Principal Component Analysis Enhances SNR for Dynamic Electron Paramagnetic Resonance Oxygen Imaging of Cycling Hypoxia In Vivo

Abstract

Purpose

Low oxygen concentration (hypoxia) in tumors strongly affects their malignant state and resistance to therapy. These effects may be more deleterious in regions undergoing cycling hypoxia. Electron paramagnetic resonance imaging (EPRI) has provided a non-invasive, quantitative imaging modality to investigate static pO₂ in vivo. However, to image changing hypoxia, EPRI images with better temporal resolution may be required. The tradeoff between temporal resolution and signal-to-noise ratio (SNR) results in lower SNR for EPRI images with imaging time short enough to resolve cycling hypoxia.

Methods

Principal component analysis (PCA) allows for accelerated image acquisition with acceptable SNR by filtering noise in projection data, from which pO₂ images are reconstructed. PCA is used as a denoising technique by including only low-order components to approximate the EPRI projection data.

Results

Simulated and experimental studies show that PCA filtering increases SNR, particularly for small numbers of sub-volumes with changing pO₂enabling an order of magnitude increase in temporal resolution with minimal deterioration in spatial resolution or image quality.

Conclusion

The SNR necessary for dynamic EPRI studies with temporal resolution required to investigate cycling hypoxia and its physiological implications is enabled by PCA filtering.

Introduction

For decades, oxygenation status of tumors has been known to have important prognostic implications (1). Low oxygen concentration (pO₂), or hypoxia, imbues cancer cells with resistance to radiation therapy (2, 3) and strong correlation has been found between electrode measurements of low pO₂ and radiotherapy treatment failure in humans (4). Hypoxic tumors are more resistant to chemotherapy as well (5). Hypoxia also leads to a more malignant state for cancers, e.g. faster tumor growth due to abnormal proliferation (6), and plays an integral role in increasing the potential for metastatic progression (7).

These implications have led to increased interest in methods for probing and, a fortiori, imaging pO₂ deep in tissues. The advances in methods to investigate/image tissue pO₂ are detailed in the literature, e.g. (8–11). In particular, electron paramagnetic resonance imaging (EPRI) has proven to be a useful modality for measuring tissue pO₂. EPRI noninvasively acquires highly-resolved, both spatially (~1mm³ voxels) and in pO₂ (1–3 torr), 3D images of in vivo pO₂ (12–16). The low electromagnetic wave excitation frequencies used in EPRI, comparable to 6T MRI, penetrate deep in tissue (>7 cm). EPR pO₂ images use an intravenously injected, non-toxic spin probe, which distributes in the extracellular compartment of tumors, to report local pO₂ (17).

There are two forms of hypoxia in tumors, resulting from different physiological processes: diffusion limited hypoxia, creating chronically hypoxic regions, too far removed from viable vasculature to receive enough oxygen (18), and perfusion limited hypoxia, creating acutely hypoxic regions. For many years, the former was believed to be the only type of hypoxia present in tumors and the most clinically relevant pO₂ parameter. More recent studies have found that perfusion limited hypoxia is also present (19–22) and may even be the major cause of hypoxia in tumors (22, 23). Studies suggest acute hypoxia (sometimes referred to as cycling or transient hypoxia) may be as important a determinant of cancer progression and patient prognosis as chronic hypoxia, although specific correlation of a quantitative measure of cycling hypoxia with treatment outcome is limited. It has been postulated that acute hypoxia may be even more deleterious than chronic hypoxia and therefore more clinically relevant because periods of reoxygenation prevent hypoxia-related cell death and select cells that can proliferate in hostile environments by abrogating normal check point signaling (23–28).

Traditional measurements of tumor hypoxia in vivo have been directed towards chronic hypoxia. Unlike chronic hypoxia studies, data relating transient hypoxia to treatment outcome are rare because methods for noninvasive quantification of transient hypoxia need further development. Current techniques include recessed-tip oxygen microelectrodes (29–31), Oxylite™ probes (32), T₂^*-weighted MRI (33), ¹⁹F MRI(34–36), ¹⁸F-FMISO positron emission tomography (PET) (37), as well as EPRI (38, 39). Phosphorescence lifetime imaging in window chamber systems (40) has sensitivity to dynamic oxygenation but is an invasive measurement and not well suited for evaluating the in vivo relationship between oxygen fluctuations and treatment outcome.

