A New Approach of Weighted Integration Technique Based on Accumulated Images Using Dynamic PET and H 50

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1 Journal of Cerebral Blood Flow and Metabolism 11: The International Society of Cerebral Blood Flow and Metabolism Published by Raven Press, Ltd., New York A New Approach of Weighted Integration Technique Based on Accumulated Images Using Dynamic PET and H 50 *ttakashi Yokoi, *Iwao Kanno, *Hidehiro Iida, *Shuichi Miura, and *Kazuo Uemura *Department of Radiology and Nuclear Medicine, Research Institute for Brain and Blood Vessels, Akita, and tdepartment of Research and Development Engineering for Nuclear Medicine, Medical Systems Division, Shimadzu Corporation, Kyoto, Japan Summary: We developed a new technique of weighted integration for the measurement of local cerebral blood flow (LCBF) and the blood-tissue partition coefficient (p) using dynamic positron emission tomography (PET) and H150. The weighted integration in the new technique is carried out on the equation of the first time integration of the Kety-Schmidt differential equation. Practically, serially accumulated images with sequentially prolonged accumulation times are weighted by two arbitrary functions. The weighting functions do not have to be differentiated because of the exclusion of the differential term in the starting equation. Consequently, the method does not require data at the end of the scan. The technique was applied to Hi50 dynamic PET performed on four normal subjects, and was verified to provide a better signal- to-noise ratio than the previously developed integrated projection (IP) technique. Computer simulations were carried out to investigate the effects of statistical noise, tissue heterogeneity, and time delay and dispersion in arterial input function. The simulation showed that the new technique provided about a 1.4 times lower statistical error in both LCBF and p at 50 ml 100 g - I min - I compared to the IP technique, and it should be noted that the new technique was less sensitive to the shape of the weighting functions. The new technique provides a new strategy with respect to the statistical error for estimation of LCBF and p. Key Words: Positron emission tomography-local cerebral blood flow-partition coefficient Weighted integration technique-single compartment model-i 50-labeled water. Various techniques have been developed to estimate local cerebral blood flow (LCBF) using a diffusible tracer and positron emission tomography (PET). These techniques are based on the Kety Schmidt single-compartment model (Kety, 1951). The C1502 steady-state method (Subramanyam et ai., 1978; Frackowiak et ai., 1980) and the H 50 autoradiographic method (Raichle et ai., 1983; Herscovitch et ai., 1983; Kanno et ai., 1984, 1987) use a Received March 26, 1990; revised September 10, 1990; accepted September 14, Address correspondence and reprint requests to Dr. I. Kanno at Department of Radiology and Nuclear Medicine, Research Institute for Brain and Blood Vessels, Senshuu Kubota-Machi, Akita City, Akita, 010, Japan. Abbreviations used: A WI, accumulation weighted-integration technique; COV, coefficient of variation; DWI, direct weightedintegration technique; FWHM, full width at half-maximum; IP, integrated projection technique; LCBF, local cerebral blood flow; PET, positron emission tomography; ROI, region of interest. fixed partition coefficient (P) for the estimation of LCBF. However, p is not a uniform value; in particular, it varies in pathological states (Herscovitch and Raichle, 1985). As a result, the use of a fixed p value gives a large error for LCBF. Dynamic analysis by nonlinear least-squares fitting (lida et ai., 1989a; Lammertsma et ai., 1989) is generally used to determine both LCBF and p simultaneously. In general, these methods require considerable calculation time for the pixel-by-pixel estimation. Koeppe et al. (1985) developed the fast estimation technique using linearized least-squares fitting with the lookup table. Huang et al. (1982) developed the integrated projection (IP) technique and succeeded in an analytical estimation of both LCBF and p simultaneously in each pixel. The IP technique was applied to the bolus input technique using decay-corrected and uncorrected projection data (Huang et ai., 1983). Carson et al. (1983, 1984, 1986) generalized the IP 492

