JOURNAL OF MAGNETIC RESONANCE IMAGING 22:549 –558 (2005)
Original Research
Mixture Model Approach to Tumor Classification Based on Pharmacokinetic Measures of Tumor Permeability Mary E. Spilker, PhD,1,2 Kok-Yong Seng, MS,1 Amy A. Yao, PhD,1 Heike E. Daldrup-Link, MD,4 David M. Shames, MD,4 Robert C. Brasch, MD,4 and Paolo Vicini, PhD1* Purpose: To categorize the disease severity of mammary tumors in an animal model through the application of a novel tumor permeability mixture model within a hierarchical modeling framework.
Conclusion: Mixture model analysis provides a robust method for subject classification without user intervention and bias. Although the present results are promising, additional research is needed to further evaluate this technique for diagnostic purposes.
Materials and Methods: Thirty-six rats with mammary tumors of varying grade were imaged via dynamic contrastenhanced (CE) MRI using albumin-(Gd-DTPA)30. Time-dependent contrast agent concentration curves for blood and tumor tissue were obtained and a mathematical model of microvascular blood–tissue exchange was developed under the hypothesis that endothelial integrity is disrupted in a manner proportional to the degree of malignancy, with benign tumors showing no disruption of the vasculature endothelium. This permeability model was incorporated into a statistical model for the benign and malignant tumor subgroups that enabled automatic subject classification. The structural and statistical models were implemented using the software Nonlinear Mixed Effects Modeling (NONMEM) to statistically separate subjects into the two subgroups.
Key Words: dynamic contrast-enhanced MRI; mixture model; NONMEM; hierarchical modeling; tumor permeability J. Magn. Reson. Imaging 2005;22:549 –558. © 2005 Wiley-Liss, Inc.
Results: Individual tumor classifications (as benign or malignant) were evaluated against the Scarff-Bloom-Richardson microscopic scoring method as applied to the tumor histology of each subject. The model-based classification resulted in 90.9% sensitivity, 92.9% specificity, and 91.7% accuracy.
1 Resource Facility for Population Kinetics, Department of Bioengineering, University of Washington, Seattle, Washington, USA. 2 Department of Nuclear Medicine, Technical University of Munich, Munich, Germany. 3 Department of Radiology, Technical University of Munich, Munich, Germany. 4 Center for Pharmaceutical and Molecular Imaging, Department of Radiology, University of California–San Francisco, San Francisco, California, USA. Contract grant sponsor: NIH; Contract grant numbers: 5P41RR012609; 5P41EB001975; 5R01CA082923. *Address reprint requests to: P.V., Resource Facility for Population Kinetics, Department of Bioengineering, Aerospace Research Building, Rm. 241, Box 352255, University of Washington, Seattle, WA 981952255. E-mail:
[email protected] Received August 16,2004; Accepted June 23, 2005. DOI 10.1002/jmri.20412 Published online 13 September 2005 in Wiley InterScience (www. interscience.wiley.com).
© 2005 Wiley-Liss, Inc.
IMAGING METHODS HAVE the potential to greatly assist clinicians and oncology researchers by providing a noninvasive assessment of tumor grade. Unfortunately, this potential has not been fully realized through the use of x-ray mammography, which has exhibited an only 10 –30% positive predictive value in mammary tumors (1). Promising results are instead being attained with contrast-enhanced (CE) MRI protocols, which have demonstrated efficacy in the detection and diagnosis of breast cancer (1). However, while various studies examining the performance of dynamic CE-MRI (DCE-MRI) mammography protocols have shown good sensitivity measures (86 –100%) for cancer, they have also exhibited varying degrees of specificity (37–97.4%) (2). Therefore, research is currently being focused on various measures that may improve the sensitivity and specificity of DCE-MRI protocols. Such methods include the use of neural networks (3), MR-based delineation of tumor architectural features (4), and interpretation contingent on quantitative and morphologic information (2). In this article we propose a novel method whereby tumors are classified as benign or malignant based on tumor microvessel permeability as assessed by a pharmacokinetic analysis of DCE-MRI data within the statistical framework of a mixture model. While other researchers have applied mixed effects modeling (MEM) techniques to DCE-MRI-derived data (5,6), this work represents the first undertaking in which a mixture model is used for automatic classification of tumor malignancy.
