Multiple reference tissue method for contrast agent arterial input function estimation

Authors


Abstract

A precise contrast agent (CA) arterial input function (AIF) is important for accurate quantitative analysis of dynamic contrast-enhanced (DCE)-MRI. This paper proposes a method to estimate the AIF using the dynamic data from multiple reference tissues, assuming that their AIFs have the same shape, with a possible difference in bolus arrival time. By minimizing a cost function, one can simultaneously estimate the parameters and underlying AIF of the reference tissues. The method is computationally efficient and the estimated AIF is smooth and can have higher temporal resolution than the original data. Simulations suggest that this method can provide a reliable estimate of the AIF for DCE-MRI data with a moderate signal-to-noise ratio (SNR) and temporal resolution, and its performance increases significantly as the SNR and temporal resolution increase. As demonstrated by its clinical application, sufficient reference tissues can be easily obtained from normal tissues and subregions segmented from a tumor region of interest (ROI), which suggests this method can be generally applied to cancer-based DCE-MRI studies to estimate the AIF. This method is applicable to general kinetic models in DCE-MRI, as well as other CE imaging modalities. Magn Reson Med, 2007. © 2007 Wiley-Liss, Inc.

Dynamic contrast-enhanced (DCE)-MRI is emerging as a cancer imaging tool for diagnosis, monitoring of treatment effect, and evaluation of anticancer drugs (1–3). In DCE-MRI, T1-weighted signals are repeatedly acquired after the injection of a contrast agent (CA), typically a low-molecular-weight gadolinium (Gd) chelate. By evaluating tissue enhancement, which is the imaging correlate of CA transfer between plasma and the tissue extravascular extracellular space (EES), one can infer clinically useful physiologic information, including tumor perfusion and/or permeability.

Although both semiquantitative and quantitative approaches to DCE-MRI kinetic analysis have been utilized, quantitative parameters, such as the CA transfer rate between the blood plasma and the EES, Ktrans, are generally believed to the most useful as DCE-MRI endpoints, such as in Ref.4 and the DCE-MRI group consensus recommendations from the NCI CIP MR Workshop on Translational Research in Cancer—Tumor Response, Bethesda, MD, USA, Nov. 22–23, 2004 (http://imaging.cancer.gov/reportsandpublications/ReportsandPresentations/MagneticResonance). To accurately extract quantitative kinetic parameters, DCE-MRI kinetic models must be constructed. These models require a precise measurement of CA concentration in the arterial blood plasma as a function of time, Cp(t), also referred to as the arterial input function (AIF). The most commonly used kinetic model, which has been recognized by consensus as the recommended model for DCE-MRI analysis, is the Tofts model (5). It assumes that the water exchange system between the different compartments of a tissue, including intra-/extracellular space and the blood, are in the fast exchange limit (FXL) so that the CA concentration vs. time curve, Ct(t), can be calculated using a linear relationship between Ct(t) and the apparent longitudinal relaxation rate constant R1(t):

equation image(1)

where R10(≡1/T10) is the longitudinal relaxation rate constant in the absence of CA, and r1 is the relaxivity coefficient. With inclusion of the blood plasma volume, vp, Ct(t) in the Tofts model is written as:

equation image(2)

where ve is the EES volume, kepKtrans/ve represents the rate constant of CA backflux from the EES to the blood plasma, Cp(t) is the AIF. The vp term in Eq. [2] is often neglected in what has been termed the “simple Tofts model.” As the CA concentration in the tissue increases, the water exchange system between the tissue compartments will deviate from the FXL, and the simple linear relation in Eq. [1] will no longer hold (6, 7). Recently, unified kinetic models, the so-called “shutter-speed models,” have been proposed to address the nonlinear effect caused by incomplete exchange averaging (8, 9).

Obtaining accurate measurements of the AIF by directly imaging the arteries or heart is a well known challenge in DCE-MRI that is complicated by T2* effects, high CA concentration (10), partial volume effects (11), low signal-to-noise ratio (SNR) when only small arteries are in the FOV, and errors due to the rapid blood flow in the artery (12). Alternative approaches that use the measured data from the tissues themselves to inversely determine the AIF are thus being used more frequently. These approaches are often referred to as “reference tissue/region-based methods.” The initial applications of reference tissue/region-based approaches either assume that the kinetic parameters of the tissues are known or have small variability, such that literature values can be utilized (13, 14), or that the functional form of the AIF is known (15, 16).

