First published online October 12, 2006
British Journal of Radiology (2007) 80, 371-375
© 2007 British Institute of Radiology
doi: 10.1259/bjr/72152927
An efficient method for calculating kinetic parameters in a dual-input single-compartment model
K Murase, PhD
S Miyazaki, MS
and
X Yang, PhD
Department of Medical Physics and Engineering, Division of Medical Technology and Science, Faculty of Health Science, Graduate School of Medicine, Osaka University, 1-7 Yamadaoka, Suita, Osaka 565-0871, Japan
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Abstract
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Quantitative measurement of hepatic perfusion has the potential to provide important information in the assessment and management of various liver diseases. The utility of hepatic perfusion characterization relies on the resolution of each component of its dual blood supply, i.e. the hepatic artery and portal vein. In this study, a linear equation was derived by integrating the differential equation describing the kinetic behaviour of contrast agent (CA) in a dual-input single-compartment model, from which the kinetic parameters can be easily obtained using the linear least-squares method. The usefulness of this method was investigated using computer simulations, in comparison with the non-linear least-squares (NLSQ) method. This method calculated the kinetic parameters faster than the NLSQ method by a factor of approximately 10, with almost the same accuracy as the NLSQ method. This method will be useful for analysing the kinetic behaviour of CA in the unique liver environment, especially by generating the functional images of kinetic parameters.
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Introduction
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Quantitative measurement of hepatic perfusion has the potential to provide important information in the assessment and management of various liver diseases and in the determination of their outcome [1–3].
Liver metastases play a major role in the morbidity and mortality associated with many cancers. The non-invasive measurement of regional blood flow within the liver is of particular clinical interest, since changes in tumour blood flow have been recognized as an important factor for assessing the treatment response of hepatic cancers [4, 5].
The utility of hepatic perfusion characterization relies on the resolution of each component of its dual blood supply, i.e. the hepatic artery and the portal vein, because contributions from each are altered in many diseases [6]. In chronic liver diseases, the increase of intrahepatic vascular resistance due to progression of hepatic fibrosis decreases the portal fraction of hepatic perfusion [7]. This decrease of portal perfusion is partially compensated by an increase of arterial inflow [7]. Therefore, separate measurement of hepatic arterial and portal perfusion is important for these diseases.
While the Kety–Schmidt model (single-compartment model) is applicable to the study of blood flow in various organs [5], it must be modified to reflect the unique delivery environment of the liver, where the tissue is fed by both arterial and portal inputs. Two methods to accomplish this include using two separate input functions or using a combined input function. The latter method has been used previously such that a flow-weighted sum of both input functions was applied to hepatic 18F-fluorodeoxy glucose (FDG) uptake, assuming a constant arterial-to-portal ratio across all regions of the liver [8]. However, such a fixed ratio system does not allow for the measurement of independent regional arterial and portal flow fluctuations. A more flexible approach is to incorporate the two separate input functions into the Kety–Schmidt model (dual-input single-compartment model), whereby the second input function reflects the tracer supplied by the portal vein. In this case, the model complexity increases and the parameter for the local portal-venous blood flow has to be estimated from the data [5].
However, the dual vascular input of the liver via the hepatic artery and the portal vein is difficult to measure separately using positron emission tomography (PET), because of the limited spatial resolution of the method. In contrast, dynamic contrast-enhanced CT has high spatial and temporal resolution, allowing for separate measurement of the arterial and portal-venous input functions [3, 7, 9, 10]. Recently, a dual-input single-compartment model has been applied to the intensity vs time curves obtained from dynamic magnetic resonance (MR) images of the liver after injection of gadolinium chelate as an alternative method to measure haemodynamic parameters [11]; there is no radiation exposure.
In general, kinetic parameter estimation is performed by model (curve) fitting using the non-linear least-squares (NLSQ) method. However, the NLSQ method is not suitable for generating functional images of physiological parameters because it is not efficient in terms of computation time. Thus, to generate functional images of physiological parameters, an efficient method for estimating kinetic parameters is necessary. The purpose of this study was to demonstrate an efficient method for calculating kinetic parameters in a dual-input single-compartment model, and to investigate its usefulness using computer simulations in comparison with the NLSQ method.
