April 2009, VOLUME 192
NUMBER 4

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April 2009, Volume 192, Number 4

Cardiopulmonary Imaging

Original Research

Automated Algorithm for Quantifying the Extent of Cystic Change on Volumetric Chest CT: Initial Results in Lymphangioleiomyomatosis

+ Affiliations:
1Department of Radiology, Cincinnati Children's Hospital Medical Center, Cincinnati, OH.

2Department of Radiology, Children's Hospital of Philadelphia, Philadelphia, PA.

3Present address: Department of Radiology, University of Virginia, Box 800170, Chalottesville, VA 22908.

4Division of Pulmonary Medicine, Cincinnati Children's Hospital Medical Center and University of Cincinnati College of Medicine, Cincinnati, OH.

5Division of Pulmonary and Critical Care Medicine, University of Cincinnati, Cincinnati, OH.

6Division of Neurology, Cincinnati Children's Hospital Medical Center, Cincinnati, OH.

7Division of Nephrology and Hypertension, Cincinnati Children's Hospital Medical Center and University of Cincinnati College of Medicine, Cincinnati, OH.

8Present address: Merck and Co., Inc., Whitehouse Station, NJ.

Citation: American Journal of Roentgenology. 2009;192: 1037-1044. 10.2214/AJR.07.3334

ABSTRACT
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OBJECTIVE. The purpose of our study was to develop a new method for quantifying the severity of cystic lung disease using chest CT and to evaluate this method in patients with lymphangioleiomyomatosis (LAM).

SUBJECTS AND METHODS. Eighteen patients with LAM (all women; mean age, 43.6 years) underwent chest CT and pulmonary function testing including diffusing capacity for carbon monoxide (DLCO). All patients were at their clinical baseline on the day of imaging. Standard quantitative CT metrics including the percentage of the lung volume < –910 HU and the 15th percentile of Hounsfield units were computed from the histogram of lung voxels. A new histogram analysis method was developed to compute the cyst volume and the volume of the remaining lung by segmenting the entire lung attenuation histogram into two underlying distributions, one from the cysts and the other from the remaining lung tissue.

RESULTS. The mean ± SD for quantitative lung metrics was 21% ± 16% for percentage < –910 HU, –915 ± 47 HU for 15th percentile of Hounsfield units, and 19% ± 13% for cyst volume. The correlation between pulmonary function tests and CT metrics was strongest for the percentage of cyst volume for all pulmonary function testing indexes, with correlations between forced expiratory volume in 1 second (FEV1) percentage predicted and the CT metrics of r = –0.52, r = 0.50, and r = –0.86 for the percentage of lung < –910 HU, the 15th percentile of Hounsfield units, and the percentage of cyst volume, respectively.

CONCLUSION. A new method for quantifying cyst volume as a percentage of total lung volume using chest CT correlates with pulmonary function parameters in patients with LAM and may have utility in the assessment of disease severity and progression of cystic lung diseases.

Keywords: chest CT, cystic lung disease, lymphangioleiomyomatosis (LAM), quantitative CT

Introduction
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Lymphangioleiomyomatosis (LAM) is a rare disease characterized by smooth muscle cell infiltration and cystic destruction of the lung [1, 2]. Although LAM primarily affects young adult women of childbearing age, it also occurs in women after menopause and has been documented in a few men. The most common presenting symptoms are dyspnea on exertion, pneumothorax, or chylothorax. Pulmonary function testing (PFT) typically shows obstructive physiology, with a disproportionate reduction in forced expiratory volume in 1 second (FEV1) percentage predicted, but restrictive patterns are also seen. Additional PFT findings include elevations in residual volume (RV) and total lung capacity (TLC) indicating air-trapping and hyperinflation, respectively [3]. Chest CT reveals characteristic findings of thin-walled parenchymal cysts, chylous pleural effusions, and recurrent pneumothoraces [46].

Findings of dyspnea and lung cysts on CT can precede findings of abnormalities on spirometry [2, 7].