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It is well known that the structural integrity of the microvessel endothelium is disrupted in malignant tumors but remains intact in benign tumors (7). When the endothelium is impaired, the tumor microvessels become permeable to macromolecules. Unlike micromolecular contrast media, which rapidly distribute throughout the vasculature and interstitial spaces, macromolecular contrast media (MMCM) can enter the tumor interstitium only when the endothelium is disrupted. Therefore, the extent of microvessel permeability is one feature of tumor physiology that can be exploited to distinguish benign from malignant tumors in dynamic imaging protocols utilizing MMCM. Toward that end, a two-compartment, unidirectional uptake pharmacokinetic model was previously developed to characterize tumor microvessel permeability to MMCM (8,9). Experiments using this model have found good correlations between image-derived pharmacokinetic measures of tumor permeability and independently determined tumor grade, as assessed by the ScarffBloom-Richardson (SBR) score. These results support the use of tumor microvascular permeability as a criterion for tumor classification. A pharmacokinetic model enables the extraction of kinetic information in terms of model parameters, such as endothelial wall permeability. When several subjects are considered at the same time, it becomes possible to move to a population or MEM environment (6,10). Applying MEM techniques to time-varying data involves the specification of mathematical expressions that describe the deterministic and stochastic (random) components present in the measurements (11). At the individual level, the random component represents the residual variability or noise in the data, which is often attributed to measurement, assay, or other similar types of error. A second level of variability is present when one moves to the population setting, where the deviation of the individual’s parameter values from the population (mean) parameter values is considered to be another random, yet quantifiable, variable. While this hierarchical construct adds an extra level of complexity to the analysis, it allows for the incorporation and estimation of other useful information by allowing each individual to differentially contribute to and gain from the predicted population parameters. For a further explanation of this method in the context of DCE-MRI, see Ref. 6. This method has a further advantage in that a mixture model can be implemented within the modeling environment. A mixture model uses the a priori knowledge that the population is not homogeneous and contains a distribution of parameter values that is multimodal, rather than unimodal, as in normally distributed statistics. In other words, the experimental subjects can be partitioned into subgroups based on pertinent parameter value(s). The mixture model then permits the structural (pharmacokinetic) and statistical models, as well as the information within each individual’s data record, to guide the individual’s assignment to predetermined subgroups. When the parameters of all of the competing subgroups have been estimated, the individuals are assigned to their respective, most probable subgroup.
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Figure 1. Distribution of the SBR scores for the 36 subjects analyzed in the present study.
The primary objective of the present study was to classify tumor malignancy status using a priori knowledge regarding the heterogeneity of disease severity. Specifically, the population was known to comprise subjects with mammary tumors that could be classified as either benign or malignant. While this information could be determined from histological measures (SBR scores), a strictly noninvasive approach based on pharmacokinetic parameter estimates characterizing DCEMRI data would be more powerful. Therefore, we hypothesized that by using the established pharmacokinetic model of microvessel permeability, we would be able to distinguish benign from malignant tumors by applying a mixture model of tumor permeability within a population modeling environment. The advantage of this approach compared to an individual pharmacokinetic modeling approach (also called a standard twostage approach) is that subject classification is fully automatic and exclusively based on the available population data. MATERIALS AND METHODS Experimental Methods Data were obtained as previously described by Daldrup et al (8). Details of the animal preparation and imaging protocol can be found in the same reference. A total of 36 subjects were involved in the present study, which included additional animals that were not utilized by Daldrup and co-authors (8). The distribution of SBR tumor grades for the subjects in this study is shown in Fig. 1. Pharmacokinetic Model A pharmacokinetic model that simultaneously characterizes contrast agent kinetics within the vasculature and tumor tissue was applied to the data. This model was previously described in detail (12). Briefly, a monoexponential decay function is fit to the vasculature time-concentration profile measured from a region of
