Physical Principles of Imaging in Medicine,

Edition 1

Editors: By Adam W. Anderson, Charles F. Caskey, John C. Gore and Todd E. Peterson

Publication Date: 05 Oct 2026

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Medical Imaging: Mathematical and Physical Principles gives an introduction to the mathematical and physical concepts and principles that underlie modern radiological imaging techniques. It explains the basic phenomena and mathematical concepts that are central to the use of physical methods for image formation in biomedicine. By focusing on the physical and mathematical principles, it provides knowledge that will continue to hold true even as technological and other advances change the nature and detail of future medical imaging applications.Written as a pedagogic textbook explaining the principles rather than a technical manual describing imaging methods, Medical Imaging: Mathematical and Physical Principles is very suitable as a foundational reference for senior undergraduates, graduate students and researchers with a background in either biomedical engineering, engineering, applied physics, computer science, and mathematics. It will be a well-thumbed resource throughout a researcher’s career.

Key Features

Written by a top team of highly respected researchers who between them have taught medical imaging for over a hundred years!
Focuses on the physical and mathematical principles to provide everlasting foundational knowledge
Balanced coverage between the modalities reflecting the authors’ areas of expertise
Provides a motivational historical overview of the field
Closely ties the physical principles to biology
Includes instructors support such as solutions and slides of the figures
Provides computational exercises

About the author

By Adam W. Anderson, Professor of Biomedical Engineering and Radiology and Radiological Sciences, Vanderbilt University, USA; Charles F. Caskey, Associate Professor of Radiology and Radiological Sciences and Biomedical Engineering, Vanderbilt University, USA; John C. Gore, University Professor of Radiology and Radiological Sciences, Biomedical Engineering, Physics and Astronomy, and Molecular Physiology and Biophysics, Vanderbilt University, USA and Todd E. Peterson, Associate Professor of Radiology & Radiological Sciences, Vanderbilt University, USA

1. Introduction to Medical Imaging
1.1 Overview of medical imaging: historical development
1.2 Types of information in imaging
1.3 Examples of relationships between physical and biological changes
1.4 Overview of energy forms used for imaging: X-ray, CT, ultrasound, MRI, nuclear, other imaging
1.5 Problems

2. Mathematics for Imaging
2.1 The Fourier Transform: introduction; inverse Fourier Transforms; symmetry properties; examples of useful transforms; theorems concerning Fourier Transforms; two dimensional transforms; relations to other integral transforms; the discrete Fourier Transform
2.2 Sampling: band limited functions; the sampling theorem; aliasing and the Nyquist rate
2.3 Linear systems: introduction; transfer functions; point spread function; modulation transfer function
2.4 Random variables: introduction; examples - Poisson statistics; random fields
2.5 Problems
2.6 Appendix: Bessel functions: integral representations; recursion relations; relation to Bessels equation; simple integrals of Bessel functions

3. Image Quality
3.1 What is image quality?
3.2 Physical measures of quality - resolution, contrast, noise
3.3 Effects of Point Spread Function and Modulation Transfer Function on image quality
3.4 Figures of merit of image quality
3.5 Signal Detection Theory model of imaging
3.6 ROC analysis
3.7 Elementary digital image processing
3.8 Summary of factors affecting image quality
3.9 Human visual psychophysics + image perception
3.10 Problems

4. Reconstruction of Objects from Projections
4.1 Formation of projections
4.2 Reconstruction from projections
4.3 Central Slice Theorem
4.4 Conventional tomography
4.5 Back projection
4.6 Filtered back projection: the Radon Transform; band limited objects; discrete filters; physical interpretation, summary
4.7 Limitations in reconstruction from projections: incomplete data, statistical limitations and propagation of noise, effects of finite number of projections, artifacts
4.8 Radial projections and reconstructions
4.9 Algebraic and model based reconstructions
4.10 Problems

