Tracking Lung Motion: Correlating Inverse Consistent Image
Registration and Spirometry
Gary E. Christensen1, Joo Hyun Song1, Issam El Naqa2, Wei Lu2, and Daniel A. Low2
1
Department of Electrical and Computer Engineering, The University of Iowa, Iowa City, IA 52242
Department of Radiation Oncology, Mallinckrodt Institute of Radiology, Washington University School of Medicine, St. Louis, MO 63110
2
Abstract
Breathing motion is one of the major limiting factors for reducing dose and irradiation of normal tissue for conventional conformal
radiotherapy. This paper describes a relationship between tracking lung motion using spirometry data and image registration of
consecutive CT image volumes collected from a multislice CT scanner over multiple breathing cycles. In this study, 15 multislice
CT image volumes were collected at the same table position over 2-3 breathing cycles. Small deformation inverse consistent linear
elastic (SICLE) image registration was used to register consecutive image scans. The log-Jacobian of these incremental
transformations was computed to give a map of the local expansion and contraction of the lung during the breathing cycle. It is
shown that there is very good correlation (R2=0.9666) between the average 3D log-Jacobian value within a transverse lung cross
section near the diaphragm with the differential spirometry data for the patient studied.
Keywords
Image registration, spirometry, multislice CT, breathing motion, 3D conformal therapy, Intensity modulated radiation therapy
Introduction
Despite the many recent technological advances that have
occurred for conformal radiation therapy1 breathing motion
remains a significant source of error in radiotherapy treatment
planning for the thorax and upper abdomen2-6. For conventional
conformal radiotherapy techniques, accounting for lung motion
requires the beam aperture to be increased relative to the
physical tumor size in order to encompass the tumor projection
during the breathing cycle. Spirometry can be used as an
indirect measurement of lung motion7. Alternatively, image
registration can be used to track lung motion by registering
sequential CT scans of the thorax taken during the breathing
cycle. This paper seeks to find a relationship between
spirometry measurements of lung motion and local expansion
and contraction measurements produced by tracking lung
motion with inverse consistent image registration.
on the expectation of 3–4 s per breathing period, and the desire
to acquire scans over 3 breathing cycles.
Spirometry
To associate the CT scans with tidal volume, simultaneous
spirometry measurements (VMM-400, Interface Associates,
Laguna Niguel, CA) were acquired using an independent
workstation. The patients were instructed to breath normally
through a mouthpiece and were fitted with a nose clip to
prevent leakage through the nostrils. The spirometer data
acquisition software (Labview, National Instruments, Austin,
TX) was written to store the time and relative tidal volume at
10 ms intervals. The patients were instructed to continue
breathing normally with the spirometer during the entire
scanning sequence, which took approximately 10 min.
SICLE Image Registration
Material and methods
CT acquisition
The process used to acquire and analyze the CT scan data is
detailed in7. In this work, a 16-slice CT scanner operated in
twelve-slice mode (Sensation 16, Siemens Medical Systems,
Iselin, NJ) was used to scan a lung cancer patient. The
scanning technique used 120 kVp and 80 mA with twelve 1.5
mm thick slices acquired using the cine´ mode (the cine´ mode
operates the scanner without couch motion). CT scans were
continuously acquired every 0.75 s with each scan (360° gantry
rotation) requiring 0.5 s acquisition time (40 mAs) followed by
0.25 s dead time. Here 15 scans were acquired at each couch
position before the couch was moved to the adjacent position.
This process was repeated until the entire thorax was scanned
(23–26 couch Positions). The selection of 15 scans was based
The small deformation inverse consistent linear elastic
(SICLE) image registration8-10 was used to nonrigidly register
adjacent 12 slice temporal image volumes. Inverse consistent
image registration jointly estimates the forward and reverse
transformations between two images while minimizing the
||x-x'||
inverse consistency error
h(x)=y
forward transformation
x
starting
coordinate
y
intermediate
coordinate
x'
final
coordinate
Image 1
g(y)=x'
reverse transformation
Image 2
Figure 1: The inverse consistency error is defined as the distance
between the starting and ending points produced by mapping a
point through the forward and then reverse transformations.
inverse consistency error between the forward and reverse
transformations (see Fig. 1). For a particular registration
algorithm, minimizing the inverse consistency error provides
more accurate correspondence between two images compared
to independently estimating the forward and reverse
transformations.
The SICLE image registration algorithm is summarized as
follows. Let T and S represent template and target image
sets, respectively, defined on the domain   [0,1)3 . The
intensities of each image T and S are scaled between 0 and
1.
The forward transformation h and the reverse
transformation g are vector-valued functions that map the
coordinate system   [0,1)3 onto itself, i.e.,
h :    and g :    . It is assumed that h and g are
continuously differentiable mappings that do not fold space so
that they preserve topology. Throughout it is assumed that
h  x   x  u  x  , h1  x   x  u  x  , g  x   x  w  x  and
Figure 2: Spirometry measurements for couch position #16.
image
g 1  x   x  w  x 
h1  h  x    x
where
and
g 1  g  x    x . The vector-valued functions u , w , u , and
w are called displacement fields since they define the
transformation in terms of a displacement from a location x .
All of the functions h , g , h 1 , g 1 , u , u , w , and w are
31 vector-valued functions defined on the  .
The SICLE image registration algorithm jointly estimates the
forward and reverse transformations h and g , respectively,
that minimize the cost function given by
C    T  h  x    S  x   S  g  x    T  x  dx
2
2

