Measurement of absolute regional lung air volumes from near-field x-ray speckles

Andrew F. T. Leong; David M. Paganin; Stuart B. Hooper; Melissa L. Siew; Marcus J. Kitchen

doi:10.1364/OE.21.027905

1. Introduction

X-ray imaging is commonly used to reveal the internal structure of the chest, differentiating materials such as soft tissue and bone by their densities. Lung diseases in their early stages, such as emphysema, cystic fibrosis and cancer are difficult to detect in a conventional x-ray image [1]. Since early diagnosis of lung disease is a critical factor for patient prognosis, improved diagnostic methods during the early stages underpins advances in the treatment of these diseases [1]. Lung functional x-ray imaging is more sensitive for detecting lung diseases than anatomical imaging, making it a prospective complementary diagnostic tool to x-ray imaging [2].

Pulmonary functional tests, such as the forced oscillation and spirometry techniques, are performed in clinical diagnostics of respiratory diseases together with x-ray imaging [3–5]. However, these tests give limited information on the location and extent of the abnormality. The need to detect regional lung pathology has produced many different tomographic-based volumetric techniques [6–8]. Some of these techniques have been combined with k-edge subtraction to help resolve small airways [9, 10]. However, they require inhalation of a contrast agent to assess regional lung ventilation except when using computed tomography (CT). Yet, CT imparts a relatively large dose of radiation particularly when several three-dimensional (3D) images need to be reconstructed from many projections in order to measure regional lung air volume (V_L) over time. The length of time involved in recording the projected images to reconstruct a three-dimensional (3D) image of the chest reduces temporal resolution significantly and so, to minimize motion blurring, breath holds or gated imaging is required. Although CT and magnetic resonance imaging (MRI) techniques can acquire and reconstruct images in real-time on the order of milliseconds, the spatial resolution achievable is only of the order of millimeters, which is insufficient to resolve the minor airway structures [11, 12].

Regional volumetric analysis techniques using two-dimensional (2D) phase contrast x-ray (PCX) imaging from a synchrotron x-ray source have emerged to offer superior temporal resolution over those that use tomography [13–15]. PCX is a class of imaging modalities that produce phase-induced intensity variations between the boundary of materials exhibiting different complex refractive indices. With a sufficiently spatially coherent x-ray source they provide high contrast images of soft tissues. When employed to image the chest, the boundaries of the conducting airways and alveoli are rendered highly visible (see Fig. 6). Thus, performing volumetric analysis on 2D PCX images simultaneously provides a detailed image of the chest and information on regional lung aeration with high spatial and temporal resolution. One major benefit of this is allowing mechanical ventilators to be adjusted in real time and on a breath-by-breath basis to avoid lung injury [16–18].

Kitchen et al. [13] developed a technique to measure changes in V_L from 2D propagation-based phase contrast x-ray (PBX) images using a single image phase retrieval algorithm (SIPRA) [19]. PBX imaging is the simplest of the PCX modalities as only a partially coherent source and a sufficiently large object-to-detector propagation distance (ODD) is required [20]. In that technique the animal is immersed upright in water and it is assumed that only the volume of air/water changes within the field of view of the detector. This is an accurate assumption for measuring the total change in V_L as the volume of all other materials, namely bone, can be made to remain constant by choosing a sufficiently large region of interest (ROI) to encompass the entire chest. Reducing the size of the ROI to measure regional changes in V_L can result in bone moving inside and outside the ROI leading to incorrect measures of V_L. Thus, that technique was shown to accurately measure changes in V_L over areas as small as one quarter of the lung, where the change in volume of air was not significantly affected by relative displacement of bones. Leong et al. [15] improved on this by removing the bone using a temporal subtraction-based algorithm to measure V_L on a pixel-by-pixel basis of micron-scale spatial resolution. That technique, however, is prone to misregistrations when the differential motion of the chest becomes overly complex at large V_L. Regional V_L has also been measured from analyzer-based phase contrast x-ray images [14]. A Laue crystal was used to split the x-ray beam to produce two complementary images that were used to create separate images of bone and soft tissue, thus enabling regional V_L measurement. That technique does not require anatomical registration, but it is experimentally more challenging and also requires the chest to be immersed in water for V_L extraction [14].

The planar imaging-type volumetric techniques described above work only when the chest is upright and enclosed in water. Moreover, these techniques are limited to measuring changes in V_L unless an image of a non-aerated lung is available, which is not always easily attainable. This restricts the type of studies that can be performed, for example, it would not be possible to analyze V_L if the animal was ventilated in a supine/prone position or determining its functional residual capacity (FRC; the volume of air at end-expiration). The attenuating medium also reduces the signal-to-noise ratio (SNR). Whilst SNR can be improved by increasing the intensity of the x-ray source, a concomitant increase in radiation dose ensues. In this study, we developed a novel approach for measuring regional V_L of PBX chest images without needing to immerse the object in a water bath. This technique relates the power spectra of lung speckle patterns directly to V_L.

1.1. Lung speckle

Spatially random samples such as particles suspended in liquid and optically rough surfaces of textured materials produce rapid complex refractive index fluctuations [21–26]. The phase of an incident wavefield is randomly altered as it traverses through or reflects off such an object [22]. Downstream of the exit surface of the object, the intensity of the randomized wavefield exhibits bright and dark spots, known as speckles, formed by constructive and destructive interferences, respectively. This is the most likely explanation, for the origin of lung speckle observed using PBX imaging, as the lung contains many air-filled alveoli that are pseudo-randomly distributed and enclosed by thin regions of tissue.

