N-Phase Local Expansion Ratio for Characterizing Out-of-Phase Lung Ventilation

A. Image Acquisition

B. Image Registration

The sum of squared tissue volume difference (SSTVD) image registration algorithm [16] was used to register lung CT volumes. This algorithm was chosen since it models the CT intensity change of the lung associated with breathing. A detailed comparison of the accuracy of the sum of squared differences, mutual information and SSTVD similarity cost functions for lung image registration can be found in [17]. The SSTVD image registration algorithm has been shown to have sub-voxel accuracy [18], [19]. A brief overview of the algorithm follows.

Let Ω ⊂ ℝ³ represent the domain or coordinate system of a 3D CT image to be registered. SSTVD image registration estimates a smooth one-to-one correspondence map ϕ : Ω → Ω between a fixed image I_f : Ω → ℝ and a moving image I_m : Ω → ℝ that minimizes the cost function

where C_{SSTV D} is the SSTVD similarity cost, Reg(ϕ) is the regularization cost and λ is the regularization weight. The CT image in Housfield unit (HU) is converted into a tissue fraction/density image by:

where the HUs of tissue and air are approximately HU_tissue = 55 and HU_air = −1000. The tissue fraction images associated with I_f and I_m are denoted by R_f and R_m, respectively, i.e., $R_f=\frac{I_f+1000}{1055}$ and $R_m=\frac{I_m+1000}{1055}$. The sum of squared tissue volume difference (SSTVD) similarity metric [16], [18], [20] is given by

The regularization cost is given by

where $\nabla ={\left[\frac{\partial }{\partial x_1},\frac{\partial }{\partial x_2},\frac{\partial }{\partial x_3}\right]}^T,\nabla$ is the divergence operator and u = ϕ - Id is the associated displacement vector field, where Id is the identity map. The values c₁ = 0.75 and c₂ = 0.25 were used in this study.

The nonrigid transformation ϕ was parameterized by uniform cubic B-splines. A multi-resolution multi-grid framework was used in the C++ implementation of the registration method. Six resolution levels were used, the final B-spline grid spacing was 4x4x4 voxels, and the final image resolution was 2x2x2 voxels. A Broyden-Fletcher-Goldfarb-Shanno (LBFGS) optimizer was used for limited memory consumption and rapid convergence.

C. 2-Phase Local Expansion Ratio (LER₂)

The 2-phase local expansion ratio (LER₂) was computed following the approach of Reinhardt et al. [12] and Ding et al. [13]. In this approach, local ventilation is estimated by first estimating a transformation (i.e., dense correspondence map) between the end inspiration and end expiration lung CT image volumes and then taking the Jacobian determinant of the transformation. SSTVD image registration was used to estimate a nonrigid pullback transformation ϕ that transforms the 100IN phase to look like the 0EX phase. The domain of the pullback transformation ϕ is the coordinate system of the 0EX phase and its range is the coordinate system of the 100IN phase, i.e., y = ϕ(x) maps a point x defined in the 0EX coordinate system to its corresponding location y in the 100IN phase. LER₂ measures the expansion of the lung at each point x and is given by

where |J_ϕ| is the determinant of the Jacobian matrix. Equation (5) is the ratio of a transformed unit volume of tissue to its original unit volume. The Jacobian matrix J_ϕ of the transformation ϕ is given by:

The derivatives in the Jacobian matrix were computed numerically by the symmetric difference.

D. N-Phase Local Expansion Ratio (LER_N)

The proposed LER_N measure uses all N phases of a 4DCT scan to estimate the LER. Calculation of the LER_N involves estimating the local lung volume in each respiratory phase. This is achieved by performing pairwise registrations from each breathing phase to the 0EX phase as shown in Fig. 2. The Jacobian determinant of the pullback transformation ϕ_i from the i^th breathing phase to the 0EX phase is denoted by J_i(x) = |J_ϕi (x)|. For i = 0, J₀(x) = |J_ϕ0 (x)| ≜ 1 is the Jacobian determinant of the identity map, i.e., the Jacobian determinant of the map from 0EX to 0EX. Note that the values of the Jacobian determinant image J_i represents the pointwise lung volume expansion in the i^th breathing phase with respect to the 0EX phase. LER_N is defined at x ∈ Ω by

where N denotes the number of breathing phases of a 4DCT scan. Note that this definition implies that LER_N ≥ LER₂ where equality holds if and only if the lung was breathing in phase at the point x.

