Pulmonary Kinematics From Image Data

The effects of certain lung pathologies include alterations in lung physiology negatively affecting pulmonary compliance. Current approaches to diagnosis and treatment assessment of lung disease commonly rely on pulmonary function testing. Such testing is limited to global measures of lung function, neglecting regional measurements, which are critical for early diagnosis and localization of disease. Increased accessibility to medical image acquisition strategies with high spatiotemporal resolution coupled with the development of sophisticated intensity-based and geometric registration techniques has resulted in the recent exploration of modeling pulmonary motion for calculating local measures of deformation. In this review, the authors provide a broad overview of such research efforts for the estimation of pulmonary deformation. This includes discussion of various techniques, current trends in validation approaches, and the public availability of software and data resources.

“Ceiinosssttuu,” or so wrote the great 17th-century scientist Robert Hooke in describing the response of a spring to an applied force in Latin anagrammatic form. 1

1 Rearranged, the letters form the Latin phrase “ut tensio sic vis,” which, roughly translated, reads “As extension, so is the force” .

Today it is easily recognized in its more ubiquitous form

F=−kΔx, F

−

where Δ x is the change in the length of a spring in response to the applied force, F , and k , which characterizes a mechanical property of the spring, is known as the spring constant. The formulation of Hooke’s law marks a seminal moment in the study of material properties and their responses to applied forces. Analogously, a similar relationship characterizing elastic properties of the lung is approximated from the pressure-volume-compliance relationship :

ΔP=1CΔV, Δ

where C is the volumetric compliance characterizing the properties of the lung (the inverse is called the elastance), and Δ P is the change in pressure, which results in the change of volume, Δ V .

The importance of characterizing biomechanical properties includes the study of pulmonary physiology. Lung disease is one of the leading causes of death in the United States, not only affecting an individual’s respiratory function but having extensive vascular sequelae. Pulmonary function testing, the gold standard for diagnosing lung disease, provides global information about lung function and depends on race, age, gender, and body habitus to interpret the normalcy of a patient’s respiratory parameters. Numerous attempts have been made to study the regional mechanics of the lung ex vivo. However, the power of medical imaging can be harnessed to noninvasively study pulmonary morphology and function. When coupled with image processing techniques such as nonrigid registration, these image data provide both global and regional quantitative assessments of the lung. This approach can be used not only to quantify motion in healthy individuals but to investigate the presence and evolution of disease in patients and populations.

Normal pulmonary anatomy and physiology

The lungs play many critical roles in physiology, such as gas exchange, rapid modulation of blood pH, thermoregulation, and immunoprotection. Structurally, the lung can be described as an elastic body, an elaborate network of fibers connecting the vasculature, airways, and pulmonary interstitium ( Fig 1 ). Normal lung tissue has a homogeneous appearance delineated by lobular boundaries and airway and vascular trees.

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Pulmonary Pathophysiology

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Pulmonary Function Testing

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Early Studies of Pulmonary Biomechanics

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Pulmonary Imaging

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Pulmonary kinematics

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Deformation Analysis

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Fair=CTwater−CTCTwater−CTair, F

air

water

−

water

−

air

where CT air is the intensity of a region of known 100% air content, such as the center of the trachea, and CT water is the intensity of a region of known 100% water content, such as the heart ventricles .

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Fair=−HU1000. F

air

−

1000

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ΔVVexp=Fins−FexpFexp⋅(1−Fins)=1000HUins−HUexpHUexp⋅(1000+HUins). Δ

exp

ins

−

exp

(

−

ins

)

1000

ins

−

exp

(

1000

ins

)

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x=μ+X,y=v+Y,z=ξ+Z. x

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F=∇xx=⎡⎣⎢⎢⎢∂x∂X∂y∂X∂z∂X∂x∂Y∂y∂Y∂z∂Y∂x∂Z∂y∂Z∂z∂Z⎤⎦⎥⎥⎥, F

∇

[

∂

]

or, in terms of the components of the transformation field,

F=⎡⎣⎢⎢⎢∂μ∂X∂ν∂X∂ξ∂X∂μ∂Y∂ν∂Y∂ξ∂Y∂μ∂Z∂ν∂Z∂ξ∂Z⎤⎦⎥⎥⎥=I, F

[

∂

]

where I is the identity matrix. Depending on the form of τ, the partial derivatives in equation 8 are either approximated using finite differences or, if τ is continuous, analytically derived.

