TRANSFER-FUNCTION ANALYSIS OF THE VENTRICULAR-FUNCTION - A NEW METHOD FOR CALCULATING RIGHT VENTRICULAR EJECTION FRACTION

1990;31:1294-1299. J Nucl Med.

Liang-Chih Wu, Shin-Hwa Yeh, Zen Chen, Por-Fen Chiou, Shyh-Jen Wang and Ren-Shyan Liu

Right Ventricular Ejection Fraction

Transfer Function Analysis of the Ventricular Function: A New Method for Calculating

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constant of the ventricular transfer function (in units of heart beat). In general, certain deconvolution proce dures have to be applied in order to measure the yen tricular transfer function. Further analysis from the convolution theory shows that the mean transit time technique can be undertaken instead. Computational complexity can be tremendously reduced. As we dem onstrate in this paper, a precise region of interest (ROI) is not necessary for getting accurate results. The sam pling time interval for TAC is ten to twenty times longer so that the counting statistics can be significantly im proved. A portion of this work has been published earlier in abstract form (1).

MATERIALS AND METHODS

Mathematical Basis of Transfer Function Analysis The radionuclidetransit ofFPRNA can be formulatedas a linear system (Fig. 1) where SVC = superior vena cava, RA = right atrium, RV = right ventricle, PA = pulmonary artery, LG = lung, LA = leftatrium, LV = left ventricle,AO = aorta, and SC = systemic circulation. Suppose that the TAC of chamber j can be described by function f3and the transfer function of chamber j is h@,then f@= f_, a h3,where a denotes the convolution operator. Let us focus on the ventricular area for the moment:

fy = f5*h@,

where f@is the ventricular output function, f5 is the atrial output function, and h@is the ventricular transfer function. Let F and F' be the Fourier transform and the inverse Fourier transform operators, respectively, then:

h@= F@ (@).

In other words,the transferfunctioncan be calculateddirectly from the known input and output functions using the Fourier transform and inverse Fourier transform functions. In prac tice, unavoidable errors in the measured values of both the input and output functions can result in physically unreason able values of the transfer function, e.g., negative values or rapid oscillatory behavior (2). Thus, indirect techniques are preferred(3-5). However,both approachessufferfrom large computational complexities. Accordingly, we analyze the yen

The relationship between the ventricular transfer function and ejectionfractionhas beeninvestigatedby the routine procedureof first-pass radionuclideangiocardiography. Ejectionfractionhas beenshownto equal1 —e_b,where b is the ratioof the R-Rintervaloverthe meantransittime difference between ventricular and atnal time-activity curves.To evaluatethe effect of regionof interest(AOl) on the rightventricularejectionfraction(AVEF),the results of the transfer function analysis(TFA) techniqueusing preciseROI,TFA using rectangularAOl, and the routine method were compared. Regression analyses among RVEFs obtained from the above ROl methods yielded good correlations. Reliable RVEFs have been obtained evenin the caseof an improperbolusinjection.Thus,the TFA techniqueis a new, simple,and reliablemethodfor calculating AVEF without needing to outline the right van tricleprecisely.

J NuclMed 1990;31:1294—1299

irst-pass radionudide angiocardiography (FPRNA) is a clinically accepted procedure for calculating the ejection fraction (EF) that is the most widely used parameter for measuring cardiac function. However, FPRNA has been criticized for its poor counting statis tics because of the generation of the ventricular time activity curves (TACs) with a sampling interval of 0.03— 0.07 sec. The difficulty in precisely outlining the yen tricular boundary and the poor counting statistics lead to large inter- and intraobserver variations in the cal culation of the EF. Therefore, an attempt was made to solve the above problems, using a simple mathematical model to analyze ventricular function. The relationship between the ventricular transfer function and the EF has been evaluated. It can be demonstrated that yen tricular EF is equal to 1 —c_b, where b is the decay

ReceivedOct. 30, 1986; revisionaccepted Feb. 8, 1990.

For reprkits contact: Peter Shin-Hwa Yeh, MD, Professor and Director,

DepartmentofNuclearMe@cine,VeteransGeneralHospftal,TaIpei,Taiwan

11217.

