Abstract
Examination of the input-output events in functioning organs by the use of the impulse-response function (IRF) for a radioactive tracer is gaining more and more ground in nuclear medicine. This study summarizes the development of deconvolution analysis, laying special stress on the ‘model-free’ approach. System linearity and time invariance are discussed, and means of eliminating noise in IRFs originating from the input and organ-time-activity curves are outlined. Typical IRFs are illustrated by flow diagrams, time-domain curves, and their representation by Laplace transforms. The cases of nondiffusible and diffusible tracers as well as parenchymally extracted and transported substances are discussed. Methods for the derivation of models and for the calculation of physiologically important parameters from theIRFs are suggested.
Similar content being viewed by others
References
Alderson PO, Douglass KH, Mendenhall KG, Guadiani VA, Watson DC, Links JM, Wagner HN Jr (1979) Deconvolution analysis in radionuclide quantitation of left-to-right cardiac shunts. J Nucl Med 20:502–506
Andrews HC (1970) Computer techniques in image processing. Academic Press, New York London
Bašsić M, Lončarić S, Popović S, Šimonović I (1984) The effect of “biological” and statistical noise on the calculated renal retention function. Eur J Nucl Med 9:A70
Bellman R (1971) Topics in pharmacokinetics. II. Identification of time-lag processes. Math Biosci 11:337–342
Bevington PR (1969) Data reduction and error analysis for the physical sciences. McGraw-Hill, New York
Boisvieux JF, Steimer JL (1979) A non-linear mathematical model for the in vivo determination of Kupffer cells number and rate of phagocytosis of radiocolloids in rats. Int J Biomed Comput 10:331–340
Branson H (1948) The use of isotopes in an integral equation description of metabolizing systems. Cold Spring Harbor Symp Quant Biol 13:35–42
Braun M (1978) Differential equations and their applications. 2nd edn. Springer, Berlin Heidelberg New York
Brigham EO (1974) The fast Fourier transform. Prentice Hall, Englewood Cliffs
Britton KE, Brown NJG (1971) Clinical renography. Lloyd Luke, London
Britton KE, Nimmon CC, Lee TY, Jarritt PH, Granowska M, Greening A, McAlister JM (1978) Carotid and cerebral blood flow. In: Brill AB, Price RR (eds): A review of information processing in medical imaging. Oak Ridge National Laboratory, Oak Ridge, pp 499–525
Bronikowski TA, Linehan JH, Dawson CA (1980) A mathematical analysis of the influence of perfusion heterogeneity on indicator extraction. Math Biosci 52:27–51
Bronikowski TA, Dawson CA, Linehan JH, Rickaby DA (1982) A mathematical model of indicator extraction by the pulmonary endothelium via saturation kinetics. Math Biosci 61:237–266
Bronikowski TA, Dawson CA, Linehan JH (1983) Model-free deconvolution techniques for estimating vascular transport functions. Int J Biomed Comput 14:411–429
Brown NJG, Britton KE (1972) The theory of renography and analysis of results. In: Blaufox MD, Funck-Brentano JL (eds) Radionuclides in nephrology. Grune and Stratton, New York, pp 315–324
Childers DG, Skinner DP, Kemerait RC (1977) The cepstrum: A guide to processing. Proc IEEE 65:1428–1443
Clements WC (1969) A note on determination of the parameters of the longitudinal dispersion model from experimental data. Chem Eng Sci 24:957–963
Clements WC, Blalock KE (1972) Comparison of method of moments and method of least-squares in analyzing tracer injectionresponse data from fluid systems. Chem Eng Sci 27:2311–2312
Clements WC, Harris TR (1967) The frequency domain evaluation of mathematical models for dynamic systems. AIChE J 13:374–378
Cobelli C, Carson ER, Finkelstein L, Leaning MS (1984) Validation of simple and complex models in physiology and medicine. Am J Physiol 246:R259-R266
CommengesD, Brendel AJ (1982) A deconvolution program for processing radiotracer dilution curves. Comput Programs Biomed 14:271–276