Progressive dyspnea and respiratory failure occur over a highly variable period of time, with an often unpredictable pattern of decline [2]. A number of studies report a high mortality rate, with a 10-year survival rate of approximately 40–80% [812], but more recent data suggest a more optimistic prognosis, perhaps related to greater awareness and earlier detection [13]. Accumulating evidence indicates that LAM is a metastatic process in which histologically benign cells of unknown origin infiltrate the lung and promote remodeling [14]. Aberrant LAM cell proliferation, migration, and infiltration result from mutations in one of two tuberous sclerosis genes, TSC1 or TSC2, both in patients with tuberous sclerosis complex (TSC-LAM) and those without germline mutations who have sporadic LAM (S-LAM) [15, 16]. The protein products of these genes, hamartin and tuberin, form a complex that regulates signaling through the mammalian target of rapamycin (mTOR) pathway [17, 18]. Sirolimus (rapamycin) is an immunosuppressive drug approved by the Food and Drug Administration that can mimic the function of the tuberin–hamartin complex by inhibiting the activity of mTOR.

In the absence of effective treatments for the disease, it is sufficient to perform a qualitative assessment of chest CT in patients with LAM to establish the diagnosis or detect complications. However, these advances in the understanding of the molecular basis of LAM form the foundation for several clinical trials of sirolimus in patients with tuberous sclerosis and LAM that are under way in the United States, Europe, Canada, and Japan. With the advent of experimental therapies, it will become vitally important to develop improved outcome measures to assess the efficacy of these treatments in clinical trials.

PFT is the standard surrogate outcome measure of lung disease but suffers from several limitations, including significant intertest variation and dependence on patient cooperation and effort. Chest CT allows greater regional resolution and can discriminate between changes in the parenchyma and changes in the airways or chest wall. In addition, quantitative chest CT is generally more sensitive than PFT to small changes in disease severity. CT has been used as an outcome measure in clinical trials of treatments for pulmonary emphysema, and CT remains the only test that has shown the efficacy of replacement therapy in patients with α-1 antitrypsin deficiency related emphysema [19]. Previous studies in LAM using human reader scoring [20, 21] and standard CT emphysema quantification measures [2123] have found moderate to strong correlations with pulmonary function. However, the quantitative chest CT measures that were developed for the evaluation of emphysema, in which there is a relative decrease in lung tissue density, may not be optimal for LAM, a disease in which very-low-attenuation lung cysts are admixed with parenchymal smooth muscle cell infiltration, which could increase the attenuation of the remaining lung parenchyma. Under these conditions, measuring only the decrease in attenuation could underestimate the severity of disease because of partial volume effects with small lung cysts.

The purpose of this study was to develop a better method for quantitatively analyzing the extent of cystic change on chest CT scans from patients with LAM and to compare the new method with standard quantitative CT metrics.

Subjects and Methods
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Patients

Eighteen patients with a history of LAM underwent CT of the thorax and PFT on the same day. All subjects were at their clinical baseline at the time of the study, and no subjects were imaged during an exacerbation of disease. Subjects with a pneumothorax or chylothorax were excluded from the study. This study was performed as part of an institutional review board (IRB)-approved protocol, and all patients gave written informed consent.

CT of the thorax was acquired from the lung apex to just below the diaphragm on a helical MDCT scanner (LightSpeed QX/i, GE Healthcare) with 0.625-mm isotropic voxel resolution at 120 kV and 80-mA tube current. For quantitative analysis, the images were reconstructed to a 5-mm section thickness. An 80-mA tube current is lower than that typically used for clinical chest CT in adults and was selected to reduce the radiation exposure associated with this examination. However, decreasing the tube current increases the amount of point noise in the images, so 5-mm reconstructed image sections were used in the analysis to reduce the amount of point noise. The reconstructed data matrices were 512 × 512 with a field of view that depended on patient size but was typically 32 × 32 × 30 cm. Approximately 60 contiguous 5-mm sections were obtained per subject to yield a large 3D data set. The standard (soft-tissue) reconstruction kernel was used. Subjects were imaged in the supine position and were instructed to take a deep breath and hold their breath for the duration of the CT.

