 Research Article
 Open Access
PET image segmentation using a Gaussian mixture model and Markov random fields
 Thomas Layer^{1, 2}Email author,
 Matthias Blaickner^{2},
 Barbara Knäusl^{3},
 Dietmar Georg^{3},
 Johannes Neuwirth^{4},
 Richard P Baum^{5},
 Christiane Schuchardt^{5},
 Stefan Wiessalla^{5} and
 Gerald Matz^{1}
https://doi.org/10.1186/s4065801501107
© Layer et al.; licensee Springer. 2015
 Received: 25 April 2014
 Accepted: 8 September 2014
 Published: 12 March 2015
Abstract
Background
Classification algorithms for positron emission tomography (PET) images support computational treatment planning in radiotherapy. Common clinical practice is based on manual delineation and fixed or iterative threshold methods, the latter of which requires regression curves dependent on many parameters.
Methods
An improved statistical approach using a Gaussian mixture model (GMM) is proposed to obtain initial estimates of a target volume, followed by a correction step based on a Markov random field (MRF) and a Gibbs distribution to account for dependencies among neighboring voxels. In order to evaluate the proposed algorithm, phantom measurements of spherical and nonspherical objects with the smallest diameter being 8 mm were performed at signaltobackground ratios (SBRs) between 2.06 and 9.39. Additionally ^{68}GaPET data from patients with lesions in the liver and lymph nodes were evaluated.
Results
The proposed algorithm produces stable results for different reconstruction algorithms and different lesion shapes. Furthermore, it outperforms all threshold methods regarding detection rate, determines the spheres’ volumes more accurately than fixed threshold methods, and produces similar values as iterative thresholding. In a comparison with other statistical approaches, the algorithm performs equally well for larger volumes and even shows improvements for small volumes and SBRs. The comparison with experts’ manual delineations on the clinical data shows the same qualitative behavior as for the phantom measurements.
Conclusions
In conclusion, a generic probabilistic approach that does not require data measured beforehand is presented whose performance, robustness, and swiftness make it a feasible choice for PET segmentation.
Keywords
 Expectation maximization
 Markov random field
 Positron emission tomography
 Radiotherapy
 Tumor segmentation
Background
The determination of the tumor volume is one of the main causes for uncertainties in dosimetry [1]. When trying to assess the volume of a tumor for the sake of treatment planning in external beam radiation therapy (EBRT) or radionuclide therapy, a very common practice is to have an expert manually draw a volume of interest (VOI) on the respective positron emission tomography (PET) or single photon emission computed tomography (SPECT) image. The resulting and inevitable interobserver variations have been reported well enough for different types of cancer [24].
Another prevalent approach is the application of a threshold. The simplest choice for the threshold is a fixed percentage of the maximum activity concentration value [5,6]. This thresholding method has been shown to predict well for big volumes but yields large errors in case of small volumes which is attributed to partial volume effects (PVE) and moreover depends on the signaltobackground ratio (SBR). Despite its questionable scientific meaningfulness, it is still a widespread method even recommended by an international experts’ report [7]. Extensions of this method are automatic [8,9] and iterative threshold approaches [1012]. However, the iterative thresholding method (ITM) requires a regression curve which has to be determined by phantom measurements for every specific imaging setting [13]. Considering the many dependencies of these regression curves such as the (i) manufacturer and the detector of the scanner, (ii) reconstruction algorithm, (iii) nuclide, (iv) SBR, and (v) volume of the lesion, ITM comes at a great expense in terms of work and time. Likewise, the inclusion of physical models of PET images such as a point spread function (PSF) is also dependent on some of the parameters listed above and therefore would require a considerable amount of calibration measurements.
Alternative methods such as watershed and edge detection are also sensitive to noise and different SBRs [14,15].
The aim of this work is to develop an automatic segmentation algorithm for PET images which is as generic as possible, i.e., threshold independent and therefore does not require systemspecific regression curves or PSF. Additionally, the volume estimates for small objects shall be improved. Therefore, in this paper, we propose an improved statistical PET image segmentation scheme. Our scheme relies on soft class assignment instead of hard class assignments and fuzzy levels. A Gaussian mixture model (GMM) is established to obtain initial estimates of the volume of the spheres. Subsequently, a MRF is obtained by declaring Markov properties for the unobserved label vector and using a Gibbs distribution to describe neighborhood dependencies. The MRF is then used to obtain the final labeling from the initial GMM labeling vector.
