Physics of medical diagnosis and treatment. Four principal sub-fields: (i) radiotherapy physics — treatment planning, dosimetry, and quality assurance for tele- and brachytherapy (linear-accelerator beams, stereotactic radiosurgery,…
medical-physics
Beer-Lambert I(x) = I₀ exp(−μx): photoelectric + Compton + pair-production
Photon attenuation in matter under narrow-beam / good-geometry conditions follows the Beer-Lambert exponential law I(x) = I₀ · exp(−μ x),…
HVL = ln 2 / μ: thickness halving narrow-beam intensity
Half-value layer (HVL) is the thickness of absorber that halves the narrow-beam intensity under Beer-Lambert attenuation: I₀ · exp(−μ ·…
H = w_R · D: w_R(photon)=1, w_R(α)=20, w_R(neutron)=f(E)
Dosimetric quantities for radiation protection. Absorbed dose D (units gray, 1 Gy = 1 J/kg) is the physical energy deposited per unit mass…
μ·HVL = ln 2; TVL/HVL = log(10)/log(2) ≈ 3.32
Sympy-exact symbolic witness of the HVL / TVL relations. Step 1 — HVL from Beer-Lambert: I(HVL) = I₀ / 2 gives exp(−μ · HVL) = 1/2, thus μ…
exp(−μ·n·HVL) = 2^{−n}: 1/2, 1/4, 1/8 at n=1,2,3; 1/1024 at n=10
Sympy-exact symbolic witness of the multiplicative Beer-Lambert attenuation evaluated at integer multiples of the HVL. Setup: substitute μ…
w_R(α) = 20: 1 Gy α absorbed ⇒ 20 Sv equivalent
Sympy-exact symbolic witness of the ICRP 103 alpha-particle radiation-weighting factor w_R = 20 applied to a 1 Gy absorbed-dose scenario. …
Beer-Lambert exponential attenuation I(x) = I_0 e^{-mu x}; first-order linear ODE
Beer-Lambert exponential-attenuation framework (Bouguer 1729 Essai d'optique sur la gradation de la lumiere; Lambert 1760 Photometria; Beer…
Compton scattering: Delta lambda = (h / m_e c)(1 - cos theta); 4-momentum conservation
Compton scattering framework (Compton 1923 Phys Rev 21, 483; Debye 1923 Z Phys 24, 161). Setup: photon of wavelength lambda scatters…
Stejskal-Tanner DWI: S(b)/S_0 = e^{-bD}; heat-kernel of Laplacian on R^3
Stejskal-Tanner pulsed-gradient spin-echo framework (Stejskal-Tanner 1965 J Chem Phys 42, 288). Setup: in diffusion-weighted MRI (DWI), a…
Theorem: dI/dx + mu I = 0 for I(x) = I_0 e^{-mu x} (Beer-Lambert ODE residual)
Theorem (Beer-Lambert-ODE canonical): for I(x) = I_0 exp(-mu x), differentiation gives dI/dx = -mu I_0 exp(-mu x) = -mu I, hence dI/dx + mu…
Theorem: Delta lambda(theta = pi/2) - h/(m_e c) = 0 (Compton wavelength at 90 deg)
Theorem (Compton-perpendicular canonical): at scattering angle theta = pi/2, cos(pi/2) = 0, so Delta lambda = (h / m_e c)(1 - 0) = h / m_e…
Theorem: dS/db + D S = 0 for S(b) = S_0 e^{-b D} (DWI signal-decay ODE)
Theorem (Stejskal-Tanner-ODE canonical): for S(b) = S_0 exp(-b D), dS/db = -D S_0 exp(-b D) = -D S, so dS/db + D S = 0 identically.…
X-ray CT (Radon transform)
Hounsfield-Cormack 1972 (Nobel 1979) X-ray-CT: tissue mu(x,y) reconstructed from line-integrals; filtered-back-projection (FBP) inverts…
MRI (Bloch equations)
Lauterbur-Mansfield 1973 (Nobel 2003) MRI: nuclear-spin Bloch equations + spatially-varying gradient B = B0 + G dot r encode position in…
PET (positron emission)
Phelps-Hoffman 1973 PET: 18F-FDG injected; positron annihilation -> 511 keV gamma-pair coincidence detection; line-of-response geometry;…
Radiation therapy (LQ model + fractionation)
Fowler 1989 linear-quadratic model: SF = exp(-alpha D - beta D^2); fractionation 2 Gy x 30 = 60 Gy maximizes tumor-control vs…
Ultrasound imaging (acoustic impedance)
Wild-Reid 1952 medical-ultrasound: tissue acoustic-impedance Z = rho c; backscatter at impedance-mismatched interfaces; B-mode + M-mode +…
Photodynamic therapy (PDT)
Dougherty 1978 PDT: photosensitizer (porphyrin) absorbs visible-light -> triplet-state -> singlet-oxygen (1O2) cytotoxic; FDA approved 1995…
Rayleigh resolution (1879)
Rayleigh-criterion θ_min ≈ 1.22 λ/D; modern super-resolution STED Hell + STORM + PALM Nobel 2014.
CT (Hounsfield 1973)
G Hounsfield-A Cormack 1973 (Nobel 1979) X-ray CT; modern dual-energy + photon-counting + AI-reconstruction.
MRI (Lauterbur-Mansfield 1973)
P Lauterbur 1973 + P Mansfield 1977 (Nobel 2003) NMR-imaging; modern 7T whole-body + ultra-fast EPI + DTI.
IMRT (Bortfeld 1990)
T Bortfeld 1990 inverse-planning IMRT; modern VMAT + Tomotherapy + adaptive-radiotherapy MRI-Linac MRIdian / Elekta-Unity.
Bragg peak (Wilson 1946)
R Wilson 1946 protons-for-cancer; modern proton-therapy + carbon-ion + FLASH ultra-high-dose-rate Vozenin 2019.
Ultrasound (Curie 1880)
P-J Curie 1880 piezoelectric effect; modern medical ultrasound + Doppler + 3D + microbubble contrast + elastography.