Coulam CM, Warner HR, Wood EH, Bassingthwaighte JB (1966) A transfer function analysis of coronary and renal circulation calculated from upstream and downstream indicator-dilution curves. Circ Res 19:879–890
Davies B (1978) Integral transforms and their applications. Springer, Berlin Heidelberg New York
De Lima JJP (1980) Dependence of an individual renogram on the other kidney through the blood activity, shown by convolution. Eur J Nucl Med 5:469–470
Dell RB, Sciacca R, Lieberman K, Case DB, Cannon PJ (1973) A weighted least-squares technique for the analysis of kinetic data and its application to the study of renal 133Xenon washout in dogs and man. Circ Res 32:71–84
Dienes JK (1976) The mathematics of recirculation and the measurement of cardiac output. Math Biosci 32:141–153
Diffey BL, Corfield JR (1976) Data-bounding technique in discrete deconvolution. Med Biol Eng Comput 14:478
Diffey BL, Hall FM, Corfield JR (1976) The 99mTc-DTPA dynamic renal scan with deconvolution analysis. J Nucl Med 17: 352–355
Doetsch G (1943) Theorie and Anwendung der Laplace-Transformation. Dover, New York
Donato L (1973) Basic concepts of radiocardiography. Semin Nucl Med 3:111–130
Feinendegen LE, Vyska K, Freundlieb C, Höck A, Machulla HJ, Kloster G, Stöcklin G (1981) Non-invasive analysis of metabolic reactions in body tissues; the case of myocardial fatty acids. Eur J Nucl Med 6:191–200
Fleming JS, Goddard BA (1974) A technique for the deconvolution of the renogram. Phys Med Biol 19:546–549
Freundlieb C, Höck A, Vyska K, Feinendegen LE, Machulla HJ, Stöcklin G (1980) Myocardial imaging and metabolic studies with [17-123I]Iodoheptadecanoic acid. J Nucl Med 21: 1043–1050
Galli G, Orlando P, Massari P, Bonifazi N, Magistrelli P, Coppola R (1983) 99mTc-diethyl-IDA: The extraction efficiency of the liver. Eur J Nucl Med 8:187–190
Gamel J, Rousseau WF, Katholi CR, Mesel E (1973) Pitfalls in digital computation of the impulse response of vascular beds from indicator-dilution curves. Circ Res 32:516–523
Garfinkel A (1983a) Nonlinear dynamics. Introduction to part I. Am J Physiol 245:R453-R454
Garfinkel A (1983b) A mathematics for physiology. Am J Physiol 245:R455-R466
Giuntini C (1971) Theoretical considerations on the measure of pulmonary blood volume and extravascular lung water in man. Bull Physiopathol Respir 7:1125–1160
Goto K, Hirano A, Hirakawa S (1981) Non-invasive estimation of the human pulmonary blood volume with gamma camera and RI-angiocardiography. Jpn Circ J 45:113–119
Gullquist R, Abrahamson H, Brandt R, Philip J, Strandell T (1983) New method to calculate the renal transit time spectrum from conventional three-probe renography. Med Biol Eng Comput 21:599–602
Harris TR (1968) The identification of recirculating systems in the frequency domain. Bull Math Biol 30:87–104
Hazelrig JB, Katholi CR, Blauenstein UW, Halsey JH, Wilson EM, Wills EL (1981) Total curve analysis of regional cerebral blood flow with 133Xe inhalation: description of method and values obtained with normal volunteers. IEEE Trans Biomed Eng 28:609–616
Herholz K, Heiss WD, Pawlik G, Ilsen HW, Wienhard K (1983) Detection of compartmental slippage in noninvasive rCBF measurements. J Nucl Med 24:1188–1191
Hoop B, Ojemann RG, Brownell GL (1971) A stochastic model of regional cerebral circulation. J Nucl Med 12:540–546
Huang TS (1975) Picture processing and digital filtering. Springer, Berlin Heidelberg New York
Hung G, Stark L (1977) The kernel identification method (1910–1977): Review of theory, calculation, application, and interpretation. Math Biosci 37:135–190
Hunt BR (1970) The inverse problem of radiography. Math Biosci 8:161–179
Johnson LE (1980) Two closed two-compartment models with time-delay. IEEE Trans Biomed Eng 27:669–671
Juni JE, Keyes JW, Carter W, Bowers R, Gross MD, Smock D, Samosik-Mast C (1983) Differentiation of obstructive from non-obstructive jaundice by deconvolution analysis of hepatobiliary scans. J Nucl Med 24:P30