Yagi et al. [27] were one of the first to encounter lung speckle, which was observed using PBX imaging, in a mouse model. They hypothesized that speckles formed as a consequence of alveoli present in the lungs. Kitchen et al. [28] investigated the origin of these speckles by simulating projected PBX images of lung tissue. Here they modeled the alveoli as spheres and generated synthetic PBX images of a lung by numerically propagating the wavefield at the contact plane to the detector surface using the angular spectrum formalism of scalar wave optics. Speckles similar to those seen in PBX chest images of a rabbit pup were observed, and it was shown that this was because the alveoli acted locally as aberrated compound refractive lenses.

Speckle possesses statistical properties that depend on that of the scattering object. The space intensity correlation function, or in Fourier space the power spectrum, is a second order statistical measure that has been used in studying spatially random objects from their speckle pattern; for our study it is the alveoli [22, 23, 29]. Approximately 90% of the total V_L can be accounted for in the alveoli [30]; the remainder is the volume of the airways of the lungs (this includes the trachea, bronchi and bronchioles). Thus, measuring the volume of air in the alveoli is a good approximation of V_L. Herein, we develop a mathematical model to show how lung speckle is related to V_L. In our study, lung speckle is quantified by the integral of its power spectrum between certain radial spatial frequencies, as explained in section 1.2.

1.2. Theory

Consider a lung being made of non-overlapping air-filled voids, where the projected thickness of each void with radius R is described by the object function $G (x, y) = 2 \sqrt{R - x^{2} - y^{2}}$ , embedded randomly in soft tissue (or equivalently in water), as shown in Fig. 1. We assume that the x-rays follow a straight path from the x-ray source to the exit surface of the object. This is known as the projection approximation and it is used to fully describe the phase and intensity changes of the x-ray wave traversing the lung at the exit surface [19]. Using Monte Carlo simulations, Kitchen et al. [28] showed that for the lungs of a mouse the projection approximation was valid at diagnostic x-ray energies (i.e. >6 keV). In the near-field regime, the altered x-ray wave produces a speckled PBX image (I(x,y,z = L)) whose power spectrum can be described as (see Appendix A):

{| F {\frac{I (x, y, z = L)}{I (x, y, z = 0)} - 1} |}^{2} = L^{2} δ_{T}^{2} k_{⊥}^{4} N {| F {G (x, y)} |}^{2},

where I(x,y,z = 0) is the absorption contrast image, T(x,y) is the projected thickness of the object function, δ_T is the refractive index decrement of lung tissue, L is the ODD,

k_{⊥} = \sqrt{k_{x}^{2} + k_{y}^{2}}

is the transverse wavenumber in the (x,y) plane, N is the number of voids, and F is the Fourier transform with respect to x and y. I(x,y,z = 0) was recovered using SIPRA [19]:

I (x, y, z = 0) = F^{- 1} {\frac{F [I (x, y, z = L)]}{1 + (\frac{δ_{l T} L}{μ_{l T}}) k_{⊥}^{2}}},

where F⁻¹ is the inverse Fourier transform with respect to x and y, and μ_T is the absorption coefficient of lung tissue. μ_T and δ_T were set to 54.7 m⁻¹ and 3.99× 10⁻⁷(24 keV), respectively. The former was calculated using the National Institute of Standards and Technology database (NIST) [31] and the latter was calculated using [32]:

δ = \frac{r_{e} λ^{2}}{2 π} \sum_{i} n_{i} {(f_{1})}_{i},

where r_e is the classical electron radius, n_i is the concentration of type i atoms per unit volume and f₁ is the real part of the atomic scattering factor in the forward direction provided by NIST [31].

Fig. 1 Alveoli in lung tissue, modeled by voids randomly embedded within an absorbing medium, is illuminated by a coherent x-ray source and a PBX image is recorded a distance L from the exit surface of the object.

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The definition of what is considered near-field depends on multiple factors. These can be succinctly summarized by the Fresnel number (N_F) [33]:

N_{F} = \frac{a}{L λ {| \nabla_{⊥} φ |}_{\max}},

where a is a characteristic length scale of the object over which the intensity changes appreciably and |∇_⊥φ|_max is the maximum phase gradient transverse to the direction of propagation. The near-field regime is defined to be at

L_{\max} < \frac{a}{λ {| \nabla_{⊥} φ |}_{\max}}

.

The alveoli were modeled as spheres with radius R. To determine the 2D power spectrum of its PBX image, the following integral was solved:

{| F {\tilde{G} (x, y, z)} |}^{2} = {| \int \tilde{G} (x, y, z) exp (2 π i k \cdot r) d r |}^{2},

where k = (k_x, k_y, k_z) and r = (x, y, z) are 3D vectors in Fourier and real space, respectively. The shape function G̃(x,y,z) = 1 for R ≤ 1 and G̃(x,y,z) = 0 elsewhere. The evaluation of Eq. (5) is simplified by the radial symmetry of G̃(x,y,z), which effectively reduces it to a one-dimensional problem. The integral in Eq. (5) can then be expressed analytically by first expanding then evaluating the exponential term as a Taylor series to give an exact solution [34]:

{| F {G (x, y, z)} |}^{2} = {| V \frac{3}{{(k R)}^{2}} [\frac{sin (k R)}{k R} - cos (k R)] |}^{2},

where

k = | k | = \sqrt{k_{x}^{2} + k_{y}^{2} + k_{z}^{2}}

and

V = \frac{4 π R^{3}}{3}

is the volume of a sphere.

According to the Fourier slice theorem, the 2D power spectrum of a projected sphere is a slice of its 3D power spectrum through the origin. Since the 3D power spectrum is radially symmetric, the equation of the 2D power spectrum is identical to Eq. (6) but the z-coordinate is dropped and k is redefined as $k_{⊥} = \sqrt{k_{x}^{2} + k_{y}^{2}}$ .

