Abstract
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 Abstract
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSION
 REFERENCES
The twocompartment Tofts model (2CTM) has had widespread use in research and clinical practice. It assumes there is no broadening associated with the bolus transit through the capillary bed of the tissue under study. This assumption is often violated, with consequences that are hard to predict intuitively. The twocompartment exchange model is a generalization of 2CTM obtained by dropping the zerobroadening hypothesis, making it suitable for estimating the impact of violating this assumption. Using data simulated on the basis of the twocompartment exchange model, the correspondence between the hemodynamic parameters serving as input for the twocompartment exchange model and the parameters resulting from fitting the data with the 2CTM was investigated. The influence of tissue type and experimental setup was studied. Generally, a large tissue and setup dependent bias of the 2CTM fitting results with respect to the twocompartment exchange model input was observed. Extreme caution is needed when interpreting 2CTM data. Magn Reson Med, 2011. © 2010 WileyLiss, Inc.
Since its introduction by Tofts et al. (1), the consensus model has been popular for pharmacokinetic modelling of DCEMRI tumour data. This socalled “Tofts model” is a onecompartment model (1CTM) that assumes the plasma volume fraction PV of the tissue to be negligible and leads to following analytical expression for the tissue response function (TRF) that describes the evolution in time of the concentration of the contrast agent in the lesion as the bolus passes through it:
 (1)
In this expression, ⊗ denotes the convolution operation and AIF the arterial input function. The parameter K^{trans} is somewhat ambiguous: it must be interpreted as the extraction flow EF (or permeabilitysurface product PS) in the permeabilitylimited regime and as the plasma flow PF in the flowlimited regime. For the intermediate, mixed regime the following expression was put forward for the relationship between K^{trans} and the two flows (1):
 (2)
In Eq. 1, k_{ep} represents the rate constant for the exchange between the blood and the extravascularextracellular compartment and is itself related to K^{trans} and the extravascularextracellular volume fraction EV. For situations where the assumption of negligible plasma volume is invalid, a twocompartment extension of the model has been formulated in which the concentration in the tissue plasma is approximated by the AIF (2):
 (3)
The approximation is equivalent to assuming there is no broadening associated with the bolus transit through the capillary bed. This extended model has often been referred to as the “modified Tofts model” (2CTM).
The ambiguity in the interpretation of the results of the Tofts models can be resolved by using the twocompartment exchange model (2CXM, see Fig. 1), first introduced for tumours by Brix et al. (3, 4), which leads to an expression for the tissue response curve of the form (5):
 (4)
In this expression, the three parameters E_{+}, T_{+}, and T_{−} are fully defined by four hemodynamic quantities: PF, PV, EF, and EV. The model is a generalization of the 2CTM obtained by dropping the assumption of negligible transit time in the plasma compartment (6) and is therefore an ideal reference for studying the validity of 2CTM results. Situations where the fundamental assumption of 2CTM is compromised are not exceptional. In the context of a study of DCE MRI of cervix cancer using both the 2CXM and 2CTM, for instance, Donaldson et al. (7) recently reported problems in interpreting their 2CTM results, attributing these problems to the fact that the mean plasma transit times (22 ± 16 sec in a cohort of 30 patients) were not negligible. Similarly long mean transit times were reported in brain tumors (12 ± 12 sec in a cohort of 15 (5)) and prostate cancer (24.3 ± 9.6 sec in 27 patients (8)).
In this light, the aim of this article was to explore the validity of 2CTM perfusion parameters on the basis of simulated DCE data. The 2CXM model with input parameters PFin, PVin, EFin, and EVin was used to simulate the measured tissue time courses and the 2CTM model was fitted to these data to retrieve the parameters PV, K^{trans}, and EV. The correspondence between the two sets of parameters was studied in terms of random and systematic deviations. The impact of the input parameters, as well as the instrumental factors temporal sampling step (TR), acquisition window (Tacq) and contrasttonoise ratio (CNR), was evaluated. The instrumental conditions needed for keeping the random and systematic deviations under control were explored.
RESULTS
 Top of page
 Abstract
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSION
 REFERENCES
The results of the simulations in which each of the factors TR, Tacq and CNR was varied separately, while keeping the other two at a level above concern, were collected in Fig. 3 for the reference tissue. Analogous results for the four limit tissues (figures not shown) confirmed the general behaviour of the deviations, however with large variations in error size between the tissues. While the precision was mainly determined by the choice of TR and CNR, Tacq was found to be decisive for the bias. The values for TR and CNR needed to ensure a precision of 10% were listed in Table 2: to meet this condition for the five tissue models, TR must not exceed 7.5 sec and CNR must be at least 29. In Table 3 the Tacq values needed to bring the bias percentage to within 1% of the ideal bias percentage were collected: for this condition to be met for the five tissue models, Tacq must be at least 2025 sec. The relative deviations encountered when combining these limits (i.e. with TR/Tacq/CNR = 7.5 sec/2025 sec/29) were plotted in Fig. 4. In all cases but one (K^{trans} for Low EV where the bias percentage differed by 2.5% from the ideal percentage) the objective of approaching the ideal bias to within 1% was reached. In all cases but two (PV for Low PV and K^{trans} for Low EV where a 5th–95th percentile spread of about 30% was encountered) the objective of a 10% precision was met. The dependence of the ideal bias on the tissues was also illustrated in Fig. 5, in which each tissue parameter was varied individually, with the other three fixed at their reference value. In Fig. 6, the ideal bias for K^{trans} with respect to PFin, EFin and the theoretical prediction for K^{trans} from Eq. 2 was plotted.
Table 2. Maximum TR and Minimum CNR Compatible With a Precision of ≤10 % for the Fitted Parameters  TR (sec)  CNR 