More recently, reference tissue/region-based approaches called “pure blind identification methods” have been proposed (17–19). These approaches use multiple reference tissues/regions to estimate the AIF without assuming a specific functional form of the AIF, or the kinetic parameters of the reference issues. By assuming that the multiple reference tissues used have the same AIF, iterative estimations can be utilized to both calculate the kinetic parameters and construct an AIF that fits the observed data. However, the existing pure blind identification approaches have important limitations. For example, Riabkov and Di Bella (17) assume that the bolus arrival times for all tissues is the same, which is unrealistic and can introduce large errors (20). Furthermore, the constructed AIF is noisy and has values only at the original measurement time points, which limits its temporal resolution and the ability to adequately resolve its shape. More importantly, the computational time for their approach is prohibitive. They utilize an iterative quadratic maximum-likelihood algorithm (IQML) that involves hundreds of iterations, with each iteration requiring the solution of a large matrix-sized linear equation. Finally, the IQML algorithm can only be used in linear system models, which means the algorithm cannot be used in the shutter-speed models or any models that require nonlinear terms. The double reference tissue method (DRTM) previously proposed by Yang et al. (19) allows a difference in bolus arrival times in the two reference tissues, is computationally efficient, and gives a robust, smooth, and continuous estimate of the AIF. However, only one tissue is used to construct the AIF, and only tissues that are adequately described by a two-compartment model can be used in the algorithm. Therefore, it also cannot be used with shutter-speed models and often cannot be used in tumors, for which multicompartment models are typically required to describe CA kinetics (21).

Here we present a novel pure blind identification method termed the “multiple reference tissue method” (MRTM). As in other approaches, the MRTM utilizes two or more reference tissues to estimate the AIF, but in this case assumes only that the AIFs in the reference tissues have the same shape, with a possible difference in bolus arrival time. Importantly, the MRTM is a mathematical framework equipped with efficient algorithms that can be applied to any kinetic model, including nonlinear models, and to any CE imaging modality. The constructed AIF is smooth and has high temporal resolution. Because there is no limitation on the kinetic models that can be utilized, the tumors themselves can be used as a source of reference tissues. In fact, the inherent heterogeneity of tumors can be exploited by utilizing individual subregions with different properties as “reference tissues.” This ability avoids the necessity of including multiple normal reference tissues in the FOV.

In this paper we describe the theoretical framework of the MRTM and use Monte Carlo simulations to investigate its performance as a function of measurement noise and temporal resolution. We also provide an example of its application to clinical DCE-MRI data, including how the multiple reference tissues can be obtained from a tumor region of interest (ROI).

THEORY

Multiple Reference Tissue Method

Generally, CE kinetic models can be expressed as:

equation image(3)

where M is the model function; n {n = 1,…,N} is the index of the reference tissues; Qn(ti) is the modeled quantity for tissue n at the measurement time points ti {i = 1,…,T}; Cp(t) is the AIF, which is assumed to be the same for all the reference tissues; Ωn are the variable parameters for reference tissue n; ln is the delay in bolus arrival time in tissue n and is one element in Ωn; and Θn are all the other known parameters for tissue n, such as the tissue native T1 value in the shutter-speed model. For the Tofts model and many other common models, the modeled quantity is Ct(t); for the first-generation shutter-speed model, the modeled quantity can be the apparent T1 relaxation rate constant R1(t) (8).

The AIF is estimated in the fashion shown by the flow chart in Fig. 1. Simply put, the MRTM involves the minimization of a cost function. By minimizing the cost function, we intend to estimate the parameters of the reference tissues from the data themselves to estimate the AIF. Given a particular set of parameters {Ωn| n = 1,…,N}, an AIF is constructed to calculate the corresponding cost function. Using a standard minimization method, the cost function is minimized to obtain the final set of parameters, which are then used to construct the final AIF. The cost function is written as:

equation image(4)

where Q̃n and Q̂n respectively denote the measured and the predicted quantities, and σ(ti) is the root mean square (RMS) noise level at time point ti. Equation [4] is the foundation of the MRTM. In Eq. [4] the first term is simply the weighted sum of squared errors (SSE) of the fit to the data. The second term P({Ωn}) is called the prior cost function, which is inspired by the prior distribution in the Bayesian inference (22). If the prior cost function is nonzero, it means that we have some prior knowledge of the parameters {Ωn}, and it generally will penalize derived parameters that are significantly different from realistic values. The prior cost function can be considered as a way to apply constraints on derived parameters to prevent convergence on unreasonable results.

Figure 1.

Algorithm flow chart for the AIF estimation with the MRTM. The Ωn, Q̃n and Q̂n respectively represent the variable parameters, the measured curve, and the predicted curve for reference tissue n. A pair of braces {} represent the entire set made up by all the reference tissues n = 1, …, N. The rectangle formed by dotted lines shows all the steps to calculate the cost function for a particular set of parameters {Ωn}. The cost function is minimized by simply varying the parameters {Ωn}.