The 2D version of Eq. (6) is combined with Eq. (1) to give:

{| F {\frac{I (x, y, z = L)}{I (x, y, z = 0)} - 1} |}^{2} = L^{2} δ_{w}^{2} k_{⊥}^{4} N {| \frac{4 π R^{3}}{{(k_{⊥} R)}^{2}} [\frac{sin (k_{⊥} R)}{k_{⊥} R} - cos (k_{⊥} R)] |}^{2} .

Equation (7) assumes there is only one alveolus size whereas a realistic lung model would have a distribution of sizes typically with a coefficient of variation as large as 0.6; however, the error introduced in assuming an average alveolus size is small (< 5%) [35]. To determine the area under the power spectrum (PS_Area), Eq. (7) is integrated over a select domain of radial frequencies, k_⊥0 ≤ |k_⊥| ≤ k_⊥_N, then ξ = k_⊥R is substituted to give:

{PS}_{Area} = 16 π^{2} L^{2} δ_{w}^{2} N R \int_{ξ_{0} / R}^{ξ_{N} / R} {| [\frac{sin (ξ)}{ξ} - cos (ξ)] |}^{2} d ξ .

The limits of the integral in Eq. (8) are dependent on R. However, under experimental conditions, the higher order peaks of the oscillatory function in the integral are suppressed by the detector point spread function (PSF) and penumbral blurring. Consequently, PS_Area is dominated by the lowest order peak (see Fig. 2(a)). This means the limits can be fixed, and be independent of R, so long as it includes the first peak. The area under the first peak bounded by the minimas ξ = 0 and ξ = 4.493 (these are the first two solutions to ( $\frac{sin (ξ)}{ξ} - cos (ξ) = 0$ ) is 2.141. Hence, the integral is approximately 2.141, and Eq. (8) can be simplified to:

{PS}_{Area} = 34 π^{2} L^{2} δ_{w}^{2} N R .

There are additional factors that we have not entirely accounted for in deriving Eq. (9): penumbral blurring, detector PSF and asymmetrically shaped alveoli. If the imaging setup remains unchanged between recordings over time then the power spectra are equally affected by penumbral blurring and the detector PSF. While alveoli are not perfectly spherical, but resemble more like polyhedra, they are randomly positioned and oriented. Consequently, the derivations above is still valid and the only aspect altered is the solution for Eq. (8). The alveoli were modelled as spheres because a simple analytic solution for Eq. (8) exists. Unless these factors are accounted for, Eq. (9) cannot be directly used to measure V_L. Instead, PS_Area can be calibrated against known V_L values as described in section 2.3. As well as accounting for the factors listed above, the calibration curve would also account for the alveoli changing shape during respiration.

Fig. 2 Lung speckle simulations. (a) Azimuthally averaged power spectra of a random distribution of 45 μm air-filled voids simulated within a 10 mm thick water-filled container with a volume packing density of 54%. Under ideal conditions (black), the power spectrum is a damped oscillatory function. To mimic our experimental conditions, ODD was increased from 0.1 m to 2 m, the PBX image was convolved with a Gaussian function, with FWHM=20 um, to simulate the PSF of a detector, and white noise was added, with a standard deviation (σ_noise) of 0.1 intensity. This yielded an image with SNR ≈ 10, which was determined from taking the ratio of the mean intensity of the image and σ_noise. This results in only one prominent peak in the power spectrum (red). (b) A plot displaying the PS_Area of the same sample, but with mean void size of 130 μm, against ODD. (for details on the simulation of lung speckles, see section 2.1)

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The relationship between V_L and PS_Area will depend on how N and R vary over time (t). That is, if N and R were parameterized to N ∝ tⁿ and R ∝ t^r, respectively, where n and r are constants, then,

V_{L} \propto {PS}_{Area}^{\frac{n + 3 r}{n + r}}

Having presented the theoretical background of near-field lung speckle and how it relates to V_L in this section, we test this model by simulating lung tissue in section 2 and present the results and analysis in section 3. Also in section 3, we evaluate how accurately our model can measure changes in V_L in a rabbit pup model. Some directions for future work are provided in section 4 and we conclude with section 5.

2. Methodology

2.1. Simulated lung tissue

To directly validate our theory, simulations were performed with conditions set similarly to that of imaging real lung tissue. The conditions for our simulations are summarized in Table 1. 11.8 mm × 11.8 mm projected lung thickness images, with pixel size 0.59μm, were simulated for two different sets of samples. The pixel size was chosen to adequately sample the phase map of the exit surface and propagated wavefield (that is, the maximum wavefield phase gradient is less than $\frac{π}{2}$ radians per pixel). In the first set of samples, spherical voids of mean radius 65μm were randomly suspended in a rectangular water-filled volume of thickness 1mm. Several non-identical 1mm thick samples were generated and stacked to achieve different sample thicknesses. This was to simulate lung tissue with varying N while R was fixed. In the second set of samples, for each of the different mean sized voids, 5900 voids were suspended in a rectangular water-filled volume of thickness 10mm. In contrast to the first set, this was to simulate lung tissue with varying R while N was fixed. A maximum volume packing fraction of 75% was achieved from the largest mean size void (70μm).

Table 1. Parameters used for simulating PBX images of lung tissue.

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In both sample sets, the position of each void was created using the Box-Muller method to generate random coordinates [36]. If the void’s volume space intersected with that of a void already in the volume then a new coordinate would be generated until an unoccupied location was found. The samples were summed along the axis of propagation to produce the projected thickness images. Despite choosing a small pixel size, |∇_⊥φ|_max was greater than $\frac{π}{2}$ radians per pixel in some parts of the images. The images were therefore filtered with a Gaussian kernel, which at a Full-Width Half-Maximum (FWHM) of 16μm, was sufficient to reduce |∇_⊥φ|_max to less than $\frac{π}{2}$ radians per pixel. Using the projection approximation, the wavefunction at the exit surface of the sample was calculated from the projected thickness image, and forward propagated using the angular spectrum method [19]. Penumbral blurring was effected by convolving the PBX sample images with a Gaussian PSF of FWHM = DR₂/R₁ (where D is the x-ray source size). D was set to 150 μm ×10 μm, which was the x-ray source size of the beamline used. The detector PSF was accounted for by increasing the FWHM of the Gaussian PSF by 20μm. PS_Area was calculated by integrating their power spectrums from k_o = 0.85 (since the zero frequency is a Dirac delta function this value is the next closest to the zero frequency) to k_N =Nyquist frequency.