1294 TheJournalof NuclearMedicine•Vol. 31 •No. 8 •August1990

Transfer Function Analysis of the Ventricular

Function: A New Method for Calculating Right

Ventricular Ejection Fraction

Liang-Chih Wu, Shin-Hwa Yeh, Zen Chen, Por-Fen Chiou, Shyh-Jen Wang, and Ren-Shyan Liu

Department ofNuclear Medicine, Veterans General Hospital; National Yang-Ming Medical College; and Institute of Computer Engineering, National Chiao-Tung University, Taipei, Taiwan

tricular function from the physiologic point of view. Two circumstances of radionucide input, as described below, are impulse response and a sequence of radionucide input.

ImpulseResponse:An InstantaneousRadionuclide Input

In the first situation, we measure the variation for the remaining radioactivity in the ventricle after an instant injec tion with a quantity, Q, of radioactivity. Four assumptions are made here: (a) a constant EF k (0 < k < 1), (b) a constant end-diastolic volume (EDV), (c) injection at the end-diastolic instant, and (d) no radioactivity in the blood entering the right ventricle during the diastolic phase. In other words, the analy sis is valid only before the recirculation occurs. Such a model is shown in Figure 2.

The quantity of radioactivity at any end-diastolic instant can be observed as follows:

Qo Q,

Q=Qo(l k),

Q2 Q@(l —

Qi = Q_@(l —k).

Thus, the quantity of the remaining radioactivityat the i-th end-diastolic time instant after the injection is:

Q=Qm-@(1 k)

= Q(l — k)tm

= Q@@ib,

where b = ln(l/l —k) > 0. As a result, it can be observed that the ventricular transfer function follows a simple expo nential decay and the decay constant b equals ln(l/l —k). The EF k is then equal to 1 —e@'.

A Sequenceof RadionuclideInput[I,]

In the second situation, the assumptionsare made as fol lows: (a) a constant EF k, (b) a constant EDV, (c) no time delaybetweenthe ventricularand atrial functions,and (d) the

atrial function as a discrete time sequence I). The subscript i is counted in number of heart beats. The radioactivity meas ured at the end-diastolic instant of each heart beat can then be expressed as follows:

Qo lo,

Qi = Q@(l —k) + Ii,

Q2 Q@(l —k) + 12,

Q=Q,_1(l—k)+I1.

Thus, the quantity of the remaining radioactivity at the i-th beat is:

As we initially suggested, the ventricular function is equal to the convolution of the atrial function and a transfer function h. Using the discrete notation, h = (1 —k)1= e@'1'@. Again, the ventricular transfer function follows a simple exponential decay and the decay constant b equals ln(l/l —k). The EF k also equals 1 - e―as is the case with the impulse response condition.

Methods of Calculating Right Ventricular Ejection

Fraction

Ninety-five patients with good bolus injection, i.e., the mean transit time of the superior vena cava being <4 see, were randomly chosen from the routine FPRNA study. In our laboratory,the routine FPRNA procedurewas modifiedfrom that ofJengo et al. (18). Data were acquired in list mode using right anterior oblique projection for 33 sec after intravenous injection of 20 mCi of technetium-99m-pertechnetate. Pa tients were divided into 2 groups. One study with 50 patients was done by linking a digital camera (Elscint APEX 410, Haifa, Israel) to a minicomputer (Informatek Simis 5, Buc, France), and the other with 45 patients was performed by linking an analog camera (Pho Gamma IV) to another mini computer (Informatek Simis 3). For the routine procedure, list mode data were reframed into one second images and the representative images for right heart, lung and left heart were chosen. Regions ofinterest were then manually defined. As in the left ventricular phase, right ventricular TAC was generated in units of 40-msec time intervals. The right ventricular ejec tion fraction(RVEF)wascalculatedby averaging2 to 3 peak valley pairs chosen from the right ventricularTAC as suggested by Jengo et al. (18).

For comparison,a rectangularROl was placed within the right ventricular area in addition to the routine precise ROI.

Qi = Q_i(1 —k) + I@

=I@(l—k)1+I1(1—k)1'+...+I1

= A I@(l — k)*_i.

FIGURE2

A transfer function analysismodelfor impulseresponse.

1295

TransferFunctionAnalysis•Wuet al

b4svcj

Ø1RAI@4RVJ @PA14wJ Ø1LAI.@LvI .@AoI ø@sc@J

FIGURE1

Linear system for transfer function analysis. See Matetials and Methods for definitions.