Fast Algorithms

One key step of the MRTM is to use the quantities measured from the reference tissues {Q̃n(ti)|n = 1,…,N} to construct an AIF for a given set of parameters {Ωn, n = 1,…,N}. This constructed AIF will be used to calculate the predicted quantities to calculate the cost function. We intend to construct the AIF at time grids dj {j = 1,…,D}, which are often selected to be denser than the original measurement time points ti {i = 1,…,T} so that we can increase the temporal resolution of the constructed AIF to better resolve its shape and to improve numerical accuracy. By adopting a nonparametric smoothing method, such as the self-adaptive local polynomial fitting (23) detailed in Appendix A, we can calculate the smoothed quantities at dj+ ln, Q̄n(dj + ln), and define them as some regridded curves Gn(dj):

equation image(5)

where we used a numerical discrete format for the model in Eq. [3]. The regridded curves Gn(dj) in Eq. [5] have identical bolus arrival times and have values at the dense time grids dj {j = 1,…, D}, and they will be directly used to construct the gridded AIF Cp(dj) {j = 1,…,D}. We notice that there is one basic but very important feature in those general models as expressed by Eqs. [3] and [5]: the modeled quantities at time t depend only on the values of the AIF before t. Inspired by this feature, we construct one Cp(dj) at a time in the sequential order. Starting from j = 1, we construct Cp(dj). Then we proceed to construct the next one Cp(dj+1) with all the Cp(dk) {kj} already constructed earlier. At each time grid dj, Cp(dj) is constructed by minimizing the following function:

equation image(6)

where we use Ĝn(dj) to denote the predicted value of Gn(dj). S is a subset of the set {n| n = 1, …, N}, which means that we have the option to use only a selected subset of the reference tissues instead of all of them to construct the AIF. Unless mentioned otherwise, in applications we often select S to be all the reference tissues. Ĝn(dj) is dependent on the unknownCp(dj) and those already known Cp(dk) {kj1} that have been constructed before Cp(dj). The solution is

equation image(7)

With this so-called “sequential AIF construction algorithm,” the construction of Cp(dj) at each grid time dj is simply a 1D root-finding problem, so the computational time is dramatically reduced. Equation [7] for the construction of the AIF is the first and most important step in the MRTM (Fig. 1). For linear system models, such as Tofts model, adiabatic model (24), and models involving three or more compartments (such as those commonly used in PET (25)), Cp(dj) can be written as a closed-form analytical equation so that no iterations are needed for its calculation. For nonlinear models, such as the shutter-speed model, only a very small number of iterations are needed to calculate Cp(dj) at each grid time dj using Eq. [7].

To calculate the predicted quantities, we recognize that many commonly used kinetic models have convolution terms with an exponential kernel similar to that in the Tofts model. For this particular class of kernel, the convolution calculation can be performed more quickly by using the following numerical algorithm rather than using the fast Fourier transform (FFT) algorithm:

equation image(8)

According to Eq. [8], we can calculate f(dj), defined as the integral over time interval [0, dj], by making use of f(dj–1). If we always save f(dj–1), the next integral f(dj) can be calculated quickly. With this “fast sequential convolution method,” the computational time to calculate the convolutions at all the time grids dj {j = 1,…,D} is proportional to D. If the FFT algorithm is used, the computational time will be proportional to 2(2D)log2(2D) because zero-padding and two FFTs have to be used (26). For example, when the number of time grids D = 512, the FFT method will be 40 times slower than the fast sequential convolution method. In addition, the time grids dj {j = 1,…, D} do not have to be equally spaced with this algorithm. For demonstration purposes, in Eq. [8] a Euler rule is used for the Cp(τ) term in the numerical integral over time interval [dj-1, dj]. A higher-order approximation with better numerical accuracy, such as a trapezoidal rule, could also be used (26), which is another advantage of this algorithm.

Freely Adjustable Parameters in the MRTM

Based on Eq. [4], the MRTM basically fits all the data points in {Q̃n(tj)|n = 1,…,N} simultaneously, or more accurately fits a vector with T*N elements composed by the T*N data points from all the reference tissues. The local polynomial smoothing in Appendix A is a linear smoother (23) so that regridded data Gn(dj) are linear combination of the measured data Q̃n(ti){i = 1,…,T}. Hence, based on Eq. [7], the constructed AIF depends only on the adjustable parameters {Ωn| n = 1,…,N} and the measured curves {Q̃n(tj)|n = 1,…,N}. This shows that there are no freely adjustable parameters from the AIF construction. The number of freely adjustable parameters in the MRTM are no greater than the number of adjustable parameters from all the reference tissues, {Ωn| n = 1,…,N}. In fact, the MRTM belongs to a class of semiparametric statistical methods. Although the parameters from the AIF construction are only intermediate or dependent parameters, they will slightly decrease the degrees of freedom of the model and expectation of the weighted SSE, as explained in Appendix B.

Applications to Tofts Model

The MRTM applies to general models. Here, as an example, we show its manifestation under the Tofts model. For the Tofts model, based on Eqs. [7] and [8] the AIF can be sequentially constructed with the following explicit form:

equation image(9)

where wn = vmath image + {1−exp[−kmath image(djdj−1)]} · vmath image. By using the algorithm in Eq. [8] to calculate the convolution, the number of computations to construct the entire AIF curve is only D, the number of time grids.

