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Bayesian Methods for Pharmacokinetic Models in Dynamic Contrast-Enhanced Magnetic Resonance Imaging Volker J. Schmid, Brandon Whitcher, Anwar R. Padhani, N. Jane Taylor, and Guang-Zhong Yang*
Abstract—This paper proposes a new method for estimating kinetic parameters of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) based on adaptive Gaussian Markov random fields. Kinetic parameter estimates using neighboring voxels reduce the observed variability in local tumor regions while preserving sharp transitions between heterogeneous tissue boundaries. Asymptotic results for standard errors from likelihoodbased nonlinear regression are compared with those derived from the posterior distribution using Bayesian estimation with and without neighborhood information. Application of the method to the analysis of breast tumors based on kinetic parameters has shown that the use of Bayesian analysis combined with adaptive Gaussian Markov random fields provides improved convergence behavior and more consistent morphological and functional statistics. Index Terms—Adaptive smoothing, Bayesian hierarchical modeling, dynamic contrast-enhanced magnetic resonance imaging, Gaussian Markov random fields, oncology, pharmacokinetic models.
I. INTRODUCTION UANTITATIVE analysis of dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) is usually achieved by applying pharmacokinetic (PK) models to signal intensity changes, or a nonlinear transformation, after contrast injection. Gadopentate dimeglumine (Gadolinium diethylenetriaminepentaacetic acid, Gd-DTPA) is a commonly used contrast agent, which has a low molecular weight and is injected after several baseline scans have been acquired. relaxation Using -weighted sequences, the reduction in time caused by the contrast agent is the dominant enhancement observed [1]. -weighted kinetic curves typically have three phases: the upslope, maximum enhancement, and washout [2]. Quantitative PK parameters can be estimated by fitting a nonlinear function—the solution of a system of linear differential equations [3]–[6]—to the observed contrast agent concentration as a function of time.
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Manuscript received May 22, 2006; revised August 7, 2006. The work of V. J. Schmid was supported by a research grant from GlaxoSmithKline. Asterisk indicates corresponding author. V. J. Schmid is with the Institute of Biomedical Engineering, Imperial College, London SW7 2AZ, U.K. B. Whitcher is with the Translational Medicine & Genetics Group, GlaxoSmithKline, Greenford UB6 0HE, U.K. A. R. Padhani and N. J. Taylor are with Paul Strickland Scanner Centre, Mount Vernon Hospital, Northwood HA6 2RN, U.K. G.-Z. Yang* is with the Institute of Biomedical Engineering, Imperial College, London SW7 2AZ, U.K. (e-mail
[email protected]). Digital Object Identifier 10.1109/TMI.2006.884210
From a statistical point of view, quantitative methods are based on the theory of nonlinear regression [7]. Nonlinear models are typically difficult to estimate due to convergence issues and consistency problems by specifying starting values for the optimization. The following quotation by Padhani [8] points out the potential difficulty of fitting nonlinear regression models in DCE-MRI: “Quantification techniques often rely on the application of a mathematical model to the data acquired. Experience shows that such models may not fit the data observed. The causes of such modeling failures are complex and often not well understood. We do not have models that fit all data and more sophisticated models that provide insights into tissue compartment behavior are needed.” In this paper, we attempt to estimate parameters in pharmacokinetic models using nonlinear regression not only within a traditional (likelihood) framework, but also with Bayesian inference as an alternative to alleviate some of the concerns stated above and provide a richer summary of the results. In likelihoodbased nonlinear regression theory, parameters are known to be asymptotically normal [7], [9], whereas the Bayesian framework produces a joint a posteriori distribution of the interesting parameters according to Bayes’ theorem [10]. Thus, asymptotic standard errors can be derived from likelihood theory, and Bayesian inference allows standard errors and confidence intervals to be computed directly from the posterior distribution, providing information about the reliability of the estimates directly from the data. The quantitative analysis of DCE-MRI data is further enhanced in this paper by incorporating spatial information to the Bayesian estimation procedure. The use of spatio-temporal models has previously been applied to functional MRI [11]–[16], but to our knowledge the quantitative analysis of DCE-MRI has only been performed on an individual voxel basis or region-of-interest in order to improve the derived signal-to-noise (SNR) ratio. To obtain an improved distribution of kinetic parameter estimates across the image, contextual information from neighboring voxels are combined in this paper by incorporating a Gaussian Markov random field (GMRF) prior [17] on the kinetic parameters into the model. It is important to note at this stage that the inclusion of a spatial prior distribution produces not clusters of voxels, but unique PK parameter estimates at each voxel, which are a weighted combination of several estimates based on the observed contrast agent concentration. In this study the neighborhood structure is
