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Departments of Radiology and Biomedical Engineering, University of Iowa, Iowa City, Iowa 52242
Submitted 2 November 2002 ; accepted in final form 2 May 2003
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ABSTRACT |
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 TOP ABSTRACT METHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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airway area; bronchial tree; lung imaging; quantitative computed tomography; deconvolution
Numerous methods have been proposed and used for the measurement of airway size and geometry from X-ray computed tomography (CT) images. Simpler, less objective methods for the assessment of airway dimensions include manual tracing and measurement either on film, computer screen, or projected image with the adjustment of window and level for better visualization (2, 4, 21, 29, 34). More sophisticated automatic and semiautomated methods have also been proposed, such as the full-width-half-maximum (FWHM) method (1, 3, 9, 37). FWHM has been used in numerous X-ray CT-based imaging studies (1, 5). The FWHM method assumes that the true edge position is halfway between the maximum and minimum values of a line profile across the edge. Region of interest-based methods using thresholding and region growing (13, 15, 18, 27, 36) select thresholds to determine edge location. Model-based approaches (24) use models for the airways and the scanner and try to match the model results to the actual measurement. Heuristic analysis and morphological filtering (7, 15, 23) use a priori information to form an intelligent guess about the geometry of airways.
Studies to date have largely limited their assessment of airway dimensions to those airways that happen to be imaged perpendicular to their local long axis so as to avoid the overestimation of wall thicknesses or luminal diameters associated with oblique sectioning of a tubular structure (1, 2, 4, 6, 17). Efforts have been underway for a number of years to devise methods of automatically extracting lung structure, including airway tree geometry, from volumetric CT images (11, 22, 24, 37). To further advance our understanding of how to best take advantage of the unique in vivo airway information offered by advanced CT imaging technology, in the present study, we have developed and tested a new method for the estimation of airway geometry from two-dimensional (2D) CT slices and have compared the results to the less computationally intense tilt-adjusted FWHM criterion. FWHM uses a simple ray-casting technique, and the wall locations are estimated on the basis of the gray-scale profile along the rays radiating from a central point outward. Our new method incorporates a three-dimensional (3D) airway model and the 3D point spread function (PSF) of the scanner, following the steps of King et al. (14) and Wang and colleagues (28, 32, 33), similar to the method used in a 2D form by Reinhardt et al. (24). Our new method is used to estimate airway inner and outer radii from 2D CT slices as well as to estimate the airway long axis tilt angle relative to the scan plane. The method allows for the quantitation of airways obliquely oriented to the scan plane. Assuming an ideal circular airway model arbitrarily oriented in space, we use mathematical deconvolution to extract the model parameters from the actual image knowing the scanner (PSF) characteristics. This is done in two phases: 1) the initialization phase; and 2) the optimization phase that generates an estimate for the inner and outer wall locations and the airway axis tilt angle.
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METHODS AND MATERIALS |
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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Phase 1
Center determination. The center of the airway is determined
manually by using a point-and-click device on a computer screen. Although, for
the purposes of this study, we did not concentrate on a "user
friendly" integration of methods, the center point could be read from a
file containing center coordinates generated automatically by a
centerline-finding software
(30,
36,
37) or by other automated
means of identifying the airway locations automatically
(15,
22). From this initial center,
equally spaced (2
/n, n = 40) rays independent of airway size are
cast out radially with enough length to cross both the inner and outer walls
of the airway. The length of each ray is set to fixed length unless no outer
wall is detected, which would increase the preset length until the outer wall
is found.