Scaling of the AIF

For all linear system models, the AIF can be estimated only within a scale factor by the MRTM. For simplicity of illustration, we will use the Tofts model as an example. If Cp(t) is a solution of the AIF at which the estimated kinetic parameters are {ven,vpn,kepn}, according to Eq. [9] ηCp(t) is also a solution with η being any coefficient and the estimated kinetic parameters at the new AIF will be {ven/η,vpn/η,kepn}. This is very similar to the situation in other related AIF estimation methods using reference tissues (18, 19). To fix the amplitude of the AIF, we can use the literature value of the ve from one normal reference tissue, such as the liver, or use measurements of tracer concentration in blood during the washout phase, which can be performed fairly accurately since the CA concentration in blood is not very high in the washout phase (19). A third method to fix the scale is to simply match the washout part of the estimated AIF with a fixed AIF, such as the Weinmann biexponential AIF (27), or a population-averaged AIF (28).

MATERIALS AND METHODS

Monte Carlo Simulations

Goals

Using Monte Carlo simulations, we investigated the effect of measurement noise and temporal resolution on the accuracy of the MRTM. The main purpose is to determine whether the MRTM works in principle. More specifically, we determine whether the minimization process in Fig. 1 can accurately estimate the reference tissues parameters and the AIF if the right kinetic model of the reference tissues is provided to the MRTM algorithm. In the simulations, the Tofts model with vp contribution is assumed for the reference tissues. Many simulation details are similar to those in Ref.19 and summarized below.

Preparation of the Simulated “True” Curves

The local AIF of the reference tissues is prepared by adding dispersion to an assumed aorta AIF:

equation image(10)

The transport function h(t) is chosen as a Gamma-variate function:

equation image(11)

where to is the bolus arrival time, Γ denotes the complete Gamma function, and we select the order α = 4 and the scale β = 0.03 min for all tissues. The aorta AIF is chosen as:

equation image(12)

where the functional form is adopted from a population-averaged AIF (28) with two parameters modified to get higher first-pass peak and slower washout rate. The aorta AIF and the local tissue AIF is shown in Fig. 2.

Figure 2.

The “true” AIF (dashed line) in the simulated tissues is prepared by convoluting an assumed aorta AIF (solid line) by a transport function h(t), which is selected to be a fourth-order Gamma-variate function with scale β = 0.03 min. Insert: the shape of h(t) when β = 0.03 [min] and to = 0.0 [min].

The concentration-vs.-time curves of four reference tissues are simulated using the Tofts model in Eq. [2] over a time period of 8 min. The curves are resampled at 2–12-s temporal resolution in different simulations. Other simulation parameters are listed in Table 1. The kinetic parameters in the first three tissues are similar to those from tumor voxels in DCE-MRI studies (29), and the parameters of tissue 4 are similar to those seen in skeletal muscle (30).

Table 1. Simulation Parameters for the Four Reference Tissues*
Tissueσto (min)vpvekep(min−1)
  • *

    The noise level σ for each tissue is selected to be time-independent. The absolute noise level in Tissue 4 (simulated muscle), σm, varies from 0.001 to 0.004 in different simulations. The temporal resolution of the simulated data ranges from 2 to 12 s.

14σm0.50.0600.301.2
24σm0.50.0150.301.0
34σm0.50.0050.300.5
4σm0.60.0020.120.35

Simulation Procedures

In each simulation realization, Gaussian noise is added to the “true” curves to produce noisy simulated data. The noise level for each tissue n is selected to be a time-independent constant σn. The simulated noisy data are then used to estimate the AIF as well as the kinetic parameters. The AIF is constructed to have 2-s temporal resolution. The estimated AIF is scaled by matching its washout phase with that of the simulated aorta AIF. In each simulation, 1000 realizations are performed and the mean and 2 SDs of the estimated parameters are calculated.

Since tissue 4 is simulated skeletal muscle, denoting the index of muscle as m, we select the prior cost function in Eq. [4] as:

equation image(13)

where I(x) represents the unit step function, equal to 0 for x < 0 and 1 for x ≥ 0. This kind of prior cost function is commonly used by us in clinical applications of the MRTM when one reference tissue is muscle. It is based on the prior knowledge that the estimated vp in muscle in DCE-MRI studies using a low-molecular-weight CA is generally very close to zero. The prior cost function is equal to zero when vp is close to zero (<0.0025), but it will penalize large vp to ensure that the final estimated vp will always be close to zero.

Applications to Clinical DCE-MRI Data

Obtaining the Multiple Reference Tissues

The reference tissues used in the MRTM should not all have the same or very similar uptake and washout patterns; otherwise, the errors in the estimated parameters will be large due to a singular or nearly-singular matrix (17, 19). To improve the accuracy, the reference tissues used should have different uptake and washout patterns. In clinical DCE-MRI studies of cancers, normal tissues (e.g., skeletal muscles) can be used as reference tissues. Multiple reference tissues can also be obtained from a single tumor ROI based on the heterogeneity of tumor voxels. Here we use Ct(t)-based models, such as the Tofts model, as an example. Those models select the concentration-vs.-time curves, Ct(t), as the modeled quantity. For these models, by using an automated cluster analysis method (31), voxels in the tumor ROI are first segmented into subsets such that all voxels within each subset have similar Ct(t) curves but are not necessarily contiguous. The tumor heterogeneity leads to many subsets having a significantly different mean Ct(t) so that each subset can be considered as a reference tissue. Heterogeneous tumors as well as normal tissues can be used as the reference tissues.