2.2. Rabbit pup lungs

All imaging experiments took place in Hutch 3 of beamline 20B2 at the SPring-8 synchrotron in Japan [37] with a Si (111) double-bounce monochromator tuned to 24 keV. The x-ray source-to-sample distance was set at 210 m. All animal procedures were conducted in accordance with the protocol approved by the Monash University Animal Ethics Committee and the SPring-8 Animal Care and Use Committee. At 31 days of gestation, pregnant New Zealand white rabbits were anesthetized initially by an intravenous injection (Rapinovet [Schering-Plough Animal Health, USA]; 12 mg kg⁻¹ bolus, 40 mg h⁻¹ infusion) and anesthesia was maintained via isoflurane inhalation (1.5– 4%; Isoflurane, Delvet Pty. Ltd., Australia). Pups were delivered by caesarean section, sedated and surgically intubated. Pups in the first group (Group-Water, n = 15) were immersed in a water-filled cylindrical poly-methyl methacrylate container (plethysmograph) with their head out and the chamber sealed with a rubber diaphragm enclosing their necks. The plethysmograph is routinely used for V_L measurement. At birth, the lungs of the newborn are completely liquid-filled and so imaging the lungs in their fluid-filled state was performed as soon as possible after the pup’s delivery. Two different detectors were used to acquire PBX images: (i) a large format (4000×2672 pixels) Hamamatsu CCD camera (C9300-124F21) with a tapered fiber optic (FOP) coupling the sensor to a 20 μm thick gadolinium oxysulfide (Gd₂O₂S : Tb⁺; P43) phosphor, and (ii) a tandem lens-coupled scientific-CMOS imaging sensor coupled to a 50 μm thick gadolinium oxysulfide (Gd₂O₂S : Tb⁺; P43) phosphor (pco.edge; 2560×2160 pixels). The effective pixel sizes were for these two detectors 16.2 μm, based on the taper ratio of 1.8:1, and 15.23 μm, respectively. Imaging sequences were respiratory gated, with timing controlled by a custom-designed pressure-controlled ventilator [38] at a frame rate of 3 Hz, with a respiratory cycle of 2.5 s and exposure time of 40 ms. All animals were humanely killed at the end of each experiment via anesthetic overdose of Nembutal (Abbott Laboratories, USA, 100 mg/kg).

Pups in the second group (Group-Air, n=3) were initially used for studying the effect of different mechanical ventilation strategies on lung aeration and humanely killed via anesthetic overdose at the end of experiment. The deceased pups were supported in an upright position and connected to a pneumotach (flowmeter) to measure differential airflow at the mouth opening throughout each respiratory cycle. The pups were imaged in air under identical conditions to the first group imaged in water but at a frame rate of 10 Hz, using only the sCMOS camera.

The optimal ODD to image the pups was determined by modeling a 10 mm thick container packed with 130 μm sized voids. The size of the void and container thickness corresponds closely to that of an alveolus [39] and a rabbit pup’s lung, measured from a lateral PBX image of the chest, respectively. Using the steps and conditions as described in section 2.1, several PBX images were simulated from the model at different ODD. Figure 2(b) shows, based on Eq. (9), the near-field regime extends up to ∼3 m but not beyond 5 m. L_max was computed to be 2.7 m. This is consistent with Fig. 2(b). PBX images of pups were therefore recorded at a propagation distance 3 m. While choosing propagation distances much less than 2.7 m would ensure the TIE approximation was well satisfied, the contrast-to-noise ratio of lung speckles would be small and weaken the correlation between V_L and PS_Area.

2.3. Image processing and analysis

Immediately before imaging each pup, multiple dark field (no x-rays) and flat field (x-rays; no sample) images were recorded then averaged to correct for the detector dark current and to normalize against the incident beam intensity, respectively. Nonlinear spatial distortions arising from the FOP camera, as a result of imperfect alignment of the fiber bundles at each end of the taper, were corrected by the use of Delaunay triangulation with bilinear interpolation [40].

Pups from Group-Water were used to generate a PS_Area − V_L calibration curve. Each PBX chest image was divided into quadrants. Quadrants were created by partitioning the chest along the spinal column to separate the left and right lungs and the seventh rib down from the neck to separate the apical and basal lobes. This increased the number of points and also helped determine whether variability in the lung thickness between quadrants affected the calibration curve. PS_Area was calculated using the LHS of Eq. (1) and integrated from k_o = 2mm⁻¹ to k_N =Nyquist frequency. Given our experimental conditions we found this k_o value optimized the correlation strength of the PS_Area − V_L curves between pups and quadrants. This is supported by Figs. 6(c) and 6(d), which shows low frequency components at up to 2mm⁻¹ are notably contaminated by remnant low frequency trends arising predominately from the bones. The technique developed by Kitchen et al. [13] was used to calculate V_L for pups from Group-Water. For those from Group-Air a flowmeter (pneumotach), measuring the rate of air flowing in and out of the lungs, was employed to validate our technique. Both PS_Area and V_L were normalized against the number of pixels so that PS_Area measured from a ROI of any size can be directly converted to V_L.