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interest (ROI) in the image corresponding to the inferior vena cava (IVC). This function then defines the input for a two-compartment unidirectional uptake model that characterizes contrast agent kinetics within the tumor tissue. A unidirectional uptake model is applicable because it is assumed that once the contrast agent enters the interstitial space, it does not return to the vasculature for the duration of the imaging protocol. The analytical solutions to the model’s differential equations are provided below: C IVC 共t兲 ⫽ Q1共t兲 ⫽ ICB 䡠 e ⫺k 01䡠t C tumor 共t兲 ⫽ Q2共t兲 ⫹ Q3共t兲 ⫽
fPV 䡠 Q1共t兲 共1 ⫺ H兲 ⫹
Kps 䡠 关ICB ⫺ Q1共t兲兴 关共1 ⫺ H兲 䡠 k 01 兴
The pharmacokinetic model consists of four primary parameters: initial concentration of the contrast agent after bolus injection (ICB: mmol mL–1), vasculature elimination rate constant (k01: min–1), endothelial transfer coefficient (Kps: mL min–1 cc–1 of tissue), and fractional plasma volume (fPV: mL cc–1 of tissue). The hematocrit, H, is fixed at a value of 0.42 (8). The main parameter of interest, Kps, provides a quantitative measure of the tumor vascular endothelium permeability (8). Kps is a unidirectional analog to the volume transfer constant, Ktrans (13). Population and Mixture Models The population approach was chosen for this application because of its power in considering multiple individuals with similar kinetics as a population, and in using information from the entire group to enhance the parameter estimates for each individual and the overall population. Similarly to an individualized kinetic analysis, the population approach utilizes a structural model, although a statistical model is also concomitantly used to describe the variability of the structural model parameters within the population. Furthermore, as mentioned above, a mixture model can be developed within this framework. In this instance, it was known a priori that the entire group of experimental subjects was classifiable into either the benign or the malignant subgroup. This study was thus an ideal candidate for a mixture-model implementation. Descriptions of the population and mixture models are provided in the following sections. Population Modeling The pharmacokinetic model was fit to the dynamic MRI data using the Nonlinear Mixed Effects Modeling (NONMEM) software (version V; NONMEM Project Group, University of San Francisco, San Francisco, CA, USA), interfaced with PDx-Pop Version 1.1j Release 4 (GloboMax LLC, Hanover, MD, USA). This software fits general statistical (nonlinear) models to data using an extended least-squares algorithm (14), and allows values from each individual to be analyzed simultaneously so that
the population, not the individual, becomes the unit of analysis. Hence, typical population values for structural model parameters are estimated along with their interindividual variability (15). The interindividual variability for each parameter was modeled logarithmically, ensuring that parameter values always remain positive. This construct is illustrated below for parameter k01. k 01,j ⫽ k 01e j where k01,j is the estimate of k01 for the jth individual, and k01 represents the population estimate (fixed effect) for this parameter. j describes the interindividual variability on k01, which is a random variable distributed normally with zero mean and variance k201. The residual variability (measurement error) associated with the data is simultaneously estimated during the modeling process. The residual (intraindividual) variability between the predicted (Cpred ij) and observed (Cobs,ij) concentration values for blood and tumor at the ith time point of the jth individual was modeled using a proportional, or constant coefficient of variation (CV), model. C obs,ij ⫽ C pred,ij 共1 ⫹ ε ij 兲 The residual variability at the ith time point of the jth individual, ⑀ij, is a normally distributed random variable with zero mean and variance 2. The variables ⑀ and define the random effects within the MEM construct and are assumed to be statistically independent from one another (16,17). Mixture Modeling The NONMEM software further allows for the implementation of a mixture model, in which it is assumed that the population of individuals is distributed multimodally with respect to a defined model characteristic. The procedures implemented in the software can then automatically separate and predict the proportion of individuals belonging to a given subgroup of an existing data set, based on the assumed distinguishing feature of the subgroups. The basic algorithm for identifying