The resolution (R
) specifies the finest division of a continuous scale that can distinguish two points without ambiguity, and is limited by the precision at which quantities represented by the scale (e.g. of image intensities) can be estimated. For spectral lines, the resolving power is given by the Rayleigh criterion (the accepted criterion for minimum resolvable detail) is said to be diffraction limited and is defined as the separation between two lines where the peak of one line occurs at the first minimum of the other (9
). For EPR imaging, the spatial resolution is given in terms of full width at half maximum (FWHM, Δ) of a Lorentzian line shape (10
). For digital images, the resolution is specified by the number of divisions (n
) per unit length where n
is a quantity equals to 1/R
. We refer to “high resolution” for larger n
or smaller R
indicating the ability to examine the image in higher detail. At low resolution, the object size artificially appears to be bigger than actual size due to large pixel size (). At higher resolutions, finer details of the image are revealed with more accurate object boundaries.
The image of the phantom in at different resolutions. A slice of MR image at size 256 × 256 was used to calculate the low-resolution images. The object size appears to be artificially larger at low resolution.
3.1 Effect of digital enhancement on low resolution
The intrinsic resolution of an image is limited by the factors inherent to the technique. In SPI, gradient strength and transverse relaxation times (T2*) limit the resolution since the signal decays by exp(−tp/T2*) where the delay time after excitation pulse, tp, and gradient magnitude are related to the obtainable resolution. In CW spatial imaging, the gradient strength to linewidth ratio is a parameter that limits the intrinsic resolution. Elimination of an inherent factor e.g. by deconvolution of gradient free line shape from raw data may enhance the resolution but is subjected to the accuracy of such algorithm. Knowledge of intrinsic resolution is particularly helpful in interpreting the resolution enhanced images, as for example mages at higher digital resolution can be produced from the low resolution images by mathematical treatments. Digital enhancement may be achieved by prediction of pixel intensities using interpolation methods such as linear, cubic, spline or zero-filled Fourier Transform (FT) techniques. Here we have used the Fourier/sinc interpolation kernel. Irrespective of the interpolation technique used, enhancement of digital resolution beyond the intrinsic resolution does not improve the image quality per se, other than rounding of sharp corners or diffusing any abrupt contrast. FT-EPR image of a phantom reconstructed from k-space data at intrinsic resolution (FOV = 25.9 mm, n = 61 and a resolution of 0.42 mm/pixel) and at higher digital resolutions (n = 128, and 256 with 0.2 and 0.1 mm/pixel) are shown in . The window-level in these images was adjusted to emphasize edge contrast. The 2D view of the phantom appears as a circular disk containing two large open circles at the top, three smaller filled circles below them and a comb-like shape. The comb-shape has teeth of different widths and different gap widths between the teeth (). In the images at n = 128 and 256, the open and closed circles appear to have much smoother edges than at n = 61 (the intrinsic resolution). The improved detail of these digitally enhanced images may not necessarily mean an increase in resolution since it depends on the algorithm used for enhancement. Fourier interpolation at a larger matrix size does not improve the intrinsic resolution of the technique since the image space is calculated at the same k-space resolution. The digital enhancement in is just a smoothening effect of interpolation, as can be seen in the noise (non-signal containing) regions of the images.
Fig. 3 Digital enhancement of a low resolution image. A 2D spin density image of the phantom filled with Oxo-63 spin label. Images were reconstructed at digital resolutions: (a) at intrinsic resolution (n = 61), (b) reconstructed at image size of 128 × (more ...)
3.2 Combining images of two different resolutions for co-registration
Digital enhancement and emulation of pixel intensities are unavoidable while co-registering two images having very different intrinsic resolutions. As an example, we show a comparison of the phantom image obtained by proton MRI at high resolution with spin density EPR images obtained at three lower spatial resolutions (). The top row in shows the images at their intrinsic resolution in which each square represents a pixel and the spin density is indicated by its brightness. The area covered by each pixel increases at lower resolution which results in blurring of the image. The open and closed circles are clearly visible at MRI resolution (R = 0.125 mm/pixel, n = 204) while the EPR images scanned at R = 0.42, 1.21 and 1.42 mm/pixel (n = 61, 23 and 21 respectively) have diffuse edges. At R=0.42, the larger circular shapes including two open and two filled circles on left are apparent, but the small filled circle at right appears as a square. At R= 1.21 and 1.42, the bigger closed circle on the left also appears as a square and the center of the open circle is blurred. The intensity profile of a row in the comb like region depicts the resolvable gaps between filled regions. In the image, the narrowest gap is the third line from left and the most narrow fill is the first bright line from the right. These are clearly visible at MRI resolution. Both are well defined with a slight blur in EPR image at R=0.42 mm, but are totally blurred at R= 1.21 and 1.42 mm. The pixel intensity line graph of a row in the comb-like regions is shown below these images in . At high resolution a sharp change of signal between maximum and minimum values depending on the presence and absence of spin probe respectively is observed. As the resolution is decreased, the profiles become broader indicating blurring of the image.