Ref  Low PF  Low PV  Low EF  Low EV  Ref  Low PF  Low PV  Low EF  Low EV 


PV  8.4  11.7  7.5  8.1  8.2  8  <5  25  7  10 
<K^{trans}  11.6  15.6  11.4  10.3  7.7  5  <5  <5  9  29 
EV  14.1  16.4  15.3  18.5  8.6  <5  <5  <5  <5  12 
Table 3. Minimum Tacq (sec) Needed to Reduce the Difference Between the Percentual Bias and the Purely ModelRelated (“ideal”) Percentual Bias to <1%  Reference  Low PF  Low PV  Low EF  Low EV 

PV  600  375  300  1575  225 
K^{trans}  750  525  225  2025  375 
EV  750  975  600  2025  375 
DISCUSSION
 Top of page
 Abstract
 MATERIALS AND METHODS
 RESULTS
 DISCUSSION
 CONCLUSION
 REFERENCES
One of the observations is that, although one can hope to restrict the random deviations by a proper choice of TR and CNR (Fig. 4), this is not the case for the systematic differences between the 2CXM input parameters and the corresponding 2CTM output. Even when all experimental factors are fixed at their value above concern, these systematic deviations remain present. Our study of these ideal bias values shows a strong dependence on the input tissue parameters (Fig. 5). While PV is found to be systematically underestimated by 35% or more, EV is overestimated by 5% or more. This can be understood from the fact that the 2CTM does not allow for broadening in the capillaries and therefore has difficulty in fitting the vascular transit of the bolus. The results obtained for K^{trans} do not show a consistent over or underestimation with respect to EFin. Instead, the bias ranges between an underestimation of about 75% and an overestimation exceeding 150%. According to (1), K^{trans} should be interpreted as a quantity that reflects PF in situations where EF is large (flowlimited regime), and EF in cases where PF is large (permeabilitylimited regime). In other cases (the mixed regime), K^{trans} is predicted to depend on both these flows according to Eq. 2. These expectations can be evaluated in Fig. 6. In the case where EFin is kept constant at 10 mL/min/100 mL and PFin is varied up to 90 mL/min/100 mL (Fig. 6a), K^{trans} is confirmed to be very nearly equal to EFin at high PF (permeabilitylimited regime). Repeating the same exercise for PFin fixed at 30 mL/min/100 mL and EFin varying over the same range (Fig. 6c) shows that, even at the highest EFin considered (90 mL/min/100 mL), K^{trans} remains about 20% short of PFin. From Fig. 6, it also appears that the fitted K^{trans} and the K^{trans} predicted by Eq. 2 for the mixed regime agree relatively well provided PVin ≪ EVin. The relative deviations obtained with TR/Tacq/CNR simultaneously set to their limit values (TR = 7.5 sec, Tacq = 2025 sec, CNR = 29) complied reasonably, though not perfectly, with the objective put forward for the selection of these limits, which was a precision of 10% or better and a bias within 1% of the ideal bias.
Although the 2CTM has been used extensively for analysing all kinds of DCEMRI data, only a few studies of the validity of the hemodynamic parameters it delivers have been undertaken. In 1998 Henderson et al. (10) reported on the temporal sampling requirements of the fit model, using a simulation approach similar to the one used here. In this pioneering work, they concluded that, with AIF and TRF sampled at the same rate, for the total relative errors at the 95% confidence level to remain within ± 10%, sampling the time courses every 1 sec or less was required, a much more stringent constraint than the one found here, probably due to several differences in study design: their tissue time courses were generated on the basis of the 2CTM itself, a wide range of AIF shapes was covered and their 10% limit referred to the total error (including bias and spread).
For a fixed experimental setup (TR = 1 sec, Tacq = 300 sec), Buckley (11) studied the systematic deviations between the physiological parameters introduced in an MMID4 model and those retrieved using three different fit models, including the 2CTM. Attention was focused on two reference tissues (breast tumor and meningioma). He found that there was a slight systematic overestimation of K^{trans} and a consistent and potentially large underestimation of PV. Our results confirm the consistent underestimation of PV. However, his observation of a slight overestimation of EF by K^{trans} is limited here to EFin ≪ PFin: for the other regimes, an underestmation is encountered (Fig. 6c). This behavior is also predicted by Eq. 2 (same figure).
This exploration of the errors associated with a 2CTM fitting analysis of DCE tissue data suffers from several limitations. The AIF used in the simulations, although typical, was kept fixed and free of noise. The tissues were assumed to be well described using the 2CXM and only a restricted set of values for the tissue parameters was investigated. Temporal sampling of both the AIF and TRF was identical. These limitations must be expected to lead to an underestimation of the variability of the fit results. However, the simulations took into account the influence of both the tissue parameters and the experimental factors related to noise and temporal sampling (including the effect of sampling uncertainty). The values for the tissue parameters covered by the various simulations were compatible with the range of values encountered in clinical practice (5, 7, 8).
The 2CXM corresponds to the most general solution of the twocompartment equations. Here it was of special interest because of its hierarchical relationship to the 2CTM. As 2CXM is a generalization of the 2CTM obtained by dropping the assumption of a negligible transit time in the capillaries, the reported bias quantifies the error introduced by applying the 2CTM in situations where its fundamental assumption is violated. As the resulting bias is not fixed (e.g. 15%), but strongly dependent on the tissue type and the experimental conditions, use of the 2CTM in these situations invites misinterpretation of the results. An important additional consequence of the results collected here is that the notion of 2CTM being a model suitable for analysing low temporal resolution acquisitions [see, for example, Armitage et al. (12)] cannot be upheld. Figure 3 shows that as TR gets larger, an increasing random deviation is superimposed on the bias that is present even in ideal measurement conditions, a result to be expected in view of the increase in discretization error with increasing TR.