# US 10,482,596 B2 · Normalization crystal efficiencies estimation for continuous motion bed acquisition > Agents: this data is available as JSON from the patentgrep API and MCP server — see https://patentgrep.com/llms.txt - Number: US 10,482,596 B2 - Document ID: US10482596B2 - Status: Granted - Kind: B2 - Application number: 15/586,485 - Filed: 2017-05-04 - Published: 2018-11-08 - Granted: 2019-11-19 - Earliest claimed: 2017-05-04 - Assignees: Siemens Medical Solutions USA, Inc. - Applicants: Siemens Medical Solutions USA, Inc. - Inventors: Vladimir Y. Panin - Web page: https://patentgrep.com/patent/US10482596B2 ## CPC - G06T 7/0012: {Biomedical image inspection} - A61B 6/037: {Emission tomography} - A61B 6/0407: {Supports, e.g. tables or beds, for the body or parts of the body} - G01T 1/202: the detector being a crystal - G01T 1/2985: {In depth localisation, e.g. using positron emitters; Tomographic imaging (longitudinal and transverse section imaging; apparatus for radiation diagnosis sequentially in different planes, steroscopic radiation diagnosis); (using external radiation sources A61B6/02)} - G06T 2207/10104: Positron emission tomography [PET] - G06T: IMAGE DATA PROCESSING OR GENERATION, IN GENERAL - A61B: DIAGNOSIS; SURGERY; IDENTIFICATION (usefulness limited to only animals A61D) - G01T: MEASUREMENT OF NUCLEAR OR X-RADIATION (radiation analysis of materials, mass spectrometry G01N23/00; tubes for determining the presence, intensity, density or energy of radiation or particles H01J47/00) ## Abstract A method and system for simultaneously monitoring a positron emission tomography scanner performance during a continuous-bed-motion acquisition is disclosed. ## Claims ### Claim 1 A method for simultaneously monitoring a positron emission tomography (PET) scanner performance during a continuous-bed-motion (CBM) acquisition, wherein the PET scanner has a patient bed that moves a patient in an axial motion through the PET scanner during the CBM acquisition, the method comprising: (a) generating time-of-flight (TOF) patient data of positron annihilation activity in a patient during the CBM acquisition, while accounting for the axial motion of the patient bed during the CBM acquisition, as a first data set; (b) generating a complimentary data set by integrating the TOF patient data over the axial motion of the patient bed during the CBM acquisition, wherein the complimentary data set is modeled by the equation: y _ ρ ⁢ ⁢ θ ⁢ ⁢ zT ⁢ = n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ B ( a ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ∑ k ⁢ C ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T , k ⁢ f k ) + n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ B ⁡ ( S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T ) + r _ ρ ⁢ ⁢ θ ⁢ ⁢ zT = n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ p _ ρ ⁢ ⁢ θ ⁢ ⁢ zT + ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ zT + r _ ρ ⁢ ⁢ θ ⁢ ⁢ zT , wherein yρθz, a motion blurring operator, is calculated by: y ρ ⁢ ⁢ θ ⁢ ⁢ z = B ⁡ ( y ρ ⁢ ⁢ θ ⁢ ⁢ ζ ) = ∫ 0 t acq ′ ⁢ dt ′ ⁢ ∑ ζ ⁢ e - λ ⁢ ⁢ t ⁢ d ρ ⁢ ⁢ θ ⁢ ⁢ z ⁡ ( t ) ⁢ y ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ δ z , Z ⁡ ( ζ , t ) ; and (c) simultaneously reconstructing the positron annihilation activity and crystal efficiency normalization coefficient from the complimentary data set. ### Claim 2 (depends on 1) The method of claim 1, wherein the step (a) comprises CBM data y with spatial projection index j=(ρ, θ, ζ) and