Methods
Phantom measurements and clinical data
Measurements of the modified NEMA sphere phantom
FG  BG  SBR 

10.94  5.30  2.06 
20.37  5.30  3.84 
26.13  5.30  4.90 
66.56  9.90  6.72 
90.90  9.68  9.39 
The device in use was a Siemens Biograph 64 TruePoint PET/CT scanner (Siemens, Erlangen, Germany). In accordance with the conditions for NEMA phantom quality assurance measurements in nuclear medicine [16], the average activity concentration never exceeded 10 kBq/ml. This way, the linearity of the scanner’s noise equivalent count rate (NECR) is preserved, and the measurements of different SBRs can be compared. The acquisition was performed using emission scans of 10 min. The images were reconstructed with an iterative OSEM2D algorithm (4 iterations on 21 subsets). A preprocessing Gaussian filter of 5 mm was applied. The dimension and volume of the voxels are 4 mm ×4 mm ×3 mm and 0.048 ml, respectively. The more advanced iterative reconstruction algorithm for the Siemens scanner, TrueX (PSF), was not taken into account since recent studies recommended cautiousness with regard to its quantitative meaningfulness [17,18]. The chosen settings correspond to the clinical routine settings at the Medical University of Vienna.
Measurements of the cylinder phantom
FG  BG  SBR 

34.20  17.40  2 
42.70  10.80  4 
53.20  8.90  6 
75.30  9.40  8 
Finally, the algorithm was applied on ^{68}GaPET data of patients suffering from disseminated neuroendocrine carcinoma that was supplied by the European Neuroendocrine Tumor Society (ENETS) Center of Excellence at the Zentralklinik Bad Berka. Lesions in the liver and lymph nodes from eight patients were segmented, and the resulting volumes were compared to the manual delineation of the experts from the ENETS Center. The imagederived SBRs of the lesions in the lymph nodes are in the order of 15, whereas for the liver they are ≤ 2.
Statistical model
With regard to the smallest object diameter under consideration, which is only twice as large as the voxel size, PVE is a very dominating effect. It is therefore necessary to formulate partial memberships of voxels, as provided by the probability theory.
The elements z _{ nk } are binary, z _{ nk }∈{0,1}, with z _{ nk }=1 indicating that voxel n belongs object k (note that z _{ nl }=0 for l≠k, i.e., the row sums of Z equal 1).
PET image segmentation amounts to estimating the unknown label matrix Z given the data x. The proposed algorithm consists of two consecutive steps: the coarse estimation step fits a basic model, yielding fairly good initial estimates. These estimates are then improved in the correction step. The coarse estimation is performed via a modified expectationmaximization algorithm for a Gaussian mixture model (EMGMM) using information from the analysis of phantom data. The correction step compensates for overestimation of small volumes by sampling from a Gaussian MRF, using Gibbs distributions to obtain the final labeling. The Gibbs interaction parameters are chosen to act only on voxels at the boundary of two objects, i.e., voxels whose neighbors are attributed to different objects in the coarse estimation step.
Here, τ _{ k } denotes the prior probabilities (normalized suchlike that \(\sum _{k=1}^{K}\tau _{k}=1\)) and \(\boldsymbol {\Theta }=\left (\begin {array}{l} \mu _{1} \dots \mu _{K} \\ \sigma _{1} \dots \sigma _{K} \end {array}\right)\)is a 2×K matrix containing all the Gaussian means and standard deviations.
Here, \(\mathcal {M}_{n}\) denotes the (firstorder) neighborhood of the nth voxel, i.e., all voxels that share a surface with the nth voxel, hence \(\mathcal {M}_{n}=6\). Furthermore, the coupling matrix is given by \(\boldsymbol {\Gamma } =\left (\begin {array}{ll} 0 \!&\! \gamma \\ \gamma \!&\! 0 \end {array}\right)\).
The specific choice of the coupling matrix Γ involves only voxels on the boundary between two objects. Since the voxels at the boundary of a FG object predominantly are part of the BG, the probability of labeling a boundary voxel as FG is decreased.
Coarse estimation step
where p(x , Z;Θ) is defined in (2).