   Lu  x   Lw  x  dx
2
together and penalizes transformations that are not inverses of
one another.
Results and discussion
Fifteen 12 slice CT scans were collected over a 11 second
period to capture 3 breathing cycles at a single couch position
for the patient in this study. Nineteen couch positions were
collected but only one couch position (number 16) near the
diaphragm was used for analysis. Figure 2 shows the
spirometry data collected for the 15 scans collected at this
couch position. This graph shows that the patient’s breathing
with respect to the tidal volume was not constant over time for
this data acquisition. The variation included the time per
breathing cycle (6 scans were required for the first cycle and 4
for the other two cycles), the incremental difference in tidal
volume between scans, and the maximum/minimum tidal
volume.
2

   u  x   w  x   w  x   u  x  dx
2
2

where  are relative weighting factors for each of imaging
modalities and  and  define the relative importance of the
bending energy minimization and the inverse consistency
terms. The constants  define the relative importance of
image data with respect to the regularization terms of the cost
function. The first integral of the cost function defines the
cumulative intensity squared error cost between the modalities
of the transformed template T h  x  and the target S  x 



and between the modalities of the transformed target S g  x 
and the template T  x  .

The second integral is used to
regularize the forward and reverse displacement fields u and
w , respectively. This term is used to enforce the displacement
fields to be smooth and continuous by penalizing large
derivatives of the displacement fields. The third integral is
called the inverse consistency constraint and is minimized
when the forward and reverse transformations h and g ,
respectively, are inverses of each other. This integral couples
the estimation of the forward and reverse transformations
The SICLE image registration algorithm was used to match
scan 1 with 2, scan 2 with 3, etc. producing 14 pairwise
volumetric image registrations. A Fourier series was used to
parameterize the transformation8. Using the Fourier series
parameterization assumes that the image volumes are periodic
in the x, y, and z coordinate directions. This is true for the x
and y coordinate directions since the transverse sections
collected included the whole thorax. However, the 12 slices in
the z-coordinate direction do not satisfy the periodicity
constraint since slice 1 and 12 of volume correspond to
different slices through the lung. This produced a mismatch
between the registration algorithm and the CT data being
registered. Another source of error in the procedure is that
portions of the lung come in and out of view during the
breathing cycle. The effect of this is that no correspondence
exists between image scans for portions of the lung that come
in and out of the field of view. The effect of the mismatched
boundary conditions and lack of field of view correspondence
is most severe at the top and bottom slices of the volume and is
decreased in the middle slices. As a result of these factors, we
only used data extracted from slice 7 of the volume for analysis.
The 3D Log-Jacobian11 was computed at each voxel location
for each of the 14 incremental transformations. The Jacobian
measures the pointwise expansion and contraction at each point
Figure 3 shows the log-Jacobian values color-coded and
superimposed on transverse slice 7 from couch position 16.
Panels A-D correspond to the expansion/contraction that
occurred from scans 5-6, 6-7, 7-8, and 8-9, respectively. The
color-coded log-Jacobian images are superimposed on the
