Lassen s equation is a good approximation of permeability-surface model: new a values for 99m Tc-HMPAO and 99m Tc-ECD

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1 Journal of erebral Blood low & Metabolism (24) 34, 57 6 & 24 ISBM All rights reserved X/4 $32. OIGINAL ATIL Lassen s equation is a good approximation of permeability-surface model: new a values for 99m Tc-HMPAO and 99m Tc-D Masashi Kameyama Brain perfusion tracers like [ 99m Tc] d,l-hexamethyl-propyeneamine oxime ( 99m Tc-HMPAO) and [ 99m Tc] ethyl-cysteinate dimer ( 99m Tc-D) underestimate regional cerebral blood flow (rb) at high flow values. To improve linearity between tracer accumulation and rb, two different models have been proposed. One is Lassen s correction algorithm for back-diffusion of tracer, and the other is based on the permeability-surface () model for correction of low first-pass extraction. Although both these models have the same goal, they have completely different forms of equation. It was demonstrated that mathematical approximation of the model equation leads to Lassen s equation. In this process, the relationship between, B values and Lassen s parameter was acquired, and how to correct both the back-diffusion and low first-pass extraction was also demonstrated. A computer simulation confirmed that the two models provided similar consequences when the parameter value is chosen according to the relationship found. Lassen s equation can be used to correct not only back-diffusion but also low first-pass extraction. To perform overall correction, the parameter value we have been using for decades may be too weak. I estimated that the parameter value for overall correction of HMPAO would be around.5, and that of D would be around.65. Journal of erebral Blood low & Metabolism (24) 34, 57 6; doi:.38/jcbfm.24.64; published online 6 April 24 Keywords: linearization correction; single photon emission computed tomography; regional cerebral blood flow; 99m Tc-D; 99m Tc-HMPAO INTODUTION The measurement of rb using single photon emission computed tomography (SPT) utilizes radiopharmaceuticals such as [ 99m Tc] d,l-hexamethyl-propyeneamine oxime ( 99m Tc-HMPAO) and [ 99m Tc] ethyl-cysteinate dimer ( 99m Tc-D). 99m Tc-HMPAO and 99m Tc-D as brain perfusion tracers suffer significant underestimation of rb, compared with [ 23 I] N-isopropyl-p-iodoamphetamine ( 23 I-IMP). The underestimation reduces tracer accumulation contrast between high-flow and low-flow regions, which may make it difficult to detect small rb changes. Two methodological approaches have been reported to correct the underestimation. The first approach focused on correction for flow-dependent back-diffusion from brain tissue to arterial plasma, which is now commonly called as Lassen s linearization correction algorithm. The second approach focused on correction for limited first-pass extraction from arterial plasma to brain tissue. 2 Although both these models improve the linearity between rb and tracer accumulation, the form of Lassen s equation is quite different from that of the equation based on the model, and their target for correction is also different. Owing to the assumption that the extraction in the region of interest () is same as that in the reference region ( r ), Lassen s method cannot be applicable under limited firstpass extraction circumstances. However, varies with variations in flow () in a region. urthermore, correction of both back-diffusion and limited first-pass extraction has never been reported to date. Despite these limitations, Lassen s correction is widely applied not only for 99m Tc-HMPAO but also for 99m Tc-D. 3 This research proposes to integrate the back-diffusion correction and the limited first-pass extraction correction. The mathematical relationship between Lassen s model and the model based on the model was examined and confirmed by simulation. MATIALS AND MTHODS Theory: Approximation of Permeability-Surface Model ¼ Z T AðtÞ dt where is regional activity of tracer, is B, is first-pass extraction, is retention fraction of extracted tracer, and A(t) is input function of artery. This equation in the reference region is also expressed as follows: r ¼ r r r Z T AðtÞ dt ð2þ So, the SPT count ratio to the reference region is as follows: ¼ r r r r Here, the effect of back-diffusion is ignored, and / r becomes. ¼ r r r ð4þ enkin 4 and rone 5 demonstrated that can be expressed as a function of and : ¼ ð5þ ðþ ð3þ Division of Nuclear Medicine, Department of adiology, School of Medicine, Keio University, 35, Shinanomachi, Shinjuku-ku, Tokyo , Japan. orrespondence: Dr, Division of Nuclear Medicine, Department of adiology, School of Medicine, Keio University, 35, Shinanomachi, Shinjuku-ku, Tokyo , Japan. -mail: kame-tky@umin.ac.jp eceived November 23; revised 23 January 24; accepted 6 March 24; published online 6 April 24