the mixture model is as follows: First, each subgroup’s structural and statistical models are fit to each individual’s data record, and a resulting objective function is computed via the maximum likelihood estimation procedure. The likelihood of the complete population model, as determined from each individual’s data, is a weighted sum of the likelihoods under each subgroup (14,17). After all population parameters are estimated, the “best” (most probable) subgroup is determined for each individual by computing Bayesian posterior probabilities for the subgroups. However, these are empirical Bayes estimates, and prior knowledge on each parameter comes from the predicted population parameters obtained as described above. For the analysis undertaken in the present study, it was assumed as previous knowledge that the endothelial transfer coefficient, Kps, was distributed bimodally, since the population comprised benign (Kps ⫽ 0) and malignant (Kps ⫽ 0) tumors. Therefore, the NONMEM
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$Mixture subroutine was used to differentiate between benign and malignant tumors based on this aspect of their pharmacokinetic behavior. This subroutine requires the specification of the number of subgroups present (two in this case), as well as specifications of the structural models that characterize each subgroup. The structural models were similar for both subgroups, although in the benign subgroup Kps was fixed to zero. Practically, this implies that the benign tumor vasculature is not permeable to the MMCA. The analysis was performed using the first-order conditional approximation (with interaction effects) to the population likelihood. With the exception of Kps, the same starting values were assigned to the population parameters for the benign and malignant subgroups in the kinetic analysis. Tissue Analysis As previously described (8), all tumors were removed postmortem and fixed in 10% formalin, embedded in paraffin, and sectioned in the same place as the MR images. Hematoxylin-eosin staining was performed for standard histological analysis. All tumors were scored in accordance with the SBR method. According to this method, each tumor is scored in three areas: gland formation, anaplasia, and mitoses. The SBR score ranges from 3 for a benign tumor to 9 for a poorly differentiated malignancy. The SBR score is further divided into the following categories: benign fibroadenomas (SBR 3), low-grade carcinomas (SBR 4 –5), moderate-grade carcinomas (SBR 6 –7), and high-grade carcinomas (SBR 8 –9). Model Assessment The NONMEM program was considered to have reached an optimal solution when both the optimization (parameter values) and covariance (parameter standard errors) steps of the analysis reached successful convergence. The performance of the analysis was evaluated by several criteria, including visual examination of the concentration-time data, which should show no significant outliers or trends. The model performance was also evaluated by comparing objective function values and by plotting the population predicted and individual Bayesian predicted concentrations (PRED and IPRED, respectively, in NONMEM notation) vs. observed concentrations, as well as inspecting weighted residuals vs. predicted value plots. The parameter values are reported as mean and standard errors unless otherwise stated. The standard errors reported for the parameters are derived from the asymptotic covariance matrix in NONMEM. Also of note is that the mean parameter values reported are geometric rather than arithmetic medians, since a logarithmic model for interindividual variability was used. Histogram plots were generated using commercial statistical software (SPSS for Windows, version 10.0.5; SPSS Inc., Chicago, IL, USA). True-positive (TP), true-negative (TN), false-positive (FP), and false-negative (FN) values were determined by comparing the model-predicted categorization of the tumors (as benign or malignant) with the histology clas-
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sification, where tumors with SBR scores ⬎ 3 were defined to be malignant. The predictive measures were calculated as follows: Sensitivity ⫽ ⫽
TP ; specificity (TP ⫹ FN)
TP ⫹ TN TN ; and accuracy ⫽ . (TN ⫹ FP) TP ⫹ TN ⫹ FP ⫹ FN