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Methods and materials
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Pharmacokinetic model
The compartment model used to describe the kinetic behaviour of the contrast agent (CA) in the liver is illustrated in Figure 1
. This model consists of a single compartment with two inputs. When using this model, the differential equation describing the kinetic behaviour of the CA in the liver region of interest is given by:
where CL(t), Ca(t) and Cp(t) are the concentrations of the CA at time t in the liver region of interest, hepatic artery or aorta, and portal vein, respectively. K1a, K1p and k2 are the rate constants for the exchanges of CA from the hepatic artery or aorta to the liver, from the portal vein to the liver, and from the liver to plasma, respectively. It should be noted that whole blood perfusion is obtained by dividing plasma perfusion by one minus the small vessel haematocrit. In this study, the arrival time of the CA from the hepatic artery or aorta to the liver region of interest and that from the portal vein to it were assumed to be zero for simplicity.
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Figure 1. Illustration of a dual-input single-compartment model. Ca(t), Cp(t) and CL(t) represent the concentrations of the contrast agent (CA) at time t in the hepatic artery or aorta, the portal vein, and the liver region of interest, respectively. K1a, K1p and k2 are the rate constants for the exchanges of CA from the hepatic artery or aorta to the liver, from the portal vein to the liver, and from the liver to plasma, respectively.
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When the kinetic model includes a vascular term, that is the contribution of the CA in the intrahepatic arterial and portal-venous plasma, Equation (1) becomes:
where V0 is the capillary plasma volume per unit volume of tissue. V0 is given by a weighted average of Va and Vp as
where Va and Vp denote the relative volume fractions of the intrahepatic arterial plasma and the portal-venous plasma, respectively. Solving Equation (2) with the assumption that the initial conditions are zero yields:
Estimation of kinetic parameters
Fitting Equation (4) to the concentration–time curve in the liver allows the estimation of K1a, K1p, k2 and V0. In general, this can be performed using the NLSQ method. Thus, we call this method the "NLSQ method". In practice, the downhill simplex method developed by Nelder and Mead [12] was used for minimization of function in this study.
Alternatively, integrating both sides of Equation (2) with the assumption that the initial conditions are zero, and expressing the result in a discrete form yields (A1) in Appendix 1. When CL(tk), Ca(tk) and Cp(tk) (k = 1, 2,•••,n) in Equation (A1) are measured, (A2) in Appendix 1 can be easily solved for the elements of 
using the conventional linear least-squares (LLSQ) method, and finally K1a, K1p, k2 and V0 can be easily obtained from the elements. We call this method the "LLSQ method".
Computer simulation
To investigate the accuracy and robustness against statistical noise of the above methods, we performed computer simulations. The details are described in Appendix 2.
In this study, K1a, K1p and k2 were assumed to be 20 ml/100 ml/min, 80 ml/100 ml/min and 4 min–1, respectively. Regarding V0, two cases with V0 of 0 and 10 ml/100 ml were considered. Figure 2 illustrates the typical examples of Ca(t), Cp(t) and CL(t). In this case, V0 and signal-to-noise ratio (SNR) were assumed to be 0 ml/100 ml and 20, respectively.
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Figure 2. Typical examples ofCa(t), Cp(t) and CL(t). In this case, K1a, K1p, k2, the capillary plasma volume per unit volume of tissue (V0) and the signal-to-noise ratio (SNR) were assumed to be 20 ml/100 ml/min, 80 ml/100 ml/min, 4 min–1, 0 ml/100 ml and 20, respectively.
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Simulations were performed using Matlab (The MathWorks Inc., Natick, MA) on a Pentium 4 (2.8 GHz) with 2 GB RAM.
Statistical analysis
A Monte Carlo simulation of 1000 runs was performed for each condition. The mean and standard deviation (SD) of the estimated K1a, K1p, k2 and V0 values for 1000 runs were calculated. The accuracy of parameter estimation was evaluated in terms of root mean square error (RMSE) defined by 
where
and N are the RMSE of the ith parameter, the estimated value of the jth data of the ith parameter, the true value of the ith parameter and the number of data (1000 in this study), respectively.
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Results
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Table 1 summarizes the results for the mean±SD for 1000 runs (a) and RMSE (b) of the estimated parameters in the case when V0 was assumed to be 0 ml/100 ml. Table 2 shows the case when V0 was assumed to be 10 ml/100 ml. When using the NLSQ method, the initial values for K1a, K1p, k2 and V0 were taken as 20 ml/100 ml/min, 20 ml/100 ml/min, 2 min–1 and 5 ml/100 ml, respectively, and the maximum allowed number of iterations and convergence criterion were set at 106 and 10–6, respectively. As shown in Tables 1 and 2, the LLSQ method estimated the K1a, K1p, k2 and V0 values with almost the same accuracy as the NLSQ method.
