An Incomplete Medical Physics Review

These are just some of the equations that may be useful in medical physics.....but certainly not all of them!

Particle interactions and basic physics

• Wave-particle
  $c = \lambda \nu$
  $E = h \nu = hc / \lambda$
  $E(keV) = 1.24 / \lambda(nm)$
• Photoelectric effect
  $E_{pc} = E_{o} + E_{b}$
• Compton Scattering
  $E_o = E_{sc} + E_{e-}$
  $\lambda' - \lambda = \frac{h}{m_e c}(1-\cos{\theta})$
  $E_{sc} = \frac{E_o}{1+(E_o/m_e c^2)(1-\cos{\theta})}$
• $E_{CE} = \frac{2E^2_\gamma}{2E_\gamma+m_ec^2}$ (compton edge)
• Klein-Nishina formula
  $ {\frac {d\sigma }{d\Omega }}={\frac {1}{2}}\alpha ^{2}r_{c}^{2}P(E_{\gamma },\theta )^{2}[P(E_{\gamma },\theta )+P(E_{\gamma },\theta )^{-1}-\sin ^{2}(\theta )]$ where $P(E_{\gamma },\theta ) = E_{sc}/E_{o}$,
• Magnetic force
  F = $q v\times B$

Radioactivity

• Attenuation
  $dN/N = \mu \, dx$ → $ N = N_0 e^{-\mu x}$
• Half-value layer and mean free path
  $HVL = \ln(2)/\mu$
  $MFP = 1/\mu = 1/(ln(2)/HVL) = 1.44 HVL$
• Change in exposure from shielding
  $I = I_0 e^{(-\ln(2) \, x / HVL)}$ where $x$ is thickness of shielding
• Radioactivity: $A = -dN/dt = \lambda N$, integrate: $N(t) = N_0 e^{-\lambda t}$ or $A(t) = A_0e^{-\lambda t}$
  $\lambda = \ln(2)/T_{1/2}$
• Effective half-life (physical + biological): $T_e = \frac{T_p \cdot T_b}{T_p + T_b} = 1 / (1/T_p + 1/T_b)$
• Effective decay constant: $\lambda_e = \lambda_p + \lambda_b$
• Cumulated activity from initial activity: $\tilde{A} = A \times T_{mean} = A \times T_{1/2} \,/\, \text{ln}(2)$
• To find the daughter activity, one can use the Bateman equation:
  ${\displaystyle A_{d}=A_{P}(0){\frac {\lambda _{d}}{\lambda _{d}-\lambda _{P}}}\times (e^{-\lambda _{P}t}-e^{-\lambda _{d}t})\times BR+A_{d}(0)e^{-\lambda _{d}t},}$
  where $A_P$ and $A_d$ are the parents and daughter activities, $T_P$ and $T_d$ are the corresponding half-lives, and $\lambda_P$ and $\lambda_d$ are the corresponding decay constants. BR is the branching ratio.
• Transient equilibrium: Production rate and decay rate of daughter are equal.
  ${\displaystyle {\frac {A_{d}}{A_{P}}}={\frac {T_{P}}{T_{P}-T_{d}}}\times BR.}$
• Activity per unit mass (Bq/g)
  $a = \frac{\lambda N_A}{m} = \frac{ln(2) N_A}{T_{1/2} \times m}$ where N_A is the Avogadro constant.
• Saturation activity from production cross section and flux
  $A(t) = \Sigma \cdot \Phi \cdot (1-e^{-\lambda\,t})$
• Fraction of saturation from number of half-lives
  $f = 1 - (1/2)^n$ 
• Exposure rate = $\frac{\Gamma\, A}{d^2}$ where $A$ is the activity, $d$ is distance from source, and $\Gamma$ is the gamma constant, specific to each radionuclide.

Exposure and Kerma

• Exposure (R or C/kg)
  $X = dQ/dm$
• Kerma (Gy)
  $K = \frac{dE_{tr}}{dm} = \Psi(\frac{\mu_{tr}}{\rho}) =\Psi(\frac{\mu_{en}}{\rho}) / (1-g)$
• Exposure from Kerma
  $X = (K^{col})_{air} \cdot (\frac{e}{\bar{W}})$
• Dose from Kerma
  $D_{air} = (K^{col})_{air}  = X \cdot \frac{\bar{W}}{e}$ (conversion of R to cGy in air is 0.876)
• Bragg-Gray cavity theory
  $D_m = D_g \cdot (\frac{S}{\rho})^m_g = \frac{Q}{m_g} \cdot \frac{W_{air}}{e} \cdot(\frac{S}{\rho})^m_g$