TOF bin index T is modeled by combining the true events modeled projection p from the emission object f, defined by voxel index k, corrected for scanner efficiency through a normalization array n for attenuation by a, and scatter estimation S, corrected for scanner efficiency, and mean random data r: y _ ρθζ ⁢ ⁢ T = a ρθζ ⁢ n ρθζ - 1 ⁡ ( ɛ ) ⁢ ∑ k ⁢ C ρθζ ⁢ ⁢ T , k ⁢ f k + n ρθζ - 1 ⁢ S ρθζ ⁢ ⁢ T + r _ ρθζ ⁢ ⁢ T = a ρθζ ⁢ n ρθζ - 1 ⁡ ( ɛ ) ⁢ p _ ρθζ ⁢ ⁢ T + n ρθζ - 1 ⁢ S ρθζ ⁢ ⁢ T + r _ ρθζ ⁢ ⁢ T ; wherein C is the geometrical projection system matrix; wherein ε is the crystal efficiency; and wherein coordinate ζ is the CBM data normalization. ### Claim 3 (depends on 2) The method of claim 2, wherein the CBM data normalization factor ζ is computed from scanner normalization factors Z accounting for the axial patient bed motion during the CBM acquisition according to the equation: n ρθζ - 1 = ∫ 0 t acq ⁢ dt ⁢ ∑ z ⁢ e - λ ⁢ ⁢ t ⁢ d ρθ ⁢ ⁢ z ⁡ ( t ) ⁢ n ρθ ⁢ ⁢ z - 1 ⁢ δ z , Z ⁡ ( ζ , t ) ; wherein λ is decay correction constant for a specific isotope, δ is Kronecker's delta, d is dead time correction factor, and tacq is CBM acquisition time. ### Claim 4 (depends on 3) The method of claim 3, wherein the scanner normalization factor Z may accommodate for the mashing and rebinning of the LORs, connecting two detector scintillation crystals i and i′ into projection bin according to the equation: n ρθ ⁢ ⁢ z - 1 ⁡ ( ɛ ) = ∑ i , i ′ ⁢ ω ρθ ⁢ ⁢ z , ii ′ ⁢ g ii ′ ⁢ ɛ i ⁢ ɛ i ′ ; wherein g is the geometrical component of the normalization array; and wherein ω is the LOR contribution factor into projection data due to mashing and spanning data compression techniques and defined by the following: φ j , ii ′ = { 1 / 2 , if ⁢ ⁢ ii ′ ⁢ ⁢ contribute ⁢ ⁢ to ⁢ ⁢ sinogram ⁢ ⁢ bin ⁢ ⁢ j 0 , otherwise . ### Claim 5 (depends on 1) The method of claim 1, wherein the step (c) comprises: maximizing the following objective function L A ⁡ ( f ) = ∑ j = { ρ ⁢ ⁢ θ ⁢ ⁢ ζ } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to f for the activity reconstruction; wherein f is activity image; wherein ordered subsets expectation maximization algorithm is applied with one iteration and 21 subsets; and maximizing the following objective function L CE ⁡ ( ɛ ) ⁢ ∑ j = { ρ ⁢ ⁢ θ ⁢ ⁢ z } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to ε, wherein ε is the crystal efficiency. ### Claim 6 (depends on 1) The method of claim 1, further comprising updating the crystal efficiency normalization coefficient by updating ɛ i ( m + 1 ) = - B i + B i 2 + 4 ⁢ A i ⁢ C i 2 ⁢ A i A i = ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁢ g ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) , ⁢ B i = ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) ⁢ ɛ i ′ ( m ) - A i ⁢ ɛ i ( m ) C i = ɛ i ( m ) ⁢ ∑ jT ⁢ y jt y _ jT ( m ) ⁡ ( f , ɛ ( m ) ) ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ jT ⁡ ( f ) + S jT ) ⁢ ɛ i ′ ( m ) j = { ρ ⁢ ⁢ θ ⁢ ⁢ z } , wherein m is iteration number for the updating the crystal efficiency ε; and wherein pj and Sj represent summation of pjT and SjT over TOF bin index T. ### Claim 7 A positron emission tomography (PET) calibration system comprising: a PET scanner having a patient bed that moves a patient in an axial motion through the PET scanner during the CBM acquisition; and a scintillation crystal efficiency calibration system that performs a method comprising: (a) generating time-of-flight (TOF) patient data of positron annihilation activity in