## Simultaneous likelihood function towards a local maxima and make

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

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