3. Results & analysis

3.1. Validation of theory using simulated lung tissue

Before validating our theory, simulated lung speckles were qualitatively compared with real lung speckles in real and reciprocal space. The intensity contrast of real speckles in Fig. 3(a) is lower than that of simulated speckles in Fig. 3(b). This is reflected in their azimuthally averaged power spectra, where for real lungs (see Fig. 3(c)) the peak is broader and weaker than that of simulated lungs (see Fig. 3(d)). We believe this could be due to incoherent scattering from the alveoli and in air between the object and detector, and that in real lungs, the alveoli are closely packed rather than randomly positioned as in our simulated lung tissue.

Fig. 3 3.24 mm × 3.24 mm PBX images of a ∼10 mm thick sample (a) of real and (b) simulated (mean diameter of 130 μm) lung tissue normalized against their phase retrieved absorption image. Their corresponding power spectra are shown in (c) and (d), respectively.

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To verify the linear relationship between N and PS_Area, PS_Area was plotted against the sample thickness of simulated lung tissue from the first sample set at 1 m and 3 m ODD (see Fig. 4(a)). The sample thickness is essentially proportional to N as the total volume fraction was made approximately constant throughout the sample. At 3 m ODD, Fig. 4(a) shows N at first varies linearly with PS_Area but begins to breakdown at N = 3000 as |∇_⊥φ|_max increases, resulting in N_F approaching and then being less than unity. Pogany et al. [41] showed that phase contrast decreases beyond the near-field region, which is reflected in reduction of the rate of increase of PS_Area in Fig. 4(a) with respect to N. At 1 m ODD, N_F remained >1 for all sample thickness, thus PS_Area was proportional to N throughout. The dependance between R and PS_Area was investigated by plotting PS_Area against R of simulated lung tissue from the second sample set packed in a 10 mm thick container at 3 m ODD, as shown in Fig. 4(b). As predicted by Eq. (9), PS_Area varies linearly with R even at this large ODD.

Fig. 4 Plots of PS_Area of simulated lung tissue versus (a) number of voids, of size 130 μm (at 1 m and 3 m ODD), and (b) radius of voids with a maximum volume packing density of 75% (at 3 m), all at energy 24 keV.

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3.2. Regional lung air volume measurements

A PS_Area − V_L calibration curve was generated from the 15 newborn rabbit pups in Group-Water. Since for each pup a PBX image of its lung in a fluid-filled state was recorded, PS_Area was calibrated against absolute V_L. Two cameras were used that had slightly dissimilar spatial frequency responses, or modulation transfer functions (MTFs). The MTF represents the camera’s ability to accurately preserve the amplitudes of each of the spatial frequencies in an image and this depends on the design of the camera. This resulted in two distinct calibration curves originating from the two cameras, which differed by a multiplicative factor of 3.35. This is likely due to the different phosphor thicknesses coupled to the two detectors. Thus, PS_Area calculated from using the sCMOS camera was multiplied by 3.35 to align with the calibration curve attained from the FOP camera. While this highlights the need to produce a calibration curve for each of the detectors, a calibration curve would also be needed for other different experimental configurations. This includes different types of animals due to variability in lung morphology and imaging setups since the power spectrum of the speckle produced would be affected by factors such as beam characteristics. A direct plot of V_L against PS_Area showed a non-linear relationship. It was found that raising V_L to the power $\frac{3}{4}$ best linearized the curve based on the chi-square goodness-of-fit test.

Figure 5(a) shows the linearized PS_Area − V_L calibration curve. A weighted linear trend ( $V_{L}^{3 / 4} = a \times {PS}_{Area} + b$ ) was fitted with coefficients a = (1.345 ± 0.001) × 10⁻⁴ and b = (−6.84 ± 0.02) × 10⁻⁷. The uncertainty in $V_{L}^{3 / 4}$ was determined by measuring the standard deviation (σ) of the volume change over time of a water-only ROI against the ROI size (N × M pixel). A rational exponent function was fitted to give σ = 3.5×10⁻⁷ ×[N × M]^3/4 +3.7×10⁻⁶ (see [15, section IIE]). The same ROIs were used to measure σ of PS_Area to determine its uncertainty. A PS_Area − V_L curve from the quadrants from one of the pups used for Fig. 5(a) is plotted in Fig. 5(b). This shows the PS_Area − V_L curve from the quadrants diverge from each other at large V_L. This divergence was observed in other pups at different V_L that depended on their total lung capacity. However, this degree of divergence is comparable to that of the points in Fig. 5(a).

Fig. 5 (a) A calibration curve between V_L and the PS_area from PBX chest images divided into quadrants consisting of (a) a subset of points (∼ 10%) from multiple pups and (b) a single pup. A weighted linear fit was performed on (a) and is shown as a red line.

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The phase gradient across the interface between water and tissue is considerably smaller than between air and tissue. When the PBX chest images of rabbit pups were divided by their absorption images to remove the absorption-induced intensity trends, the boundary of the skin was enhanced for pups imaged in air (see Fig. 6(a)) compared to those imaged in water (see Fig. 6(b)). Consequently, the azimuthally averaged power spectra of pups imaged in air showed a larger amplitude between frequencies 0 mm⁻¹ and 2 mm⁻¹ compared to that of pups imaged in water (see Fig. 6(c)). Beyond the frequency 2 mm⁻¹, the shape of the power spectra were of similar magnitude, which was the range PS_Area is calculated in. Thus, the calibration curve in Fig. 5(a) generated from pups imaged in water was used to measure the total change in V_L of 3 pups imaged in air and then compared with the volume measured using a flowmeter (see Fig. 7(a)). A straight line was fitted to Fig. 7(a) with a gradient of 1.06 ±0.06 (R²= 0.978). This shows the calibration curve to be a highly accurate tool for determining the total change in V_L without needing to immerse the animal in water. This gives us confidence for retrieving accurate regional volumetric information.

Fig. 6 A pair of 24 mm × 21 mm PCX chest images of a newborn rabbit pup in (a) a water-filled tube and in (b) air. (c) shows their respective power spectra after dividing by their absorption image reconstructed using phase retrieval.