a patient during the CBM acquisition, while accounting for the axial motion of the patient bed during the CBM acquisition, as a first data set; (b) generating a complimentary data set by integrating the TOF patient data over the axial motion of the patient bed during the CBM acquisition, wherein the complimentary data set is modeled by the equation: y _ ρ ⁢ ⁢ θ ⁢ ⁢ zT ⁢ = n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ B ( a ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ∑ k ⁢ C ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T , k ⁢ f k ) + n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ B ⁡ ( S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T ) + r _ ρ ⁢ ⁢ θ ⁢ ⁢ zT = n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ p _ ρ ⁢ ⁢ θ ⁢ ⁢ zT + ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ zT + r _ ρ ⁢ ⁢ θ ⁢ ⁢ zT , wherein Yρθz, a motion blurring operator, is calculated by: y ρ ⁢ ⁢ θ ⁢ ⁢ z = B ⁡ ( y ρ ⁢ ⁢ θ ⁢ ⁢ ζ ) = ∫ 0 t acq ′ ⁢ dt ′ ⁢ ∑ ζ ⁢ e - λ ⁢ ⁢ t ⁢ d ρ ⁢ ⁢ θ ⁢ ⁢ z ⁡ ( t ) ⁢ y ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ δ z , Z ⁡ ( ζ , t ) ; and (c) simultaneously reconstructing the positron annihilation activity and crystal efficiency normalization coefficient from the complimentary data set. ### Claim 8 (depends on 7) The system of claim 7, wherein the step (a) comprises CBM data y with spatial projection index j=(ρ, θ, ζ) and TOF bin index T is modeled by combining the true events modeled projection p from the emission object f, defined by voxel index k, corrected for scanner efficiency through a normalization array n for attenuation by a, and scatter estimation S, corrected for scanner efficiency, and mean random data r: y _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T = a ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁡ ( ɛ ) ⁢ ∑ k ⁢ C ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T , k ⁢ f k + n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + r _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T = a ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁡ ( ɛ ) ⁢ p _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + r _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T ; wherein C is the geometrical projection system matrix; wherein ε is the crystal efficiency; and wherein coordinate ζ is the CBM data normalization. ### Claim 9 (depends on 8) The system of claim 8, wherein the CBM data normalization factor ζ is computed from scanner normalization factors Z accounting for the axial patient bed motion during the CBM acquisition according to the equation: n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 = ∫ 0 t acq ⁢ dt ⁢ ∑ z ⁢ e - λ ⁢ ⁢ t ⁢ d ρ ⁢ ⁢ θ ⁢ ⁢ z ⁡ ( t ) ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁢ δ z , Z ⁡ ( ζ , t ) ; wherein λ is decay correction constant for a specific isotope, δ is Kronecker's delta, d is dead time correction factor, and tacq is CBM acquisition time. ### Claim 10 (depends on 9) The system of claim 9, wherein the scanner normalization factor Z may accommodate for the mashing and rebinning of the LORs, connecting two detector scintillation crystals i and i′ into projection bin according to the equation: n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) = ∑ i , i ′ ⁢ ω ρ ⁢ ⁢ θ ⁢ ⁢ z , ii ′ ⁢ g ii ′ ⁢ ɛ i ⁢ ɛ i ′ ; wherein g is the geometrical component of the normalization array; and wherein ω is the LOR contribution factor into projection data due to mashing and spanning data compression