Clinical DCE-MRI Data

We applied the MRTM to metastatic renal cancer patients undergoing DCE-MRI in the pelvic region during an ongoing investigation of the antiangiogenic drug Sorafenib. The study was approved by the institutional review board and all patients provided written informed consent. T1-weighted images of two slices were acquired on a 1.5-Tesla SIGNA™ scanner (General Electric Medical Systems, Waukesha, WI, USA) with 2-s temporal resolution for about 7 min. Using a power injector (Medrad, Indianola, PA, USA), 0.1 mmol/kg gadodiamide (Ominscan; GE Healthcare, Chalfont St. Giles, UK) was injected over 10 s followed by a 20-ml saline flush. A 2D fast spoiled gradient-echo (SPGR) pulse sequence was used with TR/TE = 7.8/1.7 ms, flip angle = 60°, matrix size = 256 × 128, typical FOV = 30–35 cm, slice thickness = 8 mm, and slice spacing = 1 mm.

Data Analysis

All data analysis was performed using in-house-developed software written in IDL (RSI, Boulder, CO, USA) run at a personal computer with a 2.8G Hz Pentium 4 CPU (Intel, Santa Clara, CA, USA). For each patient, ROIs in the gluteus muscle and the tumor were selected by a radiologist. Effort was made to manually exclude necrotic centers of the tumor from the ROI. The Ct(t) curves were calculated on a voxel-by-voxel basis using established methods (32). The muscle ROI and several subsets segmented from the tumor ROI were used as the multiple reference tissues. All of the tumor subsets selected had moderate to fast initial contrast uptake to reduce the possibility that a necrotic region would be inadvertently included. A prior cost function as in Eq. [13] was used for muscle. The AIF was scaled by assuming that the ve value of the skeletal muscle to be 0.12, which is consistent with the literature (30). This can be done by fixing the ve value of the muscle tissue as 0.12 in the minimization procedure of the MRTM. With the AIF estimated by the MRTM, kinetic parameters of the tumor are then calculated voxel by voxel.

RESULTS

Monte Carlo Simulations

Table 2 summarizes the statistics of estimated parameters from the 1000 realizations of the simulation using a moderate noise level (σm = 0.03) and high temporal resolution (2 s). With this set of simulation parameters, the SNR of the simulated data is about 25, which is considered moderate with typical clinical DCE-MRI data. Table 2 shows that compared to the “true” tissue parameters in Table 1, the biases and random errors are all small. For example, in the estimated kep of simulated tissue 1, the relative bias is only about 0.5% and the relative random error (2 SDs) is only about 12%. For simulated tissue 4 (muscle), which has slower uptake, the relative bias and the random error in the estimated kep are even smaller (0.2% and 7%, respectively). The estimated ve in all the reference tissues have small relative bias (<1.3%) and relative random errors (<5%).

Table 2. The Mean and 2 SD of the Fitted Parameters From the 1000 Realizations in the Simulation*
Tissueto (min)vpvekep (min−1)
  • *

    Noise level σm = 0.003 and 2-s temporal resolution. The “true” parameters of the four tissues are listed in Table 1.

  • a

    Fixed to be 0.5 min as a reference time.

10.5a0.056 ± 0.0140.304 ± 0.0141.194 ± 0.140
20.503 ± 0.0200.012 ± 0.0110.303 ± 0.0120.999 ± 0.098
30.506 ± 0.0340.0042 ± 0.00660.3009 ± 0.00690.499 ± 0.038
40.601 ± 0.0310.0015 ± 0.00180.1205 ± 0.00220.349 ± 0.025

To illustrate the performance the MRTM, out of the 1000 realizations we will show one representative example. Traditionally, the performance of a fit is measured by its weighted average squared error (WASE) (33), which is defined as

equation image(14)

where Ctn(ti) is the simulated “true” curve of tissue n, Ĉtn(ti) is the curve estimated by the MRTM, and the weight is selected to be the inverse of the square of the noise level, σn. Figure 3 shows the realization with the median WASE. An excellent fit of the concentration-vs.-time curves was achieved in all of the tissues, and the estimated curves matched the “true curves” very well. Figure 4a shows the estimated AIF for this realization, which closely resembles the “true” AIF. This simulation suggests that the MRTM is able to provide a reliable estimate of the AIF as well as the parameters for data with high temporal resolution and moderate SNR.

Figure 3.

Performance of the MRTM illustrated by the realization with the median performance of the fit from the simulation using noise level σm = 0.003 and high temporal resolution (2 s). The SNR of the simulated noisy curves (open circles) is approximately 25. In all of the simulated tissues, the curves simultaneously fitted by the MRTM (thick dotted lines) match the “true” curves (solid lines) excellently. The estimated AIF for this simulated data is shown in Fig. 4a.

Figure 4.