/

rid of nontarget activities. The gamma fitted TACs were treated as the function ofright atrium and ventricle. The mean transit times of the gamma fitted curves were then measured and the EFs were calculated from the equation described earlier in this section. The sampling interval of TACs was equal to the R-R interval, the atrial TAC was generated by a time delay of a fraction of the R-R interval, i.e., the interval of the diastolic phase, and the ventricular transfer function was describedby the simpleexponentialdecay function with decay constant b and the EF equal to 1 —

@ The decay

constant can be obtained by an exponential fit of the transfer function solved by the deconvolution technique. However, the direct solution using Fourier transform has some practical difficulties (2). Accordingly, we solve the problem in another way. According to the convolution theory, it can be proved that, if g = f * h, then MTT(g) = MTT(f) + MTT(h), where MTT(g) is the mean transit time ofthe function g. It can also be proved that, if h = ae@, then MTT(h) = 1/b. Since the TACsof both atrium and ventricleare known, calculationof mean transit times can be easily obtained (8,9). As a result, the decay constant of the ventricular transfer function can be expressed as R-R interval/MTT(h) where the R-R interval and MTT(h) were all counted in units of seconds.

RESULTS

In the experiment, the RVEF from the “precise―ROl method was compared with that from the routine man ual first-pass(MFP) method. The correlation coefficient was 0.91 for the digital camera group and 0.90 for the analog camera group. The correlation coefficients of RVEF from the “rectangular―ROI method and from the MFP procedure were all 0.90 for both groups. There were high correlations between the “rectangular―ROl method and the “precise―ROl method with r = 0.96 and 0.98 for the digital and analog groups, respectively. The regression plots of the above relationships for the digital group are shown in Figure 3 and those for the analog group are illustrated in Figure 4. After the determination of the transfer function, it could be reconvolved with the gamma fitted atrial curve.

C

C

A

gamma fitted ventricular curve. An example is shown in Figure 5.

In addition, seven patients with both rapid and slow injections were studied to show the effect of injection rate on the results. The results are shown in Table 1.

Poor correlation (r = 0.58, s.e.c.= 12.8)betweenboth

rapid and slow injections was found for MFP method whereas good correlations were obtained for both TFA methods (r = 0.98, s.e.c. = 0.30 for precise ROl TFA method and r = 0.98, s.e.c. = 0.26 for rectangular ROl TFA method).

DISCUSSION

The use of the low frequency isotope dilution curve in calculating the RVEF has been described by other authors (6,7,15—17). Their analyses have focused on the down slope of the right ventricular time-activity curve (RV TAC), and the results depend on the quality of the bolus injection. A deconvolution process was recommended in their procedures for a poor bolus injection since their model could be applied to the ideal situation only, namely, the impulse response condition. In contrast, RVEF was calculated from the transfer function instead of the RV TAC in our study. Accord ingly, RVEFs, as shown in Table 1, are relatively mdc pendent of the bolus injection since the deconvolution effect is embedded in the transfer function analysis technique. Glass et al. (15,16) used down slope transfer function analysis for calculating left ventricularejection fraction (LVEF). The deconvolution process is done with Fourier and/or Laplace transforms. Their model uses pulmonary TAC as input which is not adequate in our model. As for the compartmental model described by Konstantinow et al. (13), there is delay and spread factor for tracer in each compartment. If the input for the ventricular function is not atnal TAC, MiT differ ence will be relatively increased with a resultant under estimation of the EF. Villanueva-Meyer et al. (1 7) also

20 40 60 80 MFP RVEF 80 60 40 20 0 20 40 60 80 RVEF . p@4@ 1101 @80 C t LU > 20

B

C

Correlations of digital camera group between AVEF as calculated from (A) TFA precise AOl method and routine MFP AOl

method,(B) TFA rectangularAOl and MFP AOl, and (C) both TFA ROIs.

0 51 Mo 20

A

C

Correlations of analog camera group between RVEFs as calculated from (A) TFA precise AOl method and routine MFP AOl method, (B) TFA rectangular ROl and MFP AOl, and (C) both TFA AOls.

chose the ideal case as their model, using the superior vena cava TAC as input and pulmonary artery TAC as output. The use of pulmonary TAC as output function is not adequate in our model since the pulmonary TAC includes the transfer function of pulmonary artery, which will increase the MTT difference between input and output functions and result in the underestimation of the EF.