Fig. 4 Comparison of intrinsic resolution with digital enhancement of a 2D spin density image of a phantom filled with Oxo-63 spin label. A. MRI scanned at 0.125 mm resolution. B, C, and D are EPR images scanned at the intrinsic resolutions of 0.42, 1.21 and (more ...)
A comparison of downsizing of higher resolution images is shown in middle row of . These images were computed by summing the pixel intensities by laying out spatial grid equal to the image at smallest intrinsic resolution (R = 1.42 mm). Reducing matrix size to 21 lead to smearing of intensities of neighboring pixels but the circular shapes still appear better at high intrinsic resolution. All the four images reconstructed at a digital resolution equal to MRI are shown in the bottom row of . The circular shapes in EPR images are defined better in these illustrations than before; however, the edges still remain blurred. Comparison of the image at R=0.125 mm to R = 1.42 mm clearly points out that the anatomical shapes are not resolved any better by this mere digital enhancement.
A comparison of anatomic MR image and spin density EPR image of hind leg of a tumor bearing mouse is shown in . The intrinsic resolutions of MR and EPR images are 0.109 and 1.55 mm, respectively. The MRI and EPRI modalities, besides having very different resolutions (MRI being of a much higher resolution), capture very different aspects in their images. MRI deals with water distribution inside the animal and the proton density images that are obtained often provide nearly full anatomic details with local viscosities and dynamics providing characteristic contrasts. EPRI, on the other hand just captures the distribution of the unpaired spin and often areas with very low or negligible spin perfusion will not show up and as such the anatomical details are much less defined. The contrast in EPR images comes both from distribution of the spin by perfusion and the relaxivity of in vivo oxygen.
Fig. 5 Coregistration of EPR spin density image with MR anatomic image of a tumor bearing mouse hind leg. (a) MRI at resolution 0.109 mm. (b) Solid lines drawn at the intensity gradients trace the anatomic shape. (c) EPR image at its intrinsic resolution of (more ...)
The oval shape of the tumor is apparent in MRI () while EPR spin density image () appears to be diffuse. In spite of the low resolution of the EPR image, co-registration can be done almost at the precision of MR image by matching distantly located land marks (fiducials). The overlay of the edges from MRI on EPR image () indicates that the spin probe density around the tumor region is higher than inside the tumor. The co-registration demonstrates the differences of spin probe and water distributions inside tumor leg.
3.3 Understanding pO2 maps
Quantitative interpretation of pO2 map is important to delineate hypoxic and normoxic regions, and to evaluate oxygen levels during perfusion experiments. A pO2 image slice of a tumor bearing mouse obtained by EPRI is shown in . The image was digitally converted from a 19 × 19 × 19 matrix into 256 × 256 × 14 matrix for co-registration with MR image. A characteristic feature of digital enhancement is smoother gradients of pO2 levels (). The smearing of region boundaries due to low resolution and blurring is obscured by apparent higher digital resolution. And will be a general feature with similar trends irrespective of the exact nature of the interpolation kernel used.
