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

Abstract

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.

Volume quantification by contrast-enhanced ultrasound: an in-vitro comparison with true volumes and thermodilution

“…The center of gravity and peak-to-peak methods do not use model fitting for the MTT estimation, and they are more sensitive to low signal-to-noise ratios and contrast recirculation. Moreover, they do not provide a physical interpretation of the investigated convective diffusion process, as provided by the LDRW model [21]. The LDRW model provides better fits of skewed IDCs [5], which are present at high flows and small volumes.…”

Comparison of blood flow models and acquisitions for quantitative myocardial perfusion estimation from dynamic CT

“…The shape of a single-pass bolus in the vascular system is often well-represented by a gamma variate function (Mischi et al 2008). To reduce the effect of noise and sparse temporal sampling in the input function, we locate the peak by fitting with a gamma variate function, and the peak value of the gamma variate is used rather than the raw maximum of the input data.…”

Correction for arterial‐tissue delay and dispersion in absolute quantitative cerebral perfusion DSC MR imaging

“…In each case, the corresponding C-T curve, AIF ( t ), is computed from the AIF signal, S ( t ), as follows: $$\mathit{\text{AIF}}(t)=-\frac{k}{\text{italicTE}}\text{ln}\left(\frac{Sfalse(tfalse)}{S_0}\right)$$ where S 0 is the pre-injection signal, TE is the echo time and k reflects the contrast agent relaxivity and properties of the pulse sequence (15). An analytical expression is determined for each of the computed AIFs, by fitting them to a gamma-variate model using the Levenberg-Marquardt least squares fit MATLAB built-in function, according to the indicator dilution curve (IDC) theory: $$C_{\mathrm{normal\Gamma }}(t)=A_{\mathrm{normal\Gamma }}·t^{\alpha }·e^{-\frac{t}{\beta }}$$ where C Γ ( t ) is the gamma-variate model of the IDC, α and β are the shape and scale parameters, respectively, and the factor A Γ refers to the IDC amplitude (20–22). The initial guesses of the fit parameters( A Γ , α, and β), are determined using a χ 2 -minimization grid search algorithm to ensure robust fitting and rapid convergence of the solution.…”