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fixed and identical for all voxels, consisting of the voxel’s four adjacent neighbors in two dimensions or six adjacent neighbors in three dimensions. The weights associated with each neighbor voxel are allowed to change with the location of the center voxel in order to accommodate for the heterogeneous nature of the tissue. This is a data-adaptive approach where smoothing is only enabled in homogeneous tissue regions such that sharp boundaries between distinctive regions are maintained. The use of a compartmental model means that each kinetic parameter has a direct relationship with a key biological process of interest; e.g., transfer constants, rate constants, etc. This has a distinct advantage over the semi-quantitative or non-modelbased approach [18] where the signal intensity change is analyzed. With a semi-quantitative approach, despite the lack of direct tissue or vascular information, descriptive statistics of the kinetic curve, onset time (time from injection to the arrival of contrast agent in the tissue), gradient of the upslope of enhancement curves and washout gradient, maximum signal intensity and initial area under the time-signal curve (IAUC) or initial area under the Gadolinium concentration curve (IAUGC) are used to describe the data. Semi-quantitative parameters are relatively easy to calculate, but they can be dependent on scanner settings, which can alter baseline values. The biological relevance of semi-quantitative parameters is unclear; e.g., WalkerSamuel et al. [19] point out that IAUGC is related to a mix of biological meaningful kinetic parameters. However, several groups have shown good results using semi-quantitative parameters in clinicial applications [20]–[22]. In DCE-MRI, quantitative analysis is often performed on a whole tumor region of interest (ROI) [23], where the contrast agent concentration time curve for all voxels in the tumor is used to estimate a single set of kinetic parameters (like —volume transfer constant between blood plasma and —rate extracellular extravascular space per minute—and constant between extracellular extravascular space and blood plasma per minute [6]) for each patient study. Although this suppresses inter-voxel estimation errors, it also significantly reduces the discriminative power of the technique in assessing heterogeneous lesions. An alternative is to use only “hot spot” values. voxels [24]; i.e., those voxels with the highest In both cases, tumor ROIs or hot spots are identified folmaps, introducing potential lowing visual inspection of inter-observer variability. Our proposed method uses information from neighboring voxels to reduce estimation errors, but it produces a distinct parameter estimate for each voxel. Recent studies in DCE-MRI have concentrated on the use of voxel-based analysis of PK parameters to generate histogram plots or median kinetic values [24]. Histograms are one possibility to compare pre- and post-treatment scans and evaluate treatment responses. A recent study by O’Connor et al. [25] used principal component analysis on a kernel density estimator parameters for improved assessment treatment reof sponse. However, the method still involves manual delineation of tumor regions. We propose to use the information gained from either non-linear regression theory or Bayesian inference to mask the tumor in a semi-automatic algorithm. Potential clinical applications of DCE-MRI include screening for malignant disease, lesion characterization and monitoring
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response to treatment, and assessment of residual disease following therapy [23]. Recent applications also include prognostication, pharmacodynamic assessments of anti-vascular anticancer drugs [20], and predicting the efficacy of treatment. An overview of clinical use of MRI for breast cancer can be found in [26]. Section II introduces the basic pharmacokinetic model, including the arterial input function (AIF), data acquisition protocol, and the three statistical frameworks for parameter estimation: likelihood, Bayesian, and adaptive Bayesian Markov random field approaches. These methods are compared and contrasted on data from a breast cancer study in Section III using the distribution of estimated parameters in some cases and also the estimation of errors in model fitting. II. METHODS A. Compartment Models A standard compartmental model [27] is used in this study to describe the arterial influx of Gd-DTPA into extracellular extravascular space (EES) and its venous efflux, and we follow the standardized quantities and symbols in [6]. The time series of Gd-DTPA concentration in