Studies have found pO₂ fluctuations with periods from minutes to days (22, 28, 40–44). EPRI, heretofore, has provided a means of determining the chronically hypoxic fraction of a tumor, which has been found to predict tumor curability (45). Chronic hypoxia as determined from EPRI has also been validated at the molecular level via correlation to known hypoxia response proteins (46). Note, EPRI obtained over longer times, i.e., with lower temporal resolution, cannot differentiate chronically hypoxic regions from acutely hypoxic regions with an average hypoxic state during imaging, but improved temporal resolution will enable this distinction. Recent studies have already begun to utilize EPRI as a means for imaging temporal changes in pO₂ (38, 39, 47). However, our standard EPRI pO₂ images take 10 minutes, which may not provide adequate temporal resolution to study higher frequency pO₂ fluctuations. Therefore, to investigate spontaneously occurring cycling hypoxia in vivo, our temporal resolution must be improved. Naturally, decreasing imaging time decreases image signal-to-noise ratio (SNR). We have done extensive work to enhance temporal resolution by improving the hardware, data processing, and by using narrower-line deuterated spin probes.

We have investigated improving SNR by post-processing data to reduce noise. A data-processing method that has proven useful as a denoising and/or feature-recognizing technique is principal component analysis (PCA). Sometimes referred to as feature analysis or the Karhunen-Loève transform, PCA has been used for the enhancement and extraction of spatiotemporal features in many different fields, including geoscience (48), facial recognition (49), and astrophysics (50). In particular, PCA has been used to denoise and highlight important temporal features for dynamic medical imaging modalities, e.g. gamma camera studies (51, 52), electro/magneto-encephalography (53), PET (54–56), SPECT (57, 58), and MRI (59–61).

The focus of this paper is to show, through simulations and experiments, that PCA can be used as a pre-reconstruction, spatiotemporal filter for projection data from dynamic EPRI studies. PCA filtering produces images with higher SNR and therefore higher temporal, spatial, and pO₂ resolution. These PCA enhanced images will allow dynamic EPRI to be used to investigate important aspects of tumor physiology related to cycling hypoxia.

Methods

Pulsed EPRI Imaging Model

This paper is concerned with PCA applied as a spatiotemporal filter to pulsed EPRI dynamic studies. The pulsed imaging methods developed in our laboratory are detailed elsewhere, starting with electron spin echo or T₂-imaging (62) and, more recently, inversion recovery electron spin echo or T₁-imaging (63). T₁-imaging reduces confounding spin probe concentration dependent self-broadening and provides a near absolute pO₂ image. Signal for these EPRI techniques comes from an exogenous spin probe (trityl) (16, 64) with an unpaired electron that is remarkably difficult to reduce, particularly in the in vivo environment. Both T₂ and T₁ report on the local environment of the unpaired electron. The electron relaxation rates (R₂ = 1/T₂ and R₁ = 1/T₁) are linearly related to local pO₂. To transition from spectroscopy to imaging, linear magnetic gradients are applied to spatially encode an object. The magnetic gradients are applied along different directions in a spherical geometry to obtain projections (Fig. 1a). The relaxation information is encoded in the signal amplitude of projections obtained with different τ for T₂-imaging or T for T₁-imaging. The Fourier transform of an acquired time-domain projection (Fig. 1b) is a spatial frequency encoded projection (Fig. 1c). An inverse radon transform, e.g., filtered backprojection, of many spatial projections from different views is used to reconstruct a 3D image of spin-density. For this work, the reconstruction algorithm used was filtered backprojection with a Ram-Lak filter and frequency cutoff at half of the Nyquist frequency for high-frequency noise filtering. Taking a series of 3D images varying τ or T allows for the signal intensity in each voxel to be fit as a function of either τ, using Eq.(1), or T, using Eq.(2), for T₂- or T₁-imaging respectively, to obtain a 3D image of average voxel T₂ (R₂) or T₁R₁), thus providing pO₂ for each voxel.

$$S(\tau )={\mathit{\text{Ae}}}^{\raisebox{1ex}{$-2\tau $}\!\left/ \!\raisebox{-1ex}{$T_2$}\right.}$$ (1)

$$S(T)=A(1-2e^{\raisebox{1ex}{$-T$}\!\left/ \!\raisebox{-1ex}{$T_1$}\right.})$$ (2)

Fig. 1.