RESULTS An example of the mean concentration-time curves for the benign (SBR grade 3) and malignant (SBR grade 9) subgroups is shown in Fig. 2. The concentrations were measured as changes in the relaxation rates (⌬R1) between the pre- and postcontrast ROIs in blood and tumor, which is directly proportional to the contrast agent concentration (18). Figure 2 illustrates that the vasculature values were indistinguishable between the two subgroups, while the mean tumor ⌬R1 curves differed appreciably between the benign and malignant tumor subgroups. The benign subgroup displayed a profile that paralleled that of the blood curve, suggesting that no contrast agent was leaking from the vasculature and accumulating within the tumor’s interstitial space. The malignant subgroup, however, showed a gradual accumulation of contrast agent (the mean tumor curve steadily rose with time), suggesting that the tumor capillaries’ endothelia were permeable to the contrast agent. The standard deviations (SDs) shown in these plots serve to illustrate the large degree of variability that is present within both the malignant and benign subgroups. The NONMEM implementation of the pharmacokinetic and statistical models was performed as described in the Materials and Methods section. Figure 3 shows that while the blood curve values (Fig. 3a) were not clearly separable between the two subgroups, the tumor PRED and observed ⌬R1 values (Fig. 3b) were clearly distinct. To explain Fig. 3a and b further, the PRED values result in one set of timevarying values for the benign and malignant subgroups, respectively, which define the x-axis and hence the vertical columns at specific values along this axis. The observed data at each time point corresponding to the PRED value define the y-axis. It is evident that values are randomly scattered around the unity line and that no outliers are observed in this plot. Figure 3c and d are plots of the individual Bayesian estimated ⌬R1 vs. the observed ⌬R1 for blood and tumor tissue. Ideally, these values should lie on the unity line for optimal pharmacokinetic model performance. Although some dispersion is observed in Fig. 3, both curves exhibit a symmetric distribution around the unity line, indicating adequate model prediction and good individual fits to the DCE-MRI measurements. Figure 4 shows the observed and PRED ⌬R1 values vs. time. The fact that the temporal distributions of the mean and PRED ⌬R1s in Figs. 2 and 4 are similar is further evidence of the appropriateness of the pharmacokinetic and statistical models used in the present study. Note that
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Figure 2. Average blood (a) and tumor (b) curves for the SBR grade 3 (benign) and 9 (severely malignant) subgroups. Error bars indicate the SD. Note that a simple discrimination of the two extreme tumor grades by qualitative examination of their initial profiles is not possible due to the large amount of variability present within both populations, as illustrated by the error bars.
the PRED curves are quite similar between groups for blood data, but show a notable separation for tumor data, consistent with the existence of two distinct populations. The parameter estimates and the interindividual and residual variability estimates from the NONMEM analysis are provided in Table 1. The parameter estimates were well determined, as exhibited by the relatively small interindividual variability. The estimated blood concentration intercepts (ICB) were consistent between both subgroups, with values of 7.20 ⫾ 0.54 mmol mL–1 and 6.88 ⫾ 1.31 mmol mL–1 for the malignant and benign subgroups, respectively. The difference between the predicted k01 for both subgroups was also not significant: 0.48 ⫾ 0.54 min–1 in the malignant subgroup, and 0.41 ⫾ 0.07 min–1 in the benign subgroup. Similar to findings in a previous study (8), the predicted macromolecular permeability, Kps, for the malignant subgroup was significantly greater than zero (6.83 ⫾ 1.27
mL min–1 cc–1 of tissue). The standard error and interindividual variability for Kps of the benign group was not estimated, since this parameter was fixed to a value of zero. There was an appreciable difference between the fPVs for the benign and malignant subgroups (5.39 ⫾ 0.84 and 3.39 ⫾ 0.59 mL cc–1 of tissue respectively). The residual error in the data was fairly low, with estimated values of 5% and 9% CV for the blood and tumor data, respectively. The mixture model analysis was further evaluated by comparing the predictive ability of the NONMEM results with the independent measure provided by the SBR scores across all subjects. In this comparison, benign tumors were specified as those with an SBR score of 3 (from the histologic data) or a Kps of zero (from NONMEM analysis). Tumors that did not meet these specifications were classified as malignant. Further differentiation of the malignant tumors was not possible with the current analysis. Under
Figure 3. Observed ⌬R1 vs. PRED ⌬R1 for blood (a) and tumor (b) compartments, and observed ⌬R1 vs. IPRED ⌬R1 for blood (c) and tumor (d) compartments in both benign (SBR 3) and malignant (SBR 4 –9) subgroups. The continuous line is the unity line. All of the figure parts demonstrate good model performance, since the distribution of values is randomly scattered around the unity line with virtually no outliers. Note that whereas the blood ⌬R1 values overlap in part a, the tumor ⌬R1 values in part b are clearly distinct between subgroups.
