Higher-order singular value decomposition-based lung parcellation for breathing motion management

Abstract.

Positron emission tomography (PET) imaging of the lungs is confounded by respiratory motion-induced blurring artifacts that degrade quantitative accuracy. Gating and motion-compensated image reconstruction are frequently used to correct these motion artifacts in PET. In the absence of voxel-by-voxel deformation measures, surrogate signals from external markers are used to track internal motion and generate gated PET images. The objective of our work is to develop a group-level parcellation framework for the lungs to guide the placement of markers depending on the location of the internal target region. We present a data-driven framework based on higher-order singular value decomposition (HOSVD) of deformation tensors that enables identification of synchronous areas inside the torso and on the skin surface. Four-dimensional (4-D) magnetic resonance (MR) imaging based on a specialized radial pulse sequence with a one-dimensional slice-projection navigator was used for motion capture under free-breathing conditions. The deformation tensors were computed by nonrigidly registering the gated MR images. Group-level motion signatures obtained via HOSVD were used to cluster the voxels both inside the volume and on the surface. To characterize the parcellation result, we computed correlation measures across the different regions of interest (ROIs). To assess the robustness of the parcellation technique, leave-one-out cross-validation was performed over the subject cohort, and the dependence of the result on varying numbers of gates and singular value thresholds was examined. Overall, the parcellation results were largely consistent across these test cases with Jaccard indices reflecting high degrees of overlap. Finally, a PET simulation study was performed which showed that, depending on the location of the lesion, the selection of a synchronous ROI may lead to noticeable gains in the recovery coefficient. Accurate quantitative interpretation of PET images is important for lung cancer management. Therefore, a guided motion monitoring approach is of utmost importance in the context of pulmonary PET imaging.

1. Introduction

Respiratory motion poses a challenge to positron emission tomography (PET) quantitation for both diagnostics and therapeutics of thoracoabdominal diseases. PET plays a vital clinical role in the staging and treatment evaluation of pulmonary lesions.^1,2 Addressing the quantitation challenges posed by motion-induced blurring is critical for cancer management. Diaphragm displacement along the superior–inferior direction during normal breathing can be as high as 20 mm locally.³ A variety of methods have been developed for lung motion tracking that seek to generate PET images devoid of motion artifacts.^4–6 Image-based methods using four-dimensional (4-D) computed tomography (CT)^7–10 or 4-D magnetic resonance (MR) imaging^11–13 can generate internal (voxel-by-voxel) distributions of displacement, which can then be employed for motion-compensated image reconstruction (MCIR) of PET images.^11–17 However, 4D imaging techniques require either a costly simultaneous PET/MR scanner or complicated protocols for synchronizing breathing patterns across sequential PET and 4-D CT acquisitions. The prevalent approach for tackling respiratory motion in PET is, therefore, respiratory gating, which relies on low-cost external motion tracking instruments that generate a surrogate respiratory signal.^18–24 Available instrumentation for respiratory gating includes pneumatic bellows²⁵ and systems with pressure sensors mounted on elastic belts, such as the Anzai AZ-733.²⁶ Other gating devices, such as the Varian Real-time Position Management System (Varian Medical Systems, Palo Alto, California),^19,27 perform infrared camera-based monitoring of external optical markers placed on the chest surface. More recently, three-dimensional (3-D) camera systems based on the Microsoft Kinect have been used to derive global motion information for the torso.²⁸ These devices generate surrogate signals based on motion trajectories and/or pressure responses of the chest or abdominal surface. In light of the popularity of low-cost external motion tracking, several studies have attempted to quantify the relationship between internal and external motion measures. Strong correlations between abdominal wall marker motion and diaphragm motion have been reported in literature.²⁹ Internal–external correlations were examined in the specific context of lung tumor motion in many studies.^30–35 While these studies report high correlations between external motion and tumor motion, they also suggest a strong dependence of the correlation on the external marker position. Additionally, these results are dependent on the tumor locations in the specific clinical cases analyzed. To ensure motion tracking efficacy, it is important to position the external markers to achieve motion synchronization with the internal target.^33,36,37 Some studies have attempted to analyze the regional variation in internal–external correlations with a focus on optimal marker positioning.^37–39 However, the selection of external locations and/or regions of interest (ROIs) was subject-specific and lacked a common group-level framework for examining data from multiple subjects.