FWHM. The finite bandwidth of the CT imaging system causes edges
to appear blurred. Thus the location of an edge cannot be determined precisely
from an image, because the sharp intensity transition at the edge is replaced
by a more gradual gray-level transition. The true location is somewhere
between the maximum and minimum gray levels along a ray across the edge. In
the case of airways, there are two edges across the wall, one at the luminal
surface and one at the boundary between the airway and other pulmonary
structures that include parenchyma and associated pulmonary arteries. A line
profile across the airway wall (exclusive of regions where the airway borders
with pulmonary arteries) would look similar to the profile shown in
Fig. 2. Let the line profile be
a function Y(·) of the distance r from center of
airway, Y(r). Define the location of the inner edge to be at
a level yi between Y(rmax)
and Y(rmin1) and similarly that for the outer
edge to be yo between Y(rmax)
and Y(rmin2). The FWHM method defines the levels
yi and yo to be halfway between the
maximum and minimum observed gray-level values
(1,
3,
9,
24,
36)
 | (1a) |
 | (1b) |
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The locations of the inner and outer airway walls are estimated by using the FWHM method along each ray of the set of rays generating two sets of points: one for the inner wall and the second representing the outer wall. To date, this method has been limited to measuring airways that perpendicularly cross the image plane. The following procedures correct for tilt angle to allow estimation of tube radii that are not perpendicular to the image plane.
Ellipse fitting. In CT images of the lung, airway appearance ranges from nearly circular to elliptical. The ellipse is the general geometric shape of the cross section of an ideal circular airway intersecting with the image plane. In the special case of the ideal airway being perpendicular to the image plane, the circular cross section can be viewed as a special-case ellipse.
To mathematically describe airways, an ellipse model can be fit to the edge
points determined from the 40 rays and the FWHM estimation of edges. Ellipses
can be defined as general conic sections
(10) by using an implicit
second-order polynomial F(a, x)
 | (2) |
The fitted ellipses provide estimates that are used to calculate a refined
approximation of the inner (Ri) and outer
(Ro) radii of the airway, in-plane rotation angle
(
), and the long-axis tilt angle relative to the scan plane (
)
(see Fig. 3). In addition, it
provides an adjustment to the initial center in both the x and
y directions. The in-plane rotation and tilt angles are estimated by
using the following formulae
 | (3a) |
 | (3b) |
is the tilt angle, rmajor and
rminor are the semimajor and semiminor axes of the
ellipse, is the in-plane rotation angle, and a, b, and
c are defined in a general quadratic curve equation (Eq. 2).
In this study, we compensate for the tilt angle () by including the
estimated values in the airway model. Because the model assumes the major and
minor axes of the ellipse to be along the horizontal and vertical axes, we
compensate for the in-plane rotation angle () by rotating the original
image by (-) and applying the model on the rotation-adjusted image.
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Phase 2
FWHM method. Because the initial FWHM measurements were obtained from the actual 2D cross sections, we adjust those measurements by the tilt angle estimated from the ellipse-fitting step. To reduce the noise-induced bias in the final measurements, we assume that the airway is symmetrical around the center and along the horizontal and vertical axes and average the two horizontal estimates and the two vertical estimates of radii. Airway circularity has never been conclusively proven; however, several researchers have assumed the airways to be circular or near circular either locally or globally and have applied measures of circularity to quantitatively describe airways (12, 13, 16, 24, 27, 35). This assumption is not valid in some severe pathological states. However, we seek to devise methods for the assessment of early lung disease before dramatic disease-based disruption of normal airway geometry.
Model-based method. In the model-based method, in addition to
modeling the airway cross-sectional geometry as a circular tube that is tilted
as it intersects the scan plane, we also model the scanner and the scanning
process. We model the scanner by its 3D PSF described by a 3D Gaussian with
standard deviations in the x, y, and z directions. The
scanning process is modeled as a linear convolution of the airway and the
scanner models (see Eq. 5). However, as discussed earlier, the
cross-sectional appearance of the airway is, in general, elliptical in shape
rotated by an angle (see Fig.
3). To determine ri and ro
of the embedded tubes using the model-based method, we first generate a 3D
data set using the tilt angle parameters from ellipse fitting (see Ellipse
fitting, Eqs. 3a and 3b), using 0 Hounsfield units (HU) for
airway wall, -1,000 HU for lumen, and estimated values for lung parenchyma
(
i) outside the lumen. This sharp image is then convolved
with the 3D PSF of the scanner to produce a predicted image. The profiles
along the major and minor axes of the ellipse thus produced are compared with
the actual profiles from the original image, and a sum of squared deviations
is computed. The final values for ri and
ro are then determined by adjusting these values to
minimize the sum of squared deviations between model and actual image line
profiles (see Fig. 1) by using
the multidimensional unconstrained nonlinear minimization simplex method
(Nelder-Mead simplex method as implemented in Matlab, The Mathworks, Natick,
MA, and described in Ref.
