CT works by rotating an X-ray tube about the patient, applying a narrow X-ray beam at various angles. The scanner measures X-ray attenuation, which depends on tissue composition in the path of the beam. CT is most suitable for bones, but can also visualize soft tissue with or without contrast agents. CT scans are versatile, and faster and more cost-effective than MRI.

Mathematically, a CT scanner reconstructs a 3-D image from the raw X-ray attenuation data. There are different options for reconstruction, but generally, an algorithm first applies the Radon transform to obtain projections, represented as a sinogram. Then, the algorithm applies techniques such as the inverse Radon transform or other iterative methods to generate a 3-D image. CT can be prone to poor contrast in soft tissue, and artifacts from patient motion, metal objects, or physical limitations of the X-ray detector.

MRI works by applying a strong magnetic field and radio frequency energy that disturb hydrogen nuclei within water molecules. The scanner then measures how long the hydrogen nuclei take to return to equilibrium. This relaxation time varies between different tissues, which helps to create an image. MRI is most suitable for imaging soft tissues such as the brain, spinal cord, organs, and muscle. Different MRI sequences such as T1-weighted, T-2 weighted, Flair, diffusion, and functional MRI images vary the magnetic and RF energies to focus on different tissues.

Mathematically, an MRI scanner generates 2-D slice images in the Fourier domain, or k-space. The scanner transforms the k-space images to the spatial domain to get a 3-D image volume. MRI can be prone to artifacts from acquisition noise, undersampling, and patient motion.

PET is a type of nuclear medicine imaging that works by tracking radioactive tracers, or radiotracers, in the body. The patient inhales, ingests, or receives an injection of the radiotracer before the imaging procedure. PET imaging can be used to diagnose cancer, using radiotracers that accumulate near metabolically overactive tumors. You can also use PET with inhaled radiotracers to measure lung function, which is useful for radiation treatment planning in lung cancer patients.

Mathematically, PET imaging is a 3-D reconstruction of energy emitted from the radiotracer. As the radiotracer decays, it releases positrons that collide with electrons. Each collision releases two gamma rays, which move in opposite directions. The PET scanner detects pairs of gamma rays, and uses algorithms such as filtered back projection and iterative reconstruction to calculate a 3-D image of the radiotracer distribution. PET imaging has low spatial resolution, and scanners often combine PET with a more detailed modality like CT or MRI.

Ultrasound works using a probe that emits ultrasound waves. The probe then measures the strength of reflected echo waves, which depends on tissue composition. Ultrasound is suitable for imaging soft tissue near the surface of the body. Often, clinicians use ultrasound to monitor fetal development during pregnancy, or to image the heart, breasts, and abdomen. You can also use Doppler ultrasound to measure blood flow through the heart and blood vessels.

Mathematically, an ultrasound scanner derives a 2-D image or video sequence from the reflected ultrasound waves. Ultrasound images are prone to speckle noise, and can have a limited field of view and resolution.

X-ray imaging works by measuring X-ray attenuation on a 2-D sensor array or a radiographic film. The patient lies between the X-ray source and the detector. X-ray imaging is fast and cost-effective compared to other modalities, and is a popular option for preliminary clinical diagnosis.

Microscopy uses microscopes, which magnify objects that cannot be seen with the naked eye. There are different types of microscopes, such as light, electron scanning, and fluorescence microscopes, that apply different techniques to visualize samples. For example, light microscopes use visible light and lenses to illuminate and magnify a sample, while fluorescent microscopes use specialized stains and fluorescent light sources to highlight specific molecules or cellular structures. Some microscopes capture 2-D images, while others capture 3-D stacks of image slices through a tissue. Microscopy is useful for detecting cancer cells, infections, or other pathogens in tissue samples from patients.