Iterative deconvolution method in partial volume correction

Partial volume effect (PVE) reduces the quantitative potential of PET when the object size is less than two times the spatial resolution (FWHM) in the PET image. If the point-spread function (PSF) of the tomograph is measured, it can be used for partial volume correction (PVC) in iterative image reconstruction. Post-reconstruction image PVC (Jomaa et al., 2018) can be based on anatomical high-resolution images (MRI, CT), but iterative deconvolution method (IDM) is only based on the PET image data and measured FWHM or PSF. While the anatomy-based PVC methods are considered as gold standard, the images must be perfectly matched, which may be feasible only for the brain. Therefore, IDM PVC has been proposed for increasing quantitativity in oncological PET (Teo et al., 2007; Hoetjes et al., 2010; van Velden et al., 2011; Cysouw et al., 2016), though the results have not always been clinically significant (Hatt et al., 2012). IDM PVC has also been applied to myocardial PET (Muzic et al., 1998) and for reducing breathing motion artifacts (El Naqa et al., 2006).

PVC can improve the detection of small lesions and estimation of the lesion volume (Cysouw et al., 2017 and 2019). However, PVC generally increases the noise in images and can produce image artifacts which could be interpreted as lesions, and therefore the original, non-corrected PET images are recommended for lesion detection and drawing ROIs (Teo et al., 2007), if the noise amplification is not well regulated. Several methods for reducing the noise in IDM have been developed (Kirov et al., 2008; Boussion et al., 2009; Reilhac et al., 2015; Cysouw et al., 2019). Even in the brain PET, IDM PVC has been shown to provide reasonably good results (Tohka & Reilhac, 2006 and 2008; Thomas et al., 2011 Reilhac et al., 2015; Golla et al., 2017; Wolters et al., 2018).

In the IDM algorithms, partial volume effect is considered as the result of convolution of the true image with a 3D PSF. The van Cittert algorithm is based on the assumption of additive Gaussian noise model, and the Richardson-Lucy algorithm assumes multiplicative Poisson noise model. In each iterative step, the measured image is compared to the current version of the corrected image that is blurred using for example Gaussian smoothing). Especially the van Cittert methods tend to amplify the noise, and unless the noise amplification is regulated, the iterations must be stopped in an early phase, which may lead to incomplete recovery (Carasso, 1999; Rousset et al., 2007; Thomas et al., 2011). Noise-sensitive IDM methods may not be useful in case the lesions are very small (Soret et al., 2007).

Iterative deconvolution methods, in addition to many other partial volume correction methods, are included in PETPVC open-source software (Thomas et al., 2016).

Reblurred van Cittert method

The reblurred van Cittert method (VC) (original method was developed by van Cittert, 1931) uses additive regularization, and seeks to minimize the least-squares criterion

, where f is the measured PET image, t is the estimated true image, and h is the PSF of the PET scanner. For PVC, the PSF is usually approximated by an isotropic Gaussian. All image voxel values are constrained to be positive. Initially, the measured PET image is used as the corrected image. In each iteration k a new estimate of the true image is calculated using the residual, (f - tk ⊗ h)(x), which converges towards noise. The new image is calculated as

, or in the original Van Cittert's method,

, where α is the converging rate parameter, defining the step at each iteration (Tohka & Reilhac, 2008). The α can be set to 2 for the first iterations to speed up the process, reducing towards 1 (or even less) in subsequent iterations (Teo et al., 2007; Thomas et al., 2011). High α can be used only when signal-to-noise ratio is good (Crilly, 1987). Tohka & Reilhac proposed using the termination rule

because additional iterations would amplify noise with little or no improvement in recovery. The ability to stop iterations before artefacts become prominent is one of the befits of VC over inverse filter; with large number of iterations VC can converge to the inverse filter. Another benefit is that if it is known a priori that the data has certain characteristics such as non-negativity, then appropriate constraints can be incorporated into VC (Crilly, 1991).

Positivity of the image voxels can be guaranteed by truncation methods, such as

(Tohka & Reilhac, 2006), or by taking the pixel value for the new true image estimate from the original image times α, if the pixel value in previous estimate was negative (Schafer et al., 1981, Jansson et al., 1996), or by not changing the pixel value (by setting α=0 for that pixel) if the previous estimate is negative (Jansson et al., 1996). Relaxation-based methods ensure non-negativity without data truncation; in Jansson's method (Jansson et al., 1996) the α is replaced by relaxation function

, which leads to positive corrections if the estimate lies within its limits of 0 to β, reverse corrections if the estimate lies outside these limits, or varying positive corrections as a function of the estimates signal-to-noise ratio (Crilly, 1987).