In this paper, we present a group-level motion-guided lung parcellation technique based on higher-order singular value decomposition (HOSVD). This framework enables us to (1) identify distinct motion signatures inside the body and on the skin surface, (2) cluster the voxels both inside the volume and on the surface, and (3) delineate internal and external ROIs with the highest correlations in motion patterns. The parcellation approach generates a group-level deformation template for the lungs that can be applied to guide external marker positioning for PET motion tracking depending on which motion-guided ROI the clinical feature of interest (e.g., a tumor) belongs to. The parcellated deformation template presented in this work relies on 4-D deformation data collected using a specialized MR pulse sequence that allows motion tracking under free-breathing conditions. The parcellation result is validated by examining correlations within and across the external and internal ROIs. The robustness of the technique is assessed by means of leave-one-out cross-validation over the subject cohort and by examining the effects of choosing varying numbers of gates and singular value thresholds. Details on HOSVD-based clustering are presented in Sec. 2. Section 3 provides a description of the data, details of data analysis, and evaluation metrics used to characterize/validate the parcellation approach. The motion ROIs generated using the parcellation approach, correlation results, assessments of parcellation robustness, and examination of overall impact on PET reconstruction are presented in Sec. 4. A summary and a discussion of this method are presented in Sec. 5.

2. Theory

2.1. Overview

Deformation fields are typically computed from 4-D images by first selecting a reference phase of motion or “reference gate” and nonrigidly registering all other gates to it. Intragate motion is typically assumed to be negligible. Collating the deformations in the three Cartesian directions, an $n_x\times n_y\times n_z\times n_g$ image leads to a deformation tensor of size $n_x\times n_y\times n_z\times 3\times (n_g-1)$. For group-level motion analysis, the deformation tensors from individual subjects can be concatenated to generate a higher-dimensional motion tensor. Here, we construct such a tensor and apply HOSVD to compute a set of motion-based features or signatures which are then used to cluster the 3-D volume into a number of motion ROIs.

2.2. Higher-Order Singular Value Decomposition Basics

To elucidate the HOSVD procedure, we first introduce the $k$-mode matrix unfolding and the $k$-mode tensor product here.

2.2.1. $\mathsf{k}$-mode matrix unfolding

Matrix unfolding involves reordering the elements of a tensor array into a two-dimensional matrix. Given a tensor $\mathcal{P}\in {\mathbb{R}}^{t_1\times t_2\times \dots \times t_N}$, the $k$-mode ($k=\mathrm{1,2},\dots ,N$) matrix unfolding is a matrix ${\mathit{P}}_{(k)}\in {\mathbb{R}}^{t_k\times ({\mathrm{\Pi }}_{l,l\ne k}{t}_l)}$. For example, given a tensor $\mathcal{A}\in {\mathbb{R}}^{m\times n\times p\times q}$, the $k$-mode ($k=\mathrm{1,2},\mathrm{3,4}$) matrix unfoldings are denoted by ${\mathit{A}}_{(1)}$, ${\mathit{A}}_{(2)}$, ${\mathit{A}}_{(3)}$, and ${\mathit{A}}_{(4)}$, respectively. Then, the unfolding ${\mathit{A}}_{(1)}$ is a matrix with $m$ rows and $npq$ columns.

2.2.2. $\mathsf{k}$-mode tensor product

Given a tensor $\mathcal{P}\in {\mathbb{R}}^{t_1\times t_2\times \dots \times t_N}$ and a matrix $\mathcal{Q}\in {\mathbb{R}}^{t_k\times s}$, the $k$-mode tensor product leads to a tensor $\mathcal{T}=\mathcal{P}{\times }_k\mathcal{Q}$, where $\mathcal{T}\in {\mathbb{R}}^{t_1\times t_2\times \dots \times t_{k-1}\times s\times t_{k+1}\times \dots \times t_N}$. For example, the 1-mode product between $\mathcal{A}\in {\mathbb{R}}^{m\times n\times p\times q}$ and a matrix $\mathit{B}\in {\mathbb{R}}^{m\times s}$, denoted by $\mathcal{A}{\times }_1\mathit{B}$, is a $(s\times n\times p\times q)$ tensor whose entries are given as

$${(\mathcal{A}{\times }_1\mathit{B})}_{ijk}=\sum _i{a}_{ijk}.$$ (1)

