Synopsis

Linearization of the Kety/Tofts model for DCE analysis drastically shortens computation time. We show that addition of bolus arrival time (BAT) compensation to the linearized analysis could also allow quicker acquisition times. With BAT inclusion, 3 minute sequences yield equivalent parameter estimation accuracy to 5 minute sequences without BAT compensation. The combination of shorter acquisition and real-time analysis would reduce the general time burden of DCE, which has potential implications for increased patient turnaround, and making DCE more acceptable as a tool, for example in evaluating response to therapy or in image guided therapy.

INTRODUCTION

Dynamic Contrast-Enhanced (DCE) MRI can potentially extract information about the tissue microstructure, vascularity and neovascularity, and underlying pathology. DCE data acquisition typically takes 5 minutes or more, and requires lengthy computational analysis via curve fitting to a pharmacokinetic model.

A critical component necessary for DCE analysis is the concentration of contrast agent in the blood entering the tissue of interest, which is often calculated from a region of interest covering a large artery in the acquired images. Since the average time for blood to travel once around the body is 20 sec, the arrival of contrast agent in the tissue is often delayed compared to the large artery. Ignoring the delay, also referred to as the bolus arrival time (BAT), has been shown to degrade the fit quality in DCE modeling¹, while incorporation of a BAT shift improves reproducibility of calculated model parameters². Here we describe how BAT compensation may be incorporated into a fast linearized analysis method, and demonstrate its effectiveness in short acquisitions.

A popular model of contrast agent uptake is the Kety/Tofts model ^3,4, often used for clinical data. The contrast agent concentration time course is:

$$v_e\frac{dC_e(t)}{dt}=K_{trans}[C_p(t)−C_e(t)]$$[1]

where $$$C_e(t)$$$ is the extracellular tissue concentration of contrast agent with $$$v_e$$$ being the fraction of the tissue that is extracellular, $$$C_p(t)$$$ is the blood plasma concentration of contrast agent, often referred to as the arterial input function (AIF), and $$$K_{trans}$$$ is the transfer constant between plasma and tissue extracellular space quantifying leakage between compartments. The parameters of interest, $$$K_{trans}$$$ and $$$v_e$$$, are usually found from time-consuming nonlinear fitting of tissue contrast time courses.

Linearization of the Kety/Tofts Equation has been shown to be a very fast method of extracting pharmacokinetic parameters from DCE data, particularly at low SNR^5,6, achieving a factor of 6 improvement in computation time⁵. Linearization in this context consists of creating a set of linear equations by integrating both sides of Eq. 1 as follows:

$$C_t(t)=K_{trans}\int_{0}^{t}C_p(u)du-k_{ep}\int_{0}^{t}C_t(u)du$$[2]

where $$$k_{ep}=K_{trans}/v_e$$$, and $$$C_t(t)=v_e C_e(t)$$$ is the total tissue concentration of contrast agent.

METHODS: THEORY

If the AIF bolus arrives at the tissue of interest at time $$$t_0$$$ after it is detected in the arterial blood voxel, Eq. 2 becomes:

$$C_t(t)=K_{trans}\int_{0}^{t-t_0}C_p(u)du-k_{ep}\int_{0}^{t}C_t(u)du$$[3]

The continuous representation of concentration is measured as a sparsely sampled MR signal, i.e., images are generated at discrete timepoints $$$t_j\in\{t_1,t_2,...,t_n\}$$$ where the time between frames is $$$\Delta t=t_j-t_{j-1}$$$. Discretization of Eq. 3 results in the set of linear equations (one equation for each time point $$$t_j$$$), incorporating BAT for the 2-parameter Tofts Model:

$$C_t(t_j)=K_{trans}\left [\sum_{i=1}^{j}C_p(t_i)\Delta t-t_0C_p(t_j)\right ]-k_{ep}\sum_{i=1}^{j}C_t(t_i)\Delta t$$[4]

This set of equations can be solved very quickly in each voxel for $$$k_{trans}$$$ and $$$v_e$$$ (and $$$t_0$$$) by matrix inversion.

METHODS: SIMULATIONS

Digital reference objects (DRO) built to represent the DCE signal from typical prostate tissue and different pulse sequences with a range of values for the parameters $$$K_{trans}$$$, $$$v_e$$$, and $$$t_0$$$. The parameter values assigned were:

$$$K_{trans}\in [0.05, 0.1, 0.2, 0.3, 0.4]$$$,

$$$v_e\in [0.15, 0.25, 0.35]$$$,

$$$t_0\in[0, 2, 4]$$$.

The signal in the simulated AIF pixels shown in Figure 1 was created from the mean AIF from 7 patients in a recent prostate study⁷. The simulated T₁-FLASH pulse sequence parameters that were varied were: volume acquisition time (or frame time $$$\Delta t$$$) 3 or 5 sec, and total acquisition time: 3 or 5 minutes. After generating 100 pixels for each combination of tissue parameters , random noise was added to each voxel and each time point.

RESULTS

DISCUSSION AND CONCLUSIONS

Acknowledgements

References

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