Film/Optics

• Optical density of film
  $OD = \text{log}\frac{I_0}{I_t}$

Dose calculations (therapy)

• Correction for temperature and pressure (ion chamber)
  $P_{T,P} = 101.33 / P * (273.2 + T) / (273.2 + 22)$
• TG-51 absorbed dose to water
  $D^Q_w = M k_Q N^{^{60}Co}_{D,w}$
• Percentage depth dose (PDD)
  $\text{PDD}(d,r,SSD) = D_d / D_0 × 100%$
• Equivalent square
  $c = 4 \cdot A/P = \frac{2\,a\,\times\,b}{(a\,+\,b)}$
• Equivalent circle
  $r = 4/\sqrt{\pi} \cdot A/P$
• Mayneord F factor
  $F = \left(\frac{f_2+d_m}{f_1+d_m}\right)^2\left(\frac{f_1+d}{f_2+d}\right)^2$
• Tissue-air-ratio (TAR)
  $\text{TAR}(d,r_d) = D_d / D_{fs}$
• Scatter-air-ratio (SAR)
  $\text{SAR}(d,r_d) = TAR(r,d_r) - TAR(d,0)$ 
• Tissue maximum ratio
  $\text{TMR}(d,r_d) = (\frac{P(d,r,f)}{100}) (\frac{f+d}{f+d_{m0}})^2 (\frac{S_p(r_{m0})}{S_p(r_d)})$
• Monitory unit calculation
  $MU = \frac{D}{ D_{cal} \cdot S_c(r_c) \cdot S_p(r_d)  \cdot TPR(d,r_d) \cdot WF(d,r_d,x) \cdot  TF \cdot OAR(d,x) \cdot (SCD/SPD)^2 }$
• Adjacent fields separation
  $S = \frac{1}{2} \cdot L_1 \cdot\frac{d}{SSD_1} + \frac{1}{2} \cdot L_2 \cdot \frac{d}{SSD_2}$
• Electron field most probably energy at depth
• $(E_p)_z = (E_p)_0(1-\frac{z}{R_p})$
• Electron field most probably energy at depth
  $\bar{E_z} = \bar{E_0} (1-\frac{z}{R_p})$
• Range of heavier ions relative to proton range
  $R_{ion} = R_p \cdot M_{ion} / M_p \cdot z^2$
• Air kerma rate constant
  $\Gamma_\delta = \frac{l^2}{A}\left(dk_{air}/dt\right)$
• Air Kerma Strength
  $S_k = \dot{K_l}\cdot l^2 = \dot{X}\cdot (\bar{W}/e) \cdot l^2$

Internal dose

• Medical Internal Radiation Dose (MIRD)
  $D(r_T) = \sum_s{\tilde{A}_s(r_s) S(r_T ← r_s)} = \sum_s{\tilde{A}_s(r_s) \sum_i{\Delta_i \phi_i / m_T}}$

Shielding

• Transmission factor - primary
  $B = \frac{P \cdot d^2}{WUT}$
• Transmission factor - scatter
  $B_s = \frac{P}{\alpha W T} \cdot \frac{400}{F} \cdot d^2 \cdot d'^2$
• Transmission factor - leakage
  $B_l = \frac{P \cdot d^2}{0.001 WT}$

Counting

• Non-paralyzable count rate
  $T = M / (1 - M \, d)$
• Paralyzable count rate
  $M = T\,e^{-T d}$
• T = true, M = measured, d = dead time
• Chi-squared value:
  • $\chi^2 = \frac{\sum(n-\bar{n})^2}{\bar{n}}$ where $\bar{n}$ is the mean value and $n$ are the individual values

CT

• HU (CT number)
  $\text{HU} = 1000 \cdot \frac{\mu - \mu_w}{\mu_w - \mu_{air}}$

