SEARCH

SEARCH BY CITATION

Keywords:

  • Tofts model;
  • perfusion;
  • permeability;
  • Bolus-tracking;
  • MRI

Abstract

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. REFERENCES

The two-compartment 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 two-compartment exchange model is a generalization of 2CTM obtained by dropping the zero-broadening hypothesis, making it suitable for estimating the impact of violating this assumption. Using data simulated on the basis of the two-compartment exchange model, the correspondence between the hemodynamic parameters serving as input for the two-compartment 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 two-compartment exchange model input was observed. Extreme caution is needed when interpreting 2CTM data. Magn Reson Med, 2011. © 2010 Wiley-Liss, Inc.

Since its introduction by Tofts et al. (1), the consensus model has been popular for pharmacokinetic modelling of DCE-MRI tumour data. This so-called “Tofts model” is a one-compartment 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:

  • equation image(1)

In this expression, ⊗ denotes the convolution operation and AIF the arterial input function. The parameter Ktrans is somewhat ambiguous: it must be interpreted as the extraction flow EF (or permeability-surface product PS) in the permeability-limited regime and as the plasma flow PF in the flow-limited regime. For the intermediate, mixed regime the following expression was put forward for the relationship between Ktrans and the two flows (1):

  • equation image(2)

In Eq. 1, kep represents the rate constant for the exchange between the blood and the extravascular-extracellular compartment and is itself related to Ktrans and the extravascular-extracellular volume fraction EV. For situations where the assumption of negligible plasma volume is invalid, a two-compartment extension of the model has been formulated in which the concentration in the tissue plasma is approximated by the AIF (2):

  • equation image(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 two-compartment 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):

  • equation image(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)).

thumbnail image

Figure 1. Graphical representation of a 2-compartment exchange model (2CXM). It consists of a plasma compartment with volume fraction PV and an extravascular-extracellular (interstitial) compartment with volume fraction EV. The contrast agent enters the plasma compartment carried by the plasma flow PF. Part of the agent leaks into the extravascular compartment by a flow EF and leaves it by a backflow of the same magnitude. In the 2-compartment Tofts model (2CTM), the plasma concentration of the contrast agent in the tissue is assumed to equal the AIF at all times.

Download figure to PowerPoint

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, Ktrans, 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 contrast-to-noise ratio (CNR), was evaluated. The instrumental conditions needed for keeping the random and systematic deviations under control were explored.

MATERIALS AND METHODS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. REFERENCES

Data Generation

The computer simulations were performed in IDL (Research Systems, Boulder Colorado, USA). Data were generated on the basis of the 2CXM, with the tissue parameters PFin, PVin, EFin, EVin as input. The AIF (kept fixed) was generated using a sum of gamma-variate functions and a mono-exponentially decaying recirculation to resemble a typical patient AIF (Fig. 2a). This AIF was introduced in Eq. 4 to calculate the tissue response function corresponding to the input parameters at a temporal resolution of 0.01 sec for a time window of 2500 sec. To simulate measurement, data were sampled with temporal interval TR and time window Tacq. To allow for sampling uncertainty, sampling was started at a time point determined by generating a random number uniformly distributed between t = 0 and t = TR. The CNR of the measurement was defined as the ratio of the maximum tissue concentration to the standard deviation of the noise. Normally distributed noise with the corresponding standard deviation was added to the tissue data. As generally the CNR of the AIF is much higher than that of the TRF, no noise was added to the AIF.

thumbnail image

Figure 2. Concentration-time courses for: (a) feeding artery, (b) reference tissue, (c) low PF tissue, (d) low PV tissue, (e) low EF tissue, and (f) low EV tissue.

Download figure to PowerPoint

Fitting

The simulated data were fitted to the 2CTM to produce an estimate of PV/Ktrans/EV. Model fitting was performed using MPFIT (9) with defaults for the number of iterations and convergence threshold. Analytical formulae were used for the partial derivatives. To avoid the problem of false χ2 minima, the initial values of the fit parameters PV, Ktrans, and EV were fixed to the input values PVin, EFin, and EVin. Fitting was constrained to positive parameter values.

Error Estimation

For a given set of instrumental factors and tissue parameters, the steps of measurement simulation and fitting were repeated 1000 times, yielding an estimate of the 5th, 50th, and 95th percentiles for the fitted tissue parameters. On the basis of these percentiles, the relative deviations between the fitted and input parameters were expressed in terms of bias (deviation of the 50th percentile) and precision (distance between the 5th and 95th percentiles). The purely model-related (“ideal”) bias, characteristic for ideal measurement conditions (i.e. with all instrumental factors at their value above concern), was identified for each fit parameter. For Ktrans, the deviations with respect to EFin, PFin, and the theoretical value predicted by Eq. 2 were considered.

Tissue Models

In the first part of the study, the tissue models were restricted to a reference tissue and four limit tissues. For the reference tissue, the input parameters PFin, PVin, EFin, and EVin were equal to 30 mL/min/100 mL, 5 mL/100 mL, 10 mL/min/100 mL, and 25 mL/100 mL, respectively, corresponding to a TRF with a clear bi-phasic behavior, with time constants T+ and T situated well within the simulation ranges for TR and Tacq (Table 1). In each of the limit tissues, three of the parameters characterizing the reference tissue were left unchanged, while the remaining one was given a value at least 5 times smaller than in the reference tissue, leading to more challenging but still realistic fitting situations. The corresponding input parameter values and time constants are also listed in Table 1. Figure 2 illustrates the shapes of the AIF used for the simulations and of the TRF of the reference and four limit tissues. In the second part of the work, the tissue parameters were varied one at a time over a wide range, while keeping the other three fixed at their reference tissue value.

