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Letters to the Editor |
PET Center, Institute of Radiopharmacy, Research Center Dresden-Rossendorf, PO Box 510119, D-01314 Dresden, Germany
e-mail: j.van_den_hoff{at}fzd.de
In the June 2006 issue of Radiology, Mr Stewart and colleagues (1) compare fluorine 18 fluorodeoxyglucose positron emission tomography (PET) with computed tomography (CT)-derived blood flow in a rabbit liver tumor model. The mathematical procedure outlined in the appendix boils down to describing the contrast agent kinetics by means of a one-compartment model including a correction for the finite capillary transit time ("adiabatic model"). This approach seems to suffer from problems of parameter identifiability ("overfitting the data").
The quantitation procedure is based on an article by St Lawrence and Lee (2). In equations 14 and 15 of that article the approximation: Q(t) = HBV · I(t) + I(t) * EF exp(–EF/Vc · t) of the adiabatic model is given, which can be obtained from equations A1 and A3 of the article by Mr Stewart and colleagues (1) if the capillary transit time is approaching zero in the adiabatic model equation (while keeping Tc · F = HBV = constant); Q(t) = parenchymal contrast enhancement curve, HBV = hepatic blood volume, I(t) = contrast material input to liver, * = convolution operator, E = extraction fraction, F = liver blood flow, Vc = distribution volume of contrast material in extravascular space, Tc = capillary transit time, and t = time. Further details are found in reference 2. Not accidentally, this formula is identical to the standard formula for modeling one-compartment kinetics in PET, including a so-called fractional blood volume contribution. In reference 2 it is, moreover, correctly stated that simultaneous determination of extraction fraction and blood flow is impossible in this case: If too "broad" curves mask the effects of finite transit times (and the approximate formula above thus describes the tissue response curves adequately), only the combined quantity E · F (usually called K1) is identifiable. This is definitely the situation regularly encountered in PET.
The central question is whether the circumstances are sufficiently different in the present case: Even in the liver, capillary transit times amount to only a few seconds, and only the "microstructure" of the CT-derived response curves could potentially provide the desired flow information, if at all. For numerical/mathematical reasons I would like to express serious doubts that extraction fraction and blood flow can be simultaneously determined in the present situation. I believe that a sensitivity analysis would be mandatory. At the very least the parameter estimation errors resulting from the fit should be reported. The situation seems aggravated by the fact that a rather large number of free model parameters is used (E, F, Tc, Vc, hepatic arterial fraction [HAF]) and input function delay is not considered. Reliable determination of HAF (which is critical for the flow determination anyway), especially, seems questionable for several reasons (input function shape variable in the fitting procedure, neglect of input function delay, large covariance with other parameters). Computer simulations similar to those performed in reference 2 could help to clarify the stability, identifiability, and accuracy issues.
In conclusion, I appreciate that the presented work addresses an important question. It is possible that the numerical results (and the conclusions drawn from them) will turn out to be valid. But I do see a real need to verify the validity and numerical stability of the mathematical framework before relying on the presented data and methods. Otherwise there is a rather high risk that other groups will rely on a potentially numerically unstable (or even erroneous) approach that is not suited for clinical applications.
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Lawson Health Research Institute and Robarts Research Institute, 100 Perth Drive, London, Ontario, Canada N6A 5K8
e-mail: tlee{at}imaging.robarts.ca
Dr van den Hoff has raised a number of relevant comments concerning our study (1). Thank you for giving us the opportunity to respond to them.
Tracer kinetics modeling and scanning protocol.—The assumption that the capillary transit time is approaching zero was not made in our implementation of the adiabatic approximation of the Johnson-Wilson model (1,2). This simplification does not allow separate determination of extraction fraction and blood flow with the model. Instead, we fitted the measured aortic and portal venous concentration curves to the liver parenchymal time-density curve by using the full model as prescribed by equations A1–A4 in the appendix of our article (1). The fitting procedure gave a blood flow–scaled impulse residue function, F · H(t), in which initial (maximum) height is the total liver blood flow and its value at the (mean) transit time is the flow-extraction product, F · E (equation A1). To depict small differences in normal versus cancerous liver curves, a small time interval of 0.5 second was employed in the first (circulation) phase of CT scanning, when effects of flow and extraction dominate (figure 1c) (1).
Input function delay.—Our fitting procedure estimated the delay, T0, in appearance time of contrast material in tissue relative to that in the aorta. Specifically, the input curve (equation A4 in reference 1) used to fit liver curves was modified to include the delay parameter, T0, as: I(t) = 
A(t – T0) + (1 – )V(t – T0), where A(t) and V(t) are contrast enhancement values of the hepatic artery and the portal vein, respectively, and (or HAF) is the fraction of liver blood flow contributed by the hepatic artery. Figures 2 and 7 in our article show examples of T0 maps obtained.
Accuracy, stability, and identifiability of model parameters.—We have validated our CT measurement of the hepatic arterial blood flow (HABF) against microsphere measurement by using the same animal model (3). The Bland-Altman plot comparing the CT and microsphere measurement gave a mean difference of –13.34 mL · min–1 · (100 g)–1 (4). The limits of agreement, the interval in which 95% of the differences lie, were –28.43 and 55.11 mL · min–1 · (100 g)–1 (3). Since HABF is the product of the total liver blood flow (F) and HAF, these results demonstrate that liver blood flow can be separately determined from extraction fraction and validate the determination of HAF, as well.
In our article (1), we demonstrated that total hepatic blood flow and blood volume in the tumor core decreased significantly (in figure 3), and HAF increased significantly both in the tumor core and rim (in figure 4) as tumor grew in a small number (eight) of rabbits. These results indicate that the precision of model parameter estimates is adequate for serial study of tumor development. However, we agree that additional computer simulations as discussed in reference 2 are required to determine the stability, identifiability, and accuracy of model parameter estimates more completely. We have embarked on such a study and will report on the results in the future.
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