol B.

blood and consequent widening of the reference function is neglected. A more adequate modelling should refer to the contributions of the different delay times to blood inflow entering the sample, and probably find the distribution of delay times, which may also be of clinical value.

The basic functions simulating indicator dilution curves in our study were selected after Gamel et al. (1973) (set I in Table 1 and set I in Table 2). However, we noticed in our preliminary animal experiments that contrast echocardiographic curves are characterized by sharper slopes than those produced by the above parameters. We therefore introduced other values of parameters determining the rise and decay times of both intensity and response curves (parameter set II in Table 1 and sets II and III in Table 2). The assessment of the sensitivity of the algorithm to the input characteristics was achieved by studying different combinations of parameter sets.

Our results show that the algorithm can deal successfully with all combinations studied. Of course,

choosing curves of completely different shape may produce unexpected results, and require additional corrections and adjustments. However, our preliminary animal experiments suggest that the curves used in the present simulation work represent typical contrast-echo dilution curves. Therefore, the results of the study may be useful in the treatment of contrast- enhanced echocardiographic data.

Moreover, we believe that the functional dependence of the response function cannot be of critical importance for the evaluation of the algorithm. This function is used only to produce a time-dependent waveform, similar to experimental data, which can therefore be used as a response of a tissue sample. The entire process is numerical, and there is no attempt throughout the manuscript to restore the initial functional dependence. The time series representing the response is treated as an asymmetric "numerical bump," which could be produced by many other functions.

In real in vivo measurements, input and output

Myocardial transit time of echo contrast • V. MOR-AV! et aL 647

curves are obtained independently in two different regions of the images, which are probably subject to uncorrelated noise. On the other hand, noise in the input and output curves may also be correlated. How- ever, we chose to present the results of the analysis obtained under the most difficult conditions of truly uncorrelated noise. For this reason, we added random noise independently to both simulated input and output functions. Also, each region has its own characteristic background intensity. Thus, background intensities were added to both simulated curves to best imitate real echocardiographic data. The above algorithm was shown to successfully reconstruct the preset mean transit time under all conditions allowed by the simulation study (Fig. 6), including the most noisy data (SNR down to 2.0). The analysis was shown to be essentially nonsensitive to noise [Figs. 9(A) and 9(D)] and independent of the width of the reference function [Figs. 10(A) and 10(D)].

The response error and the reconvolution error were examined in search for an objective index for the accuracy of the algorithm performance. The response error cannot be evaluated when real experimental data are analyzed, since the true response of a sample of myocardial tissue is unknown. The reconvolution error, however, can be easily evaluated, and the question is whether it can be used to estimate the response error and the accuracy of the evaluation of the mean transit time which we derive directly from the response function. The correlation coefficients between the response error and the reconvolution error present unequivocal evidence that the reconvolution error is a poor estimate of the accuracy of the mean transit time.

The ability of the algorithm to deal with real echocardiographic noise should be studied by applying it to in vivo experimental data. However, both the curves we currently obtain in our laboratory from experimental animals, as well as those obtained by other investigators, are similar to those produced by the simulations applied here. It is thus reasonable to believe that the analysis of these curves should reach similar levels of accuracy, as the simulations. Therefore, the above algorithm, based on frequency domain analysis of the impulse response, can be used for the determination of the mean transit time of echocardiographic contrast media in myocardial tissue. Since reliable assessment of myocardial isehemia is not yet achieved by the existing techniques, such as coronary angiogra- phy (Marcus and White 1985; Marcus et al. 1986), it is of utmost importance to find a method for the direct assessment of myocardial blood supply (Kaul 1991; Watson and Belier 1991). We therefore hope

that the approach presented in this paper might con- tribute to the design of a noninvasive quantitative measurement of regional myocardial tissue blood flow.

Acknowledgements--V. M. is a recipient of the Levi-Eshkol Fel- lowship of the Israel Ministry of Science and Technology. The proj- ect was supported by the Elizabeth and Nikolas Slezak Foundation, The Abramson Center of Medical Physics, and the Ministry of Health.

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N O M E N C L A T U R E

( t ) =

/~(t) =

r,(~o) = Ip( t ) =

~ w ) = Rs( t ) =

=

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=

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m e a n t rans i t t ime o f t racer t h rough a sample o f m y o c a r d i a l t issue; t r a c e r - p r o d u c e d ( b a c k g r o u n d sub t rac ted) u l t rason ic ref lect ion m e a n in tens i ty f rom the t issue sample ; F o u r i e r t r a n s f o r m o f Is(t); t r a c e r - p r o d u c e d ( b a c k g r o u n d sub t rac ted) ref lec t ion m e a n in tens i ty f rom a h o m o g e - neous b l o o d poo l u p s t r e a m the t issue sample; F o u r i e r t r a n s f o r m o f Ip(t); t rue impu l se response func t ion o f the tissue sample ; t rue t ransfer func t ion o f the t issue s amp le [Four i e r t r a n s f o r m o f Rs(0]; eva lua t ed impu l se response func t ion o f the t issue sample ; eva lua t ed t ransfer func t ion o f the t issue s a m p l e [Four i e r t r ans fo rm o f ERs( t )] ; m i n i m a l t i m e o f passage f rom the poo l to the t issue sample ; full w id th a t h a l f m a x i m u m o f Ip(t).

Myocardial transit time of the echocardiographic contrast media

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