19).
PSF OF THE SCANNER. The PSF represents the resolution-limiting
factors that cause images not to be exact replicas of the real objects
(13). An ideal PSF is an
impulse, and any deviation from this ideal function causes the PSF to widen.
The increase in PSF width increases the blur that images incur as a result of
being generated by this nonideal system. In short, the PSF or its frequency
domain version, the modulation transfer function, specifies the resolution of
the scanner (32). The model
Reinhardt et al. (24) used
considered only perpendicular airways, and therefore the scanner can be
modeled by a 2D PSF. Because of the added degree of complexity in our airway
model, the use of a 2D PSF does not fully represent the image-formation
process. In the case of obliquely oriented airways, the effect of slice
thickness influences the 2D reconstructed image because of partial volume
effect. Using a 3D PSF better captures this phenomenon. Therefore, we modeled
the 3D PSF as a spatially invariant Gaussian function
(24,
28,
32,
33) of the form shown in
Eq. 4
 | (4) |
x, y, and
z are the standard deviations in the x, y,
and z directions, respectively. The PSF parameters
x, y, and
z must be estimated by a calibration procedure
(24,
32).
The scanning process is modeled as the 3D linear convolution of the 3D
ideal airway model with the 3D ideal scanner model
(24,
33) as shown in Eq. 5
 | (5) |
PSF ESTIMATION. System impulse response is the output of the
system when stimulated by an impulse function. In 3D, we use a small spherical
bead of a known size to simulate a 3D impulse function, and the resulting
image provides the information needed to estimate the PSF of the scanner.
Three different types of beads were used: 1.4 mm metal; 1.0 mm zirconium; and
0.794 mm sapphire (Small Parts, Miami Lakes, FL). The sapphire beads were used
to estimate the in-plane standard deviation (x and
y), and the larger beads were used to estimate the
standard deviation in the z direction (z).
A Gaussian function (Eq. 6) is used to model the scanner
characteristic function (PSF)
(14,
24,
28) as shown below
 | (6) |
x, y, and
z are the standard deviations and
x0, y0, and z0 are
the bead center offsets in the x, y, and z directions,
respectively; C is a constant offset representing the image
background (in this case to be -1,000 HU representing air); and A is
a scaling factor. To solve for the PSF parameters, we treated the system as a nonseparable system and fitted the data obtained in 3D to the model of Eq. 6. This method has the advantage of providing a more realistic model of the PSF than the separable model because the slice thickness for the Imatron electron beam CT (EBCT) scanner is dependent on in- and out-of-plane factors that render the image voxels acquired to be nonisotropic. One of these factors is the X-ray beam shape and the relative location of the targets, detectors, and object imaged relative to the gantry (8).
Because of the limited dynamic range of the scanner and the high density of the beads, the images of the metal and the zirconium beads suffered from saturation. To model this saturation, we clipped the output f(x,y,z) when the simulated image value exceeded the upper limit of the scanner (3,095 HU). The clipping occurs at the peak of the Gaussian PSF while the transitional parts are preserved. Considering the other sources of error introduced because of the geometry and the nature of the Imatron scanner, this loss of information is not significant, and the PSF of the scanner remained recoverable by using the approach mentioned above.
The beads were scanned in the Imatron C-150 scanner (Imatron, South San Francisco, CA) using the smallest pixel size of 0.1758 mm corresponding to a field of view (FOV) of 9.0 cm. The slice thickness was 1.5 mm and an overlap of 1 mm, making the effective slice thickness 0.5 mm. The scan aperture was 100 ms, and normal, sharp, and very sharp reconstruction kernels were used.