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Figure 4. Observed ⌬R1 and PRED ⌬R1 vs. time in the blood (a) and tumor (b) compartments. Note the similarity in the temporal distributions of the PRED ⌬R1 (in this figure) and the average ⌬R1 (in Fig. 2) in both compartments.
these specifications, the tumor permeability mixture model was found to yield a sensitivity of 90.9%, specificity of 92.9%, and accuracy of 91.7% (Table 2). Of the 36 subjects analyzed, three were misclassified (their tumors were actually of grades 3, 4, and 5). One benign tumor was falsely classified as malignant, while two low-grade carcinomas cases were misclassified as benign by the mixture model analysis. DISCUSSION The clinical application of DCE-MRI holds immense promise, with kinetic models of DCE-MRI data at the individual level identifying parameters that correlate
with tumor grade (8 –20). The results from the present study suggest that similar results can be attained through the application of population-based modeling techniques to imaging data. Furthermore, the accuracy (90.9%), specificity (92.9%), and sensitivity (91.7%) measures obtained herein are comparable with or superior to those obtained using alternative methods (2,3). We believe these values can substantially improve with increased population size, enhancements to the MEM method, and a more detailed differentiation of tumor malignancy than can presently be achieved with the SBR score. An initial qualitative assessment of the concentration-time profiles (presented in Fig. 2) indicates that a
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Table 1 Population Parameter Estimates From NONMEM Analysis Parametera
Notation
Estimated initial blood concentration Vasculature elimination rate constant (⫻100) Endothelial transfer coefficient (⫻100) Fractional plasma volume (⫻100) Residual error Proportional error (%)
ICB k01 Kps fPV
Est. ⫾ S.E
IIV (%)
Malignant
Benign
Malignant
Benign
7.20 ⫾ 0.54 0.48 ⫾ 0.09 6.83 ⫾ 1.27 5.39 ⫾ 0.84 Blood 4.96
6.88 ⫾ 1.31 0.41 ⫾ 0.07 0.00 ⫾ NE 3.39 ⫾ 0.59 Tumor 8.52
32.2 53.6 62.8 66.4
32.4 63.2 NE 60.7
a
Parameter units: ICB (mmol mL–1); k01 (min–1); Kps (mL min–1 cc–1 tissue); fPV(mL–1 cc–1 tissue). Est. ⫽ parameter estimate, SE ⫽ standard error, IIV ⫽ interindividual variability, NE ⫽ not estimated.
pharmacokinetic model was needed to distinguish benign from malignant tumors, not least because of the large degree of intersubject variability, as highlighted by the error bars in the plots. This variation prevents the direct classification of tumors by a model-independent examination of their concentration-time profiles alone, and therefore suggests the use of a separate model—in this case a mathematical pharmacokinetic model—as a discrimination tool. While such a pharmacokinetic analysis could be performed on an individual basis, the results would still necessitate additional interpretation to classify the tumor as either benign or malignant. Through the use of a mixture model of contrast agent kinetics embedded within the MEM environment in the present study, we show that it is possible to incorporate a measure of the population’s variability into the modeling process. This model can automatically classify the subjects by allowing the analysis to provide a better estimate as to which subgroup an individual belongs, without introducing possible human bias into the analysis. The data set utilized in the present study had low measurement noise, as shown by the low residual variability estimates of 5% and 9% CV for blood and tumor concentrations, respectively. These estimates of noise in the data suggest that the tumor curves contained a worse signal-to-noise ratio (SNR) than the blood curve. This is a reasonable finding, considering that the data were extracted from images by delineating appropriate ROIs in the blood and tumor regions. The blood curve is naturally more homogeneous and has a higher SNR than that of the tumor, which is often heterogeneous due to discrepancies in signal intensities of the enhanc-
Table 2 Tumor Classification by SBR Scores and NONMEM Analysis SBRa
NONMEM Malignant Benign a
Malignant
Benign
20 2
1 13
The SBR classification is taken to be “true.” The table can be interpreted as the following: NONMEM classified 20 malignant subjects correctly according to SBR score and two incorrectly. One benign tumor was misclassified as being malignant and 13 were correctly classified as benign by the NONMEM method. Sensitivity: 90.9%; Specificity: 92.9%; Accuracy: 91.7%.