We note that LER_N is more computationally intensive than LER₂, because it requires deformable image registration for N-1 pairs of image datasets, whereas LER₂ only requires one registration.

E. In-phase/Out-of-phase Ventilation (IOV) Threshold

Comparing LER_N to LER₂ provides a means for detecting and characterizing locations of the lung that experience out-of-phase ventilation. One may naively define out-of-phase ventilation when LER_N is greater than LER₂. However, LER_N and LER₂ both suffer from measurement error due to variation in breathing and errors in image registration. The effect of measurement error is to over estimate the amount of the lung that is out-of-phase. To reduce the problem of overestimation, we define a region of the lung to be out-of-phase if

for the IOV threshold T ≥ 1. The value of T specifies a confidence level that LER_N differs sufficiently from LER₂ to label a region as out of phase. We used an IOV threshold value of T = 1.05 for the results presented in this paper. See Section IV-B for a discussion of threshold sensitivity.

A. Image Registration Accuracy

The accuracy of LER₁₀ and LER₂ calculations depends on the accuracy of the image registration. The SSTVD image registration algorithm used in this paper has been shown to have sub-voxel accuracy [18], [19]. For quality control purpose, we investigated image registration accuracy for 5 out of the 32 subjects. We used a landmark construction tool developed by Murphy et al. [21] to automatically choose a well-distributed set of 100 landmarks in the end-exhale CT. The corresponding landmarks in the end-inhale CT were manually labeled by an expert. The mean landmark errors (MLEs) for those 5 subjects were 1.10, 0.76, 1.56, 1.35, and 1.26 in voxels, with mean MLE equal to 1.21 voxels (which were 1mm sided cubes).

B. Spatial Distribution of Out-of-Phase Breathing

The spatial distribution of out-of-phase ventilation can be visualized by displaying the ratio of LER₁₀ to LER₂. The larger this ratio is, the larger the out-of-phase ventilation is. Figure 3 shows the spatial distribution and magnitude of out-of-phase ventilation for 14 of the 32 subjects. The color bar is graduated from green to yellow to red corresponding to different IOV threshold values from 1.0 to 1.1. Regions colored green show in-phase ventilation, i.e., they shown agreement between LER₁₀ and LER₂ and indicate regions of the lung that had peak expansion from 0EX to 100IN. Regions colored yellow to orange are regions transitioning from in-phase breathing to out-of-phase breathing. Regions colored red are regions of the lung that are clearly ventilating out-of-phase. These images show that all subjects had some degree of out-of-phase ventilation. They also show that out-of-phase ventilation is subject specific and are distributed throughout the lung.

Although out-of-phase ventilation seems to more likely to occur in the basal part of the lung, a detailed analysis of the spatial distribution of out-of-phase ventilation requires further study. All the subjects studied in this work had lung cancer. Further study is required to understand the relationship between the location of the tumor and the spatial distribution of out-of-phase ventilation. Likewise, further study is required to study how the spatial distribution of pulmonary comorbidities such as emphysema, COPD, and fibrosis may affect out-ofphase lung ventilation. For example, fibrotic lung regions experience less expansion and contraction over the breathing cycle compared to healthy lung tissue and thus will affect the spatial pattern of out-of-phase breathing accordingly. In addition, further study is required to investigate what effect the motion of the heart has on out-of-phase ventilation in regions of the lung near the heart.

C. In-Phase/Out-of-Phase Ventilation (IOV) Function Plot

Figure 4 shows the in-phase/out-of-phase ventilation (IOV) function plots for 4 of the 32 subjects. An IOV plot is a 2D histogram of LER₁₀ versus LER₂. A logarithmic scale was used for visualization. The functions y = x and y = 1.05x are overlaid on the histogram to partition the plot into in-phase and out-of-phase regions. The function y = x corresponds to LER₁₀ = LER₂. By definition, all points lie above the y = x solid line since LER₁₀ ≥ LER₂. Points that lie above the y = 1.05x solid line (i.e., points above the IOV threshold of 1.05) are defined to be out-of-phase and points that lie between the two lines are defined to be in-phase.