techniques and defined by the following: φ j , ii ′ = { 1 / 2 , if ⁢ ⁢ ii ′ ⁢ ⁢ contribute ⁢ ⁢ to ⁢ ⁢ sinogram ⁢ ⁢ bin ⁢ ⁢ j 0 , otherwise . ### Claim 11 (depends on 7) The system of claim 7, wherein the step (c) comprises: maximizing the following objective function L A ⁡ ( f ) = ∑ j = { ρ ⁢ ⁢ θ ⁢ ⁢ ζ } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to f for the activity reconstruction; wherein f is activity image; wherein ordered subsets expectation maximization algorithm is applied with one iteration and 21 subsets; and maximizing the following objective function L CE ⁡ ( ɛ ) ⁢ ∑ j = { ρ ⁢ ⁢ θ ⁢ ⁢ z } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to ε, wherein ε is the crystal efficiency. ### Claim 12 (depends on 7) The system of claim 7, further comprising updating the crystal efficiency normalization coefficient by updating ɛ i ( m + 1 ) = - B i + B i 2 + 4 ⁢ A i ⁢ C i 2 ⁢ A i A i = ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁢ g ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) , ⁢ B i = ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) ⁢ ɛ i ′ ( m ) - A i ⁢ ɛ i ( m ) C i = ɛ i ( m ) ⁢ ∑ jT ⁢ y jt y _ jT ( m ) ⁡ ( f , ɛ ( m ) ) ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ jT ⁡ ( f ) + S jT ) ⁢ ɛ i ′ ( m ) j = { ρ ⁢ ⁢ θ ⁢ ⁢ z } , wherein m is iteration number for the updating the crystal efficiency ε; and wherein pj and Sj represent summation of pjT and SjT over TOF bin index T. ### Claim 13 A non-transitory, machine readable storage medium encoded with computer program software, such that when a processor executes the computer program software, the processor performs a method for simultaneously monitoring a positron emission tomography (PET) scanner performance during a continuous-bed-motion (CBM) acquisition, wherein the PET scanner has a patient bed that moves a patient in an axial motion through the PET scanner during the CBM acquisition, the method comprising: (a) generating time-of-flight (TOF) patient data of positron annihilation activity in a patient during the CBM acquisition, while accounting for the axial motion of the patient bed during the CBM acquisition, as a first data set; (b) generating a complimentary data set by integrating the TOF patient data over the axial motion of the patient bed during the CBM acquisition, wherein the complimentary data set is modeled by the equation: y _ ρ ⁢ ⁢ θ ⁢ ⁢ zT = n ρθ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ B ⁡ ( a ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ∑ k ⁢ C ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T , k ⁢ f k ) + n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ B ⁡ ( S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T ) + r _ ρ ⁢ ⁢ θ ⁢ ⁢ zT = n ρθ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ p _ ρ ⁢ ⁢ θ ⁢ ⁢ zT + n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ zT + r _ ρ ⁢ ⁢ θz ⁢ ⁢ T wherein Yρθz, a motion blurring operator, is calculated by: y ρ ⁢ ⁢ θ ⁢ ⁢ z = B ⁡ ( y ρ ⁢ ⁢ θ ⁢ ⁢ ζ ) = ∫ 0 t acq ′ ⁢ dt ′ ⁢ ∑ ζ ⁢ e - λ ⁢ ⁢ t ⁢ d ρ ⁢ ⁢ θ ⁢ ⁢ z ⁡ ( t ) ⁢ y ρ ⁢ ⁢ θζ ⁢ δ z , Z ⁡ ( ζ , t ) ; and (c) simultaneously reconstructing the positron annihilation activity and crystal efficiency normalization coefficient from the complimentary data set. ### Claim 14 (depends on 13) The non-transitory, machine readable storage medium of claim 13, wherein the step (a) comprises CBM data y with spatial projection index j =(ρ, θ, ζ) and TOF bin index T is modeled by combining the true events modeled