Comparison of four mathematical models to analyze indicator-dilution curves in the coronary circulation

Quantification of shunt fraction using contrast ultrasound and indicator dilution in an in vitro model

Transthoracic saline contrast echocardiography is commonly used to assess intrathoracic shunt flow in vivo. Though the technique has many advantages (safe, simple, repeatable), the measurement technique lacks specificity, and the contrast agent has limited stability. This study sought to determine if the indicator dilution modeling technique could be applied to ultrasound contrast data to quantify shunt fraction and to determine if buoyant force has a significant effect on microbubble pathway determination at a "vascular" bifurcation. A model of the pulmonary circuit was perfused with blood at three distinct flow rates (low, medium and high) over shunt fractions ranging from ∼2-10 %. The buoyancy effect on contrast was quantified using a simplified in vitro model of a vascular bifurcation that had an upper and lower outflow tract where saline contrast formed from carbon monoxide (CO) gas passed through the bifurcation, was collected and quantified. The indicator dilution model was found to have a mean bias of - 3.2 % for the low flow stage, - 2.6 % for the medium flow stage and - 1.4 % for the high flow stage compared to volumetric measurements, suggesting agreement increases with increasing flow rate. Investigations of the buoyant effects revealed that at lower flow rates, contrast bubbles that encounter a bifurcation will favor the upper outflow tract over the lower. However, this effect is reduced by increasing the flow rate two-fold. These data identify that application of indicator dilution theory to contrast ultrasound data and the pathway ultrasound contrast travels in a network of tubules is flow dependent.

Mathematical Models for Blood Flow Quantification in Dialysis Access Using Angiography: A Comparative Study

Blood flow rate in dialysis (vascular) access is the key parameter to examine patency and to evaluate the outcomes of various endovascular interve7ntions. While angiography is extensively used for dialysis access–salvage procedures, to date, there is no image-based blood flow measurement application commercially available in the angiography suite. We aim to calculate the blood flow rate in the dialysis access based on cine-angiographic and fluoroscopic image sequences. In this study, we discuss image-based methods to quantify access blood flow in a flow phantom model. Digital subtraction angiography (DSA) and fluoroscopy were used to acquire images at various sampling rates (DSA—3 and 6 frames/s, fluoroscopy—4 and 10 pulses/s). Flow rates were computed based on two bolus tracking algorithms, peak-to-peak and cross-correlation, and modeled with three curve-fitting functions, gamma variate, lagged normal, and polynomial, to correct errors with transit time measurement. Dye propagation distance and the cross-sectional area were calculated by analyzing the contrast enhancement in the vessel. The calculated flow rates were correlated versus an in-line flow sensor measurement. The cross-correlation algorithm with gamma-variate curve fitting had the best accuracy and least variability in both imaging modes. The absolute percent error (mean ± SEM) of flow quantification in the DSA mode at 6 frames/s was 21.4 ± 1.9%, and in the fluoroscopic mode at 10 pulses/s was 37.4 ± 3.6%. The radiation dose varied linearly with the sampling rate in both imaging modes and was substantially low to invoke any tissue reactions or stochastic effects. The cross-correlation algorithm and gamma-variate curve fitting for DSA acquisition at 6 frames/s had the best correlation with the flow sensor measurements. These findings will be helpful to develop a software-based vascular access flow measurement tool for the angiography suite and to optimize the imaging protocol amenable for computational flow applications.

Transit Time Measurement in Indicator Dilution Curves: Overcoming the Missing Ground Truth and Quantifying the Error

The vascular function of a vessel can be qualitatively and intraoperatively checked by recording the blood dynamics inside the vessel via fluorescence angiography (FA). Although FA is the state of the art in proving the existence of blood flow during interventions such as bypass surgery, it still lacks a quantitative blood flow measurement that could decrease the recurrence rate and postsurgical mortality. Previous approaches show that the measured flow has a significant deviation compared to the gold standard reference (ultrasonic flow meter). In order to systematically address the possible sources of error, we investigated the error in transit time measurement of an indicator. Obtaining in vivo indicator dilution curves with a known ground truth is complex and often not possible. Further, the error in transit time measurement should be quantified and reduced. To tackle both issues, we first computed many diverse indicator dilution curves using an in silico simulation of the indicator's flow. Second, we post-processed these curves to mimic measured signals. Finally, we fitted mathematical models (parabola, gamma variate, local density random walk, and mono-exponential model) to re-continualize the obtained discrete indicator dilution curves and calculate the time delay of two analytical functions. This re-continualization showed an increase in the temporal accuracy up to a sub-sample accuracy. Thereby, the Local Density Random Walk (LDRW) model performed best using the cross-correlation of the first derivative of both indicator curves with a cutting of the data at 40% of the peak intensity. The error in frames depends on the noise level and is for a signal-to-noise ratio (SNR) of 20 dB and a sampling rate of f_s = 60 Hz at fs-1·0.25(±0.18), so this error is smaller than the distance between two consecutive samples. The accurate determination of the transit time and the quantification of the error allow the calculation of the error propagation onto the flow measurement. Both can assist surgeons as an intraoperative quality check and thereby reduce the recurrence rate and post-surgical mortality.

Optimisation of ultrasound liver perfusion through a digital reference object and analysis tool

Background

Conventional ultrasound (US) provides important qualitative information, although there is a need to evaluate the influence of the input parameters on the output signal and standardise the acquisition for an adequate quantitative perfusion assessment. The present study analyses how the variation in the input parameters influences the measurement of the perfusion parameters.