ing and non-enhancing pixels. As such, the tumor SNR is reduced (21). The fact that the PRED ICB values of the benign and malignant subgroups overlapped can be explained by the fact that all animals in the present study received a similar albumin-(Gd-DTPA)30 dose, and ICB depends solely on the tumor-grade-independent, exogenous delivery of the contrast agent into the animal’s vascular system. Similar mean k01’s were noted for the two subgroups, suggesting that systemic pharmacokinetics do not differ between animals bearing tumors of different degrees of malignancy. Our analysis produced higher average estimates of fPV for the malignant tumor subgroup. This finding was also reported by previous investigators (8,22), who attributed the higher fPV values to a larger degree of “vascularity” in malignant tumor tissue, which has heightened angiogenesis compared to benign tumor tissue. In addition, fPV appears to be identically variable within both populations (interindividual variability ⫽ 66% and 61% for the malignant and benign subgroups, respectively). This is an interesting finding considering the fact that the malignant subgroup is composed of tumors with SBR scores ranging from 4 to 9, and that tumor vascularity (which is often measured by microvascular density) has been noted to correlate with the presence of metastases (23). It was therefore expected that the numerical value of fPV would be more sensitive to the range of tumors within this subgroup. However, our results appear to be consistent with previous studies that reported an insignificant correlation between fPV and SBR scores or microvascular density (9,20). Still, it is of interest to note that a low but significant correlation between fPV and the SBR score (r2 ⫽ 0.25) was attained by Daldrup and co-investigators (8), while van Dijke et al (19) reported a substantial linear correlation between log(fPV) and microvascular density. Our present findings of a statistically significant (P-value of ⱕ 0.05) correlation coefficient of similar magnitude between fPV and SBR score (r2 ⫽ 0.21) would appear to corroborate the previous results. In all of these studies, however, stronger correlations were found between Kps and the SBR scores. As stated above, the results of this study appear to be promising, with 90.9% accuracy, 92.9% specificity, and 91.7% sensitivity achieved. The three misclassified cases may reflect the fact that their tumor grades are located near the borderline between benign and malignant. Since low-grade tumors are the most difficult to
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classify, it is highly probable that they were not characterized correctly by the SBR score. In an earlier study, Daldrup et al (8) noted that pathologists found it difficult to differentiate benign fibroademonas (SBR score ⫽ 3) from low-grade adenocarcinomas (SBR score ⫽ 4-5). Furthermore, while the SBR score is widely used, it is subject to considerable interexpert variability (24 –27). We believe that an increased number of subjects would ultimately improve subject classification and reduce the estimation error of the interindividual variance associated with the blood parameters (i.e., the ICB and k01). Interindividual variance may not improve for tumor-specific parameters (i.e., Kps and fPV) in the malignant population due to the inherent heterogeneity of tumors. A larger population would allow for better estimates of the population parameters, which in turn should help better characterize the low-grade carcinomas. Although strong correlations have been found between tumor permeability, Kps, and tumor grade, they have been population-specific, and while these correlations are encouraging in themselves, they have not yet been consistent between studies. For example, values for the Spearman correlation coefficient, while all significant, have ranged from 0.55 (20) to 0.76 (8) and 0.88 (9). As further studies continue to evaluate the role of tumor permeability, it may be possible in a future implementation to incorporate a correlation value or other similar information into the modeling process. It may also be possible to incorporate other features as covariates in the analysis to improve the prediction of population parameters. Covariates are clinical characteristics of the subjects under investigation. These values can be further divided into two categories: categorical (e.g., treatment received and coexisting disease) and continuous (e.g. tumor size and weight). Incorporating other covariates, such as specific morphologic features, may help explain the underlying reasons for the variability of the pharmacokinetic parameters, which in turn may improve sensitivity and specificity. It has already been shown