Airway Phantom
The airway measurement techniques were validated by using a Plexiglas phantom. The phantom consists of seven Plexiglas tubes resembling airways with various inner diameters and wall thickness values. The tubes are contained in a Plexiglas cylinder and surrounded by potato flakes to simulate the lung parenchyma because potato flakes were found to have a similar HU value (24) to that of the lung parenchyma at functional residual capacity (approximately -650 HU). Figure 4 shows representative CT images of the Plexiglas phantom at two different tilt angles (0 and 45°) with respect to the scan plane.
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The five smallest tubes were used for validation. The tube dimensions are shown in Table 1. The images were acquired using the Imatron C-150 EBCT scanner. Three FOVs were reconstructed for each data set. The FOVs were 15, 26, and 35 cm, corresponding to in-plane pixel size of 0.29 x 0.29, 0.51 x 0.51, and 0.68 x 0.68 mm, respectively. Slice thickness was 3 mm for all cases, consistent with the protocol used at the time for human scans. The scan aperture was 100 ms, and normal, sharp, and very sharp reconstruction kernels were used at the 15-cm FOV and with no tilt. In addition, the phantom was scanned at three different tilt angles (0, 22.5, and 45° relative to the scan plane) while reconstructed with the normal reconstruction kernel and a 15-cm FOV. Only those at 0° tilt were reconstructed with the three different reconstruction kernels and with the three different FOVs (15, 26, and 35 cm). The FOV of the other tilt angles (22.5 and 45°) was 15 cm, and they were reconstructed with the normal kernel. Ten different slices were analyzed for each tube.
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To compare the performance of the new model-based and the tilt-adjusted
FWHM methods, the absolute error of the two methods for each combination of
imaging parameters was analyzed by using the standard paired t-test
and its nonparametric counterpart the Wilcoxon signed-rank test. Nonparametric
tests are more robust for small samples (in this case n = 10) and do
not rely on assumptions of normality for the underlying distributions. The
null hypothesis, that the absolute errors were equal, was tested against the
alternate hypothesis that the absolute errors from the model-based approach
were less than the errors from the tilt-compensated FWHM method (a one-sided
test). The results of both statistical tests were evaluated against an
alternative hypothesis with = 0.05 (probability of a type I error,
false positive, equal to 5%).
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RESULTS |
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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The error in the estimation of the tilt angle is important because of the utilization of tilt angle information in the model to estimate the airway size. In numerical simulations, we were able to estimate the tilt angle in the range of 0-90°. However, actual phantom data analyzed was for 0, 22.5 and 45° of tilt. The error in tilt angle estimation is shown in Fig. 5.
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Figure 5A shows errors in range of tilt angles, 15-cm FOV, and normal reconstruction kernel. The figure shows the error in estimating the tilt angle to be within 4.0° over the range of tilt angles considered (0-45°) for all the tubes except the first tube (the smallest), especially at the largest tilt angle (45°) where the lumen of the tubes almost disappears (Fig. 4B). The results show that the method tends to overestimate the tilt angle.
Similarly, Fig. 5B shows the effect of reconstruction kernels. The error is estimated to be <4° for all the tubes and slightly higher for the smallest tube. In this case, we notice that the tilt angle is overestimated for thin-walled tubes and slightly underestimated in the case of the thick-walled tube. In addition, it appears that the mean error differences for the same tube using all the kernels are small compared with the difference in case of different FOVs or tilt angles.
Figure 5C shows the effect of altering the FOV. The tilt angle estimation error is within 2.3° for all FOVs except for the first and the fourth tubes. It is worthy to note that at these FOVs the smallest (first) tube is covering fewer pixels as the FOV increases, meaning that there are fewer data points available for parameter estimation. However, for the fourth tube, the results are skewed because of the presence of some potato flakes inside the tube at the time of the scanning, which does not agree with the model and the assumption that the lumen is uniformly filled with air. The tilt angle estimation method seems to overestimate the degree of tilt for thin-walled tubes and underestimates the tilt for the thick-walled tube (tube 5).