Spirometry and body plethysmography using the Vmax System 6200–22 (SensorMedics) were performed in each patient. PFT including spiro metry, lung volumes, and diffusing capacity was performed in a standardized manner according to the American Thoracic Society criteria for acceptability and reproducibility [24, 25].

Automated Processing of Lung CT Images

An overview of the CT algorithm is presented here with a more detailed description included in Appendix 1 for the interested reader. In brief, the lungs were segmented from the chest wall and mediastinal structures using a density-based algorithm. The trachea and main bronchi were automatically excluded from the analysis. Histograms of the CT attenuation values from the entire lung were constructed for each subject. These histograms were then separated into two distinct underlying distributions, one thought to represent the lower-attenuation cysts and the other the higher-attenuation lung parenchyma, using an expectation-maximization (EM) algorithm, and successively optimizing the cyst and lung parenchyma distributions. One of the advantages of this methodology is that few assumptions are made about the shape of these underlying distributions. The percentage of cyst volume was determined by taking the total area under the histogram of the cyst distribution and dividing by the total number of pixels in the lung. The cyst and parenchymal distributions overlap in attenuation values, so no single threshold separates cysts from lung parenchyma. However, for the purposes of illustration, we selected the attenuation value in Hounsfield units of the point at which the two distributions cross (the attenuation value for which the posterior probability of a pixel being in a cyst is 0.5) as a threshold, coloring pixels with an attenuation value less than this threshold white to show the lung cysts. However, the percentage of cyst volume used in the statistical analysis includes the tail of the cyst distribution.

In addition, the standard quantitative chest CT measures of the percentage of the lung volume with < –910 HU and the 15th percentile of Hounsfield units were computed from the whole-lung attenuation histograms. The image analysis was performed without knowledge of the PFT results or clinical status of the patient. The correlation between the imaging measures of disease severity and the results of PFT were computed using Pearson's correlation coefficients.

Results
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The mean age ± SD of the study participants was 43.6 ± 2.7 years (age range, 19–58 years). Fourteen patients had TSC-LAM, and four had S-LAM. The spectrum of clinical disease severity represented was broad, with the ratio of forced expiratory volume in 1 second (FEV1) % predicted ranging from 23% to 97% (mean, 64%; SD, 21.6%) (Table 1). The range of disease severity on imaging was also large, with the percentage of cyst volume ranging from 1% to 49% (mean, 19%; SD, 13%) (Table 1).

TABLE 1: Study Participants with Lymphangioleiomyomatosis (LAM), Pulmonary Function Values, and CT Metrics

A representative image from a chest CT in a patient with moderate pulmonary LAM is shown in Figures 1A, 1B, 1C, and 1D. Numerous variable-sized, thin-walled cysts are present throughout the lung, as is commonly found in patients with LAM. A CT image segmented using the algorithm developed in this article shows excellent separation of the lung from the surrounding structures, including the chest wall and mediastinum (Fig. 1B). The lung attenuation histogram from the whole lung is shown in Figures 1C and 1D. Notice that the histogram has a distinctly bimodal appearance. In Figure 1C, the standard quantitative CT measures are shown. In Figure 1D, the two underlying distributions, one corresponding to the cysts and one corresponding to the noncystic lung parenchyma, are shown. The sum of the cyst and lung parenchyma distributions closely approximates the actual histogram. Using the crossover point of these two distributions as a threshold, lung pixels with a value less than the threshold are overlaid in white (Fig. 1B). Larger cysts are readily identified as cysts, but some pixels in smaller cysts are not classified as cysts at this threshold. These pixels may correspond to the tail in the cyst distribution that extends into attenuation values larger than the threshold value (Fig. 1D). Note that the percentage of cyst volume as computed in this article includes the tail of the cyst distribution and thus includes pixels not shown in white on Figure 1B.

