Xiaobing Fan^{1}, Aritrick Chatterjee^{1}, Aytekin Oto^{1}, and Gregory S. Karczmar^{1}

The Tofts pharmacokinetic model
requires contrast agent concentration as function of time (C(t)), which is normally
calculated using the non-linear model that could contribute some errors. Here,
we present signal intensity (S(t)) form of standard Tofts pharmacokinetic model
without calculating C(t). Human prostate DCE-MRI data were analyzed to compare
physiological parameters calculated from the Tofts model using S(t) and C(t).
The K^{trans} and v_{e} calculated from S(t) were correlated
strongly with the values calculated from C(t). Bland–Altman
analysis showed moderate to good agreement between for the K^{trans}
and v_{e} calculated from Tofts model with S(t) and C(t).

**INTRODUCTION**

It is important to quantitatively
analyze dynamic contrast enhanced (DCE) MRI in detection and diagnosis of
cancers. The standard and extended Tofts models^{ 1,2} are the most
common pharmacokinetic models used to extract physiological parameters (K^{trans}
and v_{e}). However, use of pharmacokinetic models requires calculation
of contrast agent concentration in tissue as a function of time (C(t)) based on
T1-weighted signal intensity (S(t)). There are several ways to calculate C(t),
including: (i) using the gradient echo signal equation (non-linear model) with
pre-contrast tissue T1 values;^{3} and (ii) using the ‘reference
tissue’ model under a simple linear approximation.^{4} Theoretically
speaking, the C(t) calculated from the non-linear model is more accurate than
the ‘reference tissue’ model, but its precision is strongly influenced by the native
T1 values. Measurements of pre-contrast tissue T1 values also contribute some error
to calculations of C(t) using non-linear model.

In this study, the standard Tofts model with signal intensity was developed without using C(t). The physiological parameters calculated from the standard Tofts model using S(t) were compared with results obtained using C(t) for human prostate DCE-MRI data.

**THEORY and METHODS**

Based on the standard Tofts model of DCE-MRI, changes of C(t) in tissue following bolus contrast agent injection is given by:

$$C(t) =K^{trans}\int_{0}^{t}C_{p}(\tau)\exp\left(-(t-\tau)K^{trans}/v_e\right)d\tau,-----[1]$$

where
K^{trans} is the volume transfer constant
between blood plasma and extravascular extracellular space (EES), v_{e} is the volume of EES per unit volume of
tissue, C_{p}(t)=C_{b}(t)/(1-Hct) is the arterial input
function (AIF), C_{b}(t) is contrast agent concentration in blood, and Hct
is the hematocrit (=0.42).

Using the ‘reference tissue’
model, C(t) can be approximated from the signal intensity S(t) when a reference
tissue with a known native T1 (T1_{ref}) is available, i.e.,

$$C(t) =\frac{1}{r_{1}\cdot{T1_{ref}}}\frac{S(t)-S(0)}{S_{ref}(0)},-----[2]$$

where
r_{1} is the
longitudinal relaxivity of the contrast agent, and S(0) and S_{ref}(0) are
the tissue and reference tissue signal intensities before contrast agent
injection, respectively. Combining Eq. [1] and Eq. [2], the following formula
can be obtained:

$$S_{r}(t) =\frac{S_{b}(0)}{(1-Hct)\cdot{S(0)}}K^{trans}\int_{0}^{t}S_{rb}(\tau)\exp\left(-(t-\tau)K^{trans}/v_e\right)d\tau,-----[3]$$

where S_{b}(t) is signal intensity in blood, $$$S_{r}(t) =\frac{S(t)-S(0)}{S(0)}$$$ and $$$S_{rb}(t) =\frac{S_b(t)-S_b(0)}{S_b(0)}$$$.