The 2D plane in Fig. 4 is divided into four regions: A, B, C and D. Region A corresponds to in-phase ventilation and regions B, C and D correspond to out-of-phase ventilation. High-function regions are defined as regions that have volume change greater than 1.1 whereas low-function regions have volume change less than 1.1. The 1.1 threshold used to define regions of high ventilation/function was chosen to match our prior work [22]. When comparing lung tissue mechanics changes following radiation therapy, lung regions with Jacobian value > 1.1 displayed significantly greater reduction in elasticity for the same radiation dose, when compared to regions with a Jacobian < 1.1. LER₂ defines points to the right of the 1.1 vertical line as high function whereas points to the left are defined as low function. Likewise, LER₁₀ defines points above the 1.1 horizontal line as high function whereas points below this line are defined as low function. Lung function is characterized as low-function by both LER₂ and LER₁₀ in region B. Lung function is characterized as low-function by LER₂ whereas high-function by LER₁₀ in region C. Finally, lung function is characterized as high-function by both LER₂ and LER₁₀ in region D. In summary, region B and region D are characterized the same by LER₂ and LER₁₀ as low-function and high-function, respectively. On the other hand, region C is characterized as high-function by LER₁₀ and mischaracterized as low function by LER₂.

Figure 5 shows the in-phase/out-of-phase ventilation (IOV) functional plot for all 32 subjects. The IOV functional plot shown in Figure 5 is the cumulative 2D histogram of LER₂ versus LER₁₀ computed from all 32 subjects. P(A), P(B), P(C), and P(D) denote the percentages of the voxels in regions A, B, C and D, respectively. On average for the 32 subjects, 78.7% of all voxels were in region A, i.e., 78.7% of the lung had in-phase ventilation using an out-of-phase threshold of 1.05. Conversely, on average 21.3% of the lung had out-of-phase ventilation. There were 3.4% of all voxels in region B, i.e., on average 16% of the out-of-phase ventilation was labeled as low-function by both LER₂ and LER₁₀. There were 10.6% of all voxels in region C, i.e., on average 50% of the out-of-phase ventilation was mislabeled as low-function by LER₂ but was correctly labeled as high-function by the LER₁₀. There were 7.3% of all voxels were in region D, i.e., on average 34% of the out-of-phase ventilation was labeled as high-function by both LER₂ and LER₁₀.

Table I summarizes the percentages of the lung volume for regions A, B, C and D for each of the 32 subjects. This table shows that all subjects had some degree of out-of-phase ventilation. The average percentages for regions A, B, C, and D reported in Table I are slightly different from 78.7% in Fig. 5. This is because percentages in Fig. 5 were calculated using all lung voxels in the population whereas the computations used to generate Table I were normalized to 100% for each subject.

D. LER_N Validation

Specific ventilation derived from Xenon-enhanced CT (Xe-CT) is considered to be a gold standard for ventilation imaging modalities [23]. We used 4DCT and Xe-CT of four sheep to evaluate the accuracy of LER₂ and LER₈ to estimate lung ventilation. For each sheep, the Spearman correlation coefficient of Xe-CT specific ventilation with LER₂ and with LER₈ in out-of-phase regions are shown in Table II. The mean Spearman correlation coefficients for LER₂ and LER₈ in out-of-phase regions are 0.436 and 0.486, respectively. Since the correlation between LER₈ and Xe-CT specific ventilation is 11.5% higher than the correlation between LER₂ and Xe-CT specific ventilation, we conclude that the LER₈ measure is more accurate than the LER₂ measure when quantifying lung function of the four sheep. Previous work reported that the mean Spearman’s correlation between LER₂ and Xe-CT specific ventilation was 0.44 [24].

One way to put the LER₈ correlation coefficient of 0.486 in perspective is to note that the LER₂ method was judged to be the most accurate method for estimating ventilation from CT images in the 2019 AAPM Computed Tomography Ventilation Imaging Evaluation 2019 (CTVIE19) Grand Challenge (publication pending). This competition was the largest competition of its kind to date consisting of 44 teams (23 of which finished the competition) from around the world and required registration of 445 inspiration and expiration CT scans collected from over 20 different research centers. We could not evaluate the LER_N approach on this challenge data set since it only consisted of pairs of CT images. To clarify, some of the challenge data sets were actually 4DCT images, however, only two phases, the end expiration and end inspiration, were released to the public as the challenge data.