Comparison of the “true” AIF (solid black lines) with the AIF estimated by the MRTM (dashed red lines) in the realizations with median performance in fit from the simulations using noise level σm = 0.003 and different temporal resolutions: (a) 2 s, (b) 8 s, (c) 12 s. The estimated AIF in a was derived from the simulated noisy data shown in Fig. 3. Overall, the AIFs estimated by the MRTM match the true AIF fairly well.

To determine whether the average performance of the MRTM changes as a function of the noise level and temporal resolution, we performed additional simulations, each with 1000 realizations. The computational time for each realization is short (on average ranging from about 5 s to 1 s), which decreases as the number of data points in the simulated data decreases. The statistics of the estimated kep of simulated tissue 1 are listed in Table 3 as a function of the temporal resolution and noise level. The relative biases of the fitted parameter are all small (<3%), and the bias increases slightly as the noise level increases and the temporal resolution decreases. The impact of the noise level and temporal resolution on the random errors of the fitted parameters—measured by 2 SDs—is more significant. As a very good approximation, the random errors of the fitted parameters are linearly proportional to the noise level of the data. The random errors also increase dramatically as the temporal resolution of the data decreases, and approximately within a square root law. For example, when the temporal resolution decreases from 2 to 8 s, the random error nearly doubles.

Table 3. The Mean and 2 SD of the Fitted kep in Tissue 1 From Simulations Using Different Noise Level and Different Temporal Resolution*
Resolutionσm = 0.001σm = 0.002σm = 0.003σm = 0.004
  • *

    Each simulation has 1000 realizations. The “true” kep in Tissue 1 is 1.2 min−1.

2 s1.198 ± 0.0461.196 ± 0.0931.194 ± 0.1401.193 ± 0.197
4 s1.203 ± 0.0721.201 ± 0.1371.201 ± 0.2041.197 ± 0.271
8 s1.191 ± 0.1021.185 ± 0.2051.178 ± 0.3111.171 ± 0.421
12 s1.192 ± 0.1161.183 ± 0.2281.176 ± 0.3441.167 ± 0.460

To demonstrate the effect of temporal resolution on the estimated AIF, in Fig. 4 we show the estimated AIFs from the realizations with median performance in fit—evaluated by WASE—from the three simulations using a moderate noise level σm = 0.003 and three different temporal resolutions. When the temporal resolution was high (2 s), the estimated AIF matched well with the “true” AIF in the first pass as well as the second pass (Fig. 4a). When the temporal resolution was moderate (8 s), the estimated AIF reproduced the first pass well, but the details in the second pass were lost. When the temporal resolution was even lower (12 s), the first pass in the estimated AIF was artificially broadened, but the overall agreement with the “true” AIF was still fair.

To illustrate the utility of a prior function such as that in Eq. [13], we repeated the simulations after setting it to zero. There was little effect on the simulated data with high temporal resolution and/or SNR, but peculiar solutions appeared with the simulation using σm = 0.004 and a 12-s temporal resolution. Specifically, in 3% of the realizations the estimated AIF had a very low first-pass peak, and the estimated vp in tissue 1 was more than two times larger than the true value (data not shown).

Applications to the Renal Cancer DCE-MRI Data

The MRTM was applied to one representative clinical example shown in Fig. 5. In this scan the arteries in the FOV could not be adequately resolved and there was significant pulsatile artifact in the artery, precluding direct measurement of the AIF. Figure 5a shows Ct(t) curves of the reference tissues used. Tissue 4 is muscle, and the first three curves are from three subsets of tumor voxels with high SNR. The AIF estimated by the MRTM shown in Fig. 5b shows the first and second passes, as well as the washout phase. It closely resembles AIFs published in the literature (34–36), and gives an excellent fit to the measured curves as shown in Fig. 5a. Figure 5c–e show maps of Ktrans, ve, and vp in the tumor ROI calculated using the estimated AIF. The maps have very high SNR and the heterogeneity of the tumor is clearly demonstrated. It should be noted that many tumor voxels have a large vp value. This corresponds to a large spike seen in the early uptake phase of Ct(t) curves, which we have commonly observed in data from the ongoing renal cell carcinoma study. This large blood plasma value in tumor is likely caused by vascular malformations, which are common in this type of cancer.

Figure 5.

Application of the MRTM to a renal cancer patient. a:Ct(t) curves (open circles) of the reference tissues. The first three curves were segmented from the tumor ROI and tissue 4 was gluteus muscle (ROI highlighted in c). The MRTM gave an excellent fit (solid lines) to the data. b: AIF estimated by the MRTM. c: Transfer constant Ktrans [min–1]. d: EES volume ve. e: Blood plasma volume vp. The estimated tumor parameter maps are superposed on the precontrast T1-weighted image.

As a comparison, we applied the previously described DRTM algorithm (19) using tissues 1 (tumor) and 4 (muscle) shown in Fig. 5a. The obtained AIF had a negative peak following a positive peak during the initial CA uptake (data not shown), which was unrealistic. The AIF also gave much poorer fit to the data—the SSE nearly doubled.