the tissue is modeled by (1) where is the observed Gd-DTPA concentration in the is the tracer concentration in plasma, tissue at time and the parameter represents the volume transfer constant is the rate between blood plasma and EES per minute, and constant between EES and blood plasma per minute. In the above equation, denotes the convolution operator. The total volume of EES per unit volume of tissue is given by (2) Recently published papers [28], [29] propose more complex pharmacokinetic models, especially the contribution to the signal from the vascular space is considered as an important issue. We have chosen to ignore additional parameters in this paper in favor of a clear focus on the concept of alternative estimation procedures. The proposed Bayesian methods can easily be extended to more complex compartmental models. , the signal To calculate Gd-DTPA concentration relaxation time values using intensity is converted into -weighted images, proton density weighted images, and data relaxation times from calibration phantoms with known [30]. The Gd-DTPA concentration can then be computed via (3) is the value before contrast agent administration, where computed as mean value of the first four images of the dynamic
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acquisition, and l s mmol [31] is the longitudinal relaxivity of Gd-DTPA. For the datasets used in this study, measurement of the arterial was not available, and an assumed input function (AIF) AIF was used for all patients. A bi-exponential function as proposed by Tofts and Kermode [3] was used, i.e.,
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Process Model: A one-compartment model as described in (1) is used as the process model. By evaluating the convolution and integrating over time, one derives
(7) (4)
where
kg l, kg l, min and min [32], and the actual dose of contrast agent for a given body weight (0.1 mmol/kg in this study). B. Statistical Models 1) Likelihood Approach: In each voxel, a nonlinear regression model was fit to the Gd-DTPA concentration time series. By carrying out the convolution in (1) with (4), along with the integration over time, the following statistical model can be derived:
In the current framework, the PK parameters, the transfer constant , and rate constant must remain positive. Reasonable values for both parameters should not exceed values of [38]. Again, we parametrize around 20 and and assume Gaussian prior probability density functions (PDFs) on the logarithmic transforms of the PK parameters for all for all
where denotes the Gaussian (Normal) distribution. So with and parameters do 99.86% probability a priori, the . not exceed 20 Prior Parameters: For the variance of the observational error, a relatively flat inverse Gamma prior PDF is used
(5)
where is the noise error at time . We assume the expected value of the error to be zero; i.e., . Inference was performed by minimizing the sum of squared . According to common methods of statistical errors inference for pharmacokinetic models [33], [34], the parameterand was used to ensure ization and . Alternative formulations are positive values for also possible but not considered here. Model parameters were estimated along with their standard errors using the Levenberg–Marquardt algorithm [35]. Software implementation relied upon the nls.lm function of the minpack.lm package [36] within the statistical computing [37]. In the likelihood framework, parameter environment estimates are known to be asymptotically normal [9]. Hence, confidence intervals and probabilities of exceeding a particular threshold value were easily constructed. 2) Bayesian Approach: An alternative approach to fitting nonlinear models is to use Bayesian theory of statistical inference. A hierarchical model was used in this study such that the model is described in three stages: the data model, the process model, and the prior parameters. Data Model: We assume the concentration of the conat each time point trast agent in each voxel takes the form of a function and additive Gaussian error with variance (6) This is the Bayesian analogue to the application of the leastsquares fitting method in the likelihood approach.
(8) (9)
(10) where IG denotes the inverse Gamma distribution, which reflects the lack of prior information about this particular parameter. The three stages of the hierarchical model reflect the a priori knowledge. To combine this with the observed data and obtain a posteriori knowledge, Bayes’ theorem can be used (11) where esting parameters and
and
denotes the vector of all interthe product of the prior PDFs, i.e.,
denotes the (Gaussian) likelihood function of from (6)
Bayesian inference principle states that conclusions can only be [39]. As the integral in drawn from the posterior PDF and thus the posterior PDF itself can not the divisor of be assessed analytically, Markov chain Monte Carlo (MCMC) methods [40] are used to obtain random samples which are distributed like this high-dimensional posterior PDF. To estimate the PK parameters, the average taken from these samples for each parameter is used.