Diagram of how a pulsed EPR image is obtained from projections. The red arrow in (a) represents a projection direction defined using linear magnetic gradients. The pulsed echo signal from this projection using one of the pulse sequences shown in (b) gives the time-domain projection and a Fourier transform gives the spatial projection in (c). This is repeated for many projections (represented by points on the sphere in (a)) and an inverse Radon transform (e.g., FBP) produces a 3D image of the spin density in the object.

PCA Algorithm

Principal component analysis is a method for defining, from an n-D data set, a new space based on the covariance of the data, such that the first few basis vectors, or principal components (PCs), form a q-D space (q < n) containing the relevant, highly-correlated features, while the space spanned by the other components contains uncorrelated noise. Therefore, PCA provides a means for low-order approximation of the data by projection onto the subspace spanned by the first q PCs. This low-order approximation using PCA acts as a spatiotemporal noise filter for a high-dimensional set of data from a dynamic EPRI study by separating correlated features from uncorrelated noise and only retaining these correlated features. This procedure will be referred to as PCA filtering and is applied to projection data, producing reduced-noise projections and, subsequently, higher SNR images.

In a dynamic EPRI study, multiple images are acquired consecutively to obtain spatial and temporal pO₂ information. For each of these n images, m projections are acquired, each with k points. All m of these projection vectors are concatenated into n total projection vectors, p⃑_i (i=1,2,…n). Mean-centered vectors are computed (π⃑_i in Eq. (3)) and arranged as the column vectors of length mk in a matrix, X, containing the mean-centered projection data for the entire dynamic study (Eq. (4)). The covariance matrix, Σ, can then be computed from X using equation (5).

$${\stackrel{\rightharpoonup }{\mathit{\pi }}}_i={\stackrel{\rightharpoonup }{\mathbf{p}}}_i=\frac{1}{n}\sum _{i=1}^n{\stackrel{\rightharpoonup }{\mathbf{p}}}_i$$ (3)

$${\mathbf{X}}_{mk\times n}={[{\stackrel{\rightharpoonup }{\mathit{\pi }}}_1,{\stackrel{\rightharpoonup }{\mathit{\pi }}}_2\cdots {\stackrel{\rightharpoonup }{\mathit{\pi }}}_n]}_{mk\times n}$$ (4)

$${\mathit{\Sigma }}_{mk\times mk}=\frac{1}{n}\sum _{i=1}^n{\stackrel{\rightharpoonup }{\mathit{\pi }}}_i{\stackrel{\rightharpoonup }{\mathit{\pi }}}_i^T={\mathit{\text{XX}}}^T$$ (5)

The next step is eigenanalysis of Σ. However, Σ is large (m~10³ and k~10³therefore mk~10⁶) and full eigenanalysis of Σ is not computationally feasible. To alleviate these computational costs, it is assumed that only eigenvectors with non-zero eigenvalues must be calculated and that, at most, there will be r such eigenvectors, where r is the rank of Σ. From equation (5), we know r ≤ min {mk, n}. Therefore, there are assumed to be only n eigenvectors with non-zero eigenvalues, because n is on the order of 10¹–10² and thus much smaller than mk (49).

The magnitude of the eigenvalues, λ_j, corresponding to the eigenvectors of Σ, µ⃑_j, indicate the relative amount of variance or information in the data lying along the direction defined by the eigenvector or PC. One can define vectors t⃑_j = X′u⃑_j and use them to express the data matrix X projected onto the orthonormal basis defined by the PCs using equation (6).

$$\mathit{X}=\sum _{j=1}^r{\stackrel{\rightharpoonup }{\mathit{t}}}_j{\stackrel{\rightharpoonup }{\mathit{u}}}_j^T$$ (6)

The eigenvectors are arranged with higher values of j corresponding to eigenvectors with smaller eigenvalues. Therefore, the sub-space defined by the first few PCs contains most of the information in the original data. This allows for an accurate representation of the data using the low-order approximation from projecting onto the first few PCs, i.e. only summing over the first q terms (q ≪ r) in equation (6). This low-order approximation selectively retains relevant information while discarding uncorrelated noise, thus filtering the data and enhancing SNR.