DISCUSSION AND CONCLUSIONS

In this paper we have presented an MRTM that is able to estimate the AIF from DCE-MRI data by utilizing two or more reference tissues. By minimizing a cost function, the MRTM is able to simultaneously estimate the parameters of the reference tissues and their common AIF. The estimated AIF is smooth and can be constructed to have higher temporal resolution than the original data so that its shape can be adequately resolved. Importantly, the algorithms are stable and efficient, taking only seconds to compute.

Monte Carlo simulations show that the MRTM is able to provide a reliable estimate of the parameters and the AIF for data with moderate SNR and high temporal resolution. On average, the estimated parameters have small random errors and nearly zero biases. The performance of the MRTM increases approximately linearly with the SNR of the data. Its performance decreases as the temporal resolution of the data decreases, although in a less significant way with the errors in the estimated parameters changing approximately within a square root law. The simulations imply that the MRTM can be reliably applied to data with temporal resolutions up to 12 s, if compensated for by high SNR, when the AIFs have the typical first-pass peak width seen in clinical DCE-MRI studies. This may be particularly useful for data acquired with 3D slab excitation, in which the SNR of the data is typically proportional to the square root of the slice number (slab volume), and the temporal resolution is inversely proportional to the slice number. Therefore, the performance of the MRTM will not vary dramatically with moderate slice number increases since the impacts from changes in the SNR and the temporal resolution will approximately cancel each other. It should be emphasized that because the number of effective adjustable parameters is relatively large in the MRTM, the method may be unreliable when the data are very sparse and hence the degrees of freedom of the fit (Appendix B) are comparable to the number of adjustable parameters.

The small bias observed in the simulations may be due to the relatively high temporal resolution of the constructed AIF. It is thus possible to even further increase the numerical accuracy by further increasing the constructed AIF temporal resolution. A higher-order approximation can also be used for the fast algorithms in Eqs. [7]–[9] to further improve numerical accuracy. However, such changes may also affect the stability and smoothness of the constructed AIF, and thus their ultimate utility will need to be carefully investigated.

We show that the heterogeneity of tumors can be utilized to obtain multiple “reference tissues” from the tumor ROI itself. By grouping tumor voxels with similar properties together to be used as one “reference tissue,” one can significantly increase the SNR of the data to obtain more accurate estimates of the AIF. In addition, normal tissue (such as skeletal muscle, which is ubiquitous) can also be used as one of the reference tissues. Hence in DCE-MRI studies of cancer, sufficient reference tissues are always available for application of the MRTM. A major benefit of using the tumor itself as the reference tissue is that the estimated AIF may better reflect the local AIF. Previous simulations have suggested that errors as large as 70% can be introduced by using a remote artery AIF to determine kinetic parameters (20).

Because the AIFs of the reference tissues used in the MRTM are assumed to have a similar shape, the muscle and tumor used should be in close anatomical proximity. In addition, voxels from necrotic regions of the tumor should be excluded as they may have a distorted input function. Currently, the necrotic center of the tumor is manually excluded based on visual inspection. Necrotic regions, however, enhance in large part by passive diffusion of CA, and thus enhancement is much slower and delayed. This feature could possibly be used for automated or semiautomated exclusion of necrotic voxels.

The assumption that the AIFs of the reference tissues have exactly the same shape may be invalid in real clinical datasets because they may undergo different degrees of dispersion as they travel from the aorta to different capillary beds. Previous simulations with the DRTM by Yang et al. (19) suggested that the resulting biases are small even when the difference in dispersion is as great as allowed by normal physiology. In addition, because the reference tissues used in clinical applications of the MRTM are generally from the same tumor and nearby muscle, the difference in AIF dispersion should be negligible. We also did not address the effect of motion in our simulations, which is another well known challenge for the quantitative analysis of DCE-MRI images. However, several image registration algorithms have been developed and some are available commercially.

The MRTM can be generally applied to different kinetic models and other imaging modalities, such as DCE-CT and PET. Recently, incomplete exchange averaging effects were observed in rodent DCE-MRI studies (6). For cases in which the tissue CA concentration is high, shutter-speed models that account for the incomplete exchange averaging effect may be a better approximation than the standard Tofts model. By applying the shutter-speed models to the MRTM, one may be able to address the incomplete exchange averaging effect and the AIF problem in the DCE-MRI simultaneously. When the shutter-speed models are used, the computation time will be several times longer than it would be with the Tofts model, mainly due to the several iterations needed to construct the AIF under Eq. [7]. Since the Tofts model takes only several seconds according to our simulations, it is estimated that the shutter-speed models would take less than a minute. When a more complex kinetic model is used, the trade-off is that the introduction of additional parameters will often result in larger errors in the parameter estimates. Therefore, when applying the MRTM, it may sometimes be more accurate to choose reference tissues, including subregions of a tumor ROI that fit simpler models in which redundant parameters are eliminated, than to apply very complicated models, such as the full version of the second-generation shutter-speed model (9). If complicated models are used in data with low SNR, proper selection of the prior cost function in Eq. [4] could possibly stabilize the solutions.