Ultrasound

• Speed of sound based on density bulk elastic modulus
  $C = (K/\rho)^{1/2}$
• Acoustic impedance
  $Z = \rho C = (K\rho)^{1/2}$ (kg/m²s = rayl)
• Relative intensity
  $\text{RI} = 10 \log(I_2/I_1) = 20 \log(A_2/A_1)$ (dB)
• Half value layer acoustics
  $I_2/I_1 = 0.5 \rightarrow 10 \log(0.5) = -3$ dB
• Reflection fraction from relative impedances
  $(Z_1 - Z_2)^2/(Z_1 + Z_2)^2$
• Snell's law
  $\sin(\theta_1)/\sin(\theta_2) = v_1/v_2 = n_2/n_1$
• Acoustic attenuation:
  $I = I_0 \exp(-\mu x)$ where $\mu = \alpha f$ in dB
  • $\alpha \approx 1$ db/cm/MHz for soft tissue.
• Spatial pulse length (SPL)
  $n\lambda$, cycles * wavelength
• Axial resolution
  $n \lambda/2 = SPL/2$  ($n$ is typically 3.)
• Doppler effect based on speed, speed of sound in medium, transducer frequency and angle of velocity wrt wave
  $f_{shift} = 2 \cdot (v/c) \cdot f_0 \cdot \cos(\theta)$
• Length of near field (Fresnel zone) relative to transducer diameter and wavelength
  $\text{L} = \frac{D^2}{4λ} = \frac{r^2}{\lambda}$
• Far field (Fraunhofer zone) divergence
  $\theta = 70 λ / D \text{(deg)} = 1.22 λ / D \text{(rad)}$

MRI

• Magnetic moment
  $\mu_z = \pm \frac{\gamma h}{4 \pi} $
• Excess spins based on Boltzman statistics
  $N_{anti}/N_{parallel} = \text{exp}(\frac{\Delta E}{kT}) = \text{exp}(\frac{\gamma h B_0}{2 \pi k T}) \approx 1 + \frac{\gamma h B_0}{2 \pi k T}$
• Larmor frequency
  $f_0 = \gamma B_0$
  hydrogen ¹H, $\gamma$ = 42.57 MHz/Tesla)
• Bloch equation
  $\frac{dM}{dt} = \gamma M x B$
• Flip angle
  $\theta = \gamma B_1 t$
• Longitudinal magnetization
  $M_z = M_0 (1-\text{exp}(-t/T1))$
• Transverse magnetization
  $M_{xy} = M_0 \text{exp}(-t/T2)$
• Spin-echo signal
  $S = M_0(1-e^{-TR/T1})e^{-TE/T2}$
• Ernst angle (flip angle that maximizes signal for a given TR)
  $\theta_E = arccos(e^{-TR/T1})$
• Magnetic susceptibility
  $B_0 \propto (1+\chi) H_0$

Imaging

• Detective Quantum Efficiency (DQE)
  $\text{DQE} = (\text{SNR}_{out}/\text{SNR}_{in})^2$
• Signal-to-Noise Ratio (SNR)
  $\text{SNR} = \mu_{sig} / \sigma_{bkgd}$
  $\text{SNR} = \mu_{sig} / \sqrt{\sigma_{bkgd}^2 + \sigma_{sig}^2}$
  $\text{SNR} = \sum_i{(x_i - \bar{x}_{bg})} / \sigma_{bkgd}$

• Contrast-to-Noise Ratio (CNR)
  $\text{CNR} = (\mu_{sig} - \mu_{bkgd}) / \sigma_{bkgd}$
  $\text{CNR} = (\mu_{sig} - \mu_{bkgd}) \,/ \sqrt{\sigma_{bkgd}^2 + \sigma_{sig}^2}$ 

Radiation biology

• Multitarget model
  $ln(n) = D_q / D_0$ 
• Linear-quadratic model surviving fraction
  $S = e^{-\alpha D - \beta D^2}$
• Dose to kill 90%
  $D_{10} = 2.3 \times D_0$
• Tumor control probability
  $\text{TCP} = e^{-(SF \times M)} = e^{-N} $ (SF is surviving fraction and M is number of clonogens, N is average # of surviving cells)
• Biologically equivalent dose
  $\text{BED} = nd(1+\frac{d}{\alpha/\beta})$
• Equivalent dose to 2 Gy/fraction
  $\text{EQD2} = D\frac{d+\alpha/\beta}{2+\alpha/\beta}$
• Adjacent field surface separation
  $S = \frac{1}{2} \cdot L_1 \cdot\frac{d}{SSD_1} + \frac{1}{2} \cdot L_2 \cdot \frac{d}{SSD_2}$
• Equivalent uniform dose for tissue with property $a$
  $\text{EUD} = \left(\sum_i v_i D_i^a\right)^{1/a}$

Constants and values to know

• h = planck's constant = 6.626e-34 J s = 4.136e-18 keV s
• 1 amu = 931.5 MeV = 1/12 mass of ¹²C
• Energy to create ion pair: $\bar{W}/e = 33.97$ J/C
• Avogadro's constant: $N_A ≡ 6.02214076×10^{23} mol^{-1}$
• barn (area/cross-section): 10⁻²⁸ m² 

Dose

• 10 mSv = 1 rem
• Absorbed Dose: 1 Gy = 1J/kg = 100 rads
• Equivalent Dose: 1 Sv = 100 rem