projection pfrom the emission object f, defined by voxel index k, corrected for scanner efficiency through a normalization array n for attenuation by a, and scatter estimation S, corrected for scanner efficiency, and mean random data r: y _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T = a ρθ ⁢ ⁢ ζ ⁢ n ρθ ⁢ ⁢ ζ - 1 ⁡ ( ɛ ) ⁢ ∑ k ⁢ C ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T , k ⁢ f k + n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + r _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T = a ρθ ⁢ ⁢ ζ ⁢ n ρθ ⁢ ⁢ ζ - 1 ⁡ ( ɛ ) ⁢ p _ ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 ⁢ S ρ ⁢ ⁢ θ ⁢ ⁢ ζ ⁢ ⁢ T + r _ ρ ⁢ ⁢ θζ ⁢ ⁢ T ; wherein C is the geometrical projection system matrix; wherein ε is the crystal efficiency; and wherein coordinate ζ is the CBM data normalization. ### Claim 15 (depends on 13) The non-transitory, machine readable storage medium of claim 13, wherein the CBM data normalization factor ζ is computed from scanner normalization factors Z accounting for the axial patient bed motion during the CBM acquisition according to the equation: n ρ ⁢ ⁢ θ ⁢ ⁢ ζ - 1 = ∫ 0 t acq ⁢ dt ⁢ ∑ z ⁢ e - λ ⁢ ⁢ t ⁢ d ρ ⁢ ⁢ θ ⁢ ⁢ z ⁡ ( t ) ⁢ n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁢ δ z , Z ⁡ ( ζ , t ) ; wherein λ is decay correction constant for a specific isotope, δ is Kronecker's delta, d is dead time correction factor, and tacq is CBM acquisition time. ### Claim 16 (depends on 13) The non-transitory, machine readable storage medium of claim 13, wherein the scanner normalization factor Z may accommodate for the mashing and rebinning of the LORs, connecting two detector scintillation crystals i and i′ into projection bin according to the equation: n ρ ⁢ ⁢ θ ⁢ ⁢ z - 1 ⁡ ( ɛ ) = ∑ i , i ′ ⁢ ω ρ ⁢ ⁢ θz , ii ′ ⁢ g ii ′ ⁢ ɛ i ⁢ ɛ i ′ ; wherein g is the geometrical component of the normalization array; and wherein ω is the LOR contribution factor into projection data due to mashing and spanning data compression techniques and defined by the following: ω j , ii ′ = { 1 / 2 , if ⁢ ⁢ ii ′ ⁢ ⁢ contribute ⁢ ⁢ to ⁢ ⁢ sinogram ⁢ ⁢ bin ⁢ ⁢ j 0 , otherwise . ### Claim 17 (depends on 13) The non-transitory, machine readable storage medium of claim 13, wherein the step (c) comprises: maximizing the following objective function L A ⁡ ( f ) = ∑ j = { ρ ⁢ ⁢ θζ } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to f for the activity reconstruction; wherein f is activity image; wherein ordered subsets expectation maximization algorithm is applied with one iteration and 21 subsets; and maximizing the following objective function L CE ⁡ ( ɛ ) = ∑ j = { ρθ ⁢ ⁢ z } , T ⁢ ( y jT ⁢ ln ⁡ ( y _ jT ) - y _ jT ) with regard to ε, wherein ε is the crystal efficiency. ### Claim 18 (depends on 13) The non-transitory, machine readable storage medium of claim 13, further comprising updating the crystal efficiency normalization coefficient by updating ɛ i ( m + 1 ) = ⁢ - B i + B i 2 + 4 ⁢ A i ⁢ C i 2 ⁢ A i A i = ⁢ ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁢ g ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) , B i = ⁢ ∑ j ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ j ⁡ ( f ) + S j ) ⁢ ɛ i ′ ( m ) - A i ⁢ ɛ i ( m ) C i = ⁢ ɛ i ( m ) ⁢ ∑ jT ⁢ y jt y _ jT ( m ) ⁡ ( f , ɛ ( m ) ) ⁢ ∑ i ′ ⁢ ω j , ii ′ ⁡ ( p _ jT ⁡ ( f ) + S jT ) ⁢ ɛ i ′ ( m ) j = ⁢ { ρ ⁢ ⁢ θ ⁢ ⁢ z } , wherein m is iteration number for the updating the crystal efficiency ε; and wherein pj and Sj represent summation of pjT and SjT over TOF bin index T.