2 NEW WEIGHTED INTEGRATION TECHNIQUE WITH PET 493 technique by introducing the concept of weighted integration using two arbitrary weighting functions (and their two differential functions). They also optimized a pair of the weighting functions in order to minimize the statistical error, and demonstrated that the error reduction was close to a theoretical minimum. Alpert et al. (1984) developed the alternative weighted-integration technique combined with the table lookup method. We propose a new approach of weightedintegration technique starting from the equation of time integration of the Kety-Schmidt differential equation in the single-compartment model. In other words, serially accumulated images with sequentially prolonged accumulation times are weighted by certain functions for estimation of LCBF and p. Simulation and human studies were performed to evaluate the characteristics of the new technique. In this report, we describe the theory of the new (!.LCi/g), respectively,fis the LCBF (ml g-i min -I), p is the tissue-blood partition coefficient (mllg), and A is the radioactivity decay constant of 150 ( min-i). The theoretical basis of the new technique is as follows: First, time integration of the Kety-Schmidt equation [Eq. (1)] is derived under the condition of Ci(O) = 0: Ci(t) = f I: Ca(s)ds - (t + A) I: Ci(s)ds (2) Subsequently integrating both sides of Eq. (2) over the scan duration T after multiplying two weighting functions W I (t) and Wit), the following simultaneous equation is obtained: technique, and show its advantages over the previous techniques. where k = f/p + A. The LCBF and k are given by the solution of Eq. (3) as follows: f= k= (T W1(t) {t Ci(s)ds dt (T W2(t) {I Ca(s)ds dt (T - W2(t) {t Ci(s)ds dt (T W1(t) {t Ca(s)ds dt Jo Jo Jo Jo Jo Jo Jo Jo (T W1(t) {t Ca(s)ds dt {T W2(t)Ci(t)dt - (T W2(t) {t Ca(s)ds dt (T W1(t)Ci(t)dt Jo Jo Jo Jo Jo Jo (4) THEORY The kinetics of a diffusible tracer such as H150 is based on the Kety-Schmidt single-compartment model, and can be described by a first-order differential equation as follows: dci(t) (f dt ) = fca(t) - p + A Ci(t) (1) where Ca(t) and Ci(t) are the radioactivity concentration in arterial blood (!.LCi/ml) and cerebral tissue The terms of JCi(s)ds and JCa(s)ds are integral data from 0 until a certain time t. The partition coefficient can be obtained from p = f/(k - A). The pair of weighting functions are imposed by the condition to be independent of each other, but not required to be differentiable (i.e., even noncontinuous functions are available). In order to compare our new technique with the IP technique, we will summarize the generalized expression of the IP technique. Performing a weighted integration on Eq. (1) directly, the general expression of LCBF and k were obtained as

3 494 T. YOKOI ET AL. f= k= JOT W1(t)Ci(t)dt [ W2(1)Ci(1) - JOT W2(t)Ci(t)dt 1- JOT W2(t)Ci(t)dt [ Wl(1)Ci(1) - JOT W1(t)Ci(t)dt JOT W2(t)Ca(t)dt - JOT W2(t)tCi(t)dt JOT W1(t)Ca(t)dt JOT WI (t)ci(t)dt 1 JOT W1(t)Ca(t)dt [ W2(1)Ci(1) - JOT W2(t)Ci(t)dt 1- JOT W2(t)Ca(t)dt [ Wl(1)Ci(1) - JOT WI (t)ci(t)dt 1 JOT W1(t)Ci(t)dt JOT W2(t)Ca(t)dt - JOT W2(t)Ci(t)dt JOT W1(t)Ca(t)dt (5) The pair of weighting functions [W I (t), Wit)) must not only be independent from each other but also differentiable because the inclusion of the differential terms [W I (t), W2(t)] of weighting functions in Eq. (5) is needed. In the original IP technique (Huang et al., 1982), the decay-corrected and uncorrected projection data were used for generating parametric images of LCBF and p. In other words, the weighting functions used were 1 and exp(o.338t) in Eq. (5). In this report, decaycorrected data are represented by the symbol with asterisk as Ci*(t). The weighted-integration procedure of Eq. (4) or (5) can be employed on either reconstructed images or projection data, and this exchangeable nature of the procedure between images and projections was previously shown by Tsui and Budinger (1978). Huang et al. used projection data in the IP technique; on the other hand, we employed a dynamic image in both