BME 386: The Physics of Medical Imaging
Estimated study time: 10 minutes
Table of contents
Sources and References
Primary texts — Bushberg, Seibert, Leidholdt, and Boone, The Essential Physics of Medical Imaging, 4th ed. (Wolters Kluwer). Prince and Links, Medical Imaging Signals and Systems, 2nd ed. (Pearson).
Supplementary texts — Nishikawa, Medical Physics of Computed Tomography (Springer). Haacke et al., Magnetic Resonance Imaging: Physical Principles and Sequence Design, 2nd ed. (Wiley). Szabo, Diagnostic Ultrasound Imaging: Inside Out, 3rd ed. (Academic Press).
Online resources — MIT OCW 22.58J Principles of Medical Imaging and HST.582J Biomedical Signal and Image Processing. IAEA Diagnostic Radiology Physics: A Handbook for Teachers and Students (open). International Commission on Radiological Protection (ICRP) publications. AAPM Task Group reports (open).
Chapter 1: Waves and Matter
1.1 Electromagnetic and Mechanical Waves
Imaging modalities exploit different wave–matter interactions. X-rays (10¹⁶–10¹⁹ Hz) ionize atoms; RF radiation (10⁶–10⁹ Hz) couples to nuclear spin; visible light interacts through electronic transitions; ultrasound (10⁶–10⁷ Hz mechanical waves) propagates through tissue elasticity. Each modality’s physics determines its strengths, limitations, and safety profile.
1.2 Electricity and Magnetism
Maxwell’s equations govern EM interactions:
\[ \nabla\times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t},\qquad \nabla\times \mathbf{B} = \mu_0\mathbf{J} + \mu_0\varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} . \]From these follow wave propagation, reflection, refraction, and absorption. Medical imaging hardware — from X-ray tubes to MRI gradient coils — is engineered Maxwell physics.
1.3 Optics Foundations
Geometric optics — Snell’s law, lens equation, apertures — underlies microscopy and endoscopy. Wave optics — diffraction, interference, coherence — sets resolution limits. The diffraction limit \( d \approx 0.61\lambda/\mathrm{NA} \) constrains optical imaging to roughly the wavelength scale; super-resolution techniques (STED, PALM, STORM) bypass this limit through optical tricks and computation.
Chapter 2: X-ray Physics and Projection Imaging
2.1 X-ray Production
X-ray tubes accelerate electrons (typically 40–150 kVp) into a tungsten target. Two mechanisms produce X-rays: bremsstrahlung (continuous spectrum from decelerating electrons) and characteristic radiation (discrete lines from inner-shell transitions). Tube output spectrum is shaped by filtration, anode angle, and heel effect.
2.2 Attenuation
Photon transmission through tissue follows Beer–Lambert:
\[ I(x) = I_0 \exp\!\left(-\int_0^x \mu(x')\, dx'\right) , \]with linear attenuation coefficient \( \mu \) dependent on photon energy and tissue composition. Major interactions below 150 keV are photoelectric absorption (strongly \( Z^3/E^3 \) dependent) and Compton scattering (weakly Z-dependent). Photoelectric absorption provides the contrast differentiating bone (high Z) from soft tissue.
2.3 Detectors and Image Quality
Flat-panel detectors (direct conversion CsI:Tl + a-Si TFT; direct amorphous selenium) replaced film and CR. Image quality is quantified by spatial resolution (MTF), noise (NPS), and their ratio (DQE). Contrast-to-noise ratio determines detectability; ALARA dictates minimizing dose to achieve diagnostic quality.
Chapter 3: Computed Tomography
3.1 Projection and Reconstruction
CT measures line integrals of attenuation along projections through the patient. The Radon transform
\[ p(\theta, s) = \int_{-\infty}^{\infty} \mu(s\cos\theta - t\sin\theta, s\sin\theta + t\cos\theta)\, dt \]captures all projections. The Fourier slice theorem asserts that the 1D Fourier transform of a projection equals a radial slice of the 2D Fourier transform of \( \mu(x,y) \). Filtered back-projection and iterative reconstruction exploit this relationship.
3.2 Filtered Back-Projection
The FBP reconstruction is
\[ \mu(x,y) = \int_0^{\pi} p_{\text{filt}}(\theta, x\cos\theta + y\sin\theta)\, d\theta , \]with \( p_{\text{filt}} \) the ramp-filtered projection. Practical implementations add apodization windows (Hann, Shepp–Logan) to control noise. Iterative reconstruction (statistical, model-based) outperforms FBP at low dose by incorporating photon statistics and system models.