2.2.3. Higher-order singular value decomposition computation

HOSVD decomposes a tensor into a core tensor multiplied by a matrix along each mode. The HOSVD of the tensor $\mathcal{A}\in {\mathbb{R}}^{m\times n\times p\times q}$ is defined as

$$\mathcal{A}=\mathcal{S}{\times }_1\mathit{U}{\times }_2\mathit{V}{\times }_3\mathit{W}{\times }_4\mathit{X},$$ (2)

where $\mathit{U}\in {\mathbb{R}}^{m\times m}$, $\mathit{V}\in {\mathbb{R}}^{n\times n}$, $\mathit{W}\in {\mathbb{R}}^{p\times p}$, and $\mathit{X}\in {\mathbb{R}}^{q\times q}$ are orthonormal factor matrices that can be thought of as the principal components of the original tensor along each mode and the tensor $\mathcal{S}\in {\mathbb{R}}^{m\times n\times p\times q}$ is called the core tensor with its elements showing the level of interaction between the different components. Given a tensor $\mathcal{A}$, its matrix factors $\mathit{U}$, $\mathit{V}$, $\mathit{W}$, and $\mathit{X}$ can be calculated as the left singular vectors of its matrix unfoldings ${\mathit{A}}_{(1)},{\mathit{A}}_{(2)},{\mathit{A}}_{(3)}$, and ${\mathit{A}}_{(4)}$, respectively.

2.3. Feature Space Generation and Clustering

We represent the deformation fields as a tensor $\mathcal{D}\in {\mathbb{R}}^{n_n\times n_g\times n_p\times 3}$, where $n_n$ represents the total number of voxels (nodes) in the 3-D image, $n_g$ represents the number of respiratory gates, $n_p$ represents the number of subjects, and the fourth dimension represents the deformation orientation. HOSVD is performed on the 1-mode unfolding of this tensor, ${\mathit{D}}_{(1)}\in {\mathbb{R}}^{n_n\times 3n_g{n}_p}$ to obtain the orthonormal basis set $\mathit{U}\in {\mathbb{R}}^{n_n\times n_n}$. In subspace analysis based on principal component analysis (PCA), $\mathit{U}\in {\mathbb{R}}^{n_n\times n_n}$ is truncated to $\tilde{\mathit{U}}\in {\mathbb{R}}^{n_n\times n_t}$, where the truncation point is determined by examining the singular value spectrum.

For data-driven generation of motion ROIs, $k$-means clustering is performed on the HOSVD-derived features over the full volume. The $k$-means++ technique was used to select $k$ centroid seeds.⁴⁰ Repeated runs of 10 replicates were used to ensure convergence. Intensity-based thresholding was performed to create a binary 3-D mask for the torso from the reference gate MR image. For a digital rendering of the outer surface where external markers are generally placed, a triangulated mesh with 4059 nodes and 7760 faces was generated from the torso mask using the MATLAB iso2mesh toolbox.⁴¹

3. Methods

3.1. Data Description

A golden-angle radial navigated gradient echo (GRANGE) pulse sequence was used in conjunction with sparsity-enforcing $k$-$t$ focal underdetermined system solver (FOCUSS) reconstruction to generate 4-D MR images used for computing group-level volumetric motion measures.^13,15 This sequence uses a fast slice-projection navigator to track the diaphragm position for each repetition time (TR). Clinical data were acquired from five subjects (three males and two females, age range: 24 to 65 years) on a Biograph mMR whole-body simultaneous PET/MR scanner (Siemens Medical Systems, Erlangen, Germany). All procedures were performed following protocol and institutional standards with prior approval from the institutional review boards at Massachusetts General Hospital, Boston, Massachusetts and SDN, Naples, Italy, the two data acquisition sites. Data were acquired with the subjects under free-breathing conditions. The MR acquisition parameters were as follows: 5-mm slice thickness, 24 coronal slices, 1-kHz/pixel bandwidth, 4096 radial lines with 256 samples/line, 3.3-ms TR, and 30-deg excitation angle. The $k$-space data were binned into six gates based on the navigator information using an optimal binning strategy as previously described.¹³ Iterative reconstruction was performed using $k$-$t$ FOCUSS, a reconstruction technique based on compressive sensing^42,43 which enables robust reconstruction in the presence of undersampling and suppresses streaking artifacts that are commonplace in undersampled radial acquisition schemes. Figures 1(a) and 1(b) show the MR images corresponding to the end-inspiration and end-expiration gates, respectively.

Fig. 1.