Fig. 6 Resolution indication on the image. (a) pO2 map at intrinsic resolution of 1.55 mm. (b) Overlay of the anatomic shape from MRI. (c) Digitally enhanced to match to MRI size of 256 × 256. The spatial resolution is shown on spatial dimension. The (more ...) 3.3.1 Spatial Resolution
The resolution of pO2
map depends on (i) spatial resolution of spin densities and (ii) spectral parameters to estimate pO2
values. Both of these depend on spectral line width of the spin probe. The effect of gradient strength relative to EPR line widths, (11
) signal to noise characteristics, (10
) and sampling of data on spatial resolution of EPR images are well documented in literature. (12
) In general, the spatial resolution of EPR images is improved by higher gradient strengths, better SNR and sensitivity of the imager, smaller line widths of spin probes and higher data sampling rate for image reconstruction. While small line widths achieve high spatial resolution, the presence of pO2
increases the line width and thereby reducing the spatial resolution. The local variations of pO2
levels influence line widths causing local variations in spatial resolution. For Oxo-63 spin probe, the line width may increase by 33% from hypoxia to a local pO2
of 5% . (14
If two objects are spatially unresolved in an image, their signals overlap with each other and the pO2
values of both objects are smeared. Consider a 1D space of spin densities divided into equal size pixels. Pixels separated by Δ (FWHM) would get about 12% of each neighboring pixel’s intensity for a Gaussian shape and 15% for a Lorentzian shape. However, this overlap becomes negligible for Gaussian (1%) at a pixel width of 2Δ. Although, the Lorentzian overlaps are considerable even at a pixel width of 4Δ (5%), background correction grossly removes their effect. (15
Gaussian overlaps of a hypothetical 1D image having five equally spaced point objects of spin densities of 6, 6, 6, 1, 1 at separations of Δ and 2Δ are shown in . The traces shown in thin lines indicate signals from individual objects and their summation shown by thick line represents observed signal. The vertical grid lines are pixel boundaries. The pixel intensities calculated from are 5.28, 6, 5.4, 1.6, 0.88 and from are 5.94, 6, 5.95, 1.05, 0.99. Note that the 4th object has intensity 1.6 instead of its original value of 1 when R = 1/Δ. The effect of overlaps on neighbors is greater when a low spin density is next to a high spin density. But the errors due to overlaps are reduced by an order of magnitude at R = 1/2Δ ().
Fig. 7 Simulation of signals using hypothetical point objects having spin densities of 6, 6, 6, 1, 1. The objects are equally spaced. (a) Spacing = Δ. Grid lines indicate the pixels boundaries at this spacing. (b) Spacing = 2Δ. (c) Effect of (more ...)
In SPI mode, the pixel-wise signal decay of the sequence of T2*-weighted images considered for oximetry, is governed by exp(−tp/T2*) where T2* is inversely related to pO2. An example of signal decay for arbitrary linewidths of Δ, 1.5Δ and 2Δ to represent oxygen free and two different pO2 levels is shown in . The horizontal line drawn at a hypothetical SNR of 1% of maximum signal level intersects the curves of faster decays at smaller tp values compared to decay for linewidth of Δ. Because of this decrease in the presence of oxygen, it is necessary to use a tp value smaller than that is suitable to oxygen free conditions for imaging both normoxic and hypoxic regions. Ignoring this condition leads to artifacts caused by lack of detection of high pO2 regions that fall within noise level.
3.3.2 Probe characteristics
Each spin probe has characteristic: (1) pO2 range that can be covered (2) sensitivity of spin probe to O2 (3) specificity.
The sensitivity of the spin probe is proportional to the magnitude of line broadening caused by O2. The line width variation of a narrow EPR line is relatively a smaller quantity than the peak height variation. In practice, the line widths and their uncertainties are estimated by fitting all the data covering entire line shape. The line width uncertainties are determined from the residuals of the fit. Therefore the precision of pO2 depends on both the sensitivity of the probe to O2 and SNR. While high oxygen sensitivity of probe is advantageous to measure low pO2 levels at better precision, it is disadvantageous to measure high oxygen levels since the line broadening decreases the spatial resolution and SNR. The spin probes sensitive to low pO2 levels are good enough for biomedical applications to delineate hypoxic and normoxic regions which deal with limited pO2 range (0 – 76 mmHg).
The line widths are also affected by spin probe concentration, viscosity, temperature and instrument settings. The contributions to line width from the factors other than O2 have to be compensated for accurate determination of pO2 values. Conventionally, pO2 values are calculated from observed line widths using calibration data generated in non-imaging experiments. The specificity of pO2 values depends on the accuracy of line width in the absence of oxygen and the proportionality constant between pO2 and linewidth. Inaccurate estimate of natural line width adds or subtracts a constant amount of pO2 value to the entire image leading to under or over estimation of hypoxic regions. Inaccurate proportionality constant leads to relative pO2 estimates instead of absolute quantities. It is important to fine tune the calibration data from non-imaging experiments to imaging conditions. The calibrations determined by in vitro measurements often require corrections to account for local concentration, viscosity and temperature variations of in vivo studies.