Simulations

Simulated dynamic EPRI studies evaluated PCA as a method of approximating/filtering the projection data. Simulations allowed for comparison of images reconstructed from approximated noiseless data to images reconstructed from the complete noiseless data, as well as the comparison of unfiltered, noisy images to PCA filtered images with the knowledge of the ideal, noiseless images. Matlab (MathWorks, Massachusetts, USA) simulations were written locally. The digital phantom consisted of a main ellipsoid body and four interior ellipsoid sub-volumes varying in spatial orientation and size. The ellipsoid sub-volumes 1–4 were 8.92%, 4.88%, 1.54%, and 0.06% of the entire phantom volume respectively. The pO₂ for each sub-volume could individually be adjusted as necessary and could be made to vary in time (Fig. 2). A second digital phantom consisting of a small cube with pO₂ oscillating in time, centered within a larger cube of constant pO₂ was also used. A forward projection algorithm simulated projection acquisition with our EPRI technology. These simulated projections were reconstructed using standard protocol.

Fig. 2.

Ellipsoid phantom used for digital simulations of dynamic EPR imaging of pO₂ changing in time with labeled ellipsoidal sub-volumes and corresponding patterns of pO₂ fluctuations.

Figures of Merit

For noiseless simulations, mean squared error (MSE) measured the fidelity of PCA filtered (approximated) images to images reconstructed from noiseless projections. MSE is defined by Eq. (7), where $I_i^{\prime }$ and I_i are the i^th voxels in the PCA filtered and noiseless images respectively, and V is the number of voxels in each image.

$$\text{MSE}=\frac{1}{V}\sum _{i=1}^V{(I_i^{\prime }-I_i)}^2$$ (7)

For noisy data, SNR was the image quality metric used to compare images with and without PCA filtering applied. Signal was defined as the voxel pO₂ from the noiseless image reconstructed using identical conditions. Noise was defined as any deviation from the noiseless image. This definition of noise accounts for random noise as well as artifacts. SNR was calculated by dividing the maximum of the signal by the standard deviation of the noise. Projection SNR and temporal fluctuation SNR were calculated similarly.

Mouse Experiments

Dynamic EPRI studies with live, tumor bearing mice determined the effect of using PCA as a filter in experiments investigating temporal pO₂ fluctuations in vivo. The pulsed imaging spectrometer used has been described previously (13). The images from these studies were 64×64×64 voxels³with a voxel size of (0.66 mm³. The EPRI images were registered with spin-echo weighted MRI images, using methods described previously (65), providing anatomic information that EPRI lacks, to delineate tumor from normal tissue.

Two female C3H type mice (Harlan Sprague Dawley Inc., Indianapolis, IN) were used in these experiments. Human mammary tumor cells, MCA4 F6m (M.D. Anderson Hospital, Houston, TX), were implanted in the mid-distal hind legs of the 8 week old mice and grown subcutaneously to an appreciable size (~2 weeks after implantation). To prevent motion during imaging, the animals were immobilized using a soft elastic vinyl polysiloxane dental mold material (GC America Inc., Alsip, IL) and anesthetized with 1–2% isofluorane mixed with medical grade air. Respiration frequency and depth were monitored continuously. A 24 gauge angiocath was used to cannulate the mouse tail vein for i.v. injection of the spin probe. A digital needle probe thermometer (Physitemp, Clifton, NJ) monitored skin temperature. The skin temperature was maintained at ~37° C using adjustable opposed heating lamps.

The first dynamic study implemented forced pO₂ fluctuations. The fluctuations were produced by having the anesthetized mouse inhale alternating gases with differing fraction of inspired oxygen (FIO₂). The experiment was 1 hour long. The mouse breathed carbogen (95% O₂ and 5% CO₂) for 5 minutes, followed by air (21% O₂) for 5 minutes, repeated for 6 periods. The mouse weighed 22.3g when imaged, and the tumor was ~400µL. Sixty, 1 minute T₂-images corrected for spin probe concentration self-broadening bias were acquired.

The second dynamic study was 2 hours long with the anesthetized mouse breathing air. The mouse weighed 25.5g when imaged, and the tumor was ~300µL. Eighty, 1.5 minute T₁-images were acquired.

These animal experiments were performed according to the US Public Health Service “Policy on Humane Care and Use of Laboratory Animals” and the protocols were approved by the University of Chicago Institutional Animal Care and Use Committee (ACUP No. 71697). The University of Chicago Animal Resources Center is an AALAC approved animal care facility.

