contingency supplement prior for joint estimation of activity and attenuation in Non–TOF PET/MR MR

Simultaneous
reconstruction of attenuation and activity (MLAA) from emission data only, suffered
from the inherent cross-talk
between the estimated attenuation and activity distributions. In this paper, we
proposed an improved MLAA algorithm by utilizing
tissue prior atlas (TPA) and a Gibbs prior as priori knowledge. TPA imposing statistical condition as a supplement for individual magnetic resonance (MR) information on the reconstruction process of attenuation map. Hence along with soft tissue distribution, provided by
segmentation of MR images, an air mask and a bone probability map (BPM) breakdown the MR low-signal class into 4 subclasses in order to favor recognitions of air and bone. Estimations on attenuation coefficients are realized as a mix of
pseudo-Gaussian distributions. The proposed algorithm evaluated using simulated
3D emission data. The proposed MLAA-TPA algorithm compared with MR-MLAA algorithm proposed by Heußer et al. Our
results demonstrate that the performance of MR-MLAA algorithm highly depends on the accuracy of MR segmentation
which is well handled by MLAA-TPA. The quantification results well illustrated that the MLAA-TPA outperformed
the MR-MLAA
algorithm,
owing
to reduction of misclassification and more
precise tissue detection.

Introduction:
Joint estimation of attenuation and activity based on the ‘maximum likelihood (ML)’ approach from the emission data only, is
an ill-posed problem due to cross-talk
between attenuation map and activity distribution. In the other hand
accurate quanti?cation reconstruction of the radiotracer activity
distribution in ‘positron emission tomography
(PET)’ mandates reliable ‘attenuation correction factors (ACF)’, in order to compensating the loss of detected photons
induced by the materials along ‘lines of response (LOR)’ ’[1]’.

Recently, it has been shown that using ‘magnetic
resonance (MR)’ partial information about distribution of soft tissue as prior
knowledge in the ‘maximum
likelihood reconstruction of activity and attenuation (MLAA)’ algorithm, derive the likelihood function towards a
local maxima and make problem less ill-posed (MR-MLAA) ’[2]’.
Although MR-MLAA compared to the
standard MR-based ‘attenuation correction (AC)’, had one step forward in PET quanti?cation by
detection of bone and air in
attenuation map, but since some misclassifications of air and bone, which can locally cause
bias in activity values is reported, the correctness of detection is more essential.
Generally, the efficiency
of the MR-MLAA algorithm can be affected by: a) the accuracy of MR segmentation, b)
the quality of registration
process between the various datasets, c) the anatomy complexity of the
reconstruction site and d) the count statistics of emission data.

In this study, we aimed at improving the
performance of non–TOF MLAA by exploiting of an air mask and a BPM,
beside patient individual soft
tissue information provided via the MR segmented images on the attenuation
estimations. The algorithm is based on joint
estimation of attenuation and activity from the PET emission data, which
alternatively updates attenuation and activity through an iterative approach. We
called the new algorithm MLAA-TPA.

Algorithm: In PET the expected counts

 for line of response (LOR)

can be expressed as:

 

 

where
µj and ?j are the values of linear attenuation
coef?cient and activity at position

. cij is the
sensitivity of detectors along LOR

 to activity in

 in a perfectly condition with no attenuation
for photons. li,j represent the effective
intersection length of voxel

 with LOR

. Considering the Poisson
nature of measured emission data, the cost function is best modeled as:

 

 

Where

,

denotes the attenuation image  

 (µ1 …. µN) and activity image

 (?1….?N) 
and yi is the measured emission data.

In a MLAA framework, optimization is done by an iterative manner. Every
iteration starts with activity update trough a ‘maximum likelihood

expectation
maximization (MLEM)’ ‘[3]’
approach, while keeping attenuation constant, and ends with the attenuation update,
using a ‘maximum likelihood
gradient ascent for transmission tomography (MLTR)’ ‘[4]’ with regards to
prior knowledge, while keeping the updated activity constant. Both MLEM and MLTR can be accelerated with ordered
subsets. Compton scatter, random coincidences are ignored in this
study.

Tissue
prior atlas and initial attenuation map: Since optimization of cost function has non-unique solutions, considering some priori knowledge about the
attenuation coef?cients in the algorithm, much improved that
situation. Toward a more
realistic circumstance, we expect estimations in
µ-map only concern a few typical continuous attenuation coefficients.

Gibbs prior RG, which defined by a Gibbs distribution as considered in MLAA, persuading local continuity between the
neighboring voxel intensities with analogous attenuation properties in µ-map.

Tissue
prior
atlas RT, imposing attenuation estimations histogram to be a mix of a few pseudo-Gaussian
distribution corresponding to each of pre-defined attenuation coefficients, as considered in MLAA. Furthermore, TPA determine the plausible region for
each of these coefficients, which in MR-MLAA only
soft tissue was taken
into account.