of the techniques [Eqs. (4) and (5)]. Hereafter, the new technique based on Eq. (4) is called the accumulation weighted-integration (A WI) technique, and the technique based on Eq. (5) is called the direct weighted-integration (DWI) technique because the weighted integration procedure is performed on the Ci(t), Ca(t) directly. In mathematical form, the IP and the DWI techniques are the same. METHODS Dynamic PET measurements The PET scanner used was the HEADTOME-IV (lida et ai., 1989b), a four-ring and seven-slice machine. A spatial resolution at the center of the field of view was 4.5 mm full width at half-maximum (FWHM) and slice thickness was 9 and 9.5 mm FWHM for cross and direct planes, respectively. The dead time and radioactivity decay corrections were performed on-line every 2.5 s (Amano et ai., 1988; Iida et ai., 1989b; Kanno et ai., 1989), and counting loss due to dead time was less than 1 % (Yamamoto et ai., 1986) in this study. The arterial blood radioactivity Ca(t) was monitored from a radial artery using a plastic scintillation beta detector. The arterial blood was withdrawn with a constant speed of 10 mumin and was sampled at I-s intervals. Dispersion occurring in the detector system was negligibly small. Details of the blood sampling system have previously been described (Iida et ai., 1986; Kanno et ai., 1987). Human studies were performed on four male normal subjects (37, 39, 43, and 54 years of age). The transmission scan was performed for 5 min for tissue attenuation correction. After the intravenous bolus injection of HfO (35 mci), dynamic scans were performed (12 scans of 5 s, 5 scans of 15 s, and 3 scans of 60 s). In each study, the calibration between the beta detector for blood sampling and the PET scanner was carried out using a well counter. Data processing The LCBF and p images were calculated by the A WI [Eq. (4)] and DWI [Eq. (5)] techniques. In the present study, the weighted-integration procedure was performed on multiple reconstructed images (with FWHM = 8 mm) measured by dynamic PET. The Ci*(t) images were weighted to the midpoint of each scan period, and the arterial input data Ca(t) were weighted to each of the sampling points. Time adjustment between the arterial and whole brain slice curves was performed according to our previous method (lida et al., 1988). Dispersion occurring in the radial arterial system has not been corrected in this study. The weighting functions used were a pair of 1 and exp( - o.t) (0. = 0.338) instead of exp(o.t) and 1, since the real-time decay correction was performed in the measured dynamic data. To determine the Ci*(I) term for the DWI [Eq. (5)], the last 60-s image was used. The total scan duration Twas 285 s for both techniques, and the last measurement [Ci*(I) measurement] was performed from T - 30 to T + 30 s for the DWI technique. Regions of interest (ROIs) were selected on LCBF and p images as follows: the gray matter of the cerebellar cortex, temporal cortex, frontal cortex, occipital cortex, and parietal cortex; the white matter of the oval center; the whole brain was defined as an average of three cross sections at 46, 59, and 72 mm above the orbitomeatal (OM) line. ROIs of the gray matter and white matter were defined in each of the two hemispheres. The characteristics of the A WI technique were investigated by the computer simulations. In all of the simulations, a typical arterial input function Ca(t) measured in a human study following the intravenous bolus injection of Hi50 (Fig. 1) was used. The scan duration was 285 s, the

4 NEW WEIGHTED INTEGRATION TECHNIQUE WITH PET 495 1x o o Time (sec) FIG. 1. A typical arterial input function (decay-uncorrected) used for all of the simulations measured from a radial artery following intravenous bolus injection of H 50. value of p was fixed at 1.0 mug, and the weighting functions used were 1 and exp(ut) (u = 0.338). The effect of statistical noise on the estimated LCBF and p was evaluated. The tissue response function Ci(t) was calculated as Ci(t) = f Ca(t) exp [ -( + x}] (6) where denotes the operation of convolution integral. Assuming that the statistical noise followed a Gaussian distribution, the noise was added on Ci(t). The magnitude of