Results

Simulations without Noise

Simulations without noise were used to determine how the number of PCs required to obtain a reasonable approximation of the data depends on the number of differently oscillating volumes. Using the ellipsoid phantom described in the methods section, noiseless simulations were done with 1, 2, or 3 of the sub-volumes fluctuating about different mean pO₂ values with different temporal patterns (Fig. 2). Note that, while the number of differently fluctuating sub-volumes was varied, all sub-volumes were present in the imaged object in fixed locations, with non-oscillating sub-volumes having constant pO₂ equal to the mean pO₂ of their respective pO₂ patterns (Fig. 2).

Figure 3 plots the eigenvalues for the three cases. The eigenvalue corresponding to a given PC is proportional to the percentage of total variance in the data accounted for by the PC and is a measure of the relevant information in the data contained along the direction of the PC. The first PC, which contains most of the information and therefore has a significantly higher eigenvalue, is not displayed in Fig. 3 for easier visualization of eigenvalues for higher PCs. For all three cases there is a steep drop in the eigenvalues for higher PCs. The dominant eigenvalues can be visually determined as those separated from the flat portion of the curve: with one volume oscillating, 2 PCs contain essentially all of the information and therefore only 2 terms are needed for the summation in Eq. (6) to sufficiently approximate the data. Similarly, for 2 and 3 oscillating volumes, only 3 and 4 terms or PCs are needed respectively.

Fig. 3.

Eigenvalue distribution for simulated EPRI data varying the number of differently oscillating volumes. The eigenvalue corresponding to the first PC has been omitted for easier visualization of the distribution. Only PCs with corresponding eigenvalues containing a significant percentage of the total variance in the data set (those significantly greater than 0) must be included in the PCA approximation.

Figure 4 shows the dependence of the MSE between noiseless PCA filtered images and noiseless unfiltered images on the number of PCs used to approximate the projection data. For one oscillating volume, the MSE with 2 PCs in the approximation is 86.3% lower than with 1 PC. With 3 PCs, the MSE is only 1.2% lower than that with 2 PCs. Therefore, the approximation for the case of one oscillating volume is sufficient with 2 PCs. Similar results can be seen in Fig. 4 for two and three oscillating volumes. For two oscillating volumes, the MSE curve flattens out after 3 PCs and for three oscillating volumes this occurs after 4 PCs.

Fig. 4.

Plots of MSE between the PCA filtered images and the true noiseless images, as a function of the number of PCs used for the approximated reconstruction for 1, 2, and 3 differently oscillating volumes. After N+1 PCs are included (N is the number of differently oscillating volumes), it can be seen that there is no appreciable decrease in the MSE.

From Fig. 3 and 4, it can be seen that N+1 PCs or terms in Eq. (6) provide the optimal PCA approximation, where N is the number of differently oscillating volumes. Figure 5 further corroborates this by comparing the pO₂ oscillations in different volumes with different numbers of PCs used in the PCA approximation to the truth (unfiltered noiseless oscillations) with 1, 2, and 3 differently oscillating volumes.

Fig. 5.

Plots of pO₂ fluctuations in the different sub-volumes of the ellipsoid digital phantom from images reconstructed using different numbers of PCs for the PCA approximation. As the number of differently oscillating volumes (N) increases, the number of PCs required for the PCA approximated fluctuations to match the true fluctuations increases as well (N+1 are required). Some curves are not visible because they lie on top of one another.

In the case where there is noise, using more than N+1 terms or PCs will mainly add noise. Therefore, when PCA is applied as a filter for the noisy simulations and real data, only N+1 PCs should be used in the approximation.

Simulations with Noise

The efficacy of PCA filtering of noisy data was investigated using simulation studies with random Gaussian noise, approximating that of an actual 1 minute EPRI scan, added to the projections. Figure 6 compares the results from a simulated dynamic EPRI study (60 one minute images) without noise, with unfiltered noise, and with PCA filtered noise. In this study, there is one oscillating volume; therefore 2 PCs are used for the PCA approximation. PCA filtering is applied to the projection data and the noise level for the projections decreases by over a factor of 5, resulting in higher quality images. The PCA filtered image is much closer to the noiseless image than the unfiltered image. The image SNR improves by a factor of 3.8 ± 0.6. Higher image quality allows for better visualization of the temporal pO₂ fluctuations as can also be seen in Fig. 6. PCA filtering of projections produces a greater improvement in projection SNR compared to the improvement in image SNR. This results from processing/filtering of the projections during reconstruction (e.g., high-frequency noise in the projections is filtered by the half-Nyquist frequency cutoff in the filtered backprojection). This filtering has a more appreciable effect on the unfiltered projections than on the PCA filtered projections and subsequently brings the SNR of unfiltered images closer to that of PCA filtered images.