FWHM Method
Figure 6, A and C, Fig. 8, A and C, and Fig. 9, A and C show the mean error ± SE in estimating the inner and outer radii of the phantom tubes listed in Table 1 using the modified FWHM method. There is a clear underestimation bias in measuring the inner radius of the tubes by this method. Figure 6A shows the mean error of the estimation of the inner radius by using images acquired at different tilt angles and reconstructed with a normal kernel at a 15-cm FOV. As the tilt angle increases, the error in estimating the inner radius increases, and in the worst case it becomes approximately one pixel (pixel size = 0.29 mm). Similarly, the same trend (underestimation of inner radius) is shown when considering the different FOV data and to a lesser degree in the case of different reconstruction kernels (see Figs. 8A and 9A). It is important to note that the estimation error is approximately the same for all the FOVs as measured in millimeters. However, as the FOV increases, the pixel size increases and the error value becomes smaller when taken relative to the pixel size. When considering the images with different reconstruction kernels, the mean error decreases as the image sharpness increase except for the thick-walled fifth tube.
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Considering the results for the estimation of the outer radius of the tubes (Figs. 6C, 8C, and 9C), the tilt-adjusted FWHM method overestimates the outer radius of thin-walled tubes and appears to underestimate the outer radius in the case of the thick-walled tube for all the tilt angles, FOVs, and normal and sharp reconstruction kernels considered. An exception to this bias pattern is images reconstructed with the very sharp reconstruction kernel. In this particular case, one cannot conclude that the tilt-adjusted FWHM method generates any specific bias pattern. As mentioned earlier, although the numerical value of the error is the same for all FOVs, the relative-to-pixel value of error is smaller as the pixel size increases.
Quantitatively, the tilt-adjusted FWHM method is able to estimate the outer
radius to within 0.13 mm (
1/2 of the smallest pixel) for all the cases.
The only exception was in the case of the smallest tube analyzed with its long
axis tilted at an angle of 45° to the scan plane. However, if errors were
reported relative to tube size instead of absolute error values, then a trend
of smaller errors occurring with larger tubes is seen
(Fig. 7 in the case of
different tilt angles). It is important to note that the radii reported here
are calculated by using the tilt-adjusted FWHM method and not raw
measurements. These results are estimations of the inner and outer radii along
the horizontal and vertical axes in the image adjusted with the estimated tilt
angle resulting from the ellipse-fitting method. Although the raw FWHM
measurements were comparable to the tilt-adjusted FWHM measurements when the
tilt was 0°, the radius estimation error becomes as large as 2 mm when
measured with other tilt angles (22.5 and 45°).
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Model-Based Method
The mean estimation error is shown in Figs. 6B, 8B, and 9B for the inner radius and Figs. 6D, 8D, and 9D for the outer radius of the phantom tubes listed in Table 1 by using the model-based method and different image parameters. This method receives its input from the previous stage (FWHM and ellipse fitting), and, by minimizing the difference between the generated model and the actual image, final estimates for the inner and outer radii of the tubes are generated.
Figure 6B shows the mean estimation error of the inner radius of the tubes obtained by using the model-based method imaged at three different tilt angles and reconstructed with a normal reconstruction filter and a 15-cm FOV. The results show a weak pattern of overestimation of the inner radius with the exception of the large-tilt case. In addition, the effect of the tilt angle shows no specific pattern on the results. However, the error estimates are in general smaller than one-third of a 0.29-mm pixel (<0.10 mm), with the exception of the thick-walled tube at all tilts and the smallest tube at 45° tilt, where the errors increase to about one-half of a pixel.
In examining the outer radius estimation results, error reported does not follow a specific pattern and is distributed on both sides of the horizontal axis. There was no specific influence for the tilt angle on the magnitude of the error except for the smallest tube, where error increases with tilt angle. Quantitatively, the error is within one-third of a 0.29-mm pixel (<0.1 mm), except for the smallest tube at the greatest tilt considered.