Table 2 contains the mean and SD of selected PFT parameters, the imaging results for the 18 patients, and the correlation coefficients between the selected PFT and the imaging measures. There was a strong correlation between PFT and the percentage of cyst volume measured on CT (FEV1, r = –0.90; FEV1% predicted, r = –0.86; FEV1/FVC (forced vital capacity), r = –0.86; and RV % predicted, r = 0.77). With the exception of TLC, the correlation between all of the PFT indexes and imaging was much stronger for the percentage of cyst volume than the conventional quantitative CT metrics (the 15th percentile of Hounsfield units, and the percentage of the lung volume < –910 HU) (Table 2). The percentage of cyst volume was moderately correlated with the conventional quantitative CT metrics (15th percentile of Hounsfield units, r = –0.74; percentage of the lung volume < –910 HU, r = 0.73). Interestingly, the two conventional quantitative CT metrics were strongly correlated (r = –0.90).

TABLE 2: Correlation Between Selected Pulmonary Function Testing Indexes and CT Metrics

For most patients, the percentage of cyst volume and the percentage < –910 HU were similar, but there were two patients in whom they were discordant (patients 11 and 14 in Table 1). For patient 14, the percentage of cyst volume was low at 3%, whereas the percentage < –910 HU was elevated at 39%. A representative CT image from this patient (Fig. 2A) shows few lung cysts. The corresponding whole-lung histogram (Fig. 2B) appears shifted to the left. Because this patient does not have an elevated RV (65% predicted), this leftward shift is likely caused by the patient taking a very large breath at the time of CT scanning and not due to persistent air trapping. Thus, the percentage of cyst volume appears to more accurately reflect the imaging findings in this patient. Patient 11 has more pronounced findings of LAM, but, again, the discordance between the percentage of cyst volume and the conventional CT measures appears to be due to a leftward shift of the whole-lung attenuation histogram, likely caused by the patient taking a large breath at the time of CT scanning.

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Fig. 1A 57-year-old woman (patient 9) with moderate lymphangioleiomyomatosis and forced expiratory volume in 1 second (FEV1) to FEV1 percentage predicted of 67%. Chest CT image (A) and same CT image segmented (B) with cysts identified by white areas and blood vessels, mediastinum, and chest wall identified by gray areas.

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Fig. 1B 57-year-old woman (patient 9) with moderate lymphangioleiomyomatosis and forced expiratory volume in 1 second (FEV1) to FEV1 percentage predicted of 67%. Chest CT image (A) and same CT image segmented (B) with cysts identified by white areas and blood vessels, mediastinum, and chest wall identified by gray areas.

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Fig. 1C 57-year-old woman (patient 9) with moderate lymphangioleiomyomatosis and forced expiratory volume in 1 second (FEV1) to FEV1 percentage predicted of 67%. Whole-lung attenuation histogram shows percentage < –910 HU (box) and 15th percentile Hounsfield units (dashed line) measures.

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Fig. 1D 57-year-old woman (patient 9) with moderate lymphangioleiomyomatosis and forced expiratory volume in 1 second (FEV1) to FEV1 percentage predicted of 67%. Whole-lung attenuation histogram separated into cyst and lung parenchymal distributions. For this patient, three CT measures show excellent concordance.

Discussion
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A new automated chest CT analysis algorithm was developed for quantitative assessment of the percentage of the lung volume that is occupied by cysts in patients with LAM. In 18 LAM patients, the percentage of cyst volume was more strongly correlated with FEV1 than were conventional quantitative chest CT measures (the percentage of the lung volume with a pixel value < –910 HU and the 15th percentile of Hounsfield units). The conventional quantitative chest CT measures were developed primarily for the assessment of emphysema [26, 27]. Emphysematous regions of the lung typically contain lung tissue of lower density than normal lung but with an attenuation that remains greater than that of air. In contrast, in the lungs of patients with LAM, air-filled cysts are surrounded by lung parenchyma that is normal or even increased in attenuation, likely caused by the infiltration of smooth muscle cells. Thus, it is not surprising that an algorithm developed for the specific imaging features of LAM performs better than the conventional quantitative chest CT measures for patients with LAM.