Fig. 6.

Summary of results from simulated dynamic EPRI study including random noise approximating that seen in experimental data. PCA filtering is applied first to the projection data (first row) and dramatically reduces the noise. The noise level is more obviously depicted in the inset. The projections are used to reconstruct 3D pO₂ images using FBP (second row). The filtered images reconstructed from the PCA approximated projection data have higher SNR and resemble the ideal noiseless image more closely than the unfiltered image. The increase in image quality from the PCA filtering results in more accurate visualization of the temporal pO₂ fluctuations (third row) seen in the region of interest (ROI) designated by a white square. The correlation between the noiseless and unfiltered noisy temporal pO₂ fluctuations is r = 0.82, whereas the correlation between the noiseless and PCA filtered temporal pO₂ fluctuations increases to r = 0.98.

Similar simulations were done with 2 and 3 differently oscillating volumes, with 3 and 4 PCs used for the PCA approximation respectively. Adding more PCs ensures that the oscillation patterns from different volumes are reconstructed properly, but includes more noise. Therefore, while SNR is enhanced, the improvement diminishes when more PCs are required for an accurate approximation. Table 1 summarizes the effectiveness of using PCA approximation as a noise filter for different numbers of oscillating volumes. Since the method of calculating SNR depends on absolute signal amplitude and weighting and/or normalization processes occur during image reconstruction, direct comparisons between the columns of Table 1 should not be made.

Table 1.

SNR enhancement from PCA filtering, varying the number of oscillating volumes. SNR is the ratio of signal amplitude to the standard deviation of the difference between noisy data and noiseless data. Qualitatively, the temporal fluctuation SNR can be used as a measure of pO₂ error and demonstrates an improvement with the noise reduction from PCA filtering. For the unfiltered data, the amount of noise in the projections and, therefore, image is independent of the number of oscillating volumes.

Number of Oscillating Volumes	Projection SNR	Image SNR	Temporal Fluctuation SNR
1 (2 PCs)	282 ± 53	48 ± 6	26 ± 3
2 (3 PCs)	224 ± 33	43 ± 5	23 ± 2
3 (4 PCs)	193 ± 26	39 ± 8	19 ± 2
Unfiltered	49.3 ± 0.4	12.8 ± 1.2	5.5 ± 0.8

Simulations were also used to determine how PCA filtering performance depends on the relative size/weight of the volume exhibiting the temporal pattern of interest. The digital phantom consisted of an outer cube with sides 20 voxels in length and a smaller cube of varying size centered inside exhibiting sinusoidal temporal pO₂ fluctuations. The results of these simulations are shown in Figure 7. The unfiltered data has SNR < 0.5, resulting in the signal being indistinguishable amongst the noise. PCA filtering results in significantly better visualization of the temporal fluctuations (higher temporal fluctuation SNR) when the volume exhibiting the temporal fluctuations is large relative to the total volume. As the fluctuating volume decreases in size so does the effectiveness of PCA filtering until the temporal fluctuation SNR with PCA filtering converges to that without filtering.

Fig. 7.

Plot of the dependence of the temporal fluctuation SNR on the relative size of the oscillating volume. The unfiltered curve does not depend on the relative size of the oscillating volume and stays relatively constant (SNR < 0.5). PCA filtering results in significant improvement of the temporal fluctuation SNR when the oscillating volume is relatively large but as the size of the oscillating volume decreases the curves converge to a common SNR.

Simulations were used to determine how PCA filtering affects image resolution by looking at the reproduction of edges in an image. Ideally, intensity values along a line perpendicular to an edge would follow a step function. In reality, the finite resolution of the imaging system results in blurring, causing intensity values along a line perpendicular to an edge to become sigmoidal. The amount of deviation from a step function is related to the image resolution. Using this as a resolution metric it was found that resolution is unaffected by PCA filtering.

Mouse Images

Dynamic EPRI studies of live mice bearing tumors were conducted to determine the ability of PCA filtering to enhance image quality for non-ideal, experimental data.