Sample GRANGE MR images and deformation fields. (a) End-inspiration gated MR image. The red line near the diaphragm has been included for position reference. (b) End-expiration gated MR image. (c) Deformation fields computed via nonrigid, intrasubject, intergate registration are visualized using blue arrows overlaid on the MR.

3.2. Motion Estimation

Nonrigid registration based on diffeomorphic demons was used to compute deformation fields and perform both intrasubject and intersubject warping.⁴⁴ The motion estimation steps are as follows:

• •The first registration step involves intrasubject and intergate nonrigid registration that maps a reference gate to all other gates. This is performed separately for each of the five subjects, for each of whom six gated images were first computed. For the intrasubject registration step, a least-squares cost function is minimized. In our studies, we used the end-expiration phase as the reference gate. This selection was guided by previous studies that have shown that, in a typical breathing cycle, patients spend the longest amount of time in an end-expiration quiescent phase.^45,46
• •To establish a common framework for group-level analysis, a reference subject is selected and all deformation data are mapped to this space by performing intersubject registration. For intersubject registration, the cost function that is minimized is normalized mutual information. Intersubject transformations are first computed by registering the reference gate MR image of any subject to the reference gate MR image of the reference subject. This intersubject (nonrigid) transformation is then applied to warp the deformation maps to a common reference space for group-level analysis.

For the clinical cohort analyzed, the observed maximum deformation varied from 11.8 to 18.1 mm across subjects. The demons algorithm utilized for nonrigid registration uses fluid-like regularization based on a Gaussian convolution kernel applied to the deformation field update. The fluid regularization parameter was set to a value of 13 for intrasubject registration and 8 for intersubject registration. Mutual information was used as the similarity metric for both the intrasubject and intersubject registration steps. Figure 1(c) shows the deformation fields computed by nonrigidly registering the GRANGE MR images of one subject shown in Figs. 1(a) and 1(b) (corresponding to the end- and end-expiration gates, respectively). The deformation directions and magnitudes are indicated using a set of arrows overlaid on the end-inspiration MR image.

3.3. Simulation Setup

We conducted a simulation experiment to assess the effect of marker placement in various data-driven ROIs on gating and final PET image quality. The experiment was based on a digital torso phantom generated from an intensity-based segmentation of the MR image of one of the five subjects. Four spherical pulmonary lesions each of diameter 12 mm were added to the lungs, two in the right lung and two in the left lung. Standardized uptake values were assigned to emulate [${}^{18}\mathrm{F}$]FDG distributions: lungs 0.5, muscle 1, and lesions 3 (all in units of g/mL). Thirteen respiratory gates were simulated by linear interpolation of previously computed deformation fields. For gate assignment, the deformation fields at four external locations, each belonging to a different ROI, were extracted, and the Euclidean norm of the displacements along three Cartesian directions was used to compute a motion trace (time series curve). Amplitude-based thresholding was applied to this motion trace to find a quiescent period corresponding to a 40% of maximum amplitude threshold.⁴⁶ The mean of the corresponding image frames was projected using the PET system model to generate sinogram data. Poisson deviates of the sinogram mimicking realistic clinical count rates were generated. The ordered subsets expectation maximization algorithm was applied to reconstruct the final set of gated PET images.

3.4. Evaluation Metrics

3.4.1. Correlation coefficient

To characterize the motion ROIs, we used the Pearson’s correlation coefficient as the figure-of-merit. We compute the mean and standard deviation values for the Pearson’s correlation coefficients between internal and external motion for each ROI. For a given volumetric (internal) node ${\mathit{x}}_i$ and a superficial (external) node ${\mathit{x}}_e$, the correlation coefficient ($\rho$) measures a linear/affine relationship defined as follows:

$$\rho ({\mathit{x}}_i,{\mathit{x}}_e)=\frac{E[(d_i-{\mu }_i)(d_e-{\mu }_e)]}{{\sigma }_i{\sigma }_e},$$ (3)

where $d_i$ and $d_e$ are the internal and external deformations in a given direction, respectively, ${\mu }_i$ and ${\mu }_e$ represent the population means, and ${\sigma }_i$ and ${\sigma }_e$ are the population standard deviations for the internal and external nodes, respectively.

3.4.2. Jaccard index

To test the robustness of the parcellation result, we employed the Jaccard index, which, for two finite sample sets $A$ and $B$, is defined as the size of the intersection divided by the size of the union:

$$J(A,B)=\frac{A\cap B}{A\cup B}.$$ (4)

This similarity measure assumes values in the range 0 to 1.

