Dynamic contrast enhanced MRI (DCE-MRI) is a widely accepted tool for detection and diagnosis of cancers. The Tofts model^[1] is commonly used to extract the pharmacokinetic parameters (K^trans and v_e). However, the tumors are extremely heterogeneous and the simple compartment models may not be compatible with tumors. In contrast to the compartmental model, analytical mathematical models are also popular for analyzing DCE-MRI data. These mathematical models make no assumptions about the underlying physiology of a tumor, but simply use one or more functions with limited parameters to characterize the important features of the tumor contrast agent concentration curves (C(t)). The use of fitted mathematic curves could greatly improve the accuracy of extracting physiological parameters and calculating area under the curve, time to peak, and initial uptake slope, etc. This is extremely important when data is noisy and/or acquired with a low temporal resolution. There were a few mathematical models that have been developed for analyzing data. Here we used numerical simulations to evaluate and compare five different mathematical models for fitting C(t) calculated by Tofts model.

Contrast agent concentration curves: Random numbers (r_n1 and r_n2) uniformly distributed between 0 and 1 were generated and mapped into the following interval to obtain practical K^trans and v_e values:

$$$K^{trans}=0.001+r_{n1}\cdot(1.0-0.001)$$$ and $$$v_{e}=0.005+r_{n2}\cdot(0.75-0.005)$$$,

i.e., 0.001≤K^trans (min^-1)≤1.0 and 0.005≤v_e≤0.75. Only values such that K^trans/v_e < 10 were used to calculate C(t) using the following equation with temporal resolution of 1.5 seconds:

$$C(t)=K^{trans }\int_{0}^{t}e^{-K^{trans}(t-\tau)/v_{e}}\cdot C_{p}(\tau)d\tau$$

where C_p(t) is a population AIF derived by Parker et al^[2]. Curves C(t) were sampled over a range of 20 min. In addition to noise free curves, we also generated curves with noise (C ̃(t)) by adding 20% of random numbers (r_n(t)) in Gaussian distribution to C(t):$$$\widetilde{C}(t)=C(t)\cdot(1+r_{n}(t))$$$.

Mathematical models: For each curve C(t) obtained above, five previously developed mathematical models were used to fit the curve. They are:

(a) Empirical mathematical model (EMM)^[3]: $$$C(t)=A\cdot(1-e^{-\alpha\cdot t})^{q}\cdot e^{-\beta\cdot t}\cdot \frac{1+e^{-\gamma\cdot t}}{2}$$$.

(b) Modified logistic model (MLM)^[4]: $$$C(t)=\frac{P_{2}+(P_{5}\cdot t)}{1+e^{-P_{4}\cdot(t-P_{3})}}+P_{1}$$$.

(c) Modified Sigmoidal functions (MSF)^[5]: $$$C(t)=\frac{a_{0}}{(1+e^{\frac{t-a_{1}}{a_{2}}})}e^{\frac{t-a_{1}}{a_{3}}}$$$.

(d) Weibull^[6]: $$$C(t)=a\cdot t\cdot e^{-\frac{t^{c}}{b}}$$$.

(e) Extended phenomenological universalities (EU1)^[7]: $$$C(t)=Me^{rt+\frac{1}{\beta}(a_{0}-r)e^{\beta t}-1}$$$.

A total of 100 curves were used to compare how good the fit was for all five mathematical models. To evaluate the accuracy of fitting for each model, first the absolute difference between C(t) and fitted curve (C_fitting(t)) were calculated for each time: D(t)=|C(t)-C_fitting(t)|. Then the top 10% (n=80) of worst fitting time points (D_i(t)) were selected to calculate the root-mean-square error (RMSE_10%). For all RSME_10% values obtained from 100 curves, one-way ANOVA and Tukey's HSD tests were performed to determine whether there was a significant difference between these five mathematical models. A p-value less than 0.05 was considered significant.

Figure 1 shows the plots of calculated C(t) (black line) for selected K^trans=0.35 (min^-1) and v_e=0.5, as well as the corresponding fits (red line) obtained from five mathematical models. We can see that some models fit the curves better than others, especially at the earlier time. For the total 100 noise fee curves C(t), all five mathematical models were used to fit all curves, and then all RMSE_10% were calculated. Figure 2 shows the box-plot of RMSE_10% for all five mathematical models for 100 noise free curves. It shows that the EMM had the smallest errors, and ELM and EU1 had larger errors than MSF and Weibull models. One-way ANOVA and Tukey's HSD test showed that there were significantly differences for RMSE_10% between mathematical models (p<0.001), except there were no significant difference between MLM and EU1 models, and between MSF and Weibull models. After adding the noises to the curves, statistical analysis showed that MLM had significantly larger (p<0.01) RMSE_10% than other models with the exception of EU1. But the EMM still had the smallest RMSE_10% than other models.Among the five models studied, the EMM proved to have the best for fitting the curves. The goodness-of-fit cannot be simply attributed to number of parameters; rather, it is mainly due to the functions used in the EMM. The MLM also had five parameters, but the fitting was worse than the EMM. Therefore, the number of parameters used in the model is one factor, and type of functions is another factor in affecting the quality of the fitting curves. In summary, analytical mathematical models were not of equal quality even when they had the same number of parameters. One should select the model based on characteristics of curves used in their studies. The EMM can be reduced to four or three parameters models by setting $$$\gamma$$$=0 and/or q=1.