The conditional expectation \(\bar {z}_{\textit {nk}}^{(i)} = E_{z_{\textit {nk}}x_{n};\theta _{k}^{(i)}}\{{z}_{\textit {nk}}\}\) is updated as:$$ \bar{z}_{nk}^{(i)} = \frac{\tau_{k}^{(i)}\,\mathcal{N} \left(x_{n};\mu_{k}^{(i)},\sigma_{k}^{(i)}\right) }{\sum_{k=1}^{2} \tau_{k}^{(i)}\, \mathcal{N}\left(x_{n};\mu_{k}^{(i)},\sigma_{k}^{(i)}\right) }. $$(7)

The parameter updates read:$$ \begin{aligned} \tau_{k}^{(i+1)} & = \sum\limits_{n=1}^{N} \frac{\bar{z}_{nk}^{(i)}}{N}, \\ \mu_{k}^{(i+1)} & = \frac{\sum\limits_{n=1}^{N} x_{n} \bar{z}_{nk}^{(i)}}{\sum\limits_{n=1}^{N} \bar{z}_{nk}^{(i)} },\\ \sigma_{2}^{(i+1)} & = \sqrt{\frac{\sum\limits_{n=1}^{N} (x_{n}  \mu_{2}^{(i+1)})^{2} \bar{z}_{n2}^{(i)} }{\sum\limits_{n=1}^{N} \bar{z}_{n2}^{(i)}}},\\ \sigma_{1}^{(i+1)} & = \sigma_{2}^{(i+1)}. \end{aligned} $$(8)

A soft labeling is achieved by using the conditional probabilities \(\bar {z}^{(i)}_{\textit {nk}}\):$$ \hat{z}^{(i)}_{nk} = \bar{z}^{(i)}_{nk}. $$(9)
Due to the usage of a prior probability for Z, the resulting parameter estimations are done according to weighted averages.
Correction step
We solve (10) numerically by Metropolis sampling according to the procedure described below using the labeling obtained in the first coarse estimation step as initialization with its parameters held fixed to obtain a refined segmentation.
denotes the local conditional probability of the labeling proposal. If \(P^{i+1}_{n} \ge {P^{i}_{n}}\), the new label is accepted. If \(P^{i+1}_{n} < {P^{i}_{n}}\), the proposal labeling is accepted with probability \(P_{n}^{i+1}/{P_{n}^{i}}\) and rejected with probability \(1P_{n}^{i+1}/{P_{n}^{i}}\).
Threshold methods
For clinical applications of volume determination in PET, values of 36% to 42% of the maximum value have been proposed as local threshold [5]. In this context ‘local’ means that the threshold is calculated using the VOI and not the whole image.
With ITM, regression curves are needed, i.e. measurements of a phantom at different SBRs. The percentage threshold yielding the true volume is calculated as function of the volume and the SBR. Using ITM for automatic segmentation, an initial threshold is applied to a VOI followed by the determination of the resulting volume (V) and SBR. With those values, the corresponding threshold from the regress function %Thr=f(V,SBR) is further applied to the VOI, resulting in an iterative update scheme. The algorithm stops when the deviation between two iterations is ≤0.1%.
For purposes of evaluation and comparison, we applied the EMGMM, the proposed GMRF algorithm, and the aforementioned thresholding methods to the phantom measurements as well as the clinical data discussed in the ‘Phantom measurements and clinical data’ section. A graphical user interface was built by the objectoriented programming language IDL to visualize and process the data and to draw VOIs around each FG object. To investigate whether the results depend on the VOI size, we used VOIs consisting of 14×14×20, 14×14×40, 22×14×20, and 22×14×40 voxels.
Results
Number of FG objects detected by 36% and 42% thresholding, ITM, EMGMM and GMRF
SBR  36%  42%  ITM  EMGMM  GMRF 

2.06  0  0  0  4  3 
3.84  0  2  4  5  5 
4.90  3  4  4  5  6 
6.72  4  4  5  5  6 
9.39  5  5  6  6  6 
Number of FG objects (cylinder phantom) detected at different SBRs by the ITM and the GMRF
SBR  ITM  GMRF 

2  0  0 
4  2  3 
6  3  3 
8  4  4 
Threshold methods
Figure 2a shows the results for local thresholding with a threshold of 42%. Clearly, for small FG objects, the activity concentration is underestimated due to PVE which leads to large volume errors. It is seen that for all FG objects the error increases with decreasing SBR. This also applies to ITM whose results are depicted in Figure 3a,b. Here, the volume error is averaged over the SBR in order to keep the amount of results readily comprehensible. No threshold method can detect FG objects at an SBR of 2.06. Furthermore only the ITM is able to detect the smallest FG object of 8mm diameter and only at the biggest SBR of 9.39.
EMGMM and GMRF
The results of the proposed GMRF method and its EMGMM initialization are shown in Figure 2c,b, respectively, for VOIs comprising 14×14×20 voxels. With regard to both spheres and cylinders, EMGMM as well as GMRF achieve a much better detection rate of FG objects than the threshold methods (see Tables 3 and 4). In particular, GMRF detects all six spherical FG objects at SBRs above 4.90 as well as the three larger spheres at SBR 2.03. The total detection rate score of the GMRF with 26/30 and 10/16 for spheres and cylinders, respectively, clearly supersedes the ITM with 19/30 and 9/16.