that the incorporation of quantitative and morphologic information can improve specificity and sensitivity in a subjective setting (2). It is possible that the same information would improve the outcome with a mixture model analysis as well. The above-mentioned findings are limited by the scope and design of the present study. While this study was performed using a prototype blood pool contrast agent, albumin-(Gd-DTPA)30, many of the contrast agents applied within the clinical setting thus far have been small-molecule contrast agents, such as Gd-DTPA (Berlex Laboratories, Wayne, NJ, USA). Gd-DTPA has been shown to be highly sensitive but nonspecific to tumor permeability (8), since the distribution of this compound into the interstitial space is independent of endothelial integrity and transendothelial transport (28). Therefore, while this contrast agent can identify the presence of a tumor, it lacks the ability to distinguish between malignant and benign tumors based on measures of permeability. However, promising results in classifying tumors based on permeability have been obtained with the use of macromolecular contrast media, such as the one used in this study. While the contrast agent we used is not currently applicable to
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human studies, other contrast agents with similar characteristics, such as USPIOs, are currently in clinical trials and may be available for such uses in the future (9). The use of such contrast agents could make the modeling methods developed in this study applicable in a clinical setting. In addition, the data in the present study were obtained in an animal model (rodent) population. We can speculate that a human population would have disease characteristics similar to those of the population studied here. While the data obtained in the present group may be more homogeneous than those acquired by clinical observation, it is evident that a fair amount of heterogeneous kinetic behavior was present within this animal population. Furthermore, a different tumor distribution may be observed clinically. However, the results of the present study suggest that similar results could be attained in a clinical setting via the mixed modeling effects method. In summary, we have presented a new method of analysis for DCE-MRI data and illustrated that the use of mixture models and population-based modeling techniques have the potential to assist radiologists and clinicians in differentiating between benign and malignant tumors. While the results of this study are encouraging, it is important to rigorously test these methods further in order to attain optimal and reproducible sensitivity and specificity values. ACKNOWLEDGMENTS The authors thank the anonymous reviewers for their helpful comments. REFERENCES 1. Williams M, Pisano E, Schnall M, et al. Future directions in imaging of breast diseases. Radiology 1998;206:297–300. 2. Liu P, Debatin J, Caduff R, et al. Improved diagnostic accuracy in dynamic contrast enhanced MRI of the breast by combined quantitative and qualitative analysis. Br J Radiol 1998;71:501–509. 3. Lucht R, Knopp M, Brix G. Classification of signal-time curves from dynamic MR mammography by neural networks. Magn Reson Imaging 2001;19:51–57. 4. Nunes L, Schnall M, Orel S, et al. Breast MR imaging: interpretation model. Radiology 1997;202:833– 841. 5. Port R, Knopp M, Hoffman U, et al. Multicompartment analysis of gadolinium chelate kinetics: blood–tissue exchange in mammary tumors as monitored by dynamic MR imaging. J Magn Reson Imaging 1999;10:233–241. 6. Port R, Knopp M, Brix G. Dynamic contrast-enhanced MRI using Gd-DTPA: interindividual variability of the arterial input function and consequences for the assessment of kinetics in tumors. Magn Reson Med 2001;45:1030 –1038. 7. Vaupel P, Hockel M. Blood supply, oxygenation status and metabolic micromilieu of breast cancers: characterization and therapeutic relevance. Int J Oncol 2000;17:869 – 879. 8. Daldrup H, Shames D, Wendland M, et al. Correlation of dynamic contrast-enhanced magnetic resonance imaging with histologic tumor grade: comparison of macromolecular and small-molecular contrast media. AJR Am J Roentgenol 1998;171:941–949. 9. Turetschek K, Huber S, Floyd E, et al. MR imaging characterization of microvessels in experimental breast tumors by using a particulate contrast agent with histopathologic correlation. Radiology 2001;218:562–569. 10. Whiting B, Kelman A, Grevel J. Population pharmacokinetics. Theory and clinical application. Clin Pharmacokinet 1986;11: 387– 401. 11. Sheiner L, Ludden T. Population pharmacokinetics/dynamics. Annu Rev Pharmacol 1992;32:185–209.
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