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

Accurately
modeling arterial input function (AIF) is important for dynamic contrast
enhanced (DCE) MRI. Simulations were performed comparing nine population AIF
models to the Parker AIF. Effects of AIF second pass with and without adding noise
onto extracted physiological parameters were evaluated with n=1,000 randomly
generated physiological parameters (K^{trans} and v_{e}) used
to calculate contrast agent concentration curves using the Tofts model and
Parker AIF. Results demonstrated that the six-parameter linear function plus
bi-exponential function AIF model was almost equivalent to Parker AIF. Effects
of the second pass were small, unless noise with signal-to-noise ratio was <10
dB.

There
are 10 parameters in Parker model: A_{n}, T_{n}, and σ_{n}
(n = 1, 2) are the scaling constant, center, and width of the Gaussian,
respectively; α and β are the amplitude and decay constant of the exponential;
and s and τ are the width and center of the sigmoid. The Parker AIF (C_{p}(t))
was calculated with average parameters, 1.5 seconds temporal resolution, and sampled
for 6 minutes. Then, the other nine population AIF models (**Table 1**) were used to fit the Parker AIF model. The closest model
to Parker model with the smallest number of parameters was selected to study
the second pass effects of AIF.

To
maximize the second pass, a new Parker AIF (C_{p2}(t)) was calculated
by setting parameters so that the first pass peak is lower and the second pass peak
is higher and wider. Therefore, Parker’s average parameters were set to be plus
or minus 2.58 times standard deviations (SD): A_{1} and T_{1}
were set to be mean-2.58*SD; A_{2}, T_{2}, σ_{2}, α and
β were set to be mean+2.58*SD; and σ_{1}, s and τ were kept as mean
values. As result, new AIF had ratios of A_{2}/A_{1}=0.62 and σ_{2}/σ_{1}=3.31,
which were much larger than the AIF calculated with average parameters.

The following steps were used in the computer simulations for a total of 1,000 contrast agent concentration curves C(t):

(i)
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:

$$K^{trans}=0.05+r_{n1}\cdot(1.0-0.05),v_e=0.05+r_{n2}\cdot(0.75-0.05).$$

To
be more realistic, only values of K^{trans} and v_{e} such that
K^{trans}/v_{e} < 10 were used in the simulations. Then C(t)
was calculated by using Tofts model with new C_{p2}(t):^{2}

^{ }

$$C(t)=K^{trans}\int_{0}^{t} C_{p2}(\tau)\cdot exp(-K^{trans}(t-\tau)/v_{e})d\tau.$$

(ii)
The simple AIF with the smallest number of parameters close to Parker AIF was
used to fit the C(t) to extract K^{trans} and v_{e} values and
compared with generated values. The effect of noise on C_{p2}(t) was
also studied by adding white Gaussian noise with signal to noise ratio (SNR) of
10, 5, and 1 dB.

**RESULTS**

1. Parker GJ, Roberts C, Macdonald A, et al. Experimentally-derived functional form for a population-averaged high-temporal-resolution arterial input function for dynamic contrast-enhanced MRI. Magn Reson Med 2006;56(5):993-1000.

2. Tofts PS, Brix G, Buckley DL, et al. Estimating kinetic parameters from dynamic contrast-enhanced T(1)-weighted MRI of a diffusable tracer: standardized quantities and symbols. J Magn Reson Imaging 1999;10:223-32.

3. Tofts PS, Kermode AG. Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts. Magn Reson Med 1991;17(2):357-367.

4. Su MY, Jao JC, Nalcioglu O. Measurement of vascular volume fraction and blood-tissue permeability constants with a pharmacokinetic model: studies in rat muscle tumors with dynamic Gd-DTPA enhanced MRI. Magn Reson Med 1994;32(6):714-724.

5. Tofts PS. Modeling tracer kinetics in dynamic Gd-DTPA MR imaging. J Magn Reson Imaging 1997;7(1):91-101.

6. Simpson NE, He Z, Evelhoch JL. Deuterium NMR tissue perfusion measurements using the tracer uptake approach: I. Optimization of methods. Magn Reson Med 1999;42(1):42-52.

7. Calamante F, Gadian DG, Connelly A. Delay and dispersion effects in dynamic susceptibility contrast MRI: simulations using singular value decomposition. Magn Reson Med 2000;44(3):466-473.

8. Mlynash M, Eyngorn I, Bammer R, et al. Automated method for generating the arterial input function on perfusion-weighted MR imaging: validation in patients with stroke. AJNR Am J Neuroradiol 2005;26(6):1479-1486.

9. Workie DW, Dardzinski BJ. Quantifying dynamic contrast-enhanced MRI of the knee in children with juvenile rheumatoid arthritis using an arterial input function (AIF) extracted from popliteal artery enhancement, and the effect of the choice of the AIF on the kinetic parameters. Magn Reson Med 2005;54(3):560-568.

10. Yankeelov TE, Luci JJ, Lepage M, et al. Quantitative pharmacokinetic analysis of DCE-MRI data without an arterial input function: a reference region model. Magn Reson Imaging 2005;23(4):519-529.

11. McGrath DM, Bradley DP, Tessier JL, et al. Comparison of model-based arterial input functions for dynamic contrast-enhanced MRI in tumor bearing rats. Magn Reson Med 2009;61 (5):1173-1184.