Applying the model-based method to images reconstructed with different FOVs, Fig. 8B shows the estimation error of the inner radius. These images were reconstructed with the normal reconstruction filter and no tilt. The error values show that no pattern of bias exists; the thick-walled tube is overestimated as previously found and only the smallest tube is underestimated. Overall, the error is estimated to be within one-half a 0.29-mm pixel for all the tubes at all the FOVs. The error pattern does not appear to be affected by the change in the FOV. However, because the increase in FOV causes the pixel dimension to increase, therefore there would be a pattern such that as FOV increases the relative error decreases.
Similarly, Fig. 8D shows the results of the model-based approach applied to images of the phantom reconstructed with different FOVs, normal reconstruction kernel, and no tilt to the direction of table motion. The results show a somewhat weak pattern of overestimation of the outer radius of the tubes, especially the thin-walled ones. In addition, the model-based method seems to perform equally well with all FOVs and generates errors that are within a half a pixel relative to the pixel size.
Figure 9B shows the inner radius estimation error when analyzing images reconstructed with three different reconstruction filters, a 15-cm FOV, and no tilt. The error values show a clear overestimation of the inner radius for all the tubes. In addition, the bias pattern exhibits an increase in error with the increase of the sharpness of the reconstruction filter to the extent of tripling the radius estimation error. Numerically, the error reported is within approximately one 0.29-mm pixel.
Lastly, Fig. 9D shows the outer radius estimation error. In the case of the model-based method with different reconstruction filters, a weak underestimation bias is noticeable, with the smallest tube as an exception. The results do not suggest any dependence of the error values on the kernel used. In all the cases, the estimation error is within one-half a 0.29-mm pixel. In general, the error estimates of the outer radii are higher when FOVs other than 15 cm or kernels other than the normal are used.
The model-based results above have been reported in terms of the absolute error. However, when these mean error values are reported relative to the tube size (see Fig. 7 for the tilt cases and in similar figures for the different FOVs and reconstruction kernels), a pattern of error is seen that demonstrates that, as the tube radius increases, the percent error value decreases. The error measured for all tilt angles slightly deviates from this pattern. In this case, with the exception of the smallest tube at the greatest tilt, the error is within 10% of the tube inner radius and within 6% of the outer radius, irrespective of the tube size (see Fig. 7).
Method Comparison
For each combination of imaging parameters (Figs. 6, 8, and 9), the tilt-adjusted FWHM and model-based methods were found to be significantly different (P < 0.05) when using both the standard paired t-test and the more robust nonparametric Wilcoxon signed-rank test when used in estimating the inner radii. This showed that the absolute measurement errors when using the model-based approach were statistically smaller than those when using the modified FWHM method. However, the absolute errors from estimating the outer radii with the two methods were not found to be statistically different.
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DISCUSSION |
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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It was noted that the tilt angle mean error variation between different reconstruction kernels for the same tube was smaller than the variation in the case of the other image parameters (FOVs or tilt angles). Considering the effect of these parameters on the appearance of the tube cross sections, the reconstruction kernel causes the least change in the overall shape compared with the effects of different FOVs or different tilt angles. As a result of using the ellipse method to estimate the tilt angle, the variation in tilt error estimation increases as the effect of tilt on shape increases.
The tilt-adjusted FWHM method was found to underestimate the inner radius independent of the tilt angle, reconstruction kernel, or size of the pixel. This has been shown earlier to be true by Reinhardt et al. (24) by theoretical simulation and experimental results. They found that, particularly for thin-walled tubes (wall thickness relative to the width of the scanner PSF) and using normal reconstruction kernel, the inner radius suffers more estimation bias than the outer radius. In an exception to this generalization, the thick-walled tube reconstructed with high-spatial-frequency kernels (sharp and very sharp) is overestimated. The combination of the thick wall and the high-spatial-frequency kernels reverses the effects of the FWHM method. By reducing the blurring caused by the finite response of the scanner and the partial volume effect, the profile across the airway wall becomes steeper and more sensitive to small estimation errors.