the statistical noise was determined by the empirical relationship obtained from a brain phantom experiment. The ROI was selected on a cortical gray matter region, and the noise coefficient of variation (COV noise) was calculated from the serial scanning experiment. In this case, the relationship between the COY noise and total counts (N) with FWHM = 8 mm for an 8-mm ROI diameter has been derived as (Kanno et ai., 1990) 1 COVnoise = 3.6 X 10 3 \IN (%) (7) Statistical errors (COVs) in LCBF and p were calculated by both techniques from 200 independent trials. The total counts N were typical of human study data. For the DWI technique, the Ci(T) value was determined by averaging over the tissue response function in 285 ± 30 s. The effect of tissue heterogeneity caused by a limited spatial resolution PET scanner was evaluated. Simulation assumed a mixture of two components corresponding to gray matter (flow = 80 ml l00 g-l min-1 and p = 1.0 mug) and white matter (flow = 20 ml 100 g -1 min - 1 and p = 1.0 mug). The fraction of the gray matter was varied from 0 to 100%. Two scan durations (T = 120 and 285 s) were examined in the A WI technique. The effect of dispersion of the measured arterial input function was simulated using a dispersion function d(t) (Iida et ai., 1986) described as 1 d(t) = - e-tit T where T is a dispersion time constant (s). In this simulation, the dispersed arterial input function was given by (8) convolution of d(t) on the input function in Fig. 1. The dispersion time constant was assumed to 4 s (lida et al., 1986, 1989a; Meyer, 1989). The effect of time delay of the arterial input function was also evaluated. The time delay.:it in the brain was assumed to be ±2 s (Iida etal., 1988). The positive delay time was defined such that the arterial input function was delayed compared to the tissue response function. The influences of the shape of the weighting functions with various exponential weighting constants u were also investigated for each simulation. RESULTS Results of human studies Typical serial time integral images in the AWl technique are shown in Fig. 2. The counts of the images become higher with increasing time, and then the counts of the last image ( = total counts) were about 2.6 x 10 6 Figure 3 shows the images of LCBF and p calculated by each technique. The calculation time for images of both LCBF and p required about 1 min for one slice (128 x 128 matrix) using a V AX 11/750 computer. Table 1 shows the LCBF and p of the gray and white matter and mean of brain obtained in four normal subjects. LCBF values estimated by each technique were basically the same, but p values estimated by the AWl technique were lower than those of the DWI technique. Figure 4 shows LCBF values (n = 2) of the gray and white matter and the mean of brain at various scan durations (T = 57.5, 97.5, 165, 225, and 285 s) in the A WI technique. LCBF values were normalized to the LCBF value of the scan duration of 285 s. LCBF values decreased as a function of the scan duration T. Simulation studies Figure 5 shows a comparison of the statistical error (%CaVs) in LCBF and p estimated by both techniques for flows ranging from 10 to 100 ml 100 g-i min-i. The Cays of LCBF in the AWl and DWI techniques were 10 and 14%, respectively, at a flow of 50 ml 100 g - I min - I. The errors in LCBF decreased with increasing flow. The Cays of p in the AWl and DWI techniques were 6.7 and 11%, respectively, at a flow of 50 ml 100 g - I min - I. The errors in p showed the same decreasing tendency to increasing flow but considerably larger at low flows for both of the techniques. The errors due to the statistical noise both in LCBF and p by the AWl technique were less than those by the DWI technique. Table 2 shows the statistical errors (CaVs) in LCBF for various exponential weighting constants