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It is worth noting that with noninformative prior PDFs Bayesian inference is known to produce the same point estimates as likelihood-based inference. Here we use informative priors, but with high a priori variance; thus, if our prior assumptions are reasonable, both the likelihood and Bayesian procedures will provide highly correlated parameter estimates. With respect to estimating the PK parameters for DCE-MRI, the distributions of the parameter estimates are only valid asymptotically for the likelihood approach. In contrast, a sample from the posterior distribution of each parameter estimate is produced from the data-driven Bayesian approach. The posterior PDF can be used to obtain additional information on the reliability of the estimation. The standard error of the sample PDF is the estimation error. Any statistic of interest may be derived from the posterior PDF; e.g., the mean, median, quantiles or the probability of exceeding a predefined threshold. Since the Bayesian framework does not depend on any optimization procedure, it will produce valid parameter estimates when the likelihood framework fails to converge. 3) Bayesian Adaptive Smoothing via Gaussian Markov Random Fields—The aBMrf Approach: The hierarchical Bayesian approach can be easily extended to a more elaborate model. In this paper, a Gaussian Markov random field (GMRF) is used as a “smoothing prior” in the hierarchical model for reducing the voxel-to-voxel uncertainty observed in more traditional analysis methods. As an alternative to uniform smoothing in space, adaptive smoothing is necessary because the tumor tissue cannot be assumed to be homogeneous from voxelto-voxel and the field of view (or even region of interest) for analysis will most likely include both normal and tumor tissue. In the proposed adaptive Bayesian Markov random field (aBMrf) approach, the data model, (6) is unchanged. However, for the process model it is assumed that functions or, more precisely the PK parameters in neighboring voxels are similar. As voxel borders are arbitrary and do not correspond the borders between tissue types, this is a reasonable assumption. Each voxel is assumed to be a mixture of different tissue types. Statements on the pharmacokinetics in a voxel, therefore, are made on the average of the proportion of tissue covered by the voxel [41]. As voxel borders are arbitrary, it is likely that neighboring voxels share similar tissue characteristics. The proposed method uses this fact and borrows strength from neighboring voxels. We assume that the amount of shared tissue and the similarity between neighboring voxel differs over the region under study, depending on underlying tissue structures. Therefore, it is not possible to specify a global smoothing parameter, but we assign a unique parameter for each neighboring voxel-to-voxel combination. As in Section II-B2, (7) still holds and we assume that the difference in the logarithmic transforms of neighboring PK parameters is a priori Gaussian distributed
where and denote neighboring voxels; i.e., “neighboring” is defined to be sharing a common border. The precisions (inand denote verse variances) of these Gaussian priors,
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the local smoothing parameters. This type of prior PDF is known as a “smoothing prior” [42]. We assume a priori that the parameter surface is smooth, i.e., no sudden changes in value from voxel to voxel. However, if the data do contain sharp features, the approach is flexible enough to accommodate this case since the variance in the prior PDFs is not constant but estimated in the Bayesian model. The model uses information from neighboring voxels to estimate the PK parameters—this is called the “borrowing strength” technique [43]—and should decrease the standard error of the estimates. When specifying the prior parameters, we have to select PDFs and . Like the varifor the local smoothing parameters ance , flat and therefore relatively uninformative PDFs were used. The local smoothing parameter is not just a single parameter for all voxels, but a double matrix of precisions describing the smoothness of the kinetic parameters. We assume these to be a priori independently gamma distributed
where and denote neighboring voxels and Ga denotes the Gamma distribution. As with the single-voxel Bayesian framework, statistical inference is made by sampling from the posterior distribution, given by Bayes’ theorem (11), via a MCMC algorithm implemented in C++. C. Data Acquisition All data used in this study were provided by Paul Strickland Scanner Centre at Mount Vernon Hospital, Northwood, U.K. The data were derived from a breast cancer study, previously reported in [44], [45]. Data from the first 12 patients were used for further analysis, these are representative of the types of lesion as whole. Each patient was scanned once at the beginning of the treatment and again after six weeks of chemotherapy (5-fluorouracil, epirubicin, and cyclophosphamide). The breasts were restrained during the scans and, therefore, no motion was visible in the scans. values, we used a two-point meaFor the calculation of surement with calibration curves as described in [46], [47]. The values are computed as ratio of a -weighted fast low-angle shot (FLASH) image and a proton-density-weighted FLASH image. The imaging parameters of the -weighted FLASH imms, ms, , the parameages were ms, ters of the proton density-weighted image were ms, . Field of view was the same for all scans, 260 260 8 mm per slice, so voxel dimensions are 1.016 1.016 8 mm. A scan consists of three sequential slices of 256 256 voxels and one slice placed in the contralateral breast as control, which we do not use for our analysis. A total of 40 to 50 scans were acquired, with one scan each 11.9 s. A dose of 0.1 mmol per kg body weight was injected after the fourth scan using a power injector with 4 ml/s with a 20 ml saline flush also at 4 ml/s. The first four scans, before contrast, were used to as the average of the values of these images. compute Regions of interest (ROIs) were drawn manually, on a scan-by-scan basis using subtraction images from the dynamic data. Although the tumor was isolated, surrounding tissue was also captured to allow for reasonable contrast between different
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Fig. 1. Gd-DTPA concentration 0, 12, 24, 36, 85, 231, 376, and 522 s after injection of the contrast agent.