2 ð6þ Theoretical analysis of Lassen s and model 58 exp (/x) can be approximated linearly using Taylor expansion: exp x exp a þ exp a ðþ ðx aþ a2 Here, x ¼/ and a ¼ r / are substituted into equation (6). As r, the linear approximation (equation (6)) makes sense. exp exp þ exp ð r Þ ð7þ r r ð8þ þ exp Within x limit, the following equation holds: b b x b þ x b ðx Þ 2 r Using equation (9), equation (8) can be rearranged as follows: þ exp ðþ, r þ exp Here, a e is defined as follows: a e ¼ exp r quation (2) is substituted into equation (): ð9þ ðþ ð2þ r þ a e þ ¼ þ a e ð3þ þ a e quation (3) is inverted: þ a e r þ a e Therefore, ¼ þ a e r r r r þ a e ð4þ ð5þ Now I have derived Lassen s equation from the model equation, which indicates that Lassen s equation can be a good approximation of the model. Note that a e has no relation to Lassen s a to correct back-diffusion ( ¼ k 3 /k 2r, igure ). To distinguish from a e, conventional Lassen s a to correct back-diffusion is named a bd in this article. K p 2 k 2 igure. Two tissue compartment model of 99m Tc-HMPAO or 99m Tc-D. p : lipophilic tracer in blood, : lipophilic tracer in brain, 2 : hydrophilic tracer in brain. K, k 2, k 3 are the rate constants for transport of the tracer between compartments. D, ethylcysteinate dimer; HMPAO, d,l-hexamethyl-propyeneamine oxime. k 3 Theory: Overall orrection Lassen et al demonstrated: ¼ þ a bd r þ a bd ð6þ But, they assumed that ¼ r. Without this assumption, the equation should be as follows. 6 ¼ þ a bd r þ a bd r quation (4) is substituted into equation (7). ð þ a bd Þð þ a eþ r ð þ a bd þ a e Þ þ a bda e Therefore, r r ð þ a e Þð þ a bd Þ ð þ a bd þ a e Þ þ a bda e ð7þ ð8þ ð9þ Here, a all is defined as follows: a bd a e a all ¼ ð2þ þ a bd þ a e / r can be described in the form of Lassen s equation. ¼ þ a all r r r r r þ a all No approximation was applied except / r. ð2þ Simulation The equation based on model and its approximation to Lassen s equation were simulated using spreadsheet software, xcel 2 (Microsoft orporation, edmond, WA, USA), on a personal computer. irst, the effect of first-pass extraction only is considered. The equation based on model ¼ ð22þ r r and Lassen s equation ¼ þ a e r r þ a e ð23þ were compared numerically. and r values were adopted from the contribution of 99m Tc-D by Tsuchida et al, 7 and a e value was obtained from equation (2). Namely, from the values of ¼ 66.2 and r ¼ 55., a e was calculated as.94. Then, the effect of both back-diffusion and first-pass extraction is considered. The equation based on model ¼ r r r þ a bd r þ a exp ¼ A r and its approximation to Lassen s equation ¼ þ a all r r þ a all ð24þ ð25þ were compared numerically. ¼ 8.7, r ¼ 5, and a bd ¼.3 were adopted. These values are based on modified Lassen s contribution of 99m Tc-HMPAO. Journal of erebral Blood low & Metabolism (24), 57 6 & 24 ISBM

3 Theoretical analysis of Lassen s and model A D uptake ratio ( / r ).5.5 B %difference 3% 2% % Lassen % rb ( / r ) rb ( / r ) 59 igure 2. Simulated relationship between rb ratio (/ r ) and tracer uptake ratio (/ r ) when the effect of first-pass extraction only is considered. The equation based on model (equation (22)) and its approximation to Lassen s equation (equation (23)) were compared numerically. and r are based on Tsuchida s 7 contribution of D. a e is obtained by equation (2). (A) The original equation based on model () and Lassen s equation (Lassen) were plotted against rb. (B) %difference between the original equation based on model and the approximated equation by Lassen s equation was plotted against rb. B, cerebral blood flow; D, ethyl-cysteinate dimer;, permeability-surface. HMPAO uptake ratio ( / r ).5 Lassen %difference 3% 2% %.5.5 rb ( / r ) 2 % rb ( / r ) igure 3. Simulated relationship between rb ratio (/ r ) and tracer uptake ratio (/ r ) when the effect of both back-diffusion and first-pass extraction is considered. The equation based on model (equation (24)) and its approximation to Lassen s equation (equation (25)) were compared numerically., r and a bd are based on modified Lassen s contribution of HMPAO. (A) The original equation based on model () and Lassen s equation (Lassen) were plotted against rb. (B) %difference between the original equation based on model and the approximated equation by Lassen s equation was plotted against rb. B, cerebral blood flow; HMPAO, d,l-hexamethyl-propyeneamine oxime;, permeability-surface. SULTS This study successfully demonstrated that the equation based on model and Lassen s equation are very close, if the parameter a e is selected according to equation (2) (igure 2). It was also demonstrated that when both back-diffusion and low first-pass extraction are corrected, Lassen s equation is still similar to the equation based on model (igure 3). The two models showed relative dissociation at low flow values. DISUSSION It was demonstrated that mathematical approximation of the equation based on model leads to Lassen s equation. This simulation confirmed that Lassen s equation is good approximation of the equation based on model. Although I assumed / r, it seems that the values of the two different correction functions are very close over a wide range of / r. Maximum %differences of the two different functions are 2.9% in HMPAO (igure 3), and 3.7% in D (igure 2), over the range of / r (.5 2). ertainly %difference is relatively large at low flow values, but the absolute difference of two models is small. urthermore, B less than 2 ml/ g per minute (/ r o.4, when r ¼ 5 ml/ g is physiologically unrealistic for viable brain tissue. We cannot solve the equation based on model (equation (22)) analytically. That is why authors from Kyoto University 2,7 obtain the / r value from the / r value by fourthorder polynomial curve fitting of the equation based on model. Their polynomial curve fitting of 99m Tc-HMPAO is certainly good (data not shown), but their curve is also an approximation. The impossibility of solving analytically is an evident limitation of the equation based on model, whereas we can obtain the inverse function of Lassen s equation very easily. This is an advantage of Lassen s equation. Now, I would like to determine suitable parameter values to make SPT images of 99m Tc-HMPAO or 99m Tc-D into rb maps. I have revealed the relationship between Lassen s parameter (a e ), and, and r (equation (2)). I also obtained the parameter for overall correction (equation (2)). These equations will be quite useful. These parameters can be estimated from the literatures (Table ). Lassen s estimation for 99m Tc-HMPAO, a ¼.5 could be too large, because they ignored the influence of change in first-pass extraction, although it has been used for around a quarter of a century. Many studies confirmed that correction with a ¼.5 is good enough, 8 but they did not confirm that the value of.5 is the best. K ( ¼ ) of 99m Tc-HMPAO was reported to be considerably underestimated in comparison with 33 Xe-B, which indicated that the single-pass extraction fraction () of & 24 ISBM Journal of erebral Blood low & Metabolism (24), 57 6

4 6 Table. Theoretical analysis of Lassen s and model Kinetic analysis of HMPAO and D Literature B (ml/ g (ml/ g K (ml/g k 2 (/minute) k 3 (/minute) a bd a e a all HMPAO Lassen a a.37 a Matsuda a a 2.9 a. a Murase 2.73 a a a.36 a Andersen a.42 a a.34 a Tsuchida 7.57 a a D iberg a 43.4 a a.39 a Ishizu a a.66 a.4 a Shishido 3.55 Yonekura 2.75 a a Tsuchida 7.7 a a B, cerebral blood flow; D, ethyl-cysteinate dimer; HMPAO, d,l-hexamethyl-propyeneamine oxime. is first-pass extraction fraction, is permeability surface area product. K, k 2, k 3 are rate constants (igure ). a bd, a e, a all are parameters for Lassen s correction algorithm. a bd is for back-diffusion, a e is for extraction, a all is for overall correction. a Denotes the author s estimation. Table 2. Kinetic estimation of HMPAO and D at B 5 ml/ g per minute Literature B (ml/ g (ml/ g K (ml/g k 2 (/min) k 3 (/min) a bd a e a all HMPAO Lassen Matsuda Murase Andersen D iberg Ishizu B, cerebral blood flow; D, ethyl-cysteinate dimer; HMPAO, d,l-hexamethyl-propyeneamine oxime. is first-pass extraction fraction, is permeability surface area product. K, k 2, k 3 are rate constants (igure ). a bd, a e, a all are parameters for Lassen s correction algorithm. a bd is for back-diffusion, a e is for extraction, a all is for overall correction. 