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∣∣F∣∣=dVxdVx, |

where dV represents an infinitesimal volume element, and | · | denotes the determinant operation (compare to equation 5 ). | F | is often referred to as the Jacobian and is useful for tabulating compliance measures (see equation 2 ). The principal focus of this review is providing an overview of the various methods that have been proposed to obtain the transformation, τ, from image data of the lungs from which these useful biomechanical measures can be calculated.

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Intensity-based Estimation of τ From Image Data

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Table 1

Sampling of Recent Intensity-based Approaches to Inferring Lung Motion

Source Similarity Regularization Modality Castillo COF HS CT Guerrero et al OF HS CT Dawood OF HS/LK PET Christensen SSD_L_ , IC CT Boldea SSD/APLDM Gaussian CT Dougherty SSD/LI LK CT Li et al ∗ SSD_L_ , IC CT Reinhardt et al SSD LE, IC CT Cook SSD SyN CT Cook MI SyN CT Cook CC SyN CT Gee CC LE MRI Sarrut SSD FFD CT Sarrut OF/APLDM Gaussian CT Voorhees CC LK t-MRI Wang Demons Gaussian CT Wu SSD FFD CT Wu Demons Gaussian CT

APLDM, a priori lung density modification; CC, cross-correlation; COF, compressible optical flow; CT, computed tomography; FFD, freeform deformation; HS, Horn-Schunck; IC, inverse consistency; L, linear operator; LE, linear elastic; LI, Laplacian-filtered images; LK, Lucas-Kanade; MRI, magnetic resonance imaging; OF, optical flow; SSD, sum of squares difference; SyN, symmetric normalization; t-MRI, tagged magnetic resonance imaging.

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Table 2

Sampling of Recent Geometric Approaches to Inferring Lung Motion

Source Similarity Regularization Feature Type Modality Gorbunova Currents Gaussian Vasculature, lung surfaces CT Hilsmann et al ED TPS Vessel bifurcations CT Plathow ED EBS Lung surface CT Li ∗ ED TPS Airway bifurcations CT Tustison JHCT DMFFD Vasculature, lung surfaces CT Cai ED LI Tagging grid cell centroids t-HHe MRI Tustison JHCT DMFFD Tagging planes t-HHe MRI

CT, computed tomography; DMFFD, directly manipulated freeform deformation; EBS, elastic-body splines; ED, Euclidean distance between corresponding features; JHCT, Jensen-Havrda-Charvat-Tsallis divergence; LI, linear interpolation; t-HHe MRI, tagged hyperpolarized 3 He magnetic resonance imaging; TPS, thin-plate splines.

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Optical Flow

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I(x,t)=I(x+δx,t+δt), I

(

)

(

)

where x denotes spatial location, and t denotes acquisition time. Expanding the right side of equation 10 using Taylor’s series expansion yields the following relationship:

I(x,t)=I(x,t)+∇I⋅δx+It⋅δt+O2. I

(

)

(

)

∇

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∇I⋅v+It=0, ∇

where v=δxδt v

t is the sought-after displacement. Because of the ill-posed nature of the problem (ie, there is only one equation for determining the multiple components of v ), explicit regularization of the velocity field is used. In the original contribution of Horn and Schunck , this involves penalization of deviations from an n -dimensional smooth field measured by

ϵsmooth=∥∇v∥2=∑ni=1∑nj=1(∂vi∂xj)2. ϵ

smooth

‖

∇

‖

∑

(

∂

)

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vn+1=vn−∇I(vn⋅∇I+Itλ2+∣∣∇I∣∣2), v

−

∇

(

∇

)

where λ is a user-specified weighting term that modulates the solution between contributions of ϵ smooth and ϵ optical flow . This technique has been used in recently reported research .

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v=It∇I∣∣∇I∣∣2+I2t. v

∇

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Compressible Optical Flow

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SSD(I,J)=∫Ω[I(x)−J(x)]2dΩ, SSD

(

)

∫

[

(

)

−

(

)

]

where Ω denotes the region over which the metric is calculated. However, for a CT acquisition of deformable tissue, such as the lung, this assumption is invalid because of density changes during deformation. If we let I ( x , t ) denote the density function within an object (which is, by definition, linearly proportional to the CT intensity) and assume conservation of mass of that object over the course of deformation, the standard continuity equation from continuum mechanics can be derived in differential form (see, eg, Chadwick ):

∇I⋅v+I∇⋅v+It=0. ∇

∇

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A Priori Lung Density Modification

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I′1(x)=I1(x)+ρ2(z′)−ρ1(z), I

′

(

)

(

)

(

′

)

−

(

)

where I′1(x) I

′

(

) is the modified pixel value at x on slice z , ρ 1 ( z ) is mean density at slice z of I 1 , and ρ 2 ( z ′) is the mean density of image I 2 at the linearly interpolated slice location z ′. The effectiveness of the APLDM preprocessing step was demonstrated by Sarrut et al and showed improvement in registering the thorax between inspiratory and expiratory scans using a modified SSD metric. Two different explicit regularization models, Gaussian and linear elastic, were compared, with the Gaussian transformation showing the biggest improvement in matching expert-segmented landmarks after registration using APLDM (from 13.1-mm average initial distance to 2.7 mm for the Gaussian model and 3.0 mm for linear elastic regularization).