5 496 T. YOKOI ET AL. FIG. 2. Typical time integral images (OM + 59 mm) in a normal subject. The images were calculated from dynamic scans that consisted of 12 scans of 5 s, 5 scans of 15 s, and 3 scans of 60 s. All images were already corrected for radioactive decay. FIG. 3. Functional images (OM + 59 mm) of LCBF (top images) and p (bohom images) by the accumulation weightedintegration technique (AWl) (A) and the direct weightedintegration technique (OWl) (8) in a normal subject. a. The Cays were less in the AWl technique than in the DWI technique for the weighting functions. Moreover, in the A WI technique, it should be noted that the Cays were less sensitive to the shape of the weighting functions, while in the DWI technique, the Cays were considerably changed. Figure 6 shows errors due to tissue heterogeneity as a function of the gray matter fraction. Underestimations of LCBF and p were observed for both techniques. The maximum errors in LCBF were -5.2 and -8.8% in the AWl and DWI techniques, respectively, and those in p were -20 and - 15%, respectively. The A WI technique gave a smaller underestimation in LCBF but a larger underestimation in p compared to the DWI technique. Both parameters revealed the largest underestimation at about a 30% gray matter fraction. Figure 7 shows the relationship between the underestimation and different scan durations (120 and 285 s, respectively) in the AWl technique. The LCBF values were more underestimated with larger T, while p values provided less error with larger T.

6 NEW WEIGHTED INTEGRATION TECHNIQUE WITH PET 497 TABLE 1. LCBF and p values calculated by each of two techniques in normal subjects LCBF (miioog-1 min-i) AWl OWl AWl p (milg) OWl Gray matter 48.4 ± ± ± ± 0.05 White matter 17.8 ± ± ± ± 0.14 Whole brain 41.0 ± ± ± ± 0.04 Values are mean ± SO. LCBF and p in normal subjects (n = 4) calculated by the accumulation weighted-integration technique (A WI) and the direct weighted-integration technique (OWl). The whole brain values were defined as an average of three slices described in the Methods section. Table 3 shows the errors in the estimated values of LCBF and p caused by the dispersion and the time delay. The dispersion and positive time delay gave an overestimation of LCBF and an underestimation of p; on the other hand, the negative time delay had an inverse tendency. With increasing flow, the error becomes larger in LCBF and smaller in p. The AWl technique was more sensitive to the dispersion and the time delay compared to the DWI technique. Table 4 shows that the biases in the estimated values of LCBF and p with various exp weighting constants a were caused by the dispersion and the time delay. The error in LCBF and p becomes smaller with larger a. DISCUSSION The weighted integration technique simultaneously provides LCBF and a partition coefficient in a short calculation time. The partition coefficient is defined as the relative solubility of water in the tissue compared to the blood, and hence may not be a uniform value, especially different between gray 1.3 r----'-----'----'----'---+ LL ID o J -g Gray matter o Z <>-- White matter... Whole brain , ,--,----t o Scan duration (sec) FIG. 4. Variation of LCBF by the AWl in normal subjects (n = 2) as a function of the scan duration. Values are normalized by LCBF of the scan duration of 285 s. The whole brain values were defined as an average of three slices described in the Methods section. and white matter, and it may also be affected by pathological states. Therefore, the use of a fixed p value gives an error for LCBF (Kanno et ai., 1987). The estimated LCBF by the weighted-integration technique is more accurate than the LCBF measured by the CI502 steady-state or the H 50 autoradiographic methods, which use a fixed value of p. In this report, we proposed a new technique based on weighting of serially accumulated images, and investigated its characteristics using dynamic PET and the computer simulations. Human studies The values of estimated LCBF by both techniques were similar to each other, i.e., 48.4 and 17.8 ml too g-i min -I in the A WI technique, and 47.5 and 17.3 ml 100 g -1 min -1 in the DWI technique for the gray and white matter, respectively. The partition coefficient values also showed similar values between the two techniques, i.e., 0.80 and 0.68 mvg in the A WI technique and 0.84 and 0.74 mvg in the DWI technique for the gray and white matter, respectively. The partition coefficient values of the gray and white matter for water were calculated as 0.98 and 0.82 mlig, respectively (Herscovitch and Raichle, 1985). A possible explanation