tissue types within the ROI. Maximum dimension is 24 mm, and not all tumor were this small. Data from this study were acquired in accordance with the recommendation given by Leach et al. [48]. Informed consent was obtained from all patients. III. RESULTS A. Data Description All three methods (likelihood, Bayesian and Bayesian adaptive smoothing via Gaussian Markov random fields, aBMrf) were applied to data obtained from the 12 patients on three slices acquired before and after treatment. Computing time was up to 10 min per scan for the likelihood technique and up to 8 h for the Bayesian techniques (to gain reliable posterior distributions, point estimates of PK parameters can be obtained much faster) on a 3.2 GHz IBM workstation with Red Hat Linux OS. Fig. 1 illustrates the concentration of the contrast agent at different time points after injection for one of the patients. Due to the rapid influx of plasma the tumor was clearly visible after 24 s, whereas the peri-tumoral rim of the tumor became visible after several minutes. B. Parameter Estimates and Standard Error Fig. 2(a)–(c) shows the parametric maps of the estimated ) for all three methods devolume transfer constant ( scribed in Section II. The point estimates are similar for the likelihood and Bayesian methods as expected for the majority of voxels. The aBMrf approach, however, shows a locally parameter map, Fig. 2(c). Different regions smoothed in the tumor with distinct borders are distinguished, with varying amount of heterogeneity. Note, the normal tissue is also locally smoothed in the aBMrf approach, but the sharp boundary between normal tissue voxels and tumor tissue voxels is preserved. Note that the obvious line of enhancement in the
south of the tumor in Fig. 1 is not picked up; further analysis showed that this is an area of not negligible vascular volume, for which this standard compartment model is not appropriate. estimates beFig. 3(a) shows a scatter plot of tween the likelihood and Bayesian methods, for the same ROI in Fig. 2. The parameter estimates are highly correlated —after removing fit failures). This is to be ( expected since relatively uninformative prior distributions were used in setting up the Bayesian method. In general, results from likelihood and Bayesian inference are very similar when there is little or no prior information incorporated into the posterior distribution, as is the case here. The Likelihood approach, however, shows a relatively large number of fit failures and nonconverged voxels. Table I list the number of nonconverged voxels and voxels with fit failures, defined according to [47]. With the Bayesian approach, nonconverged voxels and fit failures are rarely observed. The scatter plot between the estimates of from the Bayesian and aBMrf approaches in Fig. 3(b) exhibits a high degree of correlation ( ), but it also reveals the effects of smoothing. Most of the highest values from the Bayesian parameter estimates are reduced with respect to the aBMrf approach. Furthermore, the estimates with the aBMrf method prefer to lie on horizontal stripes that potentially identifies small spatial regions where similar values of were estimated in the aBMrf approach compared with much more variable estimates when considering each voxel in isolation. Important features of the tumors, such as enhanced voxels in the rim of the tumor, are still visible in Fig. 2(c) where the data really support them, but with little lower values. The estimates of the standard error of the estimated are slightly higher in the results of the Bayes approach than in those of the likelihood approach, as Fig. 2(d) and
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Fig. 2. Parametric maps (a-c) of volume transfer constant (K ) and (d-e) of standard error of = log(K tumor—from left to right—with likelihood method, with Bayesian method and with aBMrf method.
) for a region of interest around a breast
Fig. 3. Scatter plot of estimated K parameters in ROI for Bayes versus likelihood method (left) and Bayes vs. aBMrf method (right) with regression line (solid line) and line of unity (dashed line). Estimates with Bayes and likelihood method are highly correlated (R = 0:9897) and the points therefore lie mostly on the line of unity, whereas the estimates with the aBMrf method obviously differ with reductions of Bayes estimates.