Methods

A software tool with simulator of the conventional US signal was created, and the influence of the different input variables on the derived biomarkers was analysed by varying the image acquisition configuration. The input parameters considered were the dynamic range, gain, and frequency of the transducer. Their influence on mean transit time (MTT), the area under the curve (AUC), maximum intensity (MI), and time to peak (TTP) parameters as outputs of the quantitative perfusion analysis was evaluated. A group of 13 patients with hepatocarcinoma was analysed with both a commercial tool and an in-house developed software.

Results

The optimal calculated inputs which minimise errors while preserving images’ readability consisted of gain of 15 dB, dynamic range of 60 dB, and frequency of 1.5 MHz. The comparison between the in-house developed software and the commercial software provided different values for MTT and AUC, while MI and TTP were highly similar.

Conclusion

Input parameter selection introduces variability and errors in US perfusion parameter estimation. Our results may add relevant insight into the current knowledge of conventional US perfusion and its use in lesions characterisation, playing in favour of optimised standardised parameter configuration to minimise variability.

Tissue perfusion modelling in optical coherence tomography

Background

Optical coherence tomography (OCT) is a well established imaging technique with different applications in preclinical research and clinical practice. The main potential for its application lies in the possibility of noninvasively performing “optical biopsy”. Nevertheless, functional OCT imaging is also developing, in which perfusion imaging is an important approach in tissue function study. In spite of its great potential in preclinical research, advanced perfusion imaging using OCT has not been studied. Perfusion analysis is based on administration of a contrast agent (nanoparticles in the case of OCT) into the bloodstream, where during time it specifically changes the image contrast. Through analysing the concentration-intensity curves we are then able to find out further information about the examined tissue.

Methods

We have designed and manufactured a tissue mimicking phantom that provides the possibility of measuring dilution curves in OCT sequence with flow rates 200, 500, 1000 and 2000 μL/min. The methodology comprised of using bolus of 50 μL of gold nanorods as a contrast agent (with flow rate 5000 μL/min) and continuous imaging by an OCT system. After data acquisition, dilution curves were extracted from OCT intensity images and were subjected to a deconvolution method using an input–output system description. The aim of this was to obtain impulse response characteristics for our model phantom within the tissue mimicking environment. Four mathematical tissue models were used and compared: exponential, gamma, lagged and LDRW.

Results

We have shown that every model has a linearly dependent parameter on flow (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2$$\end{document}R2 values from 0.4914 to 0.9996). We have also shown that using different models can lead to a better understanding of the examined model or tissue. The lagged model surpassed other models in terms of the minimisation criterion and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R^2$$\end{document}R2 value.

Conclusions

We used a tissue mimicking phantom in our study and showed that OCT can be used for advanced perfusion analysis using mathematical model and deconvolution approach. The lagged model with three parameters is the most appropriate model. Nevertheless, further research have to be performed, particularly with real tissue.

Electronic supplementary material

The online version of this article (doi:10.1186/s12938-017-0320-4) contains supplementary material, which is available to authorized users.

Mathematical Models of Contrast Transport Kinetics for Cancer Diagnostic Imaging: A Review

Angiogenesis plays a fundamental role in cancer growth and the formation of metastasis. Novel cancer therapies aimed at inhibiting angiogenic processes and/or disrupting angiogenic tumor vasculature are currently being developed and clinically tested. The need for earlier and improved cancer diagnosis, and for early evaluation and monitoring of therapeutic response to angiogenic treatment, have led to the development of several imaging methods for in vivo noninvasive assessment of angiogenesis. The combination of dynamic contrast-enhanced imaging with mathematical modeling of the contrast agent kinetics enables quantitative assessment of the structural and functional changes in the microvasculature that are associated with tumor angiogenesis. In this paper, we review quantitative imaging of angiogenesis with dynamic contrast-enhanced magnetic resonance imaging, computed tomography, and ultrasound.

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

Background

Contrast-enhanced ultrasound (CEUS) has recently been proposed as a minimally- invasive, alternative method for blood volume measurement. This study aims at comparing the accuracy of CEUS and the classical thermodilution techniques for volume assessment in an in-vitro set-up.

Methods

The in-vitro set-up consisted of a variable network between an inflow and outflow tube and a roller pump. The inflow and outflow tubes were insonified with an ultrasound array transducer and a thermistor was placed in each tube. Indicator dilution curves were made by injecting indicator which consisted of an ultrasound-contrast-agent diluted in ice-cold saline. Both acoustic intensity- and thermo-dilution curves were used to calculate the indicator mean transit time between the inflow and outflow tube. The volumes were derived by multiplying the estimated mean transit time by the flow rate. We compared the volumes measured by CEUS with the true volumes of the variable network and those measured by thermodilution by Bland-Altman and intraclass-correlation analysis.

Results

The measurements by CEUS and thermodilution showed a very strong correlation (r_{s =} 0.94) with a modest volume underestimation by CEUS of −40 ± 28 mL and an overestimation of 84 ± 62 mL by thermodilution compared with the true volumes. Both CEUS and thermodilution showed a high statistically significant correlation with the true volume (r_{s =} 0.97 (95% CI, 0.95 - 0.98; P<0.0001) and r_{s =} 0.96 (95% CI, 0.94 - 0.98; P<0.0001, respectively).

Conclusions

CEUS volume estimation provides a strong correlation with both the true volumes in-vitro and volume estimation by thermodilution. It may therefore represent an interesting alternative to the standard, invasive thermodilution technique.