Concerning volume segmentation, GMRF achieves lower errors than the fixed threshold approaches for almost all combinations of SBR and diameter. In a direct comparison of GMRF and ITM (see Figure 3a,b), ITM performs better on smaller spheres while GRMF is better on larger ones. For cylindrical objects, ITM shows slightly smaller errors for diameters ≥15 mm. Figure 3c shows the DSC which was calculated for the phantom with spherical objects as a comparison of GMRF segmentation vs. ground truth from the CT. For objects with diameters ≥13 mm and SBR ≥3.84, the DSC ≥0.8.
Comparison to thresholdindependent algorithms
Figure 2e,f shows the results obtained by Hatt and coworkers using a fuzzy locally adaptive Bayesian (FLAB) algorithm [38] and by the maximum a posteriori estimation for a GMRF with deconvolution (MAPMRFDECON) designed by Gribben et al. [39], respectively (the data has been manually copied from the above cited publications). For further comparison, a statistical approach by DewalleVignion and coworkers acting on maximum intensity projections (PROP MIP) [40] including the fuzzy cmeans technique is shown in Figure 2d. While those papers also performed NEMA phantom measurements, FG objects with different diameter ranges were used. In particular, none of the previous works used a FG object as small as 8 mm of diameter or an SBR as low as 2.06.
Comparing GMRF with the MAPMRFDECON algorithm, the former performs better for some diameters and worse on others. However, a systematic comparison is not possible since [39] provides results only for the rather large SBR of 9. PROP MIP and FLAB yield quite accurate volumes for spheres greater than 10 mm but performs poorly when it comes to spheres smaller than 13mm diameter (the smallest sphere used in [40] and [38] is 10 mm). At such diameters, GMRF approaches have to be preferred due to their better performance.
VOI dependencies of GMRF
Finally, Figure 4 shows the results obtained with GMRF for different VOI sizes. As can be seen for all spheres with diameter ≥10 mm, the volume errors are constant over the chosen VOI range with regard to all SBRs. Concerning the 8mm sphere, no correlation can be found between the detection rate and VOI size. However, if detected, the volume error is also constant over the VOI range.
Clinical data
With regard to the clinical data, GMRF’s segmentation of the metastases in the lymph nodes yields volumes which are 15% to 20% bigger than the manual delineation. This value is constant over a volume range down to 3 ml wherefrom the overestimation gets bigger. Taking into account that the SBR for the lesions in the lymph nodes is 15, this behavior is very similar to the measurements of the NEMA spheres at higher SBRs. Likewise, the GMRF’s segmentation of lesions in the liver with very small SBRs (≤2) shows a volume which is constantly smaller by 30% and therefore reflects very well the behavior of phantom measurements at very low SBRs.
Discussion
Forcing equal standard deviations for BG and FG as discussed in the ‘Coarse estimation step’ section helped to reduce the segmentation error. While this approach appears somehow arbitrary, it can be qualitatively understood as follows: Figure 1 shows the SBR diversification for the normalized standard deviation \(\tilde \sigma _{1}\) of FG. For higher SBR, \(\tilde \sigma _{1}\) and therewith the volume estimates increase because the PVE voxels get encompassed by the FG object. By setting σ _{1}=σ _{2}, the PVE voxels are interpreted as BG such that in the subsequent estimate σ _{2} increases. As a result, also σ _{1} increases and parts of the PVE voxels are reassigned to FG, in turn lowering σ _{1}. This interplay causes a conversion towards the solutions shown in the ‘Results’ section.
Likewise, the value γ=1,000 was introduced without further explanation. Numerical calculations have shown that values for γ between 500 and 1,500 yield stable nonzero volume estimates for the FG objects instead of downgrading the volumes until they vanish (increasing the amount of neighborhood voxels \(\mathcal {M}_{n}\) leads to zero volume solutions). Since the coarse estimation step already yields good initial label estimates with a large value of γ as defined above, the system reaches equilibrium after the first three full samples have been obtained. Storing more than 70 subsequent configurations at equilibrium according to (12) does not change the solutions presented here.
When comparing the proposed GMRF method and its initializing EMGMM (see Figure 2c,b), it becomes apparent that the proposed correction step enforced in the GMRF decreases the volume error for small FG objects as intended by the choices described in the ‘Correction step’ section. This emphasizes the usefulness of MRFs as a powerful tool regarding PET image segmentation, especially in the case of small objects. In this sense, the findings presented here confirm the work of Gribben et al. [39] that also uses MRFs. Nonetheless, it can be seen in Figure 2 that the initialization step is not fully corrected for SBR diversification, especially for small FG objects. Acting with Gibbs distributions on this solution as described above is not sensible to different SBRs. Therefore, future work should aim for compensating SBR dependencies in the correction step.