(e) show. However, the standard error computed in the likelihood approach is only correct in the asymptotic sense, so the Bayesian approach is potentially more realistic since it is based on the observed data. With the aBMrf method, the standard error is noticeably reduced as expected due to “borrowing strength,” i.e., the use of information from neighboring voxels, as can be seen in Fig. 2(f). with the aBMrf Fig. 4 shows the smoothness of method, computed as for all voxels
(12)
in a smaller region of interest, to highlight the details in the tumor. The border of the tumor is clearly visible as voxels
with low smoothness (dark), as are the borders of different regions in the heterogeneous tumor. In contrast, bright areas indicate a substantial amount of smoothing was performed by the aBMrf method. It is worth noting that this figure does not consist of square pixels, but of diamonds where each diamond depicts the precision between two voxels, either in the vertical or in the horizontal direction, as schematically illustrated in Fig. 4 on the right. The out-of-plane smoothness parameters that exist between slices have been omitted in the display for parameter map are simplicity. Features visible in the evident. For example, enhancing voxels at locations around the tumor rim are surrounded by darker symbols forming double lines, indicating borders in the horizontal or vertical direction, respectively, while smoothness is present in the perpendicular direction.
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TABLE I NUMBER OF NONCONVERGED VOXELS (NA), NUMBER OF VOXELS WITH FIT-FAILURES (FF) AS DEFINED IN [47] AND TOTAL PERCENTAGE OF VOXELS IN THE ROI WITH EITHER NA OR FF FOR LIKELIHOOD AND BAYESIAN VOXEL-PER-VOXEL ANALYSIS
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Fig. 4. Left: Parametric map of smoothness , see (12), with the aBMrf method on a logarithmic scale. Each diamond depicts the smoothness between two voxels, either in vertical or horizontal direction. Sharp features like borders in the tissue show up as dark, whereas smoothed areas are bright. Right: Each voxel has four neighbors (in 2-D) and therefore four diamonds depicting . Please notice, that there are different types of smoothness diamonds: some depicting horizontal relations and some depicting vertical relations. Display of off-plane relations (between slices) is omitted for simplification.
S
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C. Estimating Tumor Volume and Effectiveness of Treatment An initial attempt to specify the exact region of the tumor into the parameter volved applying a threshold of maps, see Fig. 5(a) and (d). This helped to highlight the tumor, but many voxels not associated with the tumor remained. To produce a better specification, we took advantage of the estimated . By assuming an asymptotic normal standard error for distribution for the likelihood method, we computed the probability of each voxel exceeding the threshold. Fig. 5(b) and (c) show this probability map and the voxels where the probability of exceeding the threshold (i.e., a voxel being part of the tumor) was greater than or equal to 95%. This probabilistic thresholding produced an improved separation between tumor and non-tumor voxels. However, some voxels not associated with the tumor are still highlighted as the respective observed contrast agent concentration time curves show obvious enhancement. The same method can be applied by using the sampled posterior distribution with the aBMrf approach, as shown in Fig. 5(e) and (f). The identification of tumor voxels is further improved by including the contextual information. A potential clinical application of these methods was assessed by computing quantitative measurements for the size of
Fig. 5. Example for threshold maps and mask derived from likelihood estiof first scan of patient mates (a)–(c) and aBMrf estimates (d)-(f) of #2—from left to right—basic thresholding (a/d), probability of exceeding the threshold (b/e), probability of exceeding the threshold by at least 95%. (c/f).