Figure 8 shows the result of PCA filtering data from the study with forced FiO₂ fluctuations. Having the mouse breathing normoxic and hyperoxic gases with a fixed period of alternation provides a situation in which the expected pattern of temporal pO₂ fluctuations is known. With this knowledge, the ability of PCA filtering to improve SNR while preserving pertinent pO₂ fluctuations can be seen in the unperturbed tissue of a living animal. Assuming one dominant mode of fluctuation following the forced FiO₂ fluctuation, 2 PCs were used to filter the data. These two PCs contained 92% of the total variance from the original data. Figure 8a shows the increased image SNR with PCA filtering. In Fig. 8b, the bold curve shows the periodic FiO₂ fluctuations and the dotted curve shows the expected physiologic response to these fluctuations. The physiologic response to switching breathing gas is not expected to be instantaneous, but rather have some latency, causing exponentially increasing and decreasing tissue pO₂. Figure 8c shows an example of the temporal pO₂ fluctuation observed in the periphery of the tumor obtained by averaging over the 27 voxel region of interest (ROI). Without filtering, the SNR is too low to discern meaningful pO₂ fluctuations. However, when PCA filtered, the data shows the expected physiologic response to the fluctuating FiO₂.

Fig. 8.

PCA filtering applied to a dynamic EPRI study with the tumor bearing mouse breathing alternating normoxic and hyperoxic gases. (a) Comparison of image quality for unfiltered and PCA filtered pO₂ images. The tumor is outlined in magenta and the 27 voxel ROI is outlined by the white square. (b) Diagram of the breathing gas changing with time, i.e. the forced fluctuations in FiO₂. The bold line shows how the breathing gas was actually changed and the dotted line shows the expected response of the tissue pO₂ to the alternating breathing gas, assuming physiologic latencies. (c) Observed temporal pO₂ fluctuations averaged over the ROI shown. For the unfiltered data, the signal is lost amongst the noise (correlation with expected response: r = 0.13), whereas for the PCA filtered data, the expected fluctuations are clearly seen (correlation with expected response: r = 0.84).

Figure 9 shows the results from the study of the air breathing mouse. Two PCs were used for PCA filtering of the data. Using more than 2 PCs decreased SNR and was not found to help visualize additional patterns of temporal pO₂ fluctuations. Similar to the alternating air and carbogen experiment, these two PCs contained 93% of the total variance from the original data. In Fig. 9a the image quality enhancement can be seen. The temporal pO₂ fluctuations for the ROI at the periphery of the tumor, highlighted in Fig. 9a, can be seen in Fig. 9b with and without PCA filtering. In this region there appears to be sinusoidal fluctuation of the pO₂ with a period of ~2 hours. The unfiltered data shows some semblance of the sinusoidal behavior but PCA filtering elucidates these fluctuations.

Fig. 9.

PCA filtering applied to a dynamic EPRI study of spontaneous pO₂ fluctuations in a murine tumor. (a) Comparison of image quality for unfiltered and PCA filtered pO₂ images. The tumor is outlined in magenta and the 27 voxel ROI is outlined by the white square. (b) Observed temporal pO₂ fluctuations averaged over the ROI shown on the periphery of the tumor. PCA filtering helps to elucidate apparent sinusoidal pO₂ fluctuations that are difficult to discern in the unfiltered data. These fluctuations may be biologically relevant cycling or acute hypoxia.

Discussion and Conclusions

PCA approximation of projection data from dynamic EPRI studies is presented as a method for noise filtering and enhanced visualization of cycling hypoxia in tumors. Noiseless simulations show that to use PCA filtering effectively without losing important information, N+1 PCs are needed to approximate the data, where N is the number of differently oscillating volumes. After the first PC, each additional PC included allows accurate visualization of the next dominant pattern of temporal fluctuation, with dominance depending on the relative size of the oscillating volume. This is true regardless of which sub-volumes are oscillating, e.g. if only the smallest sub-volume is oscillating 2 PCs would still accurately approximate the data. Note, spatially separate regions exhibiting a common temporal fluctuation pattern are considered a single oscillating volume, i.e. commonly fluctuating regions need not be adjacent to increase the dominance of their pattern of fluctuation in the PCA based representation of the data. These results apply for any arbitrary temporal pattern. Any pattern represents a unique direction in the PCA defined n-D space. Therefore, for example, if the majority of an object is undergoing low-frequency oscillations and some portions are undergoing these low-frequency oscillations with high-frequency oscillations superimposed, 2 PCs will reproduce the low frequency components for both of the oscillating regions, but to reproduce the high frequency oscillations as well, 3 PCs are required.

