Tumour perfusion can be quantified by fitting a compartment model to Dynamic Contrast Enhanced (DCE) MRI data. Most models require an arterial input function (AIF), which is challenging to obtain. An alternative is to employ a reference region model that uses the tracer concentration in a reference tissue as a surrogate for the AIF [1]. One version of this model, the Linear Reference Region Model (LRRM), has been shown to be fast and robust [2]. One limitation of the LRRM is the assumption that the tissue of interest has negligible plasma volume. This assumption is not always true in tissues such as tumours, which can be highly vascularized.

This study introduces a new model, called the Extended Linear Reference Region Model (ELRRM), which accounts for the plasma volume, and evaluates its performance in simulation and in-vivo data.

The underlying assumptions of the ELRRM are that the reference tissue can be modelled by the Tofts Model, and the tissue of interest can be modelled by the Extended Tofts Model [3]. Under these conditions, the kinetic equation of the ELRRM has the form:

$$\frac{C_{TOI}(t)}{dt}=\bigg{[}\frac{K^{Trans}_{TOI}}{K^{Trans}_{RR}}+\frac{v_{p,TOI}k_{ep,TOI}}{K^{Trans}_{RR}}+\frac{v_{p,TOI}}{v_{e,RR}}\bigg{]}\frac{dC_{RR}(t)}{dt}+...$$

$$...+\bigg{[}\frac{K^{Trans}_{TOI}}{v_{e,RR}}+\frac{v_{p,TOI}k_{ep,TOI}}{v_{e,RR}}\bigg{]}C_{RR}(t)-\bigg{[}k_{ep,TOI}\bigg{]}C_{TOI}(t)+\bigg{[}\frac{v_{p,TOI}}{K^{Trans}_{RR}}\bigg{]}\frac{d^2C_{RR}(t)}{dt^2}.$$

This equation is integrated twice to remove all derivative terms, since differentiation amplifies noise. The ELRRM has four fitting parameters (in square brackets), which can be manipulated to yield relative $$$K^{Trans}$$$, defined as $$$K^{Trans}_{TOI}/K^{Trans}_{RR}$$$, and relative $$$v_e$$$, defined as $$$v_{e,TOI}/v_{e,RR}$$$.

Simulations were conducted to determine the effect of plasma volume on the accuracy of the LRRM and ELRRM. A literature-based model [4] was used to generate the AIF, with a total duration of 10 minutes and temporal resolution of 1 s. The reference region (RR) was defined by $$$K^{Trans}_{RR}=0.1$$$ min$$$^{-1}$$$, $$$v_{e,RR}=0.1$$$ and $$$v_{p,RR}=0$$$. The tissue of interest (TOI) was defined by $$$K^{Trans}_{TOI}=0.25$$$ min$$$^{-1}$$$, $$$v_{e,TOI}=0.4$$$ and $$$v_{p,TOI}$$$ ranging from 0 to 1 in 1000 equally spaced steps.

The effect of noise and temporal resolution on the TOI curves was evaluated for $$$v_{p,TOI}=\{0.001,0.005,0.05,0.10,0.15,0.20\}$$$. Gaussian noise was added to produce Concentration-to-Noise Ratios (CNR) from 5 to 50 in 10 equal steps and 10,000 realizations were simulated for each noise level. The CNR is the peak concentration in the tissue of interest divided by the standard deviation of noise [5]. The concentration-time curves were downsampled to 1, 5, and 10 s.

In-vivo evaluation was done on data from 15 soft tissue sarcoma patients acquired at 1.5 T using a 3D SPGR sequence. The temporal resolution was between 10 and 12 seconds, with a total duration of approximately 9 minutes. Comparative parameters were obtained using the Extended Tofts Model (ETM) fitted to the data using a literature-based AIF [4]. This AIF was also used to simulate the reference region using the same parameters as in the previous simulations. This approach ensures that the estimates from LRRM and ELRRM are directly comparable with the ETM.

The plasma volume has a substantial effect on the LRRM's accuracy while the ELRRM remains accurate (Fig. 1a and 1b). The ELRRM also has accuracy comparable to the ETM for the $$$v_p$$$ estimate (Fig. 1c). Figures 2 and 3 shows the effect of noise on the estimates for relative $$$K^{Trans}$$$ and $$$k_{ep,TOI}$$$ from the LRRM and ELRRM. Noise did not have a substantial impact on ELRRM's accuracy, except at the highest noise levels. The variability in the estimates increases as CNR decreases and $$$v_p$$$ increases. There is a decrease in ELRRM's accuracy when the temporal resolution decreases to 10 s (Fig. 4), but the mean percent error for relative $$$K^{Trans}$$$ remains within $$$\pm$$$20%.

Figure 5 shows percent error in $$$K^{Trans}_{TOI}$$$ for 15 soft tissue sarcoma patients using voxel-wise fitting. The large variability could be due to artifacts or poor fits caused by either the LRRM, ELRRM or ETM. A second approach using the average concentration-time curve in the tumour region of interest (ROI) was attempted and the results are shown in figure 5b. In both cases, the percent errors are smaller for the ELRRM, indicating that it is in better agreement with the ETM. The mean percent error in the voxel-wise case was -63.4% for LRRM and 14.2% for ELRRM. For the ROI approach, the mean percent error was -75.5% for LRRM and -7.4% for ELRRM.

Simulations show that blood plasma has substantial impact on the LRRM's accuracy, while the Extended LRRM (ELRRM) correctly accounts for it. Evaluation on in-vivo sarcoma data showed that the ELRRM outperformed the LRRM through better agreement with the Extended Tofts Model. These results indicate that highly vascularized tissues can be accurately characterized by the reference region approach by using the proposed ELRRM.