K
the tumor. A tumor mask was created from the estimated for each method and the number of enhanced voxels, i.e., voxel where the probability of exceeding the threshold is greater than or equal to 95%, is provided in Table II. The size is given for the 12 patients studied, both pre- and post- treatment. Each voxel is about 8.25 mm in volume. Furthermore, the masks from each of the first scans were applied to the second scans values in this (with manual registration) to compare the region before and after treatment. Table II gives the difference between mean values and the -values from the nonparametric Mann–Whitney test of for all patients. Table II also gives the pathological response of the patient to the treatment assessed after a further 12 weeks of chemotherapy. With all three proposed methods, a comparison of the mean value in the tumor—where the tumor is defined by the mask as described above—pre- and post-treatment predicts the clinical outcome correctly. That is, for the four pathological values is significantly reduced, responders the mean value is whereas for the two nonresponders the mean not reduced. In addition, the number of enhanced voxels is also reduced with the responders. So, as previously mentioned [44],
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TABLE II NUMBER OF ENHANCED VOXELS E BASED ON PROBABILITY THRESHOLD MAPS, DIFFERENCE IN MEAN K ( ) PRE- AND POST-TREATMENT ) IS LOWER IN POST-TREATMENT SCAN BASED ON PROBABILITY AND p-VALUE (p) OF THE MANN–WHITNEY TEST FOR MEAN (K THRESHOLD MAPS OF THE TUMOR IN PRE-TREATMENT SCANS
1
DCE-MRI can help to identify patients who are destined to fail to respond to treatment. The Bayesian and aBMrf approaches show the potential to improve the prediction as they provide information about the reliability of the estimated tumor size by reducing, and consequently giving the possibility to reduce the number of falsely recognized and unrecognized tumor voxels—even more by using contextual information with the aBMrf approach, as can be seen from Fig. 5. This application is purely illustrative and no difference in clinical decision making was anticipated from this study since the difference between responder and nonresponder was clearly visible without the need for statistical hypothesis testing. We anticipate that future applications with more subtle differences between subjects will benefit from the additional power offered by the aBMrf method. IV. DISCUSSION AND CONCLUSION In this paper, new statistical methods for estimating pharmacokinetic parameters from DCE-MRI data were introduced. The proposed likelihood method is a statistical formulation of the widely used nonlinear least square fitting quantification [29]. The Bayesian hierarchical model is an alternative way to estimate parameters. With weak prior information, Bayes inference will produce the same estimates as likelihood inference. The statistical theory of nonlinear models provides more information about the parameters of interest. Therefore, not only point estimates but also interval estimates along with (asymptotic) statements about the estimation error are achievable, which can lead to better tumor masks. However, these results are only asymptotically correct and the assumptions for the asymptotics seem not fulfilled here. Bayesian methods provide a (posteriori) distribution on each parameter. The posterior distribution provides instant access to valuable information about the kinetic parameters without resorting to asymptotic results. Accurate standard errors can be computed along with other statistics like interval estimators and probabilities for one parameter exceeding a given threshold. In practice, nonlinear model fitting is usually hard to optimize. Modern techniques like the Levenberg–Marquardt algorithm improve optimization, as the software used in this paper did not converge for on average 3.4% of all voxels in the ROI of all scans, and showed fit failures for another 0.6% of voxels.
However, the Bayes approach did not exhibit any optimization problems and very few fit failures. Estimates were available for all voxels along with the full posterior distributions for all parameters in the model. These improvements incur a cost of increased computation time in the MCMC algorithm. An important feature of the Bayesian technique is that it allows for the elegant inclusion of prior information into the estimation procedure. By relying on contextual information from neighboring voxels with a GMRF prior, we have shown that the estimation error is reduced significantly. The voxel-based approach uses only the mean PK parameters in a volume of interest in order to reduce the estimation error. The proposed method estimates unique kinetic parameters for each voxel by using information from a local neighborhood through a fixed-shape smoothing kernel. The aBMrf approach does not produce clusters of voxels based on the observed contrast agent concentration, but each voxel has a distinct set of kinetic parameter estimates. Each parameter estimate is a weighted combination of several parameter estimates via an application of local smoothing. The weights allowing interaction between adjacent voxels are estimated from the contrast agent concentration curves, producing a data-adaptive method that allows both the existence of locally homogeneous regions as described by the PK parameters of the model and quite sharp boundaries between drastically different tissue types; e.g., normal tissue versus tumor, but also sharp features in the tumor. This permits better background noise suppression but without smoothing over tissue borders. Due to contextual information from neighboring voxels, the error estimates decrease under the model assumptions translating into increased precision of the point estimates. We use an assumed model AIF for this study. The assumption that each patient in the study has the same clearance rate is unrealistic and may inflate the error in kinetic parameter estimates [29], [49]. For this study, no larger artery was in the field of view, hence the AIF could not be estimated through MRI of an artery [50]. The standard AIF proposed by Tofts and Kermode seems to be more appropriate for our study, rather than using the parametric AIF recently proposed by Parker [51], as the latter is based on DCE-MRI studies of abdomen or pelvis, whereas the work of Tofts and Kermode is based on Weinmann’s [32] invasive blood sampling from the left antecubital vein.