Kamiya A, Togawa T (1973) Mathematical analysis of circulatory mixing process. Bull Math Biol 35:287–300
Kenny RW, Ackery DM, Fleming JS, Goddard BA, Grant RW (1975) Deconvolution analysis of the scintillation camera renogram. Br J Radiol 48:481–486
Kloster G, Stöcklin G, Smith EF, Schrör K (1984) ω-Halofatty acids: a probe for mitochondrial membrane integrity. In vitro investigations in normal and ischaemic myocardium. Eur J Nucl Med 9:305–311
Knop J, Kröger E, Stritzke P, Schneider C, Kruse HP (1981) Deconvolution analysis of 99mTc-methylene diphosphonate kinetics in metabolic bone disease. Eur J Nucl Med 6:63–67
Kuruc A, Caldicott WJH, Treves S (1982) An improved deconvolution technique for calcutation of renal retention functions. Comput Biomed Res 15:45–46
Kuruc A, Treves S, Parker JA, Cheng C, Sawan A (1983a) Radionuclide angiocardiography: An improved deconvolution technique for improvement after suboptimal bolus injection. Radiology 148:233–238
Kuruc A, Treves S, Parker JA (1983b) Accuracy of deconvolution algorithms assessed by simulation studies: Concise communication. J Nucl Med 24:258–263
Lange RL, Horgan JD, Botticelli JT, Tsagaris T, Carlisle RP, Kuida H (1966) Pulmonary to arterial circulatory transfer function: Importance in respiratory control. J Appl Physiol 21:1281–1291
Llaurado JG (1982) Advances in computer technology: software, hardware and overall perspectives. Int J Biomed Comput 13:361–367
Marquardt DW (1963) An algorithm for least-squares estimation of nonlinear parameters. J Soc Ind Appl Math 11:431–441
Meier P, Zierler KL (1954) On thetheory of the indicator-dilution method for measurement of blood flow and volume. J Appl Physiol 6:731–744
Nakai M (1981) Computation of transport function using multiple regression analysis. Am J Physiol 240:H133-H144
Nakamura M, Suzuki Y, Tomoda H (1983) A comparison of two quantitative approaches for correction of background counts of first-pass left ventricular time-activity curves: a numerical investigation. IEEE Trans Biomed Eng 30:326–331
Newman EV, Merrell M, Genecin A, Monge C, Milnor WR, McKeever WP (1951) The dye dilution method for describing the central circulation. An analysis of factors shaping the timeconcentration curves. Circulation 4:735–746
Nyitrai L (1983) The interpretation and computation of transfer functions in perfusion studies. Izotóptechnika 26:197–214
Nyitrai L, Fodor M (1983) Use of transfer function (TF) for the study of the hepatic circulation. Eur J Nucl Med 8:A41
Nyitrai L, Szabó Z (1984) Calculation of regional cerebral blood flow (rCBF) using non-diffusible tracers. A model-free approach. In: Schmidt HAE, Vauramo E (eds) Nuklearmedizin: Proceedings of the European Nuclear Medicine Congress, 1984. Schattauer, Stuttgart New York, pp 506–509
Obrist WD, Thompson HK, King CH, Wang HS (1967) Determination of regional cerebral blood flow by inhalation of 133Xenon. Circ Res 20:124–135
Obrist WD, Thompson HK, Wang HS, Wilkinson WE (1975) Regional cerebral blood flow estimated by 133Xenon inhalation. Stroke 6:245–256
Phillips DL (1962) A technique for the numerical solution of certain integral equations of the first kind. J ACM 9:84–97
Piepsz A, Ham HR, Erbsmann F, Hall M, Diffey BL, Goggin MJ, Hall FM, Miller JA, Lumbroso J, Di Paola R, Bazin JP, Di Paola M, Fries D (1982) A co-operative study on the clinical value of dynamic renal scanning with deconvolution analysis. Br J Radiol 55:419–433
Pitkänen MA, Kuikka JT, Kiiliäinen H, Macey DJ (1982) Comparison of two optimizing algorithms for cerebral blood flow analysis using the intravenous 133Xe wash-out method. Int J Biomed Comput 13:213–220
Reeve J, Crawley JCW (1974) Quantitative radioisotope renography: The derivation of physiological data by deconvolution analysis using a single-injection technique. Clin Sci 47:317–330