The results of the GMRF also show that for spheres with diameter ≥13 mm, the coldwall effect can be accounted for. For smaller spheres, the coldwall effect gets noticeable as for all segmentation approaches in the literature [840].
The choice of the comparison methods shall be briefly outlined. Despite the existence of iterative threshold approaches (see the ‘Background’ section), fixed thresholds are very common in clinical practice. As we can see in the ‘Results’ section, GMRF clearly outreaches the fixed threshold approaches both in the detection rate and volume error, a result not very surprising. However, the fact that GMRF also outperforms ITM on detection rate and produces similar results with regard to the volume error and DSC is remarkable all the more, as GMRF uses no a priori knowledge whatsoever whereas ITM has a the luxury of a validated calibration curve. Taking into account the dependencies of these regression curves (see the ‘Background’ section) and as a consequence thereof the need of not only one but many phantom calibration measurements, GMRF seems a much more practical approach for clinics since it is equally reliable but can be directly applied for different systems, nuclide, reconstructions, etc. Furthermore, an ITM only can realize a segmentation of whole voxels whereas the inclusion of probability theory as in our approach can assign partial classifications to voxels which is inevitable in the case of small volumes. The comparison with the thresholdindependent algorithms also shows that the algorithm in this work performs equally well for larger volumes and even shows improvements for small volumes and SBRs.
When discussing the segmentation of the clinical data, one has always to recall that unlike the real volume of the phantoms the manual delineation does not represent the ground truth, i.e., it can only be a comparative analysis. Systematic overestimation as well as underestimation of the tumor volume has an extensive impact on radiation treatment planning. In practice, the socalled planning target volume (PTV) is an extension of the visible tumor taking into account microscopic spread as well as uncertainties in dose delivery and patient positioning. Therefore, a systematic overestimation will partially coincide with the PTV and be of no severe consequences as long as the overlap is not too big and thus causes widespread damage to the surrounding healthy tissue. In contrast, a systematic underestimation involves the danger of not irradiating the entire tumor and thus enhances the chances for relapse.
A mean DSC of 0.9 for high SBRs, the robustness of the results over a large volume range, and the resemblance to the results of the phantom measurements concerning the SBR dependency make the GMRF a suitable candidate for future studies that encompass patient data of higher quantity and larger diversity. Moreover, to study realistic images with inhomogeneous activity distributions and be in possession of the ground truth, simulated images have to be included as also done by various authors [11,4043].
the number of EMGMM iterations in the case of detection did not exceed 50. Given a standard laptop (dual core 2×1.8 GHz), the overall processing time stays well below 1 s even for the case of large VOI. In this time frame, it is even feasible to use the algorithm several times to repeat the processing with subVOIs to improve the detection rate for small objects.
Conclusions
Segmentation of PET data remains a very challenging issue since pertinent algorithms are very sensitive to a variety of parameters. The aim of this work was to investigate the aspects that were not sufficiently covered in the literature so far, namely the impact of the SBR, the segmentation of objects with a diameter smaller than 10 mm, and the waiver of any a priori knowledge such as a regression curve. Therefore, phantom measurements with spherical as well as nonspherical objects with SBRs ranging from 2 to 9.36 have been evaluated, including a FG sphere with a diameter of 8 mm. Additionally, lesions from clinical data have been segmented.
Combining an EMGMM with MRFs, taking advantage of Gibbs distributions to describe neighbor dependencies, results in a significant decrease of the overestimation of small volumes on the one hand and on the other hand yields vanishing volume errors in the case of bigger objects and high SBRs. The proposed algorithm has advantages over threshold methods and can be applied to any PET data, not requiring any systemspecific regression curves in order to account for the given nuclide, manufacturer, or reconstruction algorithm. The comparison with experts’ manual delineations on clinical images shows the same qualitative behavior as for the phantom measurements. In connection with its rapidness and the insensitivity towards reconstruction algorithm and lesion shape, the proposed algorithm is a suitable choice for PET segmentation, even though there is still room for improvement in future work.
Declarations
Acknowledgements
This work was cofunded by the Austrian Federal Ministry for Transport, Innovation and Technology within the program ModSim Computational Mathematics which is part of the program Research, Innovation, Technology and Information Technology.
Authors’ Affiliations
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