Antihormonal therapeutic strategies for LAM, including surgical and medical oophorectomy, became the empirical standard of care in the absence of definitive studies showing efficacy [2, 28]. New therapies directed at the underlying molecular aberrations in LAM are being developed, and upcoming randomized, placebo-controlled trials in limited numbers of patients will require consistent and sensitive surrogate outcome measures to detect or reject therapeutic candidates. Effort-independent assessments such as quantitative chest CT, as has been proposed for trials of patients with pulmonary emphysema, may be complementary to standard PFT. Whether chest CT will be an effective outcome measure for LAM is not yet known, but it seems likely that an algorithm developed for the specific imaging features of LAM, such as the percentage of cyst volume, could be more efficacious than algorithms developed for emphysema, which has strikingly different imaging findings. This methodology may also be useful for evaluation of other cystic lung diseases such as pulmonary Langerhans cell histiocytosis.

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Fig. 2A 49-year-old woman (patient 14) with tuberous sclerosis complex lymphangioleiomyomatosis. Representative CT image shows few lung cysts.

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Fig. 2B 49-year-old woman (patient 14) with tuberous sclerosis complex lymphangioleiomyomatosis. Whole-lung attenuation histogram is shifted to left, possibly due to patient taking very large breath, which causes percentage < –910 HU (box) to be elevated (39%), but percentage of cyst volume (arrow) remains low (3%) and more accurately reflects findings on CT image. Note that algorithm for calculating percentage of cyst volume can separate whole-lung histogram into two histograms, one from cysts and one from lung parenchymal structures, even in absence of obvious bimodal distribution in whole-lung histogram.

The underlying assumption of the algorithm developed to calculate the percentage of cyst volume is that there are two distinct populations within the lung, cysts and lung parenchyma, and that the underlying lung attenuation histograms for these two populations are separable. Few assumptions are made about these underlying distributions, including the attenuation of noncystic lung parenchyma. This is in contradistinction to the percentage of the lung volume less than a threshold (e.g., –910 HU), which assumes a fixed cutoff in attenuation between normal and abnormal lungs. Lung parenchyma, both normal and emphysematous, is known to vary in attenuation with the phase of respiration [29]. Thus, a fixed threshold scheme is sensitive to differences in the phase of respiration. In the two patients in this study in whom the percentage of cyst volume and the percentage < –910 HU were discordant, the discordance appears to have been caused by unusually large breaths taken by the patients at the time of CT scanning. Thus, the percentage of cyst volume appears to be more robust than the percentage < –910 HU regarding variations in the level of inspiration because the percentage of cyst volume was better correlated with spirometry. It is likely the percentage of cyst volume is less sensitive to the phase of respiration because few underlying assumptions are made about the attenuation of noncystic lung tissue. However, if the relative change in volume of the cysts is less than that of noncystic lung parenchyma with respiration, there will be at least a small dependence on the phase of respiration. In this case, the cyst volume (in liters) could be a more robust outcome measure to assess changes with time or treatment in individual patients.

One of the advantages of the percentage of cyst volume algorithm relative to a threshold-based algorithm is that the algorithm inherently compensates for partial volume effects in small cysts. Large cysts have an internal attenuation of approximately –1,000 HU because they contain pure air. Small cysts may be higher in attenuation than large cysts because of partial volume effects with adjacent lung parenchyma. The effect of the higher-attenuation lung cysts can be seen in the higher-attenuation tail of the cyst distribution, and these higher-attenuation small cysts are included in the calculation of the percentage of cyst volume (Fig. 1D).

Previous studies using human reader scoring [20, 21] and standard quantitative chest CT measures [2123] have found moderate to strong correlations between PFT and cystic change in the lung on CT in patients with LAM. Our results confirm the previous findings that the amount of cystic change in the lung is related to the degree of abnormality on PFT. However, we found that the percentage of cyst volume, which is derived using an algorithm specifically designed for LAM, had a stronger correlation with PFT than the standard quantitative chest CT metrics that were used in the previous studies.