99m Tc-HMPAO is considerably lower than unity. 2 My estimation also shows that the effect of limited first-pass extraction (a e )is almost as large as the effect of back-diffusion (a bd ) (Table ). The robustness of the correlation coefficient may be the reason why many studies confirmed the value of a as.5. Linearization correction was reported to give a stable correlation coefficient over a wide range of a (. 3.) when using the cerebellum as the reference region. The correlation coefficient changes very little, but smaller a makes correction stronger, and the contrast of the images will be also enhanced. When the a all value is needed to make SPT data into a rb map, it would be good to compare 99m Tc-HMPAO/ 99m Tc-D SPT data and B PT directly. They had to make complicated calculations to obtain the kinetic rate constants, and dynamic SPT had to be performed over a short time. These factors may cause larger errors. However, the stable correlation coefficient over a wide range of a (as discussed above) may make it difficult to obtain a suitable value by the least-squares method. Scatter radiation will make the uptake ratio (/ r ) increase at a lower range of flow, and will cause a weaker correction (larger value of the parameter). Tsuchida et al 7 compared 99m Tc-HMPAO SPT images and B PT images directly. Because they did not correct for ompton scatter, the true value of the parameter might be slightly less than.57, which I have estimated from their and r values. The a all value is estimated using kinetic analyses now. Unfortunately, the kinetic data from three kinetic analyses of 99m Tc-HMPAO,6,2 and one study with ick s principle do not provide similar outcomes (Table ). As Lassen estimated a ¼.5 at B ¼ 5 ml/ g per minute, I also estimated the kinetic parameters at B ¼ 5 ml/ g per minute (Table 2). I estimated the first-pass extraction from the value. A a all value of.5 will be suitable based on the median and the mode of Table 2. This value of.5 is also consistent with Tsuchida s 7.57 and Inugami s cerebellum reference. The parameter values of a bd, a e, a all are dependent on r. If you choose the region in which r is smaller, then the parameter will increase. Both a bd ( ¼ k 3 l/ r ( exp( / r ))) and a e (equation (2)) decrease monotonically as r increases, and a all changes in the same direction as a bd and a e, because equation (2) can be rearranged as follows: =a all þ ¼ð=a bd þ Þð=a e þ Þ ð26þ To correct 99m Tc-D images, the a value obtained by iberg et al, has been widely adopted for around two decades. It seems that few studies have evaluated this value. The a all value would be near.55 according to direct comparison studies. Shishido et al from Akita 3, and Yonekura et al and Tsuchida et al from Kyoto 2,7 compared 99m Tc-D SPT and B PT images directly. There is a discrepancy between the value from Akita (.55) and Kyoto (.9,.94). The Kyoto group did not use Lassen s model, but the model. However, I demonstrated that the equation based on model and Lassen s Journal of erebral Blood low & Metabolism (24), 57 6 & 24 ISBM

5 equation are close, so the effect of the different model would be negligible. The Kyoto group did not apply scatter correction, 7 whereas the Akita group corrected for scattered events. The study from Akita would be preferable to those from Kyoto about scatter correction. The retention fraction of 99m Tc-D is reported to remain constant irrespective of changes in rb. 4 Lassen et al assumed that (i) k 2 ¼ /l and l is constant, (ii) k 3 is constant, and (iii) is constant. 99m Tc-D might not satisfy the assumption, k 2 ¼ /l. If the retention fraction is really constant, / r will be. The a all value of 99m Tc-D is estimated using kinetic analyses now. iberg et al 3 and Ishizu et al 4 performed kinetic analysis. ortunately, outcomes from these two studies provided similar outcomes strikingly (Table ). If the back-diffusion of 99m Tc-D is constant irrespective of changes in rb, the parameter should be.65. The value did not change much if the parameter value at r ¼ 5 ml/ g per minute was estimated (Table 2). In this article, value is assumed to be constant. 