A. Potential Clinical Impact

This work has the potential to improve functional avoidance radiotherapy. Radiation therapy is used to treat nearly 75% of all lung cancers [25]. Functional avoidance RT reduces the risk of radiation-induced lung injury by avoiding irradiating high-function lung tissues (i.e., regions of high ventilation) [26]–[34]. Previous work shows that high-function lung tissues are more susceptible to radiation damage than low-function lung tissues [22]. A study conducted by Yamamoto et al. [11] showed that functional avoidance RT planning significantly reduced doses to high-function lung regions without increasing doses to other critical organs.

Figure. 6 shows transverse CT images of the lung of a patient with lung cancer. Overlaid contours show the radiation dose plans for conventional, LER₂-derived functional avoidance, and LER_N-derived functional avoidance. The smallest isodose curve (yellow) encompasses the tumor. The colored regions show the functional avoidance maps where red corresponds to high functioning regions and purple and blue corresponds to low functioning regions. This figure shows that the LER_N-derived plan is better than both the LER₂ and conventional methods. The LER_N has better coverage of the tumor than the LER₂ plan whereas at the same time having comparable coverage of the tumor to the conventional plan. Also notice that the LER_N plan delivered less radiation dose to the normal left lung than the conventional and LER₂ plans. Finally, the dose volume histograms (not shown in this paper) show that the LER_N method delivered more dose to the tumor and less dose to the anatomy to be avoided than both the conventional and LER₂ dose plans for the subject illustrated in Fig. 6.

B. In-Phase/Out-of-phase Ventilation (IOV) Threshold

The in-phase/out-of-phase (IOV) threshold value T in (8) is used to partition the lung into in-phase and out-of-phase regions. In this section, we investigate the sensitivity of T to the percentage of lung that is defined as out-of-phase. To study the sensitivity of T, we constructed the complementary cumulative distribution function (CCDF) for the 32 human subjects in this study (see Fig. 7). In this context, a CCDF is a plot of the percentage of out-of-phase lung volume versus the threshold value T. We refer to such a CCDF as a IOV threshold sensitivity plot. The thick black line in Fig. 7 corresponds to the average of the 32 plots.

By definition, LER₁₀ > T × LER₂ for values of T < 1, i.e., 100% of the lung is defined to be out-of-phase for T < 1. At T = 1, all the IOV threshold sensitivity plots have a discontinuity due to out-of-phase ventilation (See Fig. 7). On average, approximately 55% of the lung is considered as out-of-phase for T = 1. A value of T = 1 is not a useful threshold to define out-of-phase ventilation for the data sets studied because it would mean that more of the lung is out-of-phase than in-phase. This would contradict the definitions of the end inhale and exhale phases. Thus, a value of T > 1 should be chosen as the IOV threshold.

Choosing a biologically relevant value of the IOV threshold T is beyond the scope of this paper since the number of data sets we studied is too small to make any conclusions and all the data sets studied were from a population of individuals with lung cancer. In our future work, we plan to investigate this question on a larger data set that includes healthy subjects and to validate our choice of threshold with an independent measure of out-of-phase ventilation. In this work, we choose the threshold value to be 1.05 based on the average curve in Fig. 7 to reflect Jacobian determinant values that were not due to measurement error. Based on this threshold, we concluded that on average 20.2% of the lung for the individuals studied had out-of-phase ventilation. The IOV threshold determines the percentage of lung with out-of-phase ventilation. For example, if we choose T = 1.1, then on average 8.8% of the lung would be designated as out-of-phase.

C. Dependence of LER_N on Image Registration Algorithm

Jacobian measurements have been shown to be sensitive to image registration algorithms [35]–[37]. To test the sensitivity of our results to registration algorithms, we used the Elastix image registration software [38] to rerun all pairwise registrations. Instead of SSTVD, the mutual information was used as the similarity metric. We used a different multiresolution scheme and a different optimizer (standard gradient descent). Using the Elastix toolbox, we found that at a 1.05 out-ofphase threshold, 24.2% (compared to 20.2% using SSTVD) of the lung had out-of-phase ventilation. The mean Dice coefficient of the out-of-phase images computed from two different registration algorithms was 0.65. Fig. 8 shows the IOV spatial images of a typical subject computed using both image registration algorithms. Fig. 9 shows the IOV function plots for both registration algorithms. These results show that both image registration algorithms produce similar out-of phase images.