3.4.3. Recovery coefficient

To compare PET reconstruction results, we computed the recovery coefficient⁴⁷ as the ratio of the measured intensity over true intensity for a set of simulated lung lesions.

4. Results

4.1. Parcellation Result

Figure 2 shows the HOSVD spectrum obtained from the deformation tensor from five subjects. Based on this spectrum, we retain the $n_t$ largest singular values for a reduced rank representation. Subsequently, $k$-means clustering was performed. The number of clusters is decided by the number of features (singular vectors) to be captured. Figure 3 shows the result of $k$-means clustering for $k=4$ and $n_t=4$. The cluster indices were sorted according to the mean amplitude of deformation for each ROI (1 = highest and 4 = lowest).

Fig. 2.

Plot showing the first 10 singular values. An index threshold of $n_t=4$ was chosen to generate a low-rank approximation.

Fig. 3.

Internal (volumetric) and external (surface) ROIs. (a) Coronal slices from the anatomical MR image for reference (posterior end: top left and anterior end: bottom right). (b) Volumetric (internal) distribution of ROIs shown on coronal slices in correspondence with the MR slices. (c) Surface (external) distribution of the same ROIs plotted on a triangle mesh corresponding to the outer surface of the torso. The color bar indicates ROI indices for both the volume and the surface visualizations.

Figure 3(a) shows coronal slices (posterior end: top left and anterior end: bottom right) from the anatomical MR image. The corresponding coronal slices showing the volumetric (internal) distribution of the ROIs are shown in Fig. 3(b). The occurrence of these ROIs on the skin (external) surface is depicted on the outer torso mesh in Fig. 3(c). Comparison with the anatomy reveals that ROI 1 corresponds roughly to the base of the lungs and the upper abdomen, ROI 2 captures the rest of the lungs, ROI 3 spans an area to the front (anterior) part of the chest wall, and ROI 4 covers the outer extremities of the torso, including parts that exhibit very little motion such as the apex of the lungs, the carina, and the back muscles.

Figure 4 shows the distribution (mean and standard deviation) of the projection coefficients for the $n_t=4$ largest singular vectors and for the four ROIs. ROIs 1 and 2, which coincide with areas of maximum deformation along the superior–inferior direction lying in the vicinity of the diaphragm, have the largest projection coefficients. Deformation patterns corresponding to basis vectors 1 and 2 (the two largest singular vectors) are most effectively captured by these two ROIs. Singular vector 3 is mainly captured by ROI 3. Singular vector 4 is represented by ROI 4.

Fig. 4.

Projection coefficient distribution for each ROI. Mean and standard deviation (represented by error bars) of the projection coefficients of the four largest singular vectors and for the four ROIs.

4.2. Internal–External Correlation Assessment

Pearson’s correlation coefficients were computed at each surface node with respect to the mean deformation vector over all voxels belonging to a given ROI. The mean and standard deviation of the internal–external correlations for each ROI are shown in Fig. 5(a). Figure 5(b) shows the surface area on the torso occupied by each ROI. While ROIs 1 and 2 have relatively large correlation coefficients, these cover only about 2% to 8% of the body surface. In comparison, ROI 3 led to a slightly higher correlation and comprised close to 40% of the body surface making it an easier target for marker placement. Since the internal features of interest (e.g., tumors) could span multiple ROIs/clusters and since the most critical ROIs are relatively underrepresented on the surface, we also examined internal–external correlations across different ROIs. The absolute values of the cross-correlations averaged over all the surface nodes in each ROI are shown in Fig. 5(c).

Fig. 5.

(a) Mean and standard deviation (marked by error bars) of the internal–external correlation coefficients for each ROI. Statistics were computed over the surface nodes. (b) Surface area fraction for each ROI. (c) Mean of the absolute values of the between-ROI internal–external correlation coefficients.

4.3. Template Variability Assessment

4.3.1. Dependence on subject group

To assess the variability of the parcellation method over different subject groups, leave-one-out cross-validation was performed. The parcellation was generated five times, leaving out one of the five subjects at each run. The results were compared with the parcellation based on the full set. Both internal and external visualizations of the results are shown in Fig. 6. Jaccard coefficients measuring the similarity of the leave-one-out parcellations with the full parcellation are shown in Table 1. Extents of overlap as measured by this metric ranged from 61% to 89%.