SCHMID et al.: BAYESIAN METHODS FOR PHARMACOKINETIC MODELS IN DCE-MRI
Recent work was based on estimating the AIF simultaneously with the PK parameters from the contrast agent concentration time curves, including the double reference tissue method [52], the blind identification technique [53] and joint Bayesian estimation [54]; however, these techniques are beyond the scope of this paper. Recently, more elaborate PK models for DCE-MRI have been proposed [28], [55]–[57]. In this paper, a basic two-compartment model was used as a proof of concept. The water-exchange effect was generally neglected, however there are models that include this effect [58], [59]. It should be pointed out that the proposed frameworks are applicable to any PK model. In this cases, Bayesian inference has the advantage of overcoming fitting problems due to high correlation between several PK parameters, as joint posterior distributions of parameters can still be estimated. In summary, we have shown that Bayesian analysis is robust for estimating kinetic parameters for DCE-MRI. Incorporation of adaptive Gaussian Markov random fields further improves the convergence behavior and provides more consistent morphological and functional statistics. ACKNOWLEDGMENT The clinical data were graciously provided by Dr. A.R. Padhani, Paul Strickland Scanner Centre, Mount Vernon Hospital, Northwood, U.K. REFERENCES [1] G. J. M. Parker and A. R. Padhani, “T -w DCE-MRI: T -weighted dynamic contrast-enhanced MRI,” in Quantitative MRI of the Brain, P. Tofts, Ed. Chichester, U.K.: Wiley, 2003, ch. 10, pp. 341–364. [2] D. J. Collins and A. R. Padhani, “Dynamic magnetic resonance imaging of tumor perfusion,” IEEE Eng. Biol. Med. Mag., vol. 23, no. 5, pp. 65–83, Sep./Oct. 2004. [3] P. S. Tofts and A. G. Kermode, “Measurement of the blood-brain barrier permeability and leakage space using dynamic MR imaging. 1. Fundamental concepts,” Magn. Reson. Med., vol. 17, no. 2, pp. 357–367, 1984. [4] H. B. Larsson and P. S. Tofts, “Measurement of the blood-brain barrier permeability and leakage space using dynamic Gd-DTPA scanning—A comparison of methods,” Magn. Reson. Med., vol. 24, no. 1, pp. 174–176, 1992. [5] P. S. Tofts, “Modeling tracer kinetics in dynamic Gd-DTPA MR imaging,” J. Magn. Reson. Imag., vol. 7, pp. 91–101, 1997. [6] P. S. Tofts, G. Brix, D. L. Buckley, J. L. Evelhoch, E. Henderson, M. V. Knopp, H. B. W. Larsson, T.-Y. Lee, N. A. Mayr, G. J. M. Parker, R. E. Port, J. Taylor, and R. Weiskoff, “Estimating kinetic parameters from dynamic contrast-enhanced T -weighted MRI of a diffusable tracer: Standardized quantities and symbols,” J. Magn. Reson. Imag., vol. 10, pp. 223–232, 1999. [7] G. Seber and C. Wild, Nonlinear Regression. Hoboken, NJ: Wiley, 2003. [8] A. R. Padhani, “Functional MRI for anticancer therapy assessment,” Eur. J. Cancer, vol. 38, pp. 2116–2127, 2002. [9] R. Jennrich, “Asymptotic properties of nonlinear least square estimators,” Ann. Math. Stat., vol. 40, pp. 633–643, 1969. [10] B. P. Carlin and T. A. Louis, Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. London, U.K.: Chapman & Hall, 2000. [11] X. Descombes, F. Kruggel, and D. Y. von Cramen, “Spatio-temporal fMRI analysis using Markov random fields,” IEEE Trans. Med. Imag., vol. 17, pp. 1028–1039, 1998. [12] M. J. McKeown, S. Makeig, G. G. Brown, T.-P. Jung, S. S. Kinderman, A. J. Bell, and T. J. Sejnowski, “Analysis of fMRI data by blind separation into independent spatial components,” Hum. Brain Mapp., vol. 6, no. 3, pp. 160–188, 1998. [13] N. V. Hartvig and J. L. Jensen, “Spatial mixture modeling of fMRI data,” Hum. Brain Mapp., vol. 11, pp. 233–248, 2000. [14] P. L. Purdon, V. Solo, R. M. Weisskoff, and E. N. Brown, “Locally regularized spatiotemporal modeling and model comparison for function MRI,” NeuroImage, vol. 14, pp. 912–923, 2001.
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