D. Choice of LER_N Coordinate System

The definition of LER_N is independent of the choice of the reference coordinate system in which it is calculated (See Fig. 2). This can be seen with the following example. Consider a 4DCT scan of three breathing phases and suppose the volume of a voxel at those phases are 1, 2, 4, respectively. If the first breathing phase was chosen as the reference, the three Jacobian values used in (7) are 1, 2, 4, and $LE{R}_3=\frac{max\left\{1,2,4\right\}}{\min \left\{1,2,4\right\}}=\frac{4}{1}=4$. If the second breathing phase was chosen as the reference, the three Jacobian values are 0.5, 1, 2 and $LE{R}_3=\frac{max\left\{0.5,1,2\right\}}{\min \left\{0.5,1,2\right\}}=\frac{2}{0.5}=4$ similarly if the third breathing phase was chosen as the reference, we have $LE{R}_3=\frac{max\left\{0.25,0.5,1\right\}}{\min \left\{0.25,0.5,1\right\}}=\frac{1}{0.25}=4$. This example show that the computation of LER₃ at this voxel is independent of the reference coordinate system.

We now present a formal proof. Assume that ϕ_i for i = 0,…, N – 1 are given with respect to coordinate system 0 as in Fig. 2 and that we want to calculate the LER_N in the coordinate system of breathing phase j. The transformations from phase i to phase j are given by ${\psi }_{ij}={\varphi }_{i}o{\varphi }_j^{-1}$. The Jacobian determinate of transformation ψ_ij at the point y is given by $J({\psi }_{ij}(y))=J({\varphi }_{i}o{\varphi }_j^{-1}(y))$. LER_N given in the coordinate system of phase j is

Substituting y = ϕ_j(x) into the previous equation gives (7). In other words, LER_N(y) computed in phase j is just a transformed version of LER_N (x) computed in phase 0 by the transformation y = ϕ_j(x).

The statement that the definition of LER_N is coordinate system independent does not mean that there is not coordinate system bias in practice. LER_N coordinate system bias will result from the registration algorithm used, registration errors, and lack of inverse consistency and transitivity of the transformations. Bias in the LER_N calculation will also result from motion artifacts, partial volume effects, and noise in the reference phase image. Image registration from full inspiration to full expiration is preferable when registering lung images since it is easier to compress features such as airways in a digital image than to expand airways. For example small airways that are visible in the full inspiration phase are not visible in the full expiration phase. One way to think of this is that there are more samples (more voxels) of the lung when it is expanded compared to when it is compressed. In Fig. 2 all the phase images were registered to the 0EX phase so that the larger lung images were registered to the smaller lung image.

E. Lower Bounds of LER_N

LER_N for N > 2 has two lower bounds. The first lower bound is LER₂, because LER_N ≥ LER₂ by definition. The second lower bound is $\frac{1}{LE{R}_2}$ and it occurs when the maximum Jacobian determinant of the phase transformations is equal to 1. In this case, $\frac{1}{LE{R}_2}$ represents the ratio of local lung volume in the 0EX phase to local lung volume in the 100IN phase. Thus, ${\mathrm{LER}}_N\ge \frac{1}{LE{R}_2}$. Therefore, LER_N ≥ max{LER₂, 1/LER₂}, i.e., LER₂ and $\frac{1}{LE{R}_2}$ are lower bounds of LER_N. We define x ≜ LER₂ and y ≜ LER_N. The contour lines y = x and $y=\frac{1}{x}$ in the cumulative 2D histogram of LER₂ versus LER₁₀ in Fig. 5 correspond to the lower bounds LER₂ and $\frac{1}{LE{R}_2}$, respectively. Notice that when x < 1, the contour line $y=\frac{1}{x}$ looks like the −45-degree line y = −x+2. The reason for this is as follows. By Taylor expansion of the function of $y=\frac{1}{x}$ at x = 1, we have $\frac{1}{x}\approx 1+(-1)(x-1)=-x+2$ when x is close to 1.