As TPAs
derivation demonstrated in ‘Fig. 1’, MR images are segmented into outside air, soft tissue
mask, and an unknown class corresponding to MR low-signal which represent either of air cavities, cortical bones, or potential artifacts. In contrary to
Heußer’s work ‘[2]’ in this study, inside the unknown
class a BPM favouring recognition of bone, and an air mask spatially constraint the regions
susceptible to air cavities, accordingly the unknown class split
into 4 subclasses. corresponding to Air, Bone…

Tissue prior atlas is determined as combination

of the uni-modal tissue
priors air LA, bone LB, soft
tissue LST, which use single pseudo-Gaussians and bi-modal tissue priors LAB and LSTB
related to air/bone and soft tissue/bone which use double pseudo-Gaussians on the estimations of attenuation coefficients. Soft tissue mask, air mask and BPM are indicated with w(r),
w(a) and w(b) respectively.

Soft tissue mask simply
derived with a global thresholding
of MR images and smoothed for soft-transaction between two
classes. The air mask and BPM derived from the co-registered CT images of 15 patients whole head. Matching
between multimodal datasets is done by affine registration. An initial attenuation map was derived by filling the body contour
with soft tissue attenuation value (0.01 mm-1).

Results: The reconstruction results for patient 1 in low
noise scenario are presented in ‘Fig. 2a’. Estimated attenuation map with MR-MLAA aside

 

from misclassifications of air as bone (red arrows) or bone as air (blue
arrows), is clearly suffered from misclassifications of soft tissue (green
arrows), since in MR-MLAA, MR low-signal regions only can be either of air or bone.
Through a practical solution, this defect is not unavoidable due to imperfect quality of MR
images or its segmentation process. In return MLAA-TPA as regards to the MR low-signal
regions almost perfectly recover the attenuation map. Nevertheless, some
misclassification in nose (green arrow) is
obvious, because of MR low-signal. Bias in
activity distribution compared to PET-CTAC image, for the two lesions reduced from 5.2% and 5.2% for MR–MLAA to
4.9% and 1.1% for MLAA-TPA, respectively.  

 ‘Fig. 2b’shows the reconstruction results for patient
2 in low noise scenario. in MR-MLAA case misclassifications of bone as air
(blue arrows) and misclassifications of soft tissue (green arrows) related to
MR

bad quality segmentation, in reconstructed
attenuation map yields bias in activity distribution 5.5% and 5.4% for the two
lesions. for MLAA-TPA, properly recovering air and bone information as well as soft tissue lead to reduction of activity bias for two
lesions to 2.5% and 1.9% respectively. In
spite of systematically improvement of the proposed algorithm the main
challenge is still remain in the complicated region which is prone position to
both air or bone.

For quantitative comparison ‘Table 1’ and ‘Table 2’
summarizes the results of the both algorithms for high and low noise counts simulations, in ROIs defined by the MR low-signal and whole head regions. As can be seen, results illustrate
potential outperformance of the proposed algorithm in both estimated
attenuation and activity.

Table 1: Quantitative
results for reconstructed attenuation and activity distributions of the
patients 1 simulated head region. 

 

Table 2: Quantitative
results for reconstructed attenuation and activity distributions of the
patients 2 simulated head region. 

 

 

Conclusion: In this paper a non–TOF MLAA algorithm was
presented with incorporation of patient specific tissue prior atlas (TPA) as
prior knowledge.  TPA is defined by
statistical condition as a new kind of prior knowledge, as supplement for MR
partial individual information. The efficiency of proposed MLAA-TPA algorithm
compared against current state-of-the art MLAA algorithm using simulations non–TOF PET/MR.
The results illustrate systematically improvement in PET quantification for the
proposed algorithm, by suppressing misclassifications of air and bone in less
contingent/possible regions, and a more practical solution is provided due to
reduce affiliation to segmentation error introduced by MR images.

 

References

1.  Nuyts, J., Dupont, P., Stroobants, S.,
Benninck, R., Mortelmans, L., Suetens, P.: ‘Simultaneous maximum a posteriori
reconstruction of attenuation and activity distributions from emission
sinograms’, IEEE transactions on medical imaging., 1999, 18, (5), pp. 393-403,
doi: 10.1109/42.774167

2.  Heußer, T., Rank, CM., Freitag, MT.,
Dimitrakopoulou-Strauss, A., Schlemmer, HP., Beyer, T., Kachelrieß, M.:
‘MR–consistent simultaneous reconstruction of attenuation and activity for
non–TOF PET/MR’, IEEE Transactions on Nuclear Science., 2016, 63, (5), pp.
2443-2451, doi: 10.1109/TNS.2016.2515100

3.
 Nuyts, J., De Man, B., Dupont, P.,
Defrise, M., Suetens, P. and Mortelmans, L.: ‘Iterative reconstruction for
helical CT: a simulation study’, Physics in medicine and biology., 1998,  43, (4), p.729, doi: 10.1088/0031-9155/43/4/003

4.  Shepp, L.A. and Vardi, Y., 1982. Maximum
likelihood reconstruction for emission tomography. IEEE transactions on medical
imaging, 1(2), pp.113-122. Doi: 10.1109/TMI.1982.4307558

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