5. X-Radiographic Imaging; Image Formation
5.1 Overview of X-ray imaging: radiography, fluoroscopy, CT, digital radiography
5.2 Production and characteristics of X-rays
5.3 X-radiographic imaging: physical properties relevant for imaging
5.4 X-ray interactions with tissues: classical scattering, photoelectric effect, Compton effect
5.5 Origins of contrast in X-ray imaging; relation to tissue composition
5.6 Effects of scatter; spectral effects
5.7 X-ray contrast materials
5.8 Image formation in radiography: transfer functions for projection imaging, magnification, effect of focal spot, motion blurring
5.9 Image recording: PSF of films and screens; detector quantum efficiency and noise in X-ray images; model of array of photon counters, derivation of characteristic curve and effects; propagation of signal and noise in and image quality in digital radiography
5.10 Problems

6. Nuclear Imaging: Image Formation
6.1 Overview of nuclear medicine: radioactivity, radioisotopes, radiopharmaceuticals, detectors
6.2 Image formation in projection imaging; effects of collimators, countrate
6.3 Design of cameras
6.4 Image formation in SPECT and PET; effects of attenuation, scatter
6.5 Reconstruction theory
6.6 Sensitivity and resolution considerations in nuclear imaging
6.7 Problems

7. Nuclear Magnetic Resonance Imaging: Image Formation
7.1 Overview
7.2 Equilibrium and dynamics of spin systems
7.3 Image formation
7.4 Effects of relaxation
7.5 Classes of pulse sequences
7.6 Signal and noise ratio in MR imaging
7.7 Problems

8. Nuclear Magnetic Resonance Imaging: Design of Systems
8.1 The Bo magnet
8.2 Bo shims
8.3 Gradient coils
8.4 The radiofrequency coil
8.5 The NMR receiver and signal processing
8.6 Problems

9. Nuclear Magnetic Resonance Imaging: Physical Properties Relevant for Imaging [1] Relaxation
9.1 Classical description of relaxation in solutions
9.2 Relaxation by dipole-dipole interactions
9.3 Relaxation mechanisms in biological tissues
9.4 Anderson-Weiss theory
9.5 MRI contrast agents: paramagnetic relaxation effects, susceptibility effects
9.6 Chemical shift and J-coupling effects
9.7 Problems
9.8 Appendix: Simple quantum theory of spin 1/2 nuclei
9.8.1 States of a system of spins
9.8.2 Transitions between states
9.8.3 Quantum form of the Bloch equations
9.8.4 Quantum description of a spin echo
9.8.5 Quantum description of relaxation

10. Nuclear Magnetic Resonance Imaging: Physical Properties Relevant for Imaging [2] Motion and Flow
10.1 Effects of gradients, phase dispersion, q space
10.2 Effects of incoherent motion; diffusion, IVIM, turbulent flow
10.3 Spin tagging and tissue deformation
10.4 Time of flight effects, ASL, perfusion
10.5 Phase contrast effects
10.6 Problems

11. Ultrasound Imaging: Image Formation
11.1 Introduction
11.2 Overview of ultrasonic imaging
11.3 Physics of sound waves and vibrations
11.4 Image formation with ultrasound: transducers as sound radiators; sound fields from transducers; equivalent circuits of transducers; the impulse responses of a transducer
11.5 Sound propagation in uniform layers of tissue: homogeneous wave equation, reflection, refraction, behavior of pulses at smooth interfaces
11.6 Design of ultrasonic pulse echo imaging systems; limits on resolution
11.7 Image artifacts
11.8 Problems

12. Ultrasound Imaging: Physical Properties Relevant for Imaging
12.1 Characteristic Impedance, sound velocity, elastic properties of tissues
12.2 Inhomogeneous wave equation
12.3 Scattering of sound pulses: continuum description of backscattering; scattering from discrete scatterers, approaches to tissue characterization
12.4 Absorption: factors affecting absorption; wave equation with absorption
12.5 Non-linear waves; cavitation
12.6 Doppler effect: scattering from blood, signal processing and physics of flow measurements
12.7 Problems

14. Physiological Function from Imaging
14.1 Compartmental modeling and simple tracer kinetics
14.2 Measurement of blood flow and organ function using imaging:
14.2.1 Assessment of brain function
14.2.2 Assessment of cardiac function
14.3 Problem

Physical Principles of Imaging in Medicine,

Editors: By Adam W. Anderson, Charles F. Caskey, John C. Gore and Todd E. Peterson

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