Table 1. The Five Tissue Models and the Corresponding T (sec) and T+ (sec) Values
Tissue typePFPVEFEVTT+
  1. PF and EF are expressed in mL/min/100 mL, PV and EV in mL/100 mL.

Reference30510257.5143
Low PF55102520.3136
Low PV30110251.5149
Low EF3052259.4742
Low EV3051058.290

Simulations and Analysis

In the first part of the study, the influence of the instrumental parameters on the fit results was explored. For the reference and limit tissue models, the impact of each instrumental factor separately was studied by varying that factor while keeping the other two fixed at a value above concern. The values above concern were 1 sec, 2250 sec, and 1000, respectively, for TR, Tacq, and CNR. The ranges over which the factors were varied were: from 0.5 sec to 20 sec for TR, from 75 sec to 2250 sec for Tacq and from 5 to 100 for CNR. The constraints to be imposed on each instrumental factor separately in order to guarantee a precision (i.e. distance 5th–95th percentile) of at least 10% for the fit parameters of each of the five tissues were derived. The minimal instrumental conditions needed to bring the bias to within 1% of the ideal bias (i.e. IdealBias ≤ 1%) were also determined. To test the usefulness of the combination of these constraints, the relative deviations for the fit parameters were evaluated with the constraints on TR, Tacq, and CNR implemented simultaneously. In the second part, the dependence of the ideal bias on the tissue parameters was explored. For this purpose, additional simulations in which three of the tissue parameters were fixed at their reference value, while the fourth one was varied, were undertaken. The ranges over which the tissue parameters were varied were: 3–90 mL/min/100 mL for PFin and EFin, 2–60 mL/100 mL for PVin and EVin. The correspondence between the input and fitted hemodynamic parameters and between the fitted Ktrans and the value predicted by Eq. 2 was evaluated.

RESULTS

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. 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 (Ktrans 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 Ktrans 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 Ktrans with respect to PFin, EFin and the theoretical prediction for Ktrans from Eq. 2 was plotted.

thumbnail image

Figure 3. Relative deviations of the fit parameters PV/Ktrans/EV from the input parameters PVin/EFin/EVin for the reference tissue, as a function of TR, Tacq and CNR.

Download figure to PowerPoint

thumbnail image

Figure 4. Relative errors for a protocol with the instrumental factors set to the limit values TR/Tacq/CNR = 7.5 sec/2025 sec/29: (a) reference tissue, (b) low PF, (c) low PV, (d) low EF, and (e) low EV.

Download figure to PowerPoint

thumbnail image

Figure 5. Ideal percentual bias values for the fitted PV (dotted), Ktrans (full), and EV (dashed), plotted as a function of the 2CXM input parameters PFin (a), PVin (b), EFin (c), and EVin (d). These input parameters were varied one at a time, keeping the other three fixed at the reference tissue values.

Download figure to PowerPoint

thumbnail image

Figure 6. Percentual deviation of Ktrans with respect to PFin (dotted), EFin (full) and the theoretical prediction based on (2) (dashed), plotted as a function of the 2CXM input parameters PFin (a), PVin (b), EFin (c), and EVin (d). These input parameters were varied one at a time, keeping the other three fixed at the reference tissue values.

Download figure to PowerPoint

Table 2. Maximum TR and Minimum CNR Compatible With a Precision of ≤10 % for the Fitted Parameters
 TR (sec)CNR
RefLow PFLow PVLow EFLow EVRefLow PFLow PVLow EFLow EV
  1. The values listed were interpolated and the < signs refer to values outside the simulation range.

PV8.411.77.58.18.28<525710
<Ktrans11.615.611.410.37.75<5<5929
EV14.116.415.318.58.6<5<5<5<512
Table 3. Minimum Tacq (sec) Needed to Reduce the Difference Between the Percentual Bias and the Purely Model-Related (“ideal”) Percentual Bias to <1%
 ReferenceLow PFLow PVLow EFLow EV
PV6003753001575225
Ktrans7505252252025375
EV7509756002025375

DISCUSSION

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. 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 Ktrans 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), Ktrans should be interpreted as a quantity that reflects PF in situations where EF is large (flow-limited regime), and EF in cases where PF is large (permeability-limited regime). In other cases (the mixed regime), Ktrans 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), Ktrans is confirmed to be very nearly equal to EFin at high PF (permeability-limited 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), Ktrans remains about 20% short of PFin. From Fig. 6, it also appears that the fitted Ktrans and the Ktrans 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 DCE-MRI 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 set-up (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 Ktrans 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 Ktrans 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 two-compartment 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.

CONCLUSION

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. REFERENCES

Using computer simulations based on the 2CXM for data generation and 2CTM for fitting the data, the correspondence between the hemodynamic parameters used as input and the fit parameters was investigated. The simulation study shows that, while the random deviations may be limited by a proper choice of the experimental factors, the systematic deviations depend on both the tissue and the experimental factors and can range up to 200% and more. Even when keeping all the experimental factors at a level above concern (TR = 1 sec, Tacq = 2250 s, CNR = 1000), the systematic deviations between the input and fitted parameters remain large and dependent on the tissue. There is generally no simple relationship between the hemodynamic quantities entered in the 2CXM model and those retrieved using 2CTM fitting. This shows that extreme caution is needed in using the 2CTM to derive meaningful hemodynamic parameters from concentration-time courses.

REFERENCES

  1. Top of page
  2. Abstract
  3. MATERIALS AND METHODS
  4. RESULTS
  5. DISCUSSION
  6. CONCLUSION
  7. REFERENCES