The steps of the reblurred VC algorithm can be summarized as:

  1. convolve the measured PET image with the PSF,
  2. subtract the result from the measured PET image,
  3. convolve the result with the PSF,
  4. multiply the result with α,
  5. calculate an estimate of true image as the sum of this and the original image,
  6. convolve the result with the PSF,
  7. subtract the result from the measured PET image,
  8. convolve the result with the PSF,
  9. multiply the result with α,
  10. check this against the termination rule and stop if ready,
  11. calculate new estimate of true image by adding the previous estimate,
  12. go to step 6.

Reblurred van Cittert PVC can be tested with a tentative program imgidpvc.


Richardson-Lucy algorithm

In Richardson-Lucy algorithm (RL) (Richardson 1972; Lucy, 1974) the regularization step is multiplicative instead of additive. In each iteration k a new estimate of the true image t is calculated as

, where f is the measured PET image, and h is the PSF of the PET scanner (Tohka & Reilhac, 2008; Boussion et al., 2009). For PVC, the PSF is usually approximated by an isotropic Gaussian. While RL algorithm allows negative pixel values, non-negativity constraint is important for quantitative PET, so that

and

(Tohka & Reilhac, 2008). The steps of the RL algorithm can be summarized as:

  1. convolve the measured PET image with the PSF,
  2. divide the measured image with this,
  3. convolve the result with the PSF,
  4. calculate an estimate of true image by multiplying the result with measured image,
  5. convolve the result with the PSF,
  6. divide the measured image with this,
  7. convolve the result with the PSF,
  8. update the estimate of true image by multiplying it with this,
  9. repeat from step 5, until the last iteration.

See also:


Literature

Aston JAD, Cunningham VJ, Asselin M-C, Hammers A, Evans AC, Gunn RN. Positron emission tomography partial volume correction: estimation and algorithms. J Cereb Blood Flow Metab. 2002; 22: 1019-1034. doi: 10.1097/00004647-200208000-00014.

Erlandsson K, Buvat I, Pretorius PH, Thomas BA, Hutton BF. A review of partial volume correction techniques for emission tomography and their applications in neurology, cardiology and oncology. Phys Med Biol. 2012; 57: R119-R159 doi: 10.1088/0031-9155/57/21/R119.

Golla SVS, Lubberink M, van Berckel BNM, Lammertsma AA, Boellaard R. Partial volume correction of brain PET studies using iterative deconvolution in combination with HYPR denoising. EJNMMI Res. 2017; 7: 36. doi: 10.1186/s13550-017-0284-1.

Hofheinz F, Langner J, Petr J, Beuthien-Baumann B, Oehme L, Steinbach J, Kotzerke J, van den Hoff J. A method for model-free partial volume correction in oncological PET. EJNMMI Res. 2012; 2:16. doi: 10.1186/2191-219X-2-16.

Jansson PA (ed.): Deconvolution of images and spectra, 2nd ed., Academic Press, 1996. ISBN: 0-12-380222-9.

Jomaa H, Mabrouk R, Khlifa N. Post-reconstruction-based partial volume correction methods: a comprehensive review. Biomed Signal Proces. 2018; 46: 131-144. doi: 10.1016/j.bspc.2018.05.029.

Merisaari H. Algorithmic analysis techniques for molecular imaging. Turku Centre for Computer Science, TUCS Dissertations, 217, 2016.

Rousset OG, Ma Y, Evans AC. Correction for partial volume effects in PET: principle and validation. J Nucl Med. 1998; 39(5): 904-911. PMID: 9591599.

Tohka J, Reilhac A. Deconvolution-based partial volume correction in Raclopride-PET and Monte Carlo comparison to MR-based method. NeuroImage 2008; 39: 1570-1584. doi: 10.1016/j.neuroimage.2007.10.038.


Appendix

For anisotropic resolution modelling, the normalized Gaussian aperture function of the spatial function r(x,y,z) (Rousset and Zaidi, 2006) is

In one-dimensional Gaussian filtering, the truncated filter for FIR convolution is

, where r is the filter radius in pixels (Getreuer, 2013). In PET studies the size of the pixel can be large as compared to the SD of the Gaussian. To take into account the size of the pixel, we can use the definite integral of g(r) between r-0.5 and r+0.5 instead, and that is



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Updated at: 2019-11-30
Created at: 2019-11-21
Written by: Vesa Oikonen