It’s all Mathematics

Field Size Correction Factor (FSCF)

FSCF = S_cS_p

Collimator Scatter and Phantom/Patient Scatter

Field matched gap size

gap = depth / 2 * (Field1/SSD + Field2/SSD)

SAD = SSD for this procedure

Hinge Angle, Φ

θ = 90 - Φ/2

θ is wedge angle

CTDI calcs.. (CTDI, CTDI_w, CTDI_vol)

CTDI = DL/T

CTDI_w = (1/3)CTDI_center + (2/3)CTDI_periph

CTDI_vol = CTDI_w / pitch

number of field junctions for whole body

Junction # = Dose / (Tolerance - dose)

spinal cord tolerance = 4500 cGy

Sensitivity of a diagnostic test

TP / (TP + FN)

Sensitivity = True Positive Fraction

Specificity of a Diagnostic test

TN / (TN + FP)

Specificity = True Negative Fraction

Positive Predictive Value

TP / (TP + FP)

Negative Predictive Value

TN / (TN + FN)

Accuracy of a Diagnostic test

(TP + TN) / (TP + FN + FP + FN)

False Positive Fraction

FP / (FP + TN)

CNR (Contrast to Noise Ratio)

CNR = (N_s - N_bg) / σ_bg

Doppler shift

Δf = 2f_tvcosθ / c

f_t = transmitted freq

c = speed of sound

Image magnification

M = SID / SOD

SID = Source to image receptor distance

SOD = Source to object distance

Collimator Grid ratio

Grid ratio = grid height / interspace distance

Impedance of Quarter Wave Matching Layer (between skin and PZT)

Z_QWM = sqrt(Z_PZT* Z_skin)

Geometric mean of PZT and skin impedances

Linear Quadratic Model

S(D) = exp(-αD-βD²)

S(D) is surviving fraction given a dose

αD is probability of cell death due to “single hits”

βD² is probability of cell death due to “double hits” (sparsely ionizing)

Biologically Effective Dose (BED)

BED = nd[1 + d / (α/β) ]

n is number of fractions

d is dose per fractions

Equivalent Dose (EQD)

EQD_x = nd{ [d + (α/β)] / [x + (α/β)] }

x is dose per fraction

EQD₂ is most often used for fractionation schemes (“2 Gray per day”)

Bremsstrahlung efficiency

Z*E(9×10^-9)

convert Air Kerma Strength (U) from mCi

U = X_mCi ∗ Γ ∗ f

mg-Ra equivalent

mg-Ra equivalent = X_mCi ∗ (Γ_x / Γ_Ra)

Effective Diameter from rectangular field

sqrt(AP ∗ LAT)

Probability of counts (Poisson)

P(x) = μ^xe^-μ / x!

μ is mean

True count rate

n = m / (1 - mτ)

m is measured count rate

τ is dead time

relativistic energy

γmc² = KE + m_oc²

γ = 1 / sqrt(1 - β²)

β=v/c

Optical Density

OD = log(I_o/I_transmitted)

x-ray entrance skin dose

(kV)²(mA.s)

primary Barrier Transmission factor

transmission factor; B=Pd²/WUT

P is dose per week

d = dist from target to 0.3m beyond wall

W = workload, weekly dose @ isocenter

U = use factor, % of week beam is on

T = Occupancy,

Shielding Barrier thickness

#TVL= -log(B)

thickness = TVL₁ + (#TVL - 1)TVL_e

TVL₁ is 1st TVL

TVL_e is equilibrium thickness TVL for subsequent TVLs

Secondary barrier transmission factor

B_s = [P(d_sec)²(d_sac)²] / αWT ∗ [400 / field size area]

α is scatter fraction

d_sec is d from scattering surface to protected point

d_sac is d from target to scattering surface

U = 1 b/c scatter is isotropic!!!!

Head leakage transmission factor

B_L = 1000P(d_sec)² / WT

U is 1 b/c scatter is isotropic!!!

Compton Scatter photon energy

hv’ = hv / [1 + (hv / 0.511MeV)(1-cosθ)]

decibels

dB = 10 log(I₂/I₁)

Effective Half life

T_e = T_pT_b / (T_p + T_b)

λ_E = λ_b + λ_p

ultrasound far field divergence angle

sinθ = 1.22(λ/d)

d is transducer diameter

ultrasound near-field length

near field length = d² / 4λ

d is transducer diameter

ultrasound Reflection coefficient?

Transmission?

R = [(z₂ - z₁) / (z₂ + z₁)]²

ultrasound axial resolution

axial res = SPL / 2

Spatial Pulse Length

Half Value Thickness (for ultrasound)

HVT = 3 dB / [attenuation coefficient (usually in dB/cm)]

TERMA

T = Ψ(μ/ρ)

KERMA

K = Ψ(μ_tr/ρ)

Collision KERMA

K_coll = Ψ(μ_en/ρ)

Radiative KERMA

K_rad = K_coll(g/1-g)

Dose under CPE and Transient CPE

CPE: D = K_coll = Ψ(μ_en/ρ)

TCPE: D = K_coll ∗ β

Deviation Index

DI = 10log(EI/EI_target)

EI is exposure index

Dose for photons? Electrons?

photons: D = Ψ(μ_en/ρ)

electrons: D = φ(S/P)

φ is partial fluence (# elec), (S/P) is mass collisional spring stopping power

Bragg Gray equation

D_med/D_gas = (S/ρ)_med/(S/ρ)_gas

(S/ρ) is mass collisional stopping power

Spencer-Attix Cavity theory equation

D_med/D_gas = (L/ρ)_med/(L/ρ)_gas

Restricted stopping power ratio

Burlin Cavity Theory

Dmed/Dgas = d(ratio of restricted stopping powers) + (1-d)(μ_en/ρ)_med/(μ_en/ρ)_gas

d is parameter related to size of cavity → d=1 if small. d=0 if large

TG-51 Dose equation?