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Fig. 7 (a) A representation of the accuracy of using the calibration curve to measure the change in total V_L of pups imaged in air in comparison to using a flowmeter. The red line is the line of best fit. (b) Regional lung volume measurements from a lung image sequence after partitioning the images into quadrants. Note that the lower quadrant curves have been offset by 0.1 ml to better distinguish them from the upper quadrant curves.

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A quadrant-based analysis of a single pup imaged in air (see Fig. 7(b)) reveals non-uniform lung aeration. Figure 6(a) shows how the quadrants were delineated. As expected, the volumetric curves of each quadrant oscillates in phase with the mechanical ventilator. However, there are subtle but important physiological differences between quadrants. The upper left quadrant shows a lack of increase in tidal volume (i.e. volume of air entering the lungs) despite the increasing volume of air supplied by the ventilator, while the upper right quadrant is the only quadrant showing a significant increase in FRC. This quantification of non-uniform aeration using regional analysis is critical for assessing the efficacy of resuscitation strategies for newborn infants.

Next we demonstrate the ability of our technique to build a pixel-by-pixel map of V_L using 128 × 128 pixel sized ROIs to measure the volume from the speckle pattern. The calibration curve was used to convert PS_Area measured from the speckle pattern into V_L in each ROI. This was performed on pairs of images where each pair was a PBX chest image of a pup recorded at end-inspiration and at end-expiration. A volumetric map was constructed for each image of each pair and the difference was taken between them to give the tidal volume at each pixel. At a low tidal volume, Fig. 8(a) shows lung aeration to be highly localized where the greatest change in V_L is at the bifurcation of the left/right main bronchi into tertiary bronchi. As the tidal volume increased (see Figs. 8(b) and 8(c)), the flow of air became more apparent at the peripheral regions where lung expansion occurred. Summation of the pixels in Fig. 8 gave the total V_L (Fig. 8(a): 0.15 ml, Fig. 8(b): 0.27 ml, Fig. 8(c): 0.37 ml), which agreed closely with that from calculating PS_Area of the total lung (Fig. 8(a): 0.12 ml, Fig. 8(b): 0.24 ml, Fig. 8(c): 0.36 ml). These values respectively correspond to a percentage difference of 4.6%, 2.7% and 1.3%. This was predominantly attributed to having a discrete frequency domain. Since different sized ROI sample the frequency domain differently, a small error is introduced into V_L. Accuracy can be improved by interpolating the discrete spatial frequencies.

Fig. 8 14.6 mm ×14.6 mm regional volumetric maps from a mechanically ventilated rabbit pup in air ventilated using three different tidal volumes: (a) 0.12 ml, (b) 0.24 ml and (c) 0.35 ml. Map demonstrate the distribution of air when it enters the lung.

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4. Discussion

The technique outlined here presents a novel approach to measuring absolute lung air volume (V_L) regionally from PBX chest images without requiring the use of a contrast agent. Imaging modalities such as positron emission tomography or MRI require a medium to be injected inside the lungs with which to measure V_L while water acts as a contrast agent for techniques that use PCX modalities. Here, no such contrast agent was required as speckle is an inherent feature of PBX images of the lung that was related to the properties of the airway morphology to measure V_L.

There are three limitations that we foresee in our technique: (i) motion blur, (ii) the multi-valued relationship between PS_Area and V_L, and (iii) the minimum size of the region from which V_L can be measured. The motion of the chest wall causes blurring in images; which results in a reduction in PS_Area [22]. The zero spatial frequency is however largely unchanged, which is why those techniques previously discussed that measure V_L from the intensity of pixels/voxels are unaffected by motion blur. Reducing the exposure time can reduce the degree of motion blur but this coincides with a decrease in SNR. There are techniques that could measure and correct the degree of motion blur [42]. However, respiratory-induced motion blur is non-linear and difficult to correct. Motion blur can also be avoided through patient breath holds or taking measurements towards the end of inspiration/expiration when there is minimal chest motion.

The difference in the dependance of R and N with V_L and PS_Area means a dynamic relationship exists between V_L and PS_Area. The calibration curve in Fig. 5(a) showed PS_Area was approximately proportional to $V_{L}^{3 / 4}$ . From Eq. (10), the exponent ( $\frac{4}{3}$ ) is close to 1, which indicates the lungs accommodated the flow of air by changing the number of alveoli. This is expected since we are imaging the lungs from an non-aerated state, hence new alveoli are being recruited as the lungs fill with air, thereby increasing N It was found, however, a PS_Area − V_L curve of a single breath showed subtle changes in the exponent (other than $\frac{4}{3}$ ). This was masked in the calibration curve as it was made up of multiple single breath curves from pups mechanically ventilated with different positive inspiratory and positive end-expiratory pressures. This allowed a single calibration curve to approximate V_L. At large volumes (≥10 ml/kg) however, a slight divergence in the calibration curve between quadrants appears (see Fig. 5(b)). This may either be the variable relationship between V_L and PS_Area or the breakdown of the TIE approximation. The average projected thickness between quadrants is different in that the apical region the lungs is thinner than that adjacent to the diaphragm. Thus, the TIE approximation may breakdown sooner for the lower quadrants at large volumes as the lungs increase in thickness.

When performing regional V_L analysis, the region must be large enough to sufficiently sample the spatial frequencies of a power spectrum. As detector technology continues to improve, we expect that detectors with larger numbers of pixels producing lower noise levels coupled with better detector spatial resolution to be available. Hence we anticipate being able to measure V_L in smaller ROIs in the future. Another major benefit of having lower noise level detectors is a reduced ODD. In our study, the ODD was set at 3 m to produce a sufficiently strong speckle contrast-to-noise ratio (CNR), but this distance was found to be at the edge of the near-field regime. To that end, PBX chest images were also recorded at 1.5 m ODD but were not included in this manuscript because V_L weakly correlated with PS_Area due to the weak CNR of the speckle. Thus, a lower noise level will allow PBX images to be recorded at shorter ODD while still having a short exposure time to minimize chest-induced motion blur.