From the results shown in the previous section, there is a good correlation in RVEFs between the procedures of the transfer function analysis technique and the routine procedure. The advantages gained from this technique include ROI independence, good sampling statistics, simplicity, and reliable results. No obvious discrepancy in RVEF occurs between the two methods with the precise ROl and the rectangular ROl, respec

tively, as evidenced by their high correlation (r = 0.96 and 0.98 for the digital and analog groups, respectively). Since the procedure for gamma fitting has been auto mated, the good correlation here means low inter- and intraobservcr variations. With this technique, the criti cism of low count rate in the FPRNA study can be avoided because the sampling time of TACs is about 10—20times longer than that used routinely. Further more, no significant difference exists between the use of the digital cameras and analog cameras. A similar conclusion has been obtained by Aswegen et al. (6). Pitfalls of the inverse Fourier transform for deconvo lution can be avoided also because it is not necessary to consider the effect oftime delay between the ventnc ular and atrial functions in the MiT technique. The sampling time interval is not restricted to the R-R

iO,17'8'+ CHEJI @

CARDIAC CYCLE 0.96 SEC

FIGURE5

Example of transfer function analysis

of AVEF.AtrialTACon upper-leftquad rantandventricularTACon lower-left quadrantare gammafitted. One-see ond imageand ROIsdrawn are shown on upper-rightquadrant. Atrial TAC is

reconvolved (displayed in green) and

comparedwith ventticular TAC (in yel low)on lower-rightquadrant.

RV MTT = Lf,97 SEC

TransferFunctionAnalysis•Wuet al

1297

Effect of Injection on Results of RVEFs Obtained from MFP, TFA Precise ROI, and TFA RectangularAOlMethods1 StifljeCtiofl2nd

injectionPatient

SMTT EF1t EF2 EF3' SMTTEF1t

EF2EF3'1 1.9 23 26 25 4.933 29272 2.8 55 50 51 4.154 52523 2.4 43 49 53 5.127 46474 2.7 22 20 21 9.937 22245 3.4 40 38 36 23.156 42376 1.7 45 46 50 4.946 43497 3.4 55 58 54 4.545 6053.

SMTT = mean transit time of SVC.t

EF1 = RVEF from MFP procedure, r = 0.51 between injections, s.e.e. = 12.8.6

EF2 = RVEF from TFA precise ROl method, r = 0.98, s.e.e. = 3.0.I

EF3 = RVEF from TFA rectangular AOl method, r = 0.98, s.e.e. = 2.6.

interval either. This makes the algorithm fairly simple. The reliable results as shown in Table 1 can be obtained for some cases of the prolonged bolus injection. If the bolus has more than one peak, the result ofthe current method is not satisfactory either. The results would be similar to those obtained by deconvolution techniques

(10—12).

Of course, two problems can occur with the current procedure. First, the gamma fitting technique cannot remove nontarget activities; second, the current proce dure does not work for patients with severe valvular disease because of the incorrect curve fitting. To over come the effect of overlapping in organs, the algorithm described by Konstantinow et al. (13) could be helpful. It eliminates crosstalk iteratively. Factor analysis tech nique (14) could be another choice, but it is difficult to distinguish the ventricular activities from the atrial activities. Inadequate mixing of the radionucide bolus with blood introduces a potential source of error for calculating RVEF (19). Although such an effect hinders the homogeneous mixing assumption ofour model and other washout models, clinical experience has shown the feasibility and usefulness of the transfer function analysis in the assessment of RVEF (6,7,15—17).

In theory, the current analysis should work for the left heart. However, some practical difficulties do exist. In the right anterior oblique projection used by FPRNA routinely, the ROl of the left atrium cannot be reliably drawn. The use ofpulmonary TAC, as discussed above, is not adequate for the estimation of LVEF. Further more, the background factor is greater in the left heart than in the right heart. Thus, the validity of applying this technique to the determination of LVEF needs further investigation. Such an investigation is now in

progress in our laboratory.

In conclusion, the transfer function analysis tech nique is a new, simple, and reliable method for calcu lating RVEF, in which RVEF is not dependent on accurate ROl and a good bolus injection.

1298 The Journal of Nuclear Medicine •Vol. 31 •No. 8 •August 1990

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