99m Tc-D did not show any correlation between product and rb. 4 However, value of 99m Tc-HMPAO was reported to increase linearly according to rb in the rat study. 5 value may vary between regions, such as gray matter and white matter. The change of value can be a limitation of this study. DISLOSU/ONLIT O INTST The author declares no conflict of interest. AKNOWLDGMNTS The author thanks Dr Wendy Gray for editing the nglish of this manuscript. The author thanks Dr Yoshitaka Kumakura for helpful comments. NS Lassen NA, Andersen A, iberg L, Paulson OB. The retention of [ 99m Tc]-d,l-HM- PAO in the human brain after intracarotid bolus injection: a kinetic analysis. J ereb Blood low Metab 988; 8: S3 S22. 2 Yonekura Y, Tsuchida T, Sadato N, Nishizawa S, Iwasaki Y, Mukai T et al. Brain perfusion SPT with 99m Tc-bicisate: comparison with PT measurement Theoretical analysis of Lassen s and model and linearization based on permeability-surface area product model. J ereb Blood low Metab 994; 4: S58 S65. 3 iberg L, Andersen A, Lassen NA, Holm S, Dam M. etention of 99m Tc-bicisate in the human brain after intracarotid injection. J ereb Blood low Metab 994; 4: S9 S27. 4 enkin M. Transport of potassium-42 from blood to tissue in isolated mammalian skeletal muscles. Am J Physiol 959; 97: rone. The permeability of capillaries in various organs as determined by use of the indicator diffusion method. Acta Physiol Scand 963; 58: Matsuda H, Oba H, Seki H, Higashi S, Sumiya H, Tsuji S et al. Determination of flow and rate constants in a kinetic model of [ 99m Tc]-hexamethyl-propylene amine oxime in the human brain. J ereb Blood low Metab 988; 8: S6 S68. 7 Tsuchida T, Yonekura Y, Nishizawa S, Sadato N, Tamaki N, ujita T et al. Nonlinearity correction of brain perfusion SPT based on permeability-surface area product model. J Nucl Med 996; 37: Yonekura Y, Nishizawa S, Mukai T, ujita T, ukuyama H, Ishikawa M et al. SPT with [ 99m Tc]-d,l-hexamethyl-propylene amine oxime (HM-PAO) compared with regional cerebral blood flow measured by PT: effects of linearization. J ereb Blood low Metab 988; 8: S82 S89. 9 Langen KJ, Herzog H, Kuwert T, oosen N, ota, Kiwit JW et al. Tomographic studies of rb with [ 99m Tc]-HM-PAO SPT in patients with brain tumors: comparison with 5 O 2 continuous inhalation technique and PT. J ereb Blood low Metab 988; 8: S9 S94. Inugami A, Kanno I, Uemura K, Shishido, Murakami M, Tomura N et al. Linearization correction of 99m Tc-labeled hexamethyl-propylene amine oxime (HM-PAO) image in terms of regional B distribution: comparison to 5 O 2 inhalation steady-state method measured by positron emission tomography. J ereb Blood low Metab 988; 8: S52 S6. Andersen A, iberg HH, Schmidt J, Hasselbalch SG. Quantitative measurements of cerebral blood flow using SPT and [ 99m Tc]-d,l-HM-PAO compared to xenon- 33. J ereb Blood low Metab 988; 8: S69 S8. 2 Murase K, Tanada S, ujita H, Sakaki S, Hamamoto K. Kinetic behavior of technetium-99m-hmpao in the human brain and quantification of cerebral blood flow using dynamic SPT. J Nucl Med 992; 33: Shishido, Uemura K, Murakami M, Inugami A, Ogawa T, ujita H et al. erebral uptake of 99m Tc-bicisate in patients with cerebrovascular disease in comparison with B and MO 2 measured by positron emission tomography. J ereb Blood low Metab 994; 4: S66 S75. 4 Ishizu K, Yonekura Y, Magata Y, Okazawa H, ukuyama H, Tanaka et al. xtraction and retention technetium-99m-d in human brain: dynamic SPT and oxygen- 5-water PT studies. J Nucl Med 996; 37: Andersen A, iberg H, Knudsen KBM, Barry DI, Paulson OB, Schmidt J et al. xtraction of [ 99m Tc]-d,l-HM-PAO across the blood-brain barrier. J ereb Blood low Metab 988; 8: S44 S5. 6 & 24 ISBM Journal of erebral Blood low & Metabolism (24), 57 6

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REVIEW Annals of Nuclear Medicine Vol. 17, No. 6, 427 434, 2003 Spectral analysis: principle and clinical applications Kenya MURASE Department of Medical Physics and Engineering, Division of Medical Technology