Since our technique does not require the subject to be immersed in water, the much improved SNR allows significant reduction in the exposure time. The SNR of an image of a pup imaged in air was measured to be ∼ 1.4× larger than a pup imaged in water. The region chosen in calculating SNR was at the thickest (central) portion of the body below the lungs. This improvement in SNR will be important for translating this research for use with lower power laboratory-based x-ray machines [43–45].

5. Conclusions

Propagation-based phase contrast x-ray (PBX) images, of lungs modeled as a random distribution of hollow spheres, were simulated. The speckle pattern observed from the PBX image was quantified by the area under its power spectrum (PS_Area), and was found to be dependent on the absolute lung air volume (V_L). Herein we developed a simple method for measuring V_L regionally from the speckle patterns commonly seen in PBX images of the chest. This can be a useful measure of lung disease and injury. Unlike our previous techniques of measuring V_L from two-dimensional images, the subject does not need to be immersed in water, which significantly boosts the signal-to-noise ratio of the image. This has a two-fold advantage; the exposure time can be reduced to minimize both motion blur and radiation dose. Importantly, our technique is able to perform absolute measurements of V_L in select ROI; this was previously only possible if we were able to image the lungs starting from a fluid-filled state. We successfully tested this method in measuring both the total and regional changes in V_L of several newborn rabbit pups, and validated it against other proven techniques.

Appendix A: Power spectrum of a near-field 2D intensity map of a 3D random distribution of identical voids

Consider N voids randomly embedded in an absorbing medium, as shown in Fig. 1. We assume the path of the rays within the object is unperturbed, this is known as the projection approximation, and it allows the absorption (I(x,y,z = 0)) and phase shift (φ(x,y,z = 0)) induced by the object up to its exit surface to be considered projections through, respectively, the absorption index (β) and refractive index decrement (δ) with:

I_{A} (x, y, z = 0) = exp [- 2 k \int β (x, y, z) d z]

and

φ (x, y, z = 0) = - k \int δ (x, y, z) d z .

Here, k = 2π/λ is the wavenumber and λ is the wavelength of the source illuminating the object along the z direction. We assume the medium is composed of a single material of projected thickness T(x,y) along the z direction, and since any voids are non-absorbing (i.e. β_void = δ_void = 0), Eqs. (11) and (12) reduce to:

I (x, y, z = 0) = exp [- 2 k β T (x, y)]

and

φ (x, y, z = 0) = - k δ T (x, y),

where μ = 2kβ is the linear attenuation coefficient of the medium. At short distances (L) along the z direction from the exit surface of the object, the free space evolution of the intensity distribution can be described by the transport-of-intensity equation (TIE) [46]:

- k \frac{\partial I (x, y, z)}{\partial z} = \nabla_{⊥} \cdot [I (x, y, z = 0) \nabla_{⊥} φ (x, y, z = 0)],

where

\nabla_{⊥} = \hat{x} \frac{\partial}{\partial x} + \hat{y} \frac{\partial}{\partial y}

.

The TIE assumes paraxial wave propagation (i.e. $k ≫ \sqrt{k_{x}^{2} + k_{y}^{2}}$ where (k_x, k_y, k_z) is the x, y, z components of the wavevector k). The validity of the TIE is given by the Fresnel number when, $N_{F} = \frac{a}{L λ {| \nabla_{⊥} φ |}_{\max}} \geq 1$ [33]. Here, a is the characteristic length scale over which the object changes appreciably and |∇_⊥φ|_max is the maximum absolute transverse phase gradient. This form of N_F is very similar to another commonly used condition $\frac{a^{2}}{L λ} \geq 1$ , but this condition does not consider |∇_⊥φ|_max. From the ray optics perspective, the degree of deflection of the rays depends on both a and |∇_⊥φ|_max. The importance of this in the context of lung imaging is that N_F can be overestimated if |∇_⊥φ|_max > 1 radians per unit length was not considered, thus unknowingly imaging outside the near-field region.

To remove any dependency on absorption in Eq. (15), we begin with a finite difference approximation on the left hand side, $\frac{d I (x, y)}{d z} \approx \frac{I (x, y, z = L) - I (x, y, z = 0)}{L}$ , while the the right hand side (RHS) is expanded, with Eqs. (13) and (14) substituted into Eq. (15), to give:

\frac{I (x, y, z = L)}{I (x, y, z = 0)} - 1 = L δ [\nabla_{⊥}^{2} T (x, y) - μ {| \nabla_{⊥} T (x, y) |}^{2}],

where

\nabla_{⊥}^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}

denotes the Laplacian, respectively, in the xy plane. The second, and only, term on the RHS of Eq. (16) is dependent on but can be neglected if

μ {| \nabla_{⊥} T (x, y) |}^{2} ≪ | \nabla_{⊥}^{2} T (x, y) |

. To explicitly show when this is true, we make the substitutions |∇_⊥T| ≤ |ΔT|/a where |ΔT| is the maximum magnitude of the difference in projected thickness across the length a, and

| \nabla_{⊥}^{2} T | \leq | Δ T | / a^{2}

. This simplifies the inequality to μΔT(x,y) ≪ 1, that is, the object is weakly absorbing [47]. The lung tissue thickness in rabbit pups is typically on average 6 mm and at 24 keV, μ = 54.7 m⁻¹ for lung tissue, thus μ|ΔT(x,y)| ≤ 0.328. Consequently, the second term on the RHS of Eq. (16) can be ignored, hence the absolute square of the Fourier transform of Eq. (16) gives the power spectrum:

| F {\frac{I (x, y, z = L)}{I (x, y, z = 0)} - 1}^{2} | = L^{2} δ^{2} k_{⊥}^{4} {| F {T (x, y)} |}^{2} .