for underestimation is due to the tissue heterogeneity and dispersion (lida et ai., 1989a). In the original IP technique (Huang et ai., 1983), the p values of the gray and white matter were obtained as 0.85 and 0.76 ml/g, respectively. They explained the underestimation by the fact that not all brain water is freely exchangeable. However, the underestimation can be explained by the tissue heterogeneity and dispersion completely. The variation of LCBF as a function of the scan duration (Fig. 4) might be caused by the invalidity of assumptions in the single-compartment model (the tissue heterogeneity), and dispersion and time delay of the arterial input function. Similar phenomena have been discussed in previous articles about the in vivo autoradiographic method (Raichle et ai., 1983; Ginsberg et ai., 1984; Iida et ai., 1986), the weighted integration technique (Gambhir et ai., 1987; Koeppe et ai., 1987), and nonlinear leastsquares fitting (Lammert sma et ai., 1989; Iida et ai., 1989a). The variation of LCBF can be also explained by tissue heterogeneity and dispersion (lida et ai., 1989a). The effect of statistical noise The LCBF and p images (Fig. 3) calculated by the AWl technique showed a better signal-to-noise ratio than those of the DWI technique. Especially, a J Cereb Blood Flow Metab, Vol. 11, No. 3, 1991

7 498 T. YOKOI ET AL. A B 20 U. III 15 () -I > 10 o () n... ' 0'" 0 ' ' ' '0 ' ''0'''0 '' '0'''0''' Q. c: > 0 () \ \. b... \ b..... n. n.",0". FIG. 5. Comparison of COVs (%) in LCBF (A) and p (8) as a function of flow by the AWl (-e-) and the OWl ( ) technique. cavs (%) were obtained by 200 simulations in each flow. In the simulations, it was assumed that p was 1.0 milg, the scan duration was 285 s, and weighting functions were 1 and exp(at) (a = 0.338). O+- r O+-.-r--.-+ o o Flow (m1/1 OOg/min) Flow (ml/100g/min) large error appeared on p images at low flow regions calculated by the DWI technique. This result was in agreement with the simulation of the statistical noise (Fig. 5 and Table 2). One of the reasons for the lower noise level in the A WI than in the DWI technique is due to the fact that the last measurement term [Ci(1) term] is not included in the AWl technique [Eq. (4)]. In the DWI technique, the last measurement term must be measured by the integration over a few minutes around the end of the scan time. Even then, this term propagates a statistical error to LCBF and p images because of the isotope used with a short lifetime, as already pointed out previously (Blomqvist, 1984; Koeppe et al., 1985). In the IP technique, Huang et al. (1983) approximated the last measurement term by means of the average value over all pixels, so this approximation gave the systematic error of LCBF and p. Carson et al. (1986) solved this problem by eliminating the last measurement term using the weighting functions such as W\(1) = 0 and Wi1) = O. Another reason for the noise reduction is that the AWl technique uses sequentially accumulated images for the weighted-integration procedure. Koeppe et al. (1985) pointed out that the reason for the statistical error in the IP technique was due to overweighting of low count data of later scan periods. On the other hand, in our technique, the signalto-noise ratios of the integral terms are gradually higher as the time increases. The influences of the shape of the weighting function If there is no statistical noise, all weighting functions essentially give the same results. We investigated the influence of the shape of the weighting function on the COY s in LCBF using the computer simulation, and found that the COVs were less in the A WI technique compared to the DWI technique for the weighting functions (Table 2). Moreover, it should be noted that the COVs were less sensitive to the shape of the weighting functions in the A WI technique. This lesser sensitivity might be explained as the noise-smoothing effect by integration of the tissue and arterial curves. With the DWI technique, the tissue and arterial curves that contain the statistical error are multiplied by the weighting functions, and then integrated over the entire scan period. Consequently, estimated TABLE 2. The influences of the exp weighting constant 0: on the COVs (%) of LCBF AWl DWl Flow (ml 100 g-i min-i) O.I Q 0.338Q 0.7Q 1.0Q O.I Q 0.338Q 0.7Q l.oa cays (%) in LCBF due to the statistical noise varying the exp weighting constant (a). The other parameters in the simulation are the same as the notation in Fig. 5. Q exp weighting constant a in min - I. J Cereb Blood Flow Metab, Vol. 11, No. 3, 1991