Fig. 6.

Leave-one-out cross-validation. (a) Internal ROIs based on the indicated subject groups plotted on a single coronal slice from the torso. (b) The corresponding external ROIs based on the same subject groups plotted on a triangle mesh.

Table 1.

Jaccard indices for leave-one-out cross-validation.

Subject group	ROI 1	ROI 2	ROI 3	ROI 4
All minus subject 1	0.6904	0.6836	0.7478	0.8767
All minus subject 2	0.7482	0.7384	0.7758	0.8893
All minus subject 3	0.7600	0.7715	0.7416	0.8899
All minus subject 4	0.7677	0.7469	0.7299	0.8788
All minus subject 5	0.7411	0.6393	0.6148	0.8079

4.3.2. Dependence on gate number

To assess the variability of the method depending on the grouping of gates, the parcellation was computed using only the end-inspiration and end-expiration gates (one set of intergate deformation fields), three gates (two sets of fields), and all six gates (full set of fields). The results were compared with the parcellation based on the full set. Both internal and external visualizations of the results are shown in Fig. 7. Jaccard coefficients measuring the similarity of the fewer-gate parcellations with the full parcellation are shown in Table 2. Extents of overlap as measured by this metric ranged from 81% to 96%.

Fig. 7.

Comparison of 2-gate, 3-gate, and 6-gate results. (a) Internal ROIs based on the indicated gate numbers plotted on a single coronal slice from the torso. (b) The corresponding external ROIs based on the same gate numbers plotted on a triangle mesh.

Table 2.

Jaccard indices for gate dependence.

Number of gates	ROI 1	ROI 2	ROI 3	ROI 4
3	0.9109	0.8983	0.9088	0.9560
2	0.8488	0.8199	0.8062	0.9107

4.3.3. Dependence on higher-order singular value decomposition truncation

To assess the variability of the method depending on the number of HOSVD basis vectors retained after truncation ($n_t$), the parcellations computed using $n_t=3,5$, and 6 were compared with the previously shown results for $n_t=4$. Both internal and external visualizations of the results are shown in Fig. 8. Jaccard coefficients measuring the similarity of the parcellations for $n_t=3,5$, and 6 with $n_t=4$ are shown in Table 3. Extents of overlap as measured by this metric ranged from 90% to 99%.

Fig. 8.

Comparison of results for $n_t=3,4,5$, and 6. (a) Internal ROIs based on the indicated HOSVD truncation thresholds plotted on a single coronal slice from the torso. (b) The corresponding external ROIs plotted on a triangle mesh.

Table 3.

Jaccard indices comparing $n_t=4$ with $n_t=3,5$, and 6.

$n_t$	ROI 1	ROI 2	ROI 3	ROI 4
3	0.9828	0.9802	0.9028	0.9690
5	0.9909	0.9840	0.9421	0.9814
6	0.9670	0.9580	0.9150	0.9698

4.4. Impact on Positron Emission Tomography Reconstruction

The ground truth PET image corresponding to the reference frame is shown in Fig. 9(a). The reconstructed images corresponding to external markers placed in ROIs 1 through 4 are shown in Figs. 9(b)–9(e), respectively. Recovery coefficients were computed for the four simulated lesions (L1: 16-mm-diameter lower right lung, L2: 12-mm-diameter lower left lung, L1: 16-mm-diameter upper right lung, and L1: 12-mm-diameter upper right lung) and are shown in Table 4. The lower lung lesions, both located in ROI 1, showed the highest recovery coefficients with marker placement in RO1 I. For the larger lesion in the right lower lung, the recovery coefficient improved by about 5% relative to marker placement in ROI 2 and about 20% relative to marker placement in ROI 4. For the smaller lesion in the left lower lung, the recovery coefficient improved by about 4% relative to ROI 2 and about 13% relative to ROI 4. The upper lung lesions, less affected by motion, exhibited less dependence on the external marker location as indicated by recovery coefficient values.

Fig. 9.

PET simulation results. (a) True PET image. Reconstructed PET images with the external marker placed in (b) ROI 1, (c) ROI 2, (d) ROI 3, and (e) ROI 4.

Table 4.

Lesion recovery coefficients.