Here we have made use of the Fourier derivative theorem to replace $\nabla_{⊥}^{2}$ with $k_{⊥}^{2} = {(k_{x})}^{2} + {(k_{y})}^{2}$ . Returning to our object of interest, for a random distribution of N air-filled voids, described by the object function G̃(x,y,z) where G̃(x,y,z) = 1 for $\sqrt{x^{2} + y^{2} + z^{2}} \leq R$ and G̃(x,y,z) = 0 elsewhere, embedded in an absorbing medium (V(x,y,z) = 1 everywhere), the object can be expressed as a sum of convolutions:

\tilde{T} (x, y, z) = V (x, y, z) - \sum_{n = 0}^{N} δ (x - x_{n}) δ (y - y_{n}) δ (z - z_{n}) \otimes \tilde{G} (x, y, z),

where δ’s are the unit impulse functions and (x_n,y_n,z_n) represents the random position of the n^th void within the dimensions of the medium. For simplicity, the first term on the RHS of Eq. (18) will be dropped as it affects only the zero frequency in its corresponding power spectrum, which is unimportant for our analysis. Generally the random positions of the voids are unknown but the expectation value of the power spectrum of T̃(x,y,z) can be evaluated without this information [48]. With equal probability of finding a void within the medium, we have for the expectation value of the power spectrum of T̃(x,y,z):

〈 {| F {\tilde{T} (x, y, z)} |}^{2} 〉 = [N^{2} \hat{δ} (0, 0, 0) + N] {| F {\tilde{G} (x, y, z)} |}^{2},

where δ̂(0, 0, 0) has a value of unity at (0,0,0) and zero elsewhere (Kronecker delta). The expected value operator will be dropped for notational simplicity and the first term inside the square brackets of Eq. (19) will also be dropped as it only affects the zero frequency. It can then be seen that the power spectrum of the random distribution of voids is N times the power spectrum of a single void for k_⊥ ≠ 0. This remains valid if the voids are randomly positioned and do not overlap with neighboring voids [48].

To determine T(x,y) = ∫_zT̃(x,y,z)dz, we make use of the Fourier slice theorem [49],

{| F {\tilde{T} (x, y, z)} |}^{2} (k_{x}, k_{y}, 0) = {| F {\int_{z} \tilde{T} (x, y, z) d z} |}^{2} (k_{x}, k_{y}) = {| F {T (x, y)} |}^{2} (k_{x}, k_{y}),

with a similar conclusion made for |F{G(x,y)}|². If G̃(x,y,z) is azimuthally symmetric (i.e. spherical) then G(x,y) is independent of the orientation of G̃(x,y,z). However, if G̃(x,y,z) is not azimuthally symmetric, but randomly orientated, like the alveoli, then 〈|F{T̃(x,y,z)}|²〉 is approximately azimuthally symmetric with |F{G̃(x,y,z)}|² being the azimuthal average of the power spectrum of a single void replicated azimuthally. Thus, Eq. (19) can be reduced to 2D,

{| F {T (x, y)} |}^{2} = N {| F {G (x, y)} |}^{2} .

This shows that even though increasing the number of voids results in an increase in the amount of overlap between them seen in the projected thickness image, the power spectrum only changes by a factor. This may seem counter-intuitive as the more voids there are the shorter the characteristic length scale of the image, which would shift the peaks in the power spectra to higher frequencies. But this is untrue as long as there is no physical overlap between neighboring voids and the positions of the voids are sufficiently random.

Finally, substituting Eq. (21) into Eq. (17) we arrive at the power spectrum of the near-field intensity map of a 3-dimensional random distribution of identical voids normalized against its absorption image:

{| F {\frac{I (x, y, z = L)}{I (x, y, z = 0)} - 1} |}^{2} = L^{2} δ^{2} k_{⊥}^{4} N {| F {G (x, y)} |}^{2} .

Acknowledgments

The authors would like to thank Kentaro Uesugi and Naoto Yagi for assistance with the experiments. AFTL acknowledges the support of an Australian Postgraduate Award. SBH and MJK acknowledge funding from the Australian Research Council (ARC; Grant Nos. DP110101941 and DP130104913). MJK is an ARC Australian Research Fellow. SBH is a NHMRC Principal Research Fellow. This research was partially funded by the Victorian Government’s Operational Infrastructure Support Program. We acknowledge travel funding provided by the International Synchrotron Access Program managed by the Australian Synchrotron and funded by the Australian Government.

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Pixel size (μm)	0.59
Mean alveolar radius (μm)	30, 35, 40, 45, 50, 55, 60, 65, 70
Alveolar radius standard deviation (μm)	$\frac{3}{4}$ × alveolar diameter
Source-to-detector distance (m; R₁)	∞
Sample-to-detector distances (m; R₂)	3
Attenuation coefficient (μ_w [31])	54.735 m⁻¹ (at 24 keV)
Refractive index decrement (δ_w [31])	3.99 (at 24 keV)
Maximum volume packing fraction	75%
Lung thickness (mm)	10

Measurement of absolute regional lung air volumes from near-field x-ray speckles

Abstract

1. Introduction

1.1. Lung speckle

1.2. Theory

2. Methodology

2.1. Simulated lung tissue

2.2. Rabbit pup lungs

2.3. Image processing and analysis

3. Results & analysis

3.1. Validation of theory using simulated lung tissue

3.2. Regional lung air volume measurements

4. Discussion

5. Conclusions

Appendix A: Power spectrum of a near-field 2D intensity map of a 3D random distribution of identical voids

Acknowledgments

References and links

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Figures (8)

Tables (1)

Equations (22)

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