Risberg J (1980) Regional cerebral blood flow measurements by 133Xe-inhalation: Methodology and applications in neuropsychology and psychiatry. Brain Lang 9:9–34
Risberg J, Prohovnik I (1981) CBF measurement by 133Xe inhalation: Recent methodological advances. Prog Nucl Med 7:70–81
Risberg J, Ali Z, Wilson EM, Wills EL, Halsey JH (1975) Regional cerebral blood flow by 133Xenon inhalation. Stroke 6: 142–148
Ritzl F, Feinendegen LE (1971) In vivo determination of site and rate of insulin catabolism using the double tracer technique with 51Cr and 131I. In Dynamic studies with radioisotopes in medicine. IAEA, Vienna, pp 57–68
Rosenfeld A (1976) Digital picture analysis. Springer, Berlin Heidelberg New York
Rowlett RD, Harris TR (1976) A comparative study of organ models and numerical techniques for the evaluation of capillary permeability from multiple-indicator data. Math Biosci 29:273–298
Sangren WC, Sheppard CW (1953) A mathematical derivation of the exchange of a labeled substance between a liquid flowing in a vessel and an external compartment. Bull Math Biol 15:387–394
Schmitz-Feuerhake I, Falkenreck-Herbst I, Coburg AJ, Wonigeit K, Gerhard K, Prévôt H (1978) Atraumatic method of renal blood flow estimation by Xenon inhalation and its application to transplanted kidneys. Eur J Clin Invest 8:75–80
Sheppard CW (1952) Synthesis of dye dilution curves. Am J Physiol 171:767
Sheppard CW (1954) Mathematical considerations of indicator dilution techniques. Minn Med 37:93–104
Sheppard CW (1962) Stochastic models for tracer experiments in the circulation: Parallel random walks. J Theor Biol 2:33–47
Sheppard CW, Uffer MB (1969) Stochastic models for tracer experiments in the circulation. II. Serial random walks. J Theor Biol 22:188–207
Sheppard CW, Jones MP, Murphree EL (1961) Shapes of indicator-dilution curves obtained from physical and physiological labyrinths. Cire Res 9:936–944
Sherman H (1960) On the theory of indicator-dilution methods under varying blood-flow conditions. Bull Math Biol 22: 417–424
Stephenson JL (1948) Theory of the measurment of blood flow by the dilution of an indicator. Bull Math Biol 10:117–121
Strand ON, Westwater ER (1968) Statistical estimation of the numerical solution of a Fredholm equation of the first kind. J ACM 15:100–114
Stritzke P, Kröger E, Witte G, Knop J, Schneider C (1980) Darstellung charakteristischer Parameter der Tracerkinetik von Niere und Schilddrüse in Funktionsbildern. Nuklearmedizin 19: 74–79
Stritzke P, Kröger E, Knop J, Schneider C (1982) Paremeterszintigraphie unter Anwendung von Entfaltungsmethoden. In: Schmidt HAE, Rösler H (eds) Nuklearmedizin: Proceedings of the 19th Annual Meeting of the Society of Nuclearmedicine Europe, 1981. Schattauer, Stuttgart New York, pp 156–160
Szabó Z, Vosberg H, Kistenich H, Segall M, Feinendegen LE (1983) Kinetics of different radiocolloids in the liver measured by deconvolution analysis. Eur J Nucl Med 8:A41
Szabó Z, Vosberg H, Kistenich H, Feinendegen LE (1984a) Deconvolution analysis of hepatobiliary studies with 99mTc-hepatobida. In: Schmidt HAE, Adam WE (eds) Nuklearmedizin: Proceedings of the 21st Annual Meeting of the Society of Nuclearmedicine Europe, 1983. Schattauer, Stuttgart New York, pp 576–577
Szabó Z, Vosberg H, Kistenich H, Feinendegen LE (1984c) Presentation of renal transplant dysfunction in parametric spaces. In: Schmidt HAE, Adam WE (eds) Nuklearmedizin: Proceedings of the 21st Annual Meeting of the Society of Nuclearmedicine Europe, 1983. Schattauer, Stuttgart New York, pp 578–579
Szabó Z, Nyitrai L, Vosberg H, Wilcke C, Feinendegen LE (1984d) Quantitative measurement of renal transplant perfusion using transfer function and a random walk model. In: Schmidt HAE, Vauramo E (eds) Nuklearmedizin: Proceedings of the European Nuclear Medicine Congress 1984. Schattauer, Stuttgart New York, pp 416–419
Taylor TP, Macfarlane PW (1974) Digital filtering of the e.c.g. — a comparison of low pass digital filters on a small computer. Med Biol Eng 12:493–502
