Using numerical simulations to compare and evaluate different mathematical models for analyzing dynamic contrast enhanced MRI data

Dianning He^{1,2}, Wei Qian^{1,3}, Lisheng Xu^{1,4}, and Xiaobing Fan^{2}

** Contrast agent
concentration curves: **Random numbers (r

$$$K^{trans}=0.001+r_{n1}\cdot(1.0-0.001)$$$ 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.

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2. Parker GJ, Roberts C, Macdonald A, Buonaccorsi GA, Cheung S, Buckley DL, Jackson A, Watson Y, Davies K, Jayson GC. Experimentally-derived functional form for a population-averaged high-temporal-resolution arterial input function for dynamic contrast-enhanced MRI. Magnetic resonance in medicine 2006;56(5):993-1000.

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5. Orth RC, Bankson J, Price R, Jackson EF. Comparison of single-and dual-tracer pharmacokinetic modeling of dynamic contrast-enhanced MRI data using low, medium, and high molecular weight contrast agents. Magnetic resonance in medicine 2007;58(4):705-716.

6. Gal Y, Mehnert A, Bradley A, McMahon K, Crozier S. An evaluation of four parametric models of contrast enhancement for dynamic magnetic resonance imaging of the breast. 2007. IEEE. p 71-74.

7. Gliozzi A, Mazzetti S, Delsanto PP, Regge D, Stasi M. Phenomenological universalities: a novel tool for the analysis of dynamic contrast enhancement in magnetic resonance imaging. Physics in medicine and biology 2011;56(3):573.

Proc. Intl. Soc. Mag. Reson. Med. 24 (2016)

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