Simulated data with random noise approximating that of experimental data showed that incorporating more PCs in the PCA approximation resulted in worse SNR. This is to be expected since noiseless simulations found that using N+1 PCs resulted in an approximation of the data essentially containing the entirety of the signal and therefore, when noise is present, using PCs higher than N+1 should almost solely add noise. This implies that, as N increases and the number of necessary PCs increases, the noise reduction seen from PCA filtering diminishes. However, for the example shown in Fig. 6 with a single oscillating volume, 2 PCs are used and the image SNR is increased by a factor of 3.8. Image SNR is proportional to the square root of the imaging time. Therefore, for this example, using PCA filtering allows for an improvement in temporal resolution by over an order of magnitude while maintaining the image quality of an unfiltered image. PCA filtering provides a means for increasing temporal resolution of dynamic EPRI without decreasing image quality. For the simulations presented here, the noise used was random and uncorrelated. In real EPRI experiments this may not be the case and the noise may have some correlation. Preprocessing methods that could help whiten the noise and remove correlation could improve the efficacy of PCA filtering.

The usefulness of PCA filtering decreases as the relative sizes of the oscillating volumes decrease (Fig. 7). As less of the volume exhibits a certain pattern, the given pattern begins to be considered noise rather than signal by the PCA filtering and is erroneously discarded.

Simulations showed that PCA filtering, unlike many commonly used filters, did not result in image resolution degradation. This suggests that the apparently higher contrast in the unfiltered images seen in Figures 8a and 9a is an artifact of noise. However, while image resolution is unaffected by PCA filtering in the sense that the modulation transfer function is unaffected, dimension reduction may not always be possible without loss of information. Relatively small volumes with important pO₂ fluctuations can be considered noise by PCA filtering and may not be resolved. In this respect, PCA filtering affects spatial resolution. These implications must be considered before applying PCA filtering. For situations with dominant regions of signal amongst less-correlated noise, there is no loss of resolution, but in the case of actual physiology there may be small but real temporal fluctuations (represented in the higher PCs) that are unresolved in PCA filtered images.

Presently, how many oscillating volumes to expect in vivo or how large these volumes might be is not definitively known. However, studies suggest that many tumors contain large regions undergoing cycling hypoxia and that, while the spatial distribution and temporal pattern of these fluctuations vary significantly from tumor to tumor, for a single tumor the acutely hypoxic regions tend to have pO₂ fluctuations of a common pattern (40, 66). Therefore PCA filtering is particularly suited to enhance dynamic EPRI studies investigating cycling hypoxia, as PCA filtering requires no a priori knowledge of the temporal or spatial pO₂ patterns and works well for situations with large portions of the data having similar features and when there are a small number of these features, or in this case, modes of pO₂ fluctuation. Preliminary results of PCA filtering applied to dynamic EPRI studies of temporal pO₂ fluctuations in murine tumors are promising. PCA filtering works exceptionally well for experiments involving forced fluctuations in pO₂ by alternating breathing gas for the mouse between normoxic and hyperoxic (Fig. 8). This experimental paradigm results in pO₂ fluctuations in regions with functional vasculature following the periodicity of the controlled FiO₂ fluctuations. This is an ideal situation for the application of PCA filtering because there is a single dominant mode of pO₂ fluctuation distributed over a large portion of the imaged volume. PCA filtering for the forced pO₂ fluctuation experiment appears to be successful in removing noise while preserving the expected pO₂ fluctuations.

PCA filtering of dynamic EPRI studies investigating spontaneous cycling hypoxia also has potential. The example presented in this paper (Fig. 9) is an indication that PCA filtering can enhance the visualization of spontaneous temporal pO₂ fluctuations using dynamic EPRI, so that cycling hypoxia can be imaged in vivo without perturbing the biologic system being studied. One of the limits to the entire concept of cycling hypoxia has been the limitation of current techniques to quantify the amplitudes of pO₂ fluctuations and the volume of the tumor undergoing such fluctuations. This has prevented correlation of these parameters with the effectiveness of anti-cancer therapy. There may even be other pertinent characteristics of cycling hypoxia in addition to amplitude and spatial extent of pO₂ fluctuations. The advances presented in this work further enable the evaluation of these characteristics of cycling hypoxia. This evaluation can be obtained non-invasively, allowing for correlation of cycling hypoxia and therapeutic outcome. This, in turn, will begin to disentangle the relationship between chronic and cycling hypoxia, and provide a means for evaluating the role of both forms of hypoxia in therapeutic outcome as well as therapeutic optimization.