Eighteen patients with
biopsy-confirmed prostate cancer were included in this IRB-approved study (mean
age=60 years old). MRI data were acquired on a Philips Achieva 3T-TX scanner.
After T2-weighted and diffusion-weighted imaging, baseline T1 mapping was
performed with variable flip angles. Subsequently, DCE 3D T1-FFE data were
acquired pre- and post-contrast media injection (0.1 mmol/kg DOTAREM; TR/TE =
3.5/1.0 ms, FOV = 180×180 mm^{2}, matrix size = 160×160, flip angle =
10°, slice thickness = 3 mm, typical number of slices = 24, SENSE factor = 3.5,
half scan factor = 0.625) for 150 dynamic scans with typical temporal
resolution of 2.2 sec/image (1.0-4.3 sec).

Regions-of-interest (ROIs) for
prostate cancer, normal tissue in different prostate zones and gluteal muscle
were drawn on T2W images and transferred to DCE images. ROI’s for blood vessels
were manually traced on the iliac artery on a slice with cancer. For each ROI,
the average S(t) was calculated, and then C(t) was calculated from the
non-linear model using the gradient echo signal equation.^{3}

Pearson’s correlation coefficient
was calculated between physiological parameters (K^{trans} and v_{e})
obtained from C(t) and S(t). Bland-Altman analysis was performed to evaluate
the agreement of two methods calculated physiological parameters. A p-value
less than 0.05 was considered statistically significant.

**RESULTS**

**DISCUSSION**

DCE-MRI
data from human prostates was used to validate the Tofts model with S(t) for
estimation of physiological parameters. Overall correlation between physiological parameters (K^{trans}
and v_{e}) calculated from C(t)
and S(t) was good. On average, the K^{trans} calculated from S(t) was
about 30% larger than calculated from C(t). The concept used here can be easily
applied to the extended Tofts model including estimation of fractional plasma
volume v_{p}, such as:

$$S_{r}(t) =\frac{S_{b}(0)}{(1-Hct)\cdot{S(0)}}K^{trans}\int_{0}^{t}S_{rb}(\tau)\exp\left(-(t-\tau)K^{trans}/v_e\right)d\tau+{\frac{S_{b}(0)}{(1-Hct)\cdot{S(0)}}}S_{rb}(t)v_p$$

The main advantage of using the Tofts model with S(t) is that it avoids error propagation associated with calculation of C(t). Implementation of signal intensity form of Tofts model in clinical practice may facilitate quick estimation of physiological parameters.

**CONCLUSION**

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

2. Tofts PS, Brix G, Buckley DL, Evelhoch JL, Henderson E, Knopp MV, Larsson HB, Lee TY, Mayr NA, Parker GJ, Port RE, Taylor J, Weisskoff RM. 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(3):223-232.

3. Dale BM, Jesberger JA, Lewin JS, Hillenbrand CM, Duerk JL. Determining and optimizing the precision of quantitative measurements of perfusion from dynamic contrast enhanced MRI. J Magn Reson Imaging 2003;18(5):575-584.

4. Medved M, Karczmar G, Yang C, Dignam J, Gajewski TF, Kindler H, Vokes E, MacEneany P, Mitchell MT, Stadler WM. Semiquantitative analysis of dynamic contrast enhanced MRI in cancer patients: Variability and changes in tumor tissue over time. J Magn Reson Imaging 2004;20(1):122-128.

Figure
1. Scatter plots (top row) of K^{trans} calculated from the Tofts model with C(t) vs.
S(t) for (a) muscle, (b) prostate tissue, and (c)
prostate cancers. The gray lines represent the linear correlations. The
corresponding Bland–Altman plots are in the bottom row (d-f). The solid line
represents the mean difference (K^{trans}-S(t) – K^{trans}-C(t))
and the dashed gray lines represent the lower and upper limits of agreement,
defined by a range of ±1.96 standard deviations (SD) (95% confidence interval)
around the mean.

Figure
2. Scatter plots (top row) of v_{e} calculated from the Tofts model with C(t) vs.
S(t) for (a) muscle, (b) prostate tissue, and (c)
prostate cancers. The gray lines represent the linear correlations. The
corresponding Bland–Altman plots are in the bottom row (d-f). The solid line
represents the mean difference (v_{e}-S(t) – v_{e}-C(t)) and
the dashed gray lines represent the lower and upper limits of agreement,
defined by a range of ±1.96 standard deviations (SD) (95% confidence interval)
around the mean.