Lesion index	ROI 1	ROI 2	ROI 3	ROI 4
L1	0.7711	0.7347	0.6773	0.6428
L2	0.6897	0.6645	0.6455	0.6087
L3	0.7761	0.7722	0.7792	0.7784
L4	0.7091	0.6862	0.6649	0.6829

5. Conclusion

Motion tracking systems for PET gating and MCIR assume high correlations between external marker and internal target motion. Here, we have presented a framework for group-level motion-based lung parcellation that applies HOSVD-based clustering to deformation fields extracted from 4-D MR data from five subjects. Using this framework, we are able to identify potential locations for marker placement which efficiently capture the key signatures of internal motion. Unlike previous studies, which were limited to some specific external marker sites and internal anatomical landmarks, our method allows us to examine correlations across different external and internal ROIs determined by the deformation patterns from a higher-dimensional group-level tensor.

Typically, the detection of cancerous lesions in the lungs relies on a diagnostic CT scan. PET, on the other hand, is the modality of choice for staging and treatment evaluation of lung lesions. The clinical application of this group-level parcellation approach will, therefore, rely on the availability of a 3-D CT scan to pinpoint the location of the lesion or target an area of interest. Nonrigid registration can then be used to warp the reference 3-D MR template from this work to the diagnostic CT image for an individual subject. This would enable mapping of group-level respiratory motion information and data-driven clusters generated from group-level HOSVD to an individual subject. Our correlation analysis results could then guide external marker placement for a subsequent PET scan. The acquired PET raw data can be optimally binned for gating. The envisioned overall data acquisition and processing workflow is shown in Fig. 10.

Fig. 10.

Potential process flow for utilizing the group-level parcellated template for optimizing motion tracking.

It should also be noted that the clinical utility of this MR-based template will rely on effective MR-CT coregistration. Existing literature shows that deformable registration based on elastic regularization coupled with common cross-modality similarity measures, such as normalized mutual information^48,49 or the normalized correlation coefficient,⁵⁰ can be successfully used for MR-CT torso or whole-body registration. However, in this case, the deformable registration problem involves a deep-inspiration diagnostic CT and a free-breathing MR and could be challenging. Whereas our free-breathing MR template is relatively artifact-free, practical strategies for lung MR-CT registration may benefit from multimodal registration examples that involve cine-MRI and 3-D CT images.^51,52 Also, of potential relevance are liver imaging studies (another scenario where respiratory motion poses challenges), in which coregistration of a preprocedural high-contrast MR to an intraoperative CT is commonly needed.^53,54 A potential intermediate route for simplifying the registration problem could involve the creation of a pseudo-CT template from the free-breathing MR template using one of the many recently published image synthesis techniques.^55,56

Although this analysis is based on a unique 4-D MR dataset obtained using a specialized pulse sequence, our current dataset is limited to only five subjects. As future work, we plan to extend this work to a larger population of subjects so as to ultimately generate a statistical atlas for the respiratory motion fields. Another limitation of this work is that it is based on data collected from subjects with healthy lungs. Additional investigation is needed to analyze and model the change in lung motion trajectories in the presence of lesions. Finally, our research efforts work will target a wider variety of subjects and breathing pattern types, including both “chest breathers” and “tummy breathers.” For this analysis to be more complete, other aberrant patterns such as breathing amplitude variations, phase shifts, and baseline drifts between internal and external motion signals⁵⁷ need to be factored into the data-driven framework and the internal–external correlation calculations. With a larger and more varied cohort, it will be possible to perform more robust motion analysis, use deep neural networks to learn a more sophisticated motion model, and potentially even attempt to predict voxelwise internal displacements from external displacement measures.

Biographies

Samadrita Roy Chowdhury received her MS degree in physics from the Indian Institute of Technology Chennai and her PhD in physics from Duke University in 2006. She worked with Xerox Innovation Group from 2006 to 2011. Currently, she is a postdoctoral trainee in electrical and computer engineering at the University of Massachusetts Lowell. She is a recipient of the 2018 M. Hildred Blewett fellowship from the American Physical Society.

Joyita Dutta is an assistant professor in electrical and computer engineering at the University of Massachusetts Lowell and an instructor in radiology at Massachusetts General Hospital. She completed her BTech. from the Indian Institute of Technology Kharagpur in 2004 and her MS and PhD degrees from the University of Southern California in 2006 and 2011, respectively. Her contributions to medical imaging have been recognized by the SNMMI Faber Memorial Award and the IEEE Bruce Hasegawa Award.

Disclosure

No conflicts of interest, financial or otherwise, are declared by the authors.