The primary limitation of this study is the small number of patients—a feature that is common to all rare diseases. As it happens, the number of patients in this study is similar to those of previous studies investigating the use of CT in LAM [2123]. One mitigating factor is that we included patients with a wide range of disease severity in this study. A second limitation was the use of only the two most commonly used emphysema measures for comparison with the percentage of cyst volume. Also, we used relatively thick CT slices (5 mm) for the analysis because we found the performance of our algorithm degraded with the increase in noise on thinner slices at the relatively low tube current used in this study. However, the use of a larger tube current or a noise-reducing filter [30, 31] might mitigate this effect and permit the use thinner slices for the analysis. This use might reduce partial volume effects and further improve the performance of the algorithm. The performance of the algorithm might improve on expiration phase CT images as well. It is not yet known whether changes in percentage of cyst volume over time will correlate with changes in PFT parameters or clinical outcome. However, the purpose of this study was to develop a quantitative metric for the assessment of cystic change on chest CT in LAM, and given the promising results of this baseline study, a longitudinal study in patients with LAM would be the next step in the evaluation of the percentage of cyst volume as an outcome metric for LAM.

To summarize, a new automated analysis algorithm was developed for quantifying the extent of cystic change in patients with LAM. A better correlation with pulmonary function parameters, including FEV1 and FVC, was found with the percentage of cyst volume derived from this algorithm than with conventional quantitative chest CT measures. The percentage of cyst volume may be useful as an outcome measure in future trials to follow disease progression, assess new treatments for LAM, or for the evaluation of other cystic lung diseases, such as pulmonary Langerhans cell histiocytosis.

APPENDIX 1: CT Quantification Algorithm
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The CT images were transferred to a PC Windows (Microsoft) workstation. All processing was done via routines written in IDL (Research Systems). The data were transformed from the scanner-native DICOM format into raw floating-point data readable in IDL.

Automated Masking

The lung parenchyma was separated from the rest of the image on the CT scans via an automated masking algorithm. A threshold of –200 HU was used to separate air from soft tissue. Air outside of the lungs was identified by using region-growing algorithms using seed points at the corners of the image. A threshold of –400 HU was used to separate lung parenchyma from blood vessels. An additional region-growing algorithm was used to separate the lung from the trachea. Because the trachea was not always completely separable from the lung (due to partial volume effects) before applying the region-growing, an erosion filter (with a width of 7 pixels) was applied to the mask of voxels inside the body with < –400 HU. The lung parenchyma was identified as those regions with at least 2,000 contiguous pixels; a dilation filter was then applied to the mask of voxels inside the lung to restore the previous boundaries. The method was verified by visual inspection to reliably segment the lung parenchyma from the rest of the image.

Automated Segmentation of Cysts from Remaining Lung Parenchyma

A histogram of the lung density was created for all pixels inside the lung (with < –400 HU to exclude blood vessels). Pixels with –1,024 HU were excluded because of floor effects (the reconstruction algorithm is limited to 12-bit resolution). Although air (present in cysts) should have –1,000 HU, noise in the data would result in some air pixels actually displaying a lower Hounsfield unit value. Including those pixels would result in an undesirable spike in the histogram. The histogram was modeled as the mixture of two densities: cysts and lung parenchyma. The Hounsfield unit distribution of lung parenchymal tissue is not well described by a gaussian distribution, so a gaussian mixture model was not used. Instead, a more flexible mixture model was used, constrained by previous information on Hounsfield unit distributions. An expectation-maximization (EM) algorithm was used, successively optimizing the cyst and lung parenchyma distributions.

