E i Y i ( r T r S ; E i ;t ) i S ( r T r s ;t ) = (2) m r T ( t - - PDF document

• 1. Absorbed Dose

The absorbed dose in a target region rT at time t from the activity contained in a source region rs is given by, DrT(t)= ~ ArS⋅S(rT ←RS;t) (1) where ~ Ar S is the cumulated activity - total number of nuclear disintegrations that have occurred in the source region up to time t. The “S-Factor” relates the absorbed dose in the target region to the cumulated activity in the source region, S(r T←r s;t)=

∑

i

EiY iφ(rT←rS; Ei;t) mrT(t) (2) where the summation is over all nuclear decay channels, Ei is the energy of the ith decay, Yi is the yield of that channel, Φ is the absorbed fraction of the energy Ei emitted at time t in rs which is absorbed by rT and mrT(t) is the mass of the target region at time t. The total absorbed dose to rT is given by, DrT(T exp)=∫

Texp dD rT(t)

dt dt (3) =∫

Texp

ArS(t)⋅S(rT ←RS;t)dt In practice the target region will irradiated by multiple source regions (Ns), DrT(T exp)=∑

rS=1 N s

∫

T exp

ArS(t) ⋅S(rT←RS;t)dt (4)

• 2. Activity quantification

The activity in a source region rS is a function of the initial activity in the region and the decay of that activity. Ar S(t)=∫

T exp

A0e

(−λ t)dt

(5)

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For MRT the decay constant, λ, has a biological and physical component, Ar S(t)=∫

T exp

A0e

(−(λphysical+λbiological)t)dt

(6) The initial activity, A0, is determined from SPECT imaging using a phantom scan with a known activity as a calibration standard. A0=cf⋅N γ (7) where the calibration factor, cf, is defined as, cf = A

Ref

N γ

Ref

(8) Nγ is determined from SPECT projections (EM) which have been scatter corrected (SC). Nγ=EM (r ,φ)−SC(r ,φ) (9) Attenuation correction (AC) is also commonly applied. For a non uniform attenuating medium this is an iterative process using OSEM. For uniform attenuation it can be simplified using the Chang technique. EM AC(r ,φ)= EM (r ,φ) 1 M ∑

i=1 M

e

−μ li

(10) where M is the total number of projections used to acquire the data and li is the distance to the

• bject in the ith projection.

The reference activity is based on a decay corrected calibrator standard. A Ref= Acalibrator(m4−m3)2

−Δt/t( 1/2)

(m2−m1) (11) And hence, A0= Acalibrator(m4−m3)2

−Δt/t(1/2)

EM Ref(r ,φ) 1 M ∑

i=1 M

e

−μEM li

− SC Ref (r ,φ) 1 M ∑

i=1 M

e

−μSCli

⋅ EM (r ,φ) 1 M ∑

i=1 M

e

−μEM li

− SC (r ,φ) 1 M ∑

i=1 M

e

−μSCli

(12)

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• 3. Expression for total absorbed dose

DrT(T exp)=∑

rS=1 N s

∫

T exp

A0e

(−(λphysical+λbiological)t)⋅S(r T←RS;t)dt

(13) DrT(T exp)=cf⋅(EM AC(r ,φ)−SCAC(r ,φ))∑

rS=1 N s

∫

T exp

e

(−(λphysical+λbiological)t)⋅S(r T← RS;t)dt

(14) DrT(T exp)=cf⋅(EM AC(r ,φ)−SCAC(r ,φ))∑

rS=1 N s

∫

T exp

e

(−(λphysical+λbiological)t)⋅S(r T← RS;t)dt

(15) We can split this into four main components which effect dose (highlighted in colour here). DrT(T exp)=cf⋅(EM AC(r ,φ)−SCAC(r ,φ))∑

rS=1 N s

∫

T exp

e

(−(λphysical+λbiological)t)⋅

∑

i

EiY iφ(r T←rS; Ei;t) mrT(t) dt (16) The key parts of each expression are now highlighted in bold. DrT(T exp)=cf⋅ (EM AC(r ,φ)−SC AC(r ,φ))∑

rS=1 N s

∫

Texp

e

(−(λ physical+ λbiological)t)⋅

∑

i

EiY iφ(rT←r S;Ei;t) mrT(t) dt (17)

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