On the physical and stochastic representation of an indicator dilution curve as a gamma variate

The analysis of intravascular indicator dynamics is important for cardiovascular diagnostics as well as for the assessment of tissue perfusion, aimed at the detection of ischemic regions or cancer hypervascularization. To this end, indicator dilution curves are measured after the intravenous injection of an indicator bolus and fitted by parametric models for the estimation of the hemodynamic parameters of interest. Based on heuristic reasoning, the dilution process is often modeled by a gamma variate. In this paper, we provide both a physical and stochastic interpretation of the gamma variate model. The accuracy of the model is compared with the local density random walk model, a known model based on physics principles. Dilution curves were measured by contrast ultrasonography both in vitro and in vivo (20 patients). Blood volume measurements were used to test the accuracy and clinical relevance of the estimated parameters. Both models provided accurate curve fits and volume estimates. In conclusion, the proposed interpretations of the gamma variate model describe physics aspects of the dilution process and lead to a better understanding of the observed parameters, increasing the value and credibility of the model, and possibly expanding its diagnostic applications.

[1]  Two comments on the estimation of blood flow and central volume from dye-dilution curves. , 1959, Journal of applied physiology.

[2]  S. Feinstein,et al.  The powerful microbubble: from bench to bedside, from intravascular indicator to therapeutic delivery system, and beyond. , 2004, American journal of physiology. Heart and circulatory physiology.

[3]  Kenneth H Norwich,et al.  Molecular Dynamics in Biosystems , 1977 .

[4]  Hessel Wijkstra,et al.  Quantitative microbubble enhanced transrectal ultrasound as a tool for monitoring hormonal treatment of prostate carcinoma , 2002, The Prostate.

[5]  Frederick S. Hillier,et al.  Introduction of Operations Research , 1967 .

[6]  L L Huntsman,et al.  Noninvasive Doppler Determination of Cardiac Output in Man: Clinical Validation , 1983, Circulation.

[7]  P. Tofts.  Modeling tracer kinetics in dynamic Gd‐DTPA MR imaging , 1997, Journal of magnetic resonance imaging : JMRI.

[8]  G. Fegler,et al.  MEASUREMENT OF CARDIAC OUTPUT IN ANÆSTHETIZED ANIMALS BY A THERMO‐DILUTION METHOD , 1954 .

[9]  W. F. Hamilton,et al.  STUDIES ON THE CIRCULATION: I. Injection Method: Physical and Mathematical Considerations , 1929 .

[10]  L. Crum,et al.  Thresholds for inertial cavitation in Albunex suspensions under pulsed ultrasound conditions , 2001, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[11]  G. J. G. Upton,et al.  An Introduction to Statistical Modelling , 1983 .

[12]  M. Wise,et al.  Tracer dilution curves in cardiology and random walk and lognormal distributions. , 1966, Acta physiologica et pharmacologica Neerlandica.

[13]  A. Weyland,et al.  Changes in cardiac output and intrathoracic blood volume: a mathematical coupling of data? , 2001, Acta anaesthesiologica Scandinavica.

[14]  W. Hamilton,et al.  SIMULTANEOUS DETERMINATION OF THE PULMONARY AND SYSTEMIC CIRCULATION TIMES IN MAN AND OF A FIGURE RELATED TO THE CARDIAC OUTPUT , 1928 .

[15]  C. Starmer,et al.  Indicator Transit Time Considered as a Gamma Variate , 1964, Circulation research.

[16]  D. Marquardt.  An Algorithm for Least-Squares Estimation of Nonlinear Parameters , 1963 .

[17]  A. Versprille,et al.  Physiological interpretation of the skewness of indicatordilution curves; theoretical considerations and a practical application , 1984, Basic Research in Cardiology.

[18]  D M Band,et al.  A new method of analysing indicator dilution curves. , 1995, Cardiovascular research.

[19]  E. Korsten,et al.  Contrast echocardiography for pulmonary blood volume quantification , 2004, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control.

[20]  K. Zierler.  Indicator Dilution Methods for Measuring Blood Flow, Volume, and Other Properties of Biological Systems: A Brief History and Memoir , 2000, Annals of Biomedical Engineering.

[21]  J M Bland,et al.  Statistical methods for assessing agreement between two methods of clinical measurement , 1986 .

[22]  R. Stow,et al.  An empirical formula for indicator-dilution curves as obtained in human beings. , 1954, Journal of applied physiology.

[23]  D. Altman,et al.  STATISTICAL METHODS FOR ASSESSING AGREEMENT BETWEEN TWO METHODS OF CLINICAL MEASUREMENT , 1986, The Lancet.

[24]  Gerhard Ransmayr,et al.  Brain perfusion scintigraphy with99mTc-HMPAO or99mTc-ECD and123I-β-CIT single-photon emission tomography in dementia of the Alzheimer-type and diffuse Lewy body disease , 1997, European Journal of Nuclear Medicine.

[25]  G. Fegler,et al.  Measurement of cardiac output in anaesthetized animals by a thermodilution method. , 1954, Quarterly journal of experimental physiology and cognate medical sciences.

[26]  A R Jayaweera,et al.  Quantification of myocardial blood flow with ultrasound-induced destruction of microbubbles administered as a constant venous infusion. , 1998, Circulation.

[27]  C. Keyl,et al.  Assessment of intrathoracic blood volume as an indicator of cardiac preload: single transpulmonary thermodilution technique versus assessment of pressure preload parameters derived from a pulmonary artery catheter. , 2001, Journal of cardiothoracic and vascular anesthesia.

[28]  Ton Kalker,et al.  Videodensitometric Methods for Cardiac Output Measurements , 2003, EURASIP J. Adv. Signal Process..

[29]  H. Anger.  SCINTILLATION CAMERA WITH MULTICHANNEL COLLIMATORS. , 1964, Journal of nuclear medicine : official publication, Society of Nuclear Medicine.

[30]  Jan de Koning,et al.  Relationships between volume and pressure measurements and stroke volume in critically ill patients , 2000, Critical care.

[31]  G Brix,et al.  Multicompartment analysis of gadolinium chelate kinetics: Blood‐tissue exchange in mammary tumors as monitored by dynamic MR imaging , 1999, Journal of magnetic resonance imaging : JMRI.

[32]  R. Millard,et al.  Indicator-dilution dispersion models and cardiac output computing methods. , 1997, The American journal of physiology.

[33]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

[34]  Xucai Chen,et al.  A mathematical model for the assessment of hemodynamic parameters using quantitative contrast echocardiography , 1998, IEEE Transactions on Biomedical Engineering.

[35]  Wise Me,et al.  Tracer dilution curves in cardiology and random walk and lognormal distributions. , 1966 .