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Other Similarity Metrics

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CC(I,J)=⟨I¯∥∥I¯∥∥,J¯¯∥∥J¯¯∥∥⟩, CC

(

)

〈

‖

〉

where ⟨⋅,⋅⟩ 〈

〉 and ∥⋅∥ ‖

‖ denote the inner product and Euclidean norm, respectively, calculated within the voxel neighborhood.

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Regularization Models and Symmetric Transformation Considerations

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Modeling Four-Dimensional Lung Trajectories

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Geometric Estimation of τ From Image Data

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Landmark Matching

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Matching of Curve and Surface Geometric Primitives

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MRI Tagging

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Validation

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Intensity-based Similarity Measures

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Overlap Assessment of Anatomic Contours

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Jaccard(S,T)=|S∩T||S∪T|, Jaccard

(

)

∩

∪

whereas the Dice coefficient is calculated from

Dice(S,T)=2|S∩T||S|+|T|. Dice

(

)

∩

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Expert-defined Landmarks

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TRE(τ)=1n∑ni=1[τ(pi)−qi]2−−−−−−−−−−√. TRE

(

)

∑

[

(

)

−

]

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STE(T1,T2)=1t2−t1∫t2t1dist{T1[s(t)],T2[s(t)]}dt, STE

(

)

−

∫

dist

{

[

(

)

]

[

(

)

]

}

where s ( t ) defines the curve length along the trajectory between the initial time t 1 and t . For multiple-phase data, this allows one to weight the error in the intermediate phases less than the error at the extreme phases.

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Miscellaneous Approaches

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Public resources

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Table 3

Available Software and Data Resources

Web Site Description Software http://www.itk.org Collection of image analysis algorithms, including image registration sponsored by the NIH (158) http://picsl.upenn.edu/ANTS/ Suite of image registration, segmentation, and other analysis tools (159) http://biocomp.cnb.csic.es/∼iarganda/bUnwarpJ/ Bidirectional spline-based image registration (160); available as an ImageJ plug-in http://elastix.isi.uu.nl Suite of general registration tools, including multiple similarity metrics and transforms (rigid, affine, and spline-based nonrigid) (161) http://www.fmrib.ox.ac.uk/fsl/ FMRIB Software Library: contains FLIRT (linear) and FNIRT (nonrigid) registration algorithms (162,163) Data http://www.dir-lab.com CT image and corresponding landmark evaluation data for 10 subjects across multiple phases of the respiratory cycle https://imaging.nci.nih.gov/ncia/ LIDC of the NCI (164)

CT, computed tomographic; FLIRT, FMRIB’s Linear Image Registration Tool; FMRIB, Functional Magnetic Resonance Imaging of the Brain; FNIRT, FMRIB’s Non-Linear Image Registration Tool; LIDC, Lung Image Database Consortium; NCI, National Cancer Institute; NIH, National Institutes of Health.

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Discussion

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Acknowledgment

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Pulmonary Kinematics From Image Data

Normal pulmonary anatomy and physiology

Pulmonary Pathophysiology

Pulmonary Function Testing

Early Studies of Pulmonary Biomechanics

Pulmonary Imaging

Pulmonary kinematics

Deformation Analysis

Intensity-based Estimation of τ From Image Data

Optical Flow

Compressible Optical Flow

A Priori Lung Density Modification

Other Similarity Metrics

Regularization Models and Symmetric Transformation Considerations

Modeling Four-Dimensional Lung Trajectories

Geometric Estimation of τ From Image Data

Landmark Matching

Matching of Curve and Surface Geometric Primitives

MRI Tagging

Validation

Intensity-based Similarity Measures

Overlap Assessment of Anatomic Contours

Expert-defined Landmarks

Miscellaneous Approaches

Public resources

Discussion

Acknowledgment

References

Further Reading

ACR and the Federal Government

Analysis of 2-[Fluorine-18 ] -Fluoro-2-deoxy-D-glucose Uptake Kinetics in PET Studies of Pulmonary Inflammation

Assessment of Pulmonary Hypertension