Every point in the measured spin density image is essentially an EPR line characterized by a shape and height. When the line shape is constant, the observed image can be described by a convolution of spin densities with line shape function.(16
) Assessment of oxygen levels from the line shape are associated with uncertainties arising from noise characteristic to the technique including motion effects, gradient imperfections and random noise. Both the intrinsic resolution specified by spectral line width and gradient strengths, and other sources of noise contribute to the point spread function of the pO2
image. Supposing that observed and true images are O
respectively and P
is a point spread function (PSF), the intensity of O
at point z
is expressed by convolution as
represents convolution. When EPR line shape (P
) is known, the true spin densities I
can be recovered from observed image O
by deconvolution. The effect of a Gaussian PSF on a hypothetical pO2
map is shown in . This image depicts tumor bearing and normal hind legs of a mouse and two fiducials placed in between and at the side of the legs. The pO2
regions of tumor include hypoxic core surrounded by high oxygen layer, while the remaining including fiducials exhibit normoxic levels. When P
is a Gaussian PSF () the hypothetical pO2
is blurred (). The hypoxic regions with higher pO2
values are smeared with lower pO2
values at normoxic regions at the boundaries. We see that the hypoxic core becomes smaller and the oxygen rich layer surrounding the tumor becomes narrow by diffusing both into normoxic and hypoxic regions. Additionally, the fiducial diameter increases, and its pO2
level falls close to hypoxic values.
Fig. 8 (a) A hypothetical pO2 image schematically representing tumor (left) with hypoxic core (0%), surrounding high oxygen region (5%) and normoxic regions (4%). The two dots represent fiducials. The thin white lines indicate boundaries of pO2 regions. (b) (more ...)
) of a pO2
map to remove this blurring is often complicated because line widths vary with changes in oxygen level. Deconvolution of approximate PSF’s including contributions of the natural EPR line shape and random noise can partially sharpen the image. In the single point imaging (SPI) technique, k
-space sampling interval is determined as a function of tp
which is proportional to the image field of view and thus intrinsic resolution. In other words, a series of images at different resolutions are naturally obtained in SPI. Considering that a low resolution image is a convolution of high resolution image and a PSF, one can estimate PSF to use with deconvolution within a single SPI dataset. We have estimated PSF by this procedure and found that it is approximately Gaussian. The deconvolution of spin density image by this procedure is shown in . The deconvolution clearly shows sharpening of the object dimensions similar to the original object proposed in . The pO2
calculation using these spin densities leads to a sharpened pO2
map where the hypoxic core appears to be much larger as expected. However, deconvolution does not necessarily improve resolution. When the PSF is estimated within a set of SPI images, the images are similarly sharpened, except in this case the linewidths are uniformly scaled down by a factor in proportion to the ones obtained from un-deconvolved images, without affecting, however, the spatially resolved relaxivity information. These are summarized in .
Spin density 3D EPR image of a tumor bearing mouse leg. (a) The positioning of the tumor-bearing leg on a 17 mm resonator. Surface-rendered images at identical angles and cut-off intensity (b) before and (c) after deconvolution.
Fig. 10 pO2 map of a slice from 3D EPR image of a tumor bearing mouse leg at the intrinsic resolution of 1.55 mm. (a) Before deconvolution. (b) pO2 map calculated using spin densities after deconvolution as shown in . (c) pO2 map obtained by the deconvolution (more ...)
3.5 Indication of Intrinsic Resolution
The data presentation as an image can be a limiting factor in some situations. For example, the spatial resolution may be much higher than depicted in an illustration. At the other extreme, a computer can present the number of shades of colors or grays much higher than human eye can distinguish. For example, in , only a few shades of pO2 levels are spatially distinguishable by the eye though it contains 64 color shades. Therefore explicit indication of resolution of the scale as shown in is helpful in interpreting pO2 maps. The hypoxic core inside tumor region appears as a dark square and its surrounding regions at higher pO2 levels appear bright at different levels (). Co-registration with MRI facilitates the identification of anatomic locations in EPRI accurately both at its intrinsic resolution () and digital enhancement (). An illustration of pO2 map as in without resolution indication exaggerates the precision at which the hypoxic regions can be outlined. This ambiguity is removed either by using illustrations at intrinsic resolution as in or by explicit indication of resolution as in .