Eur J Nucl Med (1988) 14:403-407 European M I I~lt~cir Journal of I ~ilU~t~/l~,f..4/ Medicine © Springer-Verlag 1988 The appended curve technique for deconvolutional analysis- method and validation Jack E. Juni l, James H. Thrall 2, Jerry W. Froelich 2 Roger C. Wiggins 3, Darrell A. Campbell, Jr 3, and Michael Tuscan 3 1 William Beaumont Hospital, Department of Nuclear Medicine, 44201 Deqmndre Road, Troy, MI 48098-1198, USA 2 Henry Ford Hospital, Detroit. Michigan, USA 3 University of Michigan Medical Center, Departments of Internal Medicine and Surgery, Ann Arbor, Michigan, USA Abstract. Deconvolutional analysis (DCA) is useful in cor- rection of organ time activity curves (response function) for variations in blood activity (input function). Despite enthusiastic reports of applications of DCA in renal and cardiac scintigraphy, routine use has awaited an easily im- plemented algorithm which is insensitive to statistical noise. The matrix method suffers from the propagation of errors in early data points through the entire curve. Curve fitting or constraint methods require prior knowledge of the ex- pected form of the results. DCA by Fourier transforms (FT) is less influenced by single data points but often suffers from high frequency artifacts which result from the abrupt termination of data acquisition at a nonzero value. To reduce this artifact, we extend the input (i) and re- sponse curves to three to five times the initial period of data acquisition (P) by appending a smooth low frequency curve with a gradual taper to zero. Satisfactory results have been obtained using a half cosine curve of length 2 3P. The FTs of the input and response I and R, are computed and R/I determined. The inverse FT is performed and the curve segment corresponding to the initial period of acquisi- tion (P) is retained. We have validated this technique in a dog model by comparing the mean renal transit times of 13li_iodohippuran by direct renal artery injection to that calculated by deconvolution of an intravenous injection. The correlation was excellent (r= 0.97, P< 0.005). The extension of the data curves by appending a low frequency "tail" before DCA reduces the data termination artifact. This method is rapid, simple, and easily imple- mented on a microcomputer. Excellent results have been obtained with clinical data. Key words: Deconvolution - Pharmacokinetics - Transit time - Filters - Algorithm Quantitative measurements of tracer kinetics are an impor- tant part of nuclear medicine. The dynamic pattern of tracer activity over a given organ may be influenced by many factors other than the function of that organ. Quality of bolus injection, systemic recirculation of tracer, and multi- compartmental removal of blood pool activity all may po- tentially alter or distort the temporal pattern of activity seen at the organ of interest. These complicating factors Offprints requests to. J.E. Juni hamper, and may defeat, attempts to accurately assess iso- lated organ function. Deconvolutional analysis is a mathematical technique which can correct an organ's time activity curve for the dynamically changing pattern of blood pool activity being presented to that organ. Several techniques of deconvolu- tion have been described in the medical literature (Valentin- uzzi and Montaldo 1975; Williams 1979; Kenney etal. 1975; Alderson et al. 1979; Gamel et al. 1973; Nakamura etal. 1982; Kuruc et al. 1983) each with disadvantages which have hindered routine clinical use. We have developed and validated a method of deconvo- lutional analysis which provides excellent results with clini- cal scintigraphic data. We have also investigated the corre- lation of organ impulse response functions (IRF) obtained in-vivo by deconvolution following intravenous and direct intra-arterial injection of tracer. Methods Deconvolution of scintigraphic data by the division of Fourier transforms has been reported by several groups (Alderson et al. 1979; Gamel et al. 1973; Kuruc et al. 1983). Unfortunately, the abrupt termination of data collection while organ radioactivity remains at a nonzero value results in an sharp discontinuity in the data curves. This discontin- uity results in high frequency artifacts in the computed IRF. In order to reduce this artifact, we have developed a modifi- cation of the Fourier transform technique. This technique is illustrated graphically in Figs. 1 and 2. Figure 1 a shows an example pair of time activity curves derived from a 99mTc-disofenin hepatobiliary study. Data was collected at = 1 min intervals for 32 min. The blood pool or input function curve was derived from a region of interest over the heart. The liver time activity curve is used as the response func- tion. Figure I b shows the original data curves from Fig. 1 a on an expanded time scale of 0-256 min. The abrupt discon- tinuity at the end of data collection is clearly seen._ : A smoothly tapering, low frequency curve is appended to each of the original data curves in such a manner as to make the curves gradually and smoothly taper to zero (Fig. 2a). This lengthens the original curve several fold. The shape of the appended curve or tail is not particularly criti- cal and is choosen to consist primarly of very low frequen- cies relative to the original data and to provide a smooth transition with the terminal points of the original data. We have found that a raised one-half cosine wave with an initial