The simulation studies described here did not evaluate the bias of the MRTM when one (true) kinetic model was used to simulate the data but another (wrong) kinetic model was used to solve the problem. The Tofts model evaluated here may introduce systematic errors due to its FXL assumption (9). It will be interesting to compare the Tofts model with other models, and to apply statistics-based model selection criteria to evaluate the behavior of the MRTM under different kinetic models. Those important issues will be investigated in follow-up studies.

Monte Carlo simulations and preliminary clinical application of the MRTM suggest that it is feasible to apply this method to cancer-based DCE-MRI studies to estimate a regional AIF. Future studies will focus on development of computer-aided selection of tumor subregions to act as reference tissues, and clinical validation of the method. This will include evaluation of intra-/interpatient and interobserver variability, as well as the comparison of estimated parameters and AIF with the results from other well-established methods. If the promise of a more accurate and reproducible AIF suggested by our simulations is realized, then the potential for obtaining more reproducible quantitative DCE-MRI parameters to assess tumor perfusion and permeability can also be realized. This will greatly reduce inter- and intrapatient variability in DCE-MRI studies, and will likely further optimize and standardize the approach for DCE-MRI-based cancer diagnostics and therapeutic monitoring.

APPENDIX A

Simply put, a pth-order local polynomial fitting (LPF) estimates the value of a smooth function at x by locally fitting the data that fall within the interval [x − h, x + h] with a weighted pth-order polynomial regression (23). The data farther away from x are assigned less weight by using a kernel function and the bandwidth parameter h, which controls how many local data points to use for the local regression. When h is small enough, the LPF is simply interpolation; when h is infinite, the LPF becomes the commonly used polynomial fit. In the application of the MRTM, for DCE-MRI data we generally use a third-order LPF and an Epanechnikov kernel function, which have a theoretical foundation (23) and work excellently in practice as well.

We use a variable-bandwidth selection scheme that selects the bandwidth h as a function of x (23) according to the following procedure: First, we split the entire measurement time interval into subintervals, say Ik, according the rule below. The time interval before the contrast uptake is selected as one subinterval, the first 2 min right after the contrast uptake is evenly split into two subintervals, and the rest are evenly divided approximately 2 min a part. Then for each subinterval Ik, a bandwidth ĥk that minimizes the integrated residual squares criterion (IRSC) of Ik is obtained to get a bandwidth step function. Finally, the bandwidth step function is averaged locally using an Epanechnikov kernel and 0.5-min bandwidth to get the smooth bandwidth function ĥ(x).

Because we want to select the smooth bandwidth function to be the same for the data from all the reference tissues, to search for the common ĥk for all the reference tissues we use a composite IRSC that is the weighted sum of the individual IRSCs from different reference tissues:

equation image(A.1)

where IRSC(h)n is calculated from the data in reference tissue n using Eq. [4.18] in Ref.23, and σn is the noise level in tissue n.

The performance of the residual squares criterion (RSC)-based bandwidth selector can sometimes be unsatisfactory when the density of the data is low (23). When the temporal resolution of the data is low (>5 s), instead of using the RSC selector we simply reset the ĥk in the first subinterval after the contrast uptake to be the maximum bandwidth that interpolates the data. Our experience shows that this maneuver gives stable results in the application of the MRTM. Generally we use the automatic bandwidth selection scheme described above. It is possible to manually select the bandwidth function, for example, by manually modifying the ĥk recommended by the RSC. The manual manipulation of ĥk might be useful in some situations, and further exploration is warranted.

APPENDIX B

The MRTM is a semiparametric method. Many concepts from traditional parametric methods cannot be directly used in semiparametric methods. For semiparametric models, various definitions for the degrees of freedom can be found in Refs.37 and38. The expectation of the weighted SSE (alternatively called the residual sum of squares (RSS)) is equal to the degrees of freedom for errors (37). In the MRTM, the total number of data points is T*N, and we can break down the degrees of freedom as

equation image(B.1)

where nintermediate denotes the number of intermediate or dependent parameters from the AIF construction step, and nadjustable denotes the number of freely adjustable parameters that are solely contributed by the variable parameters of the reference tissues. One method to calculate the generalized degrees of freedom in the semiparametric method is described in Ref.38. Calculation shows that the number of intermediate AIF parameters is small in the MRTM. For example, nintermediate of the examples in Figs. 3, 4a–c, and 5 are respectively 18.41, 18.41, 14.77, 12.46, and 19.18. The degrees of freedom in semiparametric methods are generally not an integer. It should be noted that the number of intermediate AIF parameters in the MRTM is very small and does not increase much as the temporal resolution of the data increases. This is because the constructed AIF in the MRTM is smooth and only weakly subject to the effect of measurement noises.

For the pure blind identification method proposed by Riabkov and Di Bella (17), the number of intermediate AIF parameters is equal to T, the number of measurement time points. It is large and increases dramatically as the temporal resolution of the data increases. This can result in a noisy estimated AIF and sometimes unstable results.

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