Spline Smoothing

To generate the starting values used for the EM algorithm, the histogram was spline-smoothed using a locally varying objective function because previous information is available from CT scans of healthy volunteers showing that more curvature is present near the peak of the histogram than at the tails. Specifically, the following objective function was optimized: where i is an index representing the range of Hounsfield unit values in the histogram, Si is the value of the spline-smoothed histogram, Si is the 2nd derivative of the spline-smoothed histogram, Hi is the value of the original histogram, and α is a parameter (common to all spline-smoothing algorithms) that determines the relative weights of the accuracy and curvature terms in the objective function. Empirically, a value of α = 0.2 was found to yield reliable results for the spline smoothing (although it was not critical). To avoid division by 0, the denominator in the first term was set equal to 1 for Hounsfield unit values for which Hi was equal to 0. The optimization of the objective function may be done in closed-form by computing and setting it to 0 for each Sj. The resulting set of linear equations may be readily solved via Cholesky or LU decomposition.

Local maxima were found from the spline-smoothed histogram by looking for a shift in the first derivative from positive to negative. Local maxima not at least 0.25 times the global maximum were discarded. If two local maxima were found, they were taken to be the peaks of the cyst and lung parenchyma distributions, with the maximum from the lowest Hounsfield unit taken to be that of the cyst distribution. If only one local maximum was found, it was assumed to be of lung parenchyma and the peak of the cyst distribution estimated by Pc = (PL – 1,024) / 2, where Pc is the peak of the cyst distribution, and PL is the peak of the lung parenchyma distribution. The starting estimate of the cyst distribution was taken as a bilinear function, rising from the value of the smoothed histogram at the minimum Hounsfield unit (–1,023) to the peak of the cyst distribution at its found location, and diminishing from there to 0 at the location of the peak of the lung parenchyma distribution. The starting estimate of the lung parenchyma distribution was also taken as bilinear, rising from 0 at the location of the peak of the cyst distribution to the value of the peak of the lung parenchyma distribution at its found location, and diminishing from there to the value of the smoothed histogram at the maximum Hounsfield unit. The cyst and lung parenchyma distributions were modified by multiplicative coefficients, determined by least-squares optimization to best fit the actual histogram.

EM Optimization of Cyst and Lung Parenchyma Distribution

Although EM algorithms for mixture models are robust to errors in specifications of the starting estimates, correctly estimating the peaks of the cyst and lung parenchyma distributions before the EM algorithm will produce savings in computational time. The EM model assumes similar smoothing characteristics for the cyst and lung parenchyma distributions, as was assumed for the whole-lung distribution in the spline smoothing. Specifically, the objective function to optimize is where Ci is the cyst distribution and Li is the lung distribution. The cyst and lung parenchyma distribution are subject to the following constraints depending on the Hounsfield unit value:

These correspond to applying a 0 previous probability of any pixel with > –824 HU being in a cyst and a 0 previous probability of any pixel with –1,023 HU of being in the lung. We note that this model is amenable to a later modification involving the use of soft constraints (e.g., nonzero values of previous probability) by the incorporation of additional terms in the objective function, but we found the listed hard constraints to yield sufficiently robust results in the segmentation of lung tissue into healthy lung and cysts.

The EM algorithm successively optimizes the model for Ci (given the previous solution for Li) and then for Li (given the previous solution for Ci) until convergence, empirically seen to occur after approximately 50 iterations. As shown for the spline-smoothing algorithm, the optimization may be performed in closed form by setting the partial derivatives of the objective function equal to 0 and solving the resulting set of linear equations by LU decomposition. Constraints may be applied by eliminating all equations pertaining to the constrained values in the linear equation set. Computational time is approximately 1 minute or less on a Pentium IV (Intel) 2-GHz workstation. The author of this algorithm has offered to provide advice and technical assistance to those who wish to use it in the future.

D. N. Franz, F. X. McCormack, and J. J. Bissler supported by grants from the Rare Lung Disease Consortium, National Institutes of Health (RR019498); National Cancer Institute (CA103486); LAM Foundation; and Tuberous Sclerosis Alliance.

Address correspondence to T. A. Altes ()

We thank the LAM patients who participated in the study.

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