D_w,Q=M_raw∗(N_Co-60/N_D,w)∗K_Q∗P_TP∗P_elec∗P_ion∗P_pol

D_w,Q is dose to water for beam quality Q

M_raw is raw chamber reading

N ratio is cal factor in a Co-60 beam in water

P_elec is cal factor for different electrometers

K_Q relates chamebr response in C0-60 to your beam of quality Q (K_Q has all the corrections from the theoretical equation)

K_Q in TG-51?

K_Q = (Ratio of L/ρ for water to air ∗P_fl∗P_gr∗P_wall∗P_cel∗P_ion)_Q divided by (Ratio of L/ρ for water to air ∗P_fl∗P_gr∗P_wall∗P_cel∗P_ion)_Co-60

LET

LET = dE/dx

Probability of Coherent scattering

Z/E²

Probability of Photoelectric

(Z/E)³

Probability of Compton

1/E

Probability of Pair Production

Zln(E)

Half life? Mean life?

T_1/2 = ln2/λ

τ = 1/λ = 1.443 * T_1/2

Number of interactions in a unit mass?

proportional to (C/v)²

C is charge, v is velocity

(K_coll)_air

(K_coll)_air = X*(W_air/e)

How to calculate image size (MB)?

Pixel array x bit depth

eg. CT with 8 bit depth→ 512 × 512 × 8 = 2,097,152 bits = 262,144 bytes → (Convert) = 0.25 MB per image and ~60 slices = 15 MB per scan

What is Lower Limit of Detectability? Equation?

Minimum number of counts that can be reliably detected above background

LLD = 4.66σ_B + 3

→ only depends on background counts

Minimum Detectable Activity

Min # of disintegrations per unit time that can be detected

MDA = LLD / ε_iε_gt

geometrical efficiency? Solid Angle?

How many counts, of those incident on the detector, are counted

ε_g = Ω / 4π

Ω = πr²/d² is solid angle, d from detector, r radius of detector

Intrinsic Efficiency

ε_i = # counts / # incident

Depends on type of radiation and energy

Error for total counts, N, from recorded counts, C, and background, B?

σ_N = sqrt(σ_c² + σ_B²)

Sieverts Integral equation (for Brachy seed)

Pay attention to the terms here. θ₁ vs θ₂, ACTIVE LENGTH not seed length, etc..

Mean electron energy at patient surface?

E_o = 2.33R₅₀

Average electron energy at depth z?

E_z = E_o (1 - Z/R_P)

E_o = 2.33R₅₀

R_P = practical range

electron %DD

%DD = %I * ((L/p)_air^water)_dmax^d

%DD

%DD = D_d / D_dmax

TAR

TAR = Dose @ depth d in water / Dose @ d in air with buildup cap

TMR

TMR = Dose @ depth / Dose @ d_max

TMR is ~ as PDD but the chamber is stationary and the water level rises

TPR

TPR = Dose @ depth d / Dose @ depth d_ref

Chamber is stationary, water height varies

d_ref usually 10 cm

Patient Scatter factor

S_P = S_CP/S_C

Inverse Square Correction factor (PDD and TMR calcs)

ISC = (Reference distance / calculation point)²

Mayneords

Corrects measured %DD’s at different SSDs.

“Old and deep” (old SSD+d) times “new and shallow”, over the opposite and squared

%DD to TMR equation

TAR to TMR

TMR = TAR / BSF

BSF is backscatter factor = TAR @ d_max

TMR to TPR

TPR = TMR_d / TMR_dref

TMR_drefis TMR value for TPR depth d_ref (usuallyt 10 cm)

TMR to %DD

PDD = TMR [(SSD + d_max) / SAD]²

Electron applicator and cutout factors (AF and CF)

AF = Output with applicator / output without applicator

CF = Output with cutout / output without cutout

Given Dose (GD) for electrons

GD = Tumor Dose / Treat-to

delivered at d_max

MU calcs

MU = Dose / factors

use %DD for SSD setups, and TXR for SAD setups

equivalent square field

a = 4A/P

How to match divergence of cranial field with upper spinal field?

rotate cranial field collimator by θ = arctan [(L/2) / SAD]

L is field size along spine

How to match divergence of cranial fields (Parallel opposed fields)? 2 ways

1. Half beam block → no divergence

2. Rotate (“kick”) the couch through θ = arctan[(F/2) / SAD] in the inferior/superior direction

F is cranial field size

End Effect correction time factor?

tau = (R_n - R₁)t / (nR₁ - R_n)

R1 is taken with long exposure for time t

Rn is take with series of exposures (of equal times) that sum to time t

Subtract correction time, tau, from treatment time (tau is a positive value)

Beam Flatness

Flatness = (D_max - D_min) / (D_max + D_min) x100

P_ion

P_pol

abs[(M⁺ - M^-) / 2M]

P_rp

P_rp = 1/avg normalized profile value at reference depth

flat fields: P_rp = 1

FFF fields: P_rp > 1

Effective Path Length

Eff Path Length = L∗(ρ_tissue / ρ_water)

Standard Uptake Value

SUV = tissue radiation concentration / body injection concentration

i.e. (kBq per gram of desired tissue) / (kBq per gram of whole body at time of injection)