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Table 1. (a) Mean±standard deviation (SD) for 1000 runs and (b) root mean square error (RMSE) (%) of the K1a, K1p and k2 values estimated using the linear least-squares (LLSQ) method and the non-linear least-squares (NLSQ) method as a function of signal-to-noise ratio (SNR). In these cases, V0 was assumed to be 0 ml/100 ml
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Table 2. Mean±SD for 1000 runs (a) and RMSE (%) (b) of the K1a, K1p, k2 and V0 values estimated using the LLSQ method and the NLSQ method as a function of SNR. In these cases, V0 was assumed to be 10 ml/100 ml
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Regarding computation time, the LLSQ method was faster than the NLSQ method by a factor of approximately 10 (46 s and 473 s on average for 1000 runs in the LLSQ method and the NLSQ method, respectively).
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Discussion
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In this study, we demonstrated an efficient method for calculating the kinetic parameters in the liver from dynamic contrast-enhanced CT data using a dual-input single-compartment model, and investigated its usefulness using computer simulations in comparison with the NLSQ method. As shown in Tables 1
and 2, the LLSQ method can estimate the kinetic parameters in the liver with almost the same accuracy as the NLSQ method. Furthermore, the LLSQ method can estimate them faster than the NLSQ method by a factor of approximately 10. These results suggest that the LLSQ method is useful especially when generating the functional images of kinetic parameters by applying it pixel by pixel. For routine clinical use, the faster the better. The computation time would be further improved by using other speed-up techniques such as parallel processing.
As previously described, a dual-input single-compartment model has also been applied to dynamic MR images of the liver after injection of gadolinium chelate to measure haemodynamic parameters [11]. The LLSQ method will be applicable to this MR imaging method with minor modification.
In conclusion, the LLSQ method presented here will be useful for kinetic analysis of CA in the unique liver environment, especially by generating the functional images of kinetic parameters.
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Appendix 1
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Integrating both sides of Equation (2) with the assumption that the initial conditions are zero, and expressing the result in a discrete form yields:
Equation (A1) can be given in a matrix form as
where
In this study, the elements of 
in Equation (A2) were generated by numerical integration using the trapezoidal rule [12].
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Appendix 2
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In general, when calculating kinetic parameters from dynamic contrast-enhanced CT data, the first several time points of the concentration–time curves obtained after injection of the CA are averaged and subtracted from the subsequent time points to ensure that the initial baseline concentration of the CA is zero [3]. After this normalization, the post-contrast density measurement using CT is directly proportional to the concentration of CA [13, 14], i.e. C(t) = K•
HU(t), where C(t) is the concentration of CA, HU(t) is the relative measure of contrast enhancement, and K is a constant of proportionality. Since this proportionality constant is the same for the blood and liver tissue curves [3, 14], the constant K will cancel out in Equation (1) and thus does not need to be calculated or known. Therefore, in this study, it was assumed that the contrast enhancement was equal to the concentration of CA, i.e. K = 1, and that the initial baseline concentration of the CA was zero.
First, the noise-free time-dependent concentration in the hepatic artery or aorta [Ca(t)] was generated by fitting actually measured data [15] using gamma-variate functions. For CP(t), we assumed a finite series of basis functions, where each basis function is formed by the convolution of an exponential function with Ca(t), as done by Becker et al [5], i.e.
, where 
denotes the convolution operator. In this study, m, b1 and 
1 were taken as 1, 1 min–1 and 2 min–1, respectively, for simplicity [5]. Simulated concentration–time curves were generated in the form of dynamic images with 120 frames with a time step of 1.0 s. Gaussian noise was then added to give SNRs ranging from 10 to 100 with an increment of 10. The SNR was given by the concentration of the CA generated above, divided by the SD of the noise generated from normally distributed random numbers with zero mean and unit variance.
The noise-free concentration–time curve in the liver region of interest [CL(t)] was given by Equation (4). As in the simulation of Ca(t) and Cp(t), Gaussian noise was added to give SNRs ranging from 10 to 100 with an increment of 10 as previously described.
Received for publication April 24, 2006.
Revision received July 22, 2006.
Accepted for publication August 15, 2006.
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A quantitative method for estimating hepatic blood flow using a dual-input single-compartment model
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Abstract
Introduction