Thron CD (1978) Some remarks on compartmental analysis of uptake and washout data. Life Sci 22:1287–1304
Twomey S (1963) On the numerical solution of Fredholm integral equations of the first kind by the inversion of the linear system procedure by quadrature. J ACM 10:97–101
Twomey S (1965) The application ofnumerical filtering to the solution of integral equations encountered in indirect sensing measurements. J Franklin Inst 279:95–109
Valentinuzzi M, Valentinuzzi ME (1973) Newman's chamber models in the study of blood circulation through the liver and heart. Bull Math Biol 35:19–29
Valentinuzzi M, Townsend AA, Kelso AF (1971) A study of the liver circulation by indicator dilution method with a conductivity cell. J Am Osteopath Assoc 70:1093–1096
Valentinuzzi ME (1971) A mathematical model of the hepatic portal system. Med Biol Eng 9:213–220
Van Straalen NM (1983) Physiological time and time-invariance. J Theor Biol 104:349–357
Vaughan CL (1982) Smoothing and differentiation of displacement — time data: an application of splines and digital filtering. Int J Biomed Comput 13:375–386
Von Niederdrenk K (1981) Mathematische Grundlagen zur Bildverarbeitung. Rechenzentrum RWTH, Aachen
Vosberg H, Szabó Z, Nase D, Feinendegen LE (1983) Mean transit time parametric images in brain tumors. Eur J Nucl Med 8:A33
Vosberg H, Szabó Z, Nase D, Feinendegen LE (1984a) Mean transit time parametric imaging in cerebrovascular diseases. In: Schmidt HAE, Adam WE (eds) Nuklearmedizin: Proceedings of the 21 st International Annual Meeting of the Society of Nuclearmedicine Europe, 1983. Schattauer, Stuttgart New York, pp 599–602
Vosberg H, Szabó Z, Nase D, Feinendegen LE (1984b) Optimierung der quantitativen Hirnperfusionsszintigrafie durch Parameterbilder der mittleren Transitzeiten. In: Höfer R, Bergmann H (eds) Radioaktive Isotope in Klinik und Forschung, Egermann, Bad Gastein, pp 43–52
Wagner HN Jr, Walton WW Jr, Jacquez J (1968) Mathematics. In: Wagner HN Jr (ed) Principles of nuclear medicine. WB Saunders, Philadelphia, pp 23–74
Weinstein H, Fu BS, Acosta R, Bernstein B, Schaffer AB (1973) Estimation of circulating blood volume from a tracer-response curve. IEEE Trans Biomed Eng 20:269–277
Williams DL (1979) Improvement in quantitative data analyses by numerical deconvolution techniques. J Nucl Med 20: 568–570
Yamamoto K, Tanada S, Kawamura M, Ishine M, Hamamoto K (1982) Method of determining transfer functions in the case of the intravenous injection of radioactive tracers. Med Biol Eng Comput 20:215–222
Yamamoto WS (1962) Transmission of information by the arterial blood stream with particular reference to carbon dioxide. Biophys J 2:143–159
Yamamura Y, Yamamoto K, Sekiya T (1972) Extended Greuling Goertzel kernels for elastically and inelastically slowing down neurtrons. Atomkernenergie 2:249–254
Zelter M, Capderou A, Mallet A, Boisvieux JF, Silbert D, Mensch-Dechêne J (1982) Distribution of transit times of a bolus of 99mTc sodium pertechnetate in human pulmonary circulation. In: Schmidt HAE, Rösler H (eds) Proceedings of the 19th International Annual Meeting of the Society of Nuclearmedicine Europe, 1981. Schattauer, Stuttgart New York, pp 121–124
Zierler KL (1965) Equations for measuring blood flow by external monitoring of radioisotopes. Circ Res 16:309–321
Author information
Authors and Affiliations
Additional information
At present, a guest scientist at the Institute for Medicine, Nuclear Research Center Jülich, Jülich, Federal Republic of Germany
Rights and permissions
About this article
Cite this article
Szabó, Z., Vosberg, H., Sondhaus, C.A. et al. Model identification and estimation of organ-function parameters using radioactive tracers and the impulse-response function. Eur J Nucl Med 11, 265–274 (1985). https://doi.org/10.1007/BF00279082
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00279082