Considering the model-based method, a less defined overestimating pattern of bias is present for the inner radius. The model-based method shows smaller error values compared with the tilt-adjusted FWHM method. The model-based results for the inner radius are individually significantly different from the results generated by using the modified FWHM method at the significance level P < 0.05 and are more accurate when compared using Wilcoxon signed-rank test. This shows that the model devised represents the blurring of the interface of the lumen and the inner wall with good accuracy. On the other hand, the performance of the model-based method declines when analyzing images reconstructed with high-spatial-frequency kernels. This is to be expected because the sharpening kernels change the scanner system model from that of the proposed Gaussian model. Notice that the error in estimating the inner radius more than triples (only doubles for tube 5) when analyzing images reconstructed with very sharp compared with normal reconstruction kernel. This suggests that sharpening filters are more useful for visualization and image display rather than model-based quantitative assessment.
Considering the bias patterns for the estimation of the outer radius of the phantom tubes, the tilt-adjusted FWHM method shows an overestimation bias for all the thin-walled tubes and an underestimation bias for the thick-walled tube. In addition, the outer radius estimation bias was less pronounced than the inner radius estimation bias. This is, again, in agreement with what Reinhardt et al. (24) have shown. This bias pattern for the outer radius estimation becomes less prominent when considering images with different reconstruction kernels. The thick-walled tube is unaffected and the method underestimates the outer radius. For the thin-walled tubes, the bias cannot be presumed to follow a certain pattern.
The outer radius estimation using the model-based approach appears to
produce good results for the 15-cm FOV and normal reconstruction kernel. It
does not show any specific bias patterns compared with the modified FWHM
method. The model-based method, however, did not perform as well when applied
to images that were reconstructed with other FOVs. In addition to the fewer
sampling points represented by larger pixel size, this could be attributed to
the process by which the background gray scale (i) is
selected in phase 1 in combination with the compensation for the
in-plane rotation. The background gray scale is taken as the mean of the pixel
gray levels comprising the outermost 8% of the profile through the center of
the tube. Depending on the size of the tube, the 8% is at least one pixel
wide, which with the heterogeneity of the background (which is expected to be
more pronounced in actual lung images) might not be an appropriate
representative value. The effect of the rotation is more prominent around the
edges of the image than closer to the center. Therefore, when an image is
rotated, the 8% of the profile would have been interpolated, leading to
somewhat altered profile. This could explain why the Wilcoxon signed-rank test
shows that the tilt-compensated FWHM method is more accurate than the
model-based method. Despite the fact that the error is larger for the
multiple-FOV data set, it is still within one-half a pixel relative to the
pixel size of the image analyzed. It whether unclear if there is a pattern of
bias for the outer radius estimation using the model-based method.
To estimate the scanner characteristics, a scanner-specific calibration
step was performed using sapphire 0.4-mm radius beads to estimate
x and y, and the larger
metal and zirconium beads where used to estimate z.
Simulations have shown that the error in estimating the sigma values decreases
as the size of the beads becomes smaller. This is in agreement with systems
theory, which requires an impulse to be infinitesimally narrow to enable
system identification. The scanner resolution and partial volume effects limit
the smallest practical bead size. Because the scanner does not provide a slice
thickness comparable to the in-plane pixel size, the use of the sapphire beads
is limited to the estimation of x and
y. There were only a few sample points along the
z-axis for the sapphire beads, making an estimation of a
z unreliable. Therefore, the metal and zirconium
beads were used to estimate z because they provided
greater numbers of samples along the z direction
(20). Simulations show that
beads of the size used in estimating the PSF parameters overestimate in-plane
values by 8%. The experimental results show that the PSF is
slightly anisotropic
(x
y). The geometry
and the design of the scanner used (Imatron EBCT C-150) cause the PSF to be
anisotropic. Chang (8) has
shown that the slice characteristics of the EBCT are nonuniform and vary
depending on the position within the image field and on the mode under which
the scanner is operated.