8 NEW WEIGHTED INTEGRATION TECHNIQUE WITH PET o.... W cf AWl technique OWl technique IT = 285sec) %Gray matter FIG. 6. Errors due to the tissue heterogeneity in both techniques as a function of the gray matter fraction. Blood flows of 80 and 20 ml 100 g -1 min -1 were assumed to correspond to gray and white matter, respectively, and p was 1.0 mlig, the scan duration was 285 s, and weighting functions were 1 and exp(at) (a = 0.338). LCBF values depend on the weighting functions, while in the A WI technique, the noise of the tissue and arterial curves is gradually smoothed as the time increases, so the statistical errors are less sensitive to the weighting functions. We also examined the weighting functions of other forms (1 and t, 1 and vt), the same results were observed. Carson et ai. (1986) performed an optimization of the weighting functions with respect to statistical noise, and the errors were decreased in the optimized IP technique than in the original IP technique at high LCBF values. In our technique, the optimization is less important because of the lesser sensitivity to the shape of the weighting functions. The effect of tissue heterogeneity The error of the tissue heterogeneity was smaller in the A WI technique with regard to LCBF but larger regarding p compared to the DWI technique.. o.... w o 0.. T = 120 sec _ T=285 sec -30 L-'--' J o %Gray matter FIG. 7. Errors due to the tissue heterogeneity in the AWl technique with different scan durations of 120 and 285 s. The other parameters are the same as the notations in Fig. 6. (Fig. 6). The error of LCBF was less sensitive to the tissue heterogeneity than that of p, which was also observed in nonlinear least-squares fitting (!ida et ai., 1989a). As for the error with two scan durations (Fig. 7), the LCBF values were more underestimated and the p values approach their true values for increasing T. In a previous study (Koeppe et ai., 1985), the error of the tissue heterogeneity was larger in the table lookup weighted-integration technique than in the DWI technique with regard to LCBF for H 50 intravenous bolus injection. The effect of dispersion and time delay In general, the dispersion (Iida et ai., 1986) and the time delay (Carson et ai., 1986; Dhawan et ai., 1986; Koeppe et ai., 1987) give an important error for estimated parameters. The A WI technique was more sensitive to the accuracy of the arterial input function compared to the DWI technique (Table 3). The high sensitivity might arise from the integral term of Ca(t). According to our simulation (Table 4), the reduction in these errors could be accomplished by applying a small weight to the early data and a heavy weight to the later data in the A WI technique. The time delay between the arterial and whole brain slice curves (measured by the detector ring total coincidence counts) was usually determined by means of dynamic curve fitting (!ida et ai., 1988). Recently, the technique considering not only the time delay but also dispersion for adjustment of the time axis has been developed (Lammert sma et ai., 1989; Meyer, 1989). These techniques minimize the error due to the dispersion and time delay, so this problem is minor. Practical advantages in the new technique In our technique, it is unnecessary to measure the last measurement [Ci(1) measurement] since this is eliminated by the first time integration of the Kety Schmidt equation; on the other hand, in the DWI technique, the last measurement is needed. The practical advantages in weighting functions are that differential terms of weighting functions are not needed; therefore, they are arbitrary, and even noncontinuous functions are acceptable. The alternative weighted-integration technique developed by Alpert et ai. (1984) also has the same advantages in our technique, but the drawbacks are that for this technique it is necessary to compute two lookup tables for each study, and, in addition, it is difficult to extend the technique to the three-compartment model for estimation of the four rate constants. On the other hand, in our technique, the preparation of the lookup table is unnecessary, and there is no