3.3 Modern CT
Helical, multi-slice, dual-energy, and photon-counting CT expand the design space. Dual-energy separates materials by their energy-dependent attenuation ratio; photon-counting enables spectral imaging without detector-based energy binning degradation.
Chapter 4: Magnetic Resonance Imaging
4.1 Nuclear Magnetic Resonance
A nucleus with spin in magnetic field \( B_0 \) precesses at the Larmor frequency
\[ \omega_0 = \gamma B_0 , \]with gyromagnetic ratio \( \gamma/2\pi = 42.58 \) MHz/T for ¹H. An RF pulse at \( \omega_0 \) tips magnetization into the transverse plane, where it precesses and induces signal in receiver coils. Relaxation returns the magnetization to equilibrium: longitudinal \( T_1 \) and transverse \( T_2, T_2^* \).
4.2 Spatial Encoding
Gradient coils generate position-dependent fields:
\[ B(\mathbf{r},t) = B_0 + \mathbf{G}(t)\cdot \mathbf{r} . \]Frequency and phase encoding map the signal to k-space; the image is the inverse Fourier transform of the k-space data. Pulse sequences (spin-echo, gradient-echo, EPI) balance speed, contrast, and artifact susceptibility.
4.3 Contrast
\( T_1 \)-weighted images highlight fat as bright, fluids as dark; \( T_2 \)-weighted invert this, making fluid bright. Diffusion-weighted imaging senses water motion, with apparent diffusion coefficient
\[ S = S_0 \exp(-b\,\mathrm{ADC}) , \]revealing cytotoxic edema in acute stroke within minutes. Functional MRI uses the BOLD signal from deoxyhemoglobin’s paramagnetism to map neural activity indirectly.
Chapter 5: Ultrasound
5.1 Acoustic Propagation
Ultrasound waves propagate at tissue-specific speeds (\( c \approx 1540 \) m/s average in soft tissue). Acoustic impedance \( Z = \rho c \); reflection coefficient at normal incidence
\[ R = \left(\frac{Z_2 - Z_1}{Z_2 + Z_1}\right)^2 . \]Attenuation increases with frequency (\( \approx 0.5 \) dB/cm/MHz in soft tissue), constraining penetration at high resolution.
5.2 Transducers and Beam Formation
Piezoelectric array transducers convert electrical pulses to acoustic pulses and vice versa. Phased arrays electronically steer and focus the beam by controlling element-specific delays. Axial resolution scales with pulse length; lateral resolution with aperture and focus.
5.3 Doppler and Modes
Doppler ultrasound measures blood flow via frequency shift \( f_d = 2 f_0 v \cos\theta/c \). B-mode (brightness), M-mode (motion over time), spectral Doppler, colour Doppler, and power Doppler serve complementary clinical questions. Harmonic imaging and contrast-enhanced imaging with microbubble agents extend capability.
Chapter 6: Optical Microscopy and Integrative Imaging
6.1 Bright-Field, Phase, and Fluorescence
Bright-field reveals absorption contrast; phase-contrast and differential-interference-contrast microscopy convert refractive-index differences to intensity. Fluorescence microscopy uses excitation/emission spectra and epifluorescence geometry. Confocal microscopy rejects out-of-focus light through a pinhole; two-photon microscopy extends depth by using low-energy photons whose nonlinear excitation localizes at focus.
6.2 Super-Resolution
STED shrinks the effective point-spread function by stimulated emission depletion. PALM/STORM exploit single-molecule localization with sequential activation. Structured-illumination doubles resolution through spatial frequency mixing. These methods routinely achieve 20–50 nm resolution, opening cellular machinery to direct imaging.
6.3 Multimodality and Image Fusion
Combined PET-CT, PET-MR, and SPECT-CT fuse anatomic and functional information. Co-registration — rigid or deformable — aligns datasets; display uses overlays or parametric maps. Quantitative imaging biomarkers require calibration, reproducibility studies, and bias correction before clinical use.
6.4 Safety
Each modality has its own safety profile: ionizing radiation (X-ray, CT, PET), static and RF field interactions (MRI — RF heating, gradient-induced nerve stimulation, projectile risk), acoustic heating and cavitation (ultrasound), photon flux (laser-based imaging). Regulatory limits (FDA, ICRP, IEC) translate physics into operational constraints. Engineers designing imaging hardware or protocols own responsibility for keeping operation within safe bounds.