transverse section of the second of the two image volumes,
respectively. The spirometry data in Fig. 2 shows that the tidal
volume increased from scans 5-6 and 6-7 and decreased from
scans 7-8 and 8-9. The colors in Fig. 3 are normalized with
each other and green corresponds to zero expansion/contraction.
Red corresponds to expansion from the current image to the
next and magenta corresponds to contraction. Notice that lungs
in panels A and B are colored red corresponding to expansion
of the lungs. Likewise, the log-Jacobian values of lungs in
panels C and D are magenta corresponding to contraction of
the lungs. All four of these results are consistent with the
spirometry data shown in Fig. 2. The backgrounds of these
images are mostly green serving as a qualitative check that
there is no expansion or contraction outside the body.
volumes rather than the CT images directly. The plot in Fig. 4.
shows the difference in tidal volume between two consecutive
image scans plotted against the average log-Jacobian value
across the lung segmentation in slice 7 of the volume. The 14
data points are shown to have good correlation with each other
as evident by the regression line. The simple linear regression
analysis provides a coefficient of determination of 0.9666 for
this patient. We have applied this analysis to image scans from
a second patient and found similar results.
Log-Jacobian vs. Spirometry Difference
0.80
0.60
0.40
0.20
Log-Jac
in the image volume. Taking the logarithm of the Jacobian
produces a linearized metric for expansion and contraction.
For example, a unit cube of material that expands in volume by
a factor of 2 has a Jacobian of 2, that is reduced in volume by a
factor of 2 has a Jacobian of 0.5, and that has no change in
volume has a Jacobian of 1. It is seen by taking the base-10
logarithm of these numbers log(2)=0.301, log(0.5)=-0.301, and
log(1)=0 gives a linear scale for expansion and contraction.
0.00
-300
-200
-100
0
100
200
300
400
-0.20
-0.40
y = 0.0017x + 0.0065
R2 = 0.9666
-0.60
Spirom etry Difference
Figure 4: Correlation between the average 3D log-Jacobian values
in the lung for slice 7 of couch position #16 and the difference in
the spirometry measurement.
Conclusion
This paper presented preliminary results that show a good
correlation (R2=0.9666) between expansion/contraction
determined from inverse consistent image registration of
consecutive multislice CT image scans and spirometry data.
This result suggests that the SICLE image registration
algorithm is able to retro-actively track breathing motion of the
lung across consecutive CT image acquisitions. Future work is
needed to verify these findings and to use image registration to
improve real-time conformal radiation treatment.
Figure 3: Color coding superimposed on a temporal sequence of
transverse CT images shows the 3D expansion and contraction of
the lung during normal breathing predicted by the SICLE image
registration algorithm. 12 slice CT data volumes were used to
estimate the motion shown. Red corresponds to maximum
expansion, green to no expansion/contraction, and magenta to
maximum contraction.
The correlation between the spirometry data and the average
log-Jacobian values in the lungs for this study is shown in
Figure 4. The transformations used for this graph were
computed by registering segmented lung volume images rather
than the volumetric CT data. There was virtually no difference
in the Jacobian pattern across the lung using the segmented
Acknowledgments: This work supported in part by NIH R01
96679, NIH R24 HL64368, and a corporate grant from
Computerized Medical Systems.
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