9 500 T. YOKOI ET AL. TABLE 3. Percentage errors in LCBF and p due to the effects of dispersion and time delay of the artery input curve with various flow values Flow AWl (miioo g-i min-i) Dispersion % Error in LCBF % Error inp Time delay % Error in LCBF +2 s s % Error inp +2 s s II.! The dispersion constant T was assumed to 4 s. The positive time delay was defined such that the arterial input function was delayed compared to the tissue response function. The scan duration was 285 s, p was 1.0 mljg, and weighting functions used were I and exp ( nt) (n = 0.338). DWI mathematical problem with applying the threecompartment model with four parameters. CONCLUSION In summary, a new technique based on weighting of serially accumulated images for estimation of LCBF and p is presented. The new technique has some practical advantages as mentioned above. The characteristics of it were investigated using dynamic PET and computer simulations. In the new technique, the simulation shows that the statistical error decreased about 1. 4 times in LCBF and p at a flow of 50 ml 100 g -1 min - 1 compared to the DWI technique, and was less sensitive to the shape of the weighting functions. The new technique provides a new strategy with respect to the statistical error for estimation of LCBF and p. TABLE 4. Percentage errors in LCBF and p due to the effects of dispersion and time delay of the artery input curve with various exp weighting constant Dispersion % Error in LCBF % Error inp Time delay % Error in LCBF +2 s 4.94 n s % Error inp +2 s s Flow value was 50 ml 100 g -I min - I. The other parameters in the simulation are the same as in Table 3 except for the exp weighting constant n. Acknowledgment: We thank Dr. Ian Law-Kung-Sam for invaluable discussions and advice in preparing this manuscript. We are greatly indebted to the technical staff of the Department of Radiology and Nuclear Medicine, Research Institute for Brain and Blood Vessels, Akita. REFERENCES Alpert NM, Ericksson L, Chang JY, Bergstrom M, Litton JE, Correia JA, Bohm C, Ackerman RH, Taveras JM (1984) Strategy for the measurement of regional cerebral blood flow using short-lived tracers and emission tomography. J Cereb Blood Flow Metab 4:28-34 Amano M, lida H, Kanno I, Miura S, Hirose Y, Yamamoto S (1988) A large-scale realtime-operation cache memory system installed in high resolution PET: Preliminary results of realtime corrections of deadtime and decay and application to weighted integral method. J Nucl Med 29 :P878 Blomqvist G (1984) On the construction of functional maps in positron emission tomography. J Cereb Blood Flow Metab 4: Carson RE, Huang S-C, Phelps ME (1983) Optimization of the 0-15 water bolus injection technique for quantitative local cerebral blood flow measurements using positron emission tomography. J Cereb Blood Flow Metab 3(suppll) :S II-SI 2 Carson RE, Huang S-C, Phelps ME (1984) Error analysis of the integrated projection technique and the weighted integration method for measurement of local cerebral blood flow with positron emission tomography. J Nucl Med 25 :P88 Carson RE, Huang S-C, Green MV (1986) Weighted integration method for local cerebral blood flow measurements with positron emission tomography. J Cereb Blood Flow Metab 6: Dhawan V, Conti J, Memyk M, Jarden JO, Rottenberg DA (1986) Accuracy of PET RCBF measurements : effect of time shift between blood and brain radioactivity curves. Phys Med Bioi 31 : Frackowiak RSJ, Lenzi G-L, Jones T, Heather JD (198 0) Quantitative measurement of regional cerebral blood flow and oxygen metabolism in man using 150 and positron emission tomography : Theory, procedure, and normal values. J Comput Assist Tomogr 4: Gambhir SS, Huang S-C, Hawkins RA, Phelps ME (1987) A study of the single compartment tracer kinetic model for the measurement of local cerebral blood flow using 150-water and positron emission tomography. J Cereb Blood Flow Metab 7: 13-20

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