The model we are using assumes to be constant throughout the image
field and is represented by its value near the isocenter, which simplifies the
model and introduces error as the region of interest moves away from the
center. This intrinsic spatial variability in the scanner PSF and its
interaction with the effect of the finite bead size has not been studied.
Therefore, the obtained values for the PSF parameters were not corrected for
the finite bead size. Because of the unique construction of the Imatron
scanner used, the previous assumptions are valid approximations to the actual
scanner characteristics provided that target "C" is being used
(8). In our case all the images
obtained were collected using the C target. Another source of error in
estimating the standard deviation in the z direction is the problem
arising from not having enough samples along the z-axis. This is
caused by the relative thickness of the slice compared with the finer size of
the in-plane pixel. State-of-the-art MDCT scanners can produce isotropic image
voxels, allowing a uniform sampling of the PSF
(33). These nonuniformity
problems might not be as apparent if a different scanner is used, which would
require the calibration step to be repeated. Calibration would also be needed
even if the same scanner is used but the image is reconstructed with a
different reconstruction kernel.
Other factors may be affecting the accuracy of our methods. Error could be
attributed, in part, to the slight eccentricity of the phantom tubes. This
source of error causes the tubes to seem more elliptical than they are and
will result in a higher error especially in the 0°-tilt case. The apparent
ellipticity of the tubes is also affected by the error in estimating the tilt
angle. To the extent that actual airways are noncircular in cross section, the
same error will be present when this method is applied to the in vivo human
lung. Another source of error is the approximations implied in the airway
model and the scanner PSF. The simple linear model used may not accommodate
for nonlinearities in the system function owing to asymmetry in the 3D PSF of
the scanner or any proprietary algorithms used in reconstructing the images.
The inaccuracies in the estimation of the parameters due to noise or
the finite size of the beads used are also another possible source of
error.
The model-based method is not limited to the analysis of airways that happen to be perpendicular to the scan plane. It allows us to use those airways that are scanned tilted to the scan plane. Thus one is able to evaluate a much larger sampling of airway segments in a 2D CT slice. The 2D evaluation of the airway tree remains of importance even in the presence of newer 3D methods, because of the continued and growing worry of X-ray dose to subjects entered into study protocols requiring assessments to be done repeatedly over prolonged time periods. This is of particular concern in young children. This method can be automated by adding methods for airway tree detection and centerline determination (22, 25, 30). This method not only helps with the 2D assessment of the airway tree, but it can work well in combination with volumetric image sets of the airway (30).
In conclusion, in this paper, we have presented a new method for the measurement of airway geometry from 2D HRCT images with simultaneous estimation of airway tilt angle relative to the scan plane. The new model-based method models the scanning process as a linear convolution between an airway model and a scanner model. The airways are assumed to be circular in cross section and have a uniform lumen and wall composition, and the background parenchyma is assumed to be of varying density. The scanner is modeled as a spatially invariant 3D Gaussian function and is described by its PSF. A calibration step is required to estimate the scanner characteristics. The new method was compared against a modified FWHM method. Correcting for the tilt angle and averaging the measurements along the horizontal and vertical axes of the elliptical pattern representing the airway modified the FWHM technique.
The model-based method is able to estimate airway geometry to within one-half a 0.29-mm pixel or less in size and to within 4° in tilt for all FOVs and tilt angles reconstructed with the normal kernel. One disadvantage is the longer time it takes compared with the near instantaneous results obtained with the tilt-adjusted FWHM method. However, this method eliminates the need for reslicing the data set and increases the number of airways available for evaluation by allowing us to be able to measure airways not previously possible to analyze directly. As computers become faster and with code optimization, the computational cost could be reduced greatly. A hybrid system that combines the tilt-adjusted FWHM, for its better outer radius accuracy, and our model-based method, for its greater inner radius accuracy, will enhance the overall system accuracy as well as decrease the execution time.
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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ACKNOWLEDGMENTS |
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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TOPABSTRACTMETHODS AND MATERIALSRESULTSDISCUSSIONDISCLOSURESACKNOWLEDGMENTSREFERENCES |
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