Chapter 19: PET/SPECT & Emission Tomography

Welcome to the nuclear-medicine part. CT and MRI both image the body's structure. PET (positron emission tomography) and SPECT (single-photon emission computed tomography) image function: where an administered radioactive tracer accumulates and what metabolism is happening, measured as radiation emitted from the body. The device is different, but the reconstruction mathematics returns to the tools built for CT — and the MLEM of Chapter 8 finally shows its true form.

Emission vs transmission

CT measures photons transmitted through the body from an external X-ray source and images the attenuation coefficient $\mu$. In PET/SPECT the source is inside the body: we measure line integrals of the photons emitted by the tracer, i.e. the distribution of activity. Emission, as opposed to transmission.

Attenuation cannot be avoided here either. Photons emitted inside the body are attenuated by $\mu$ on their way out. Photons from deep locations arrive less often, so without correction the image sinks in the center. Ironically, the $\mu$ map needed for this correction is provided by CT. PET/CT scanners became widespread precisely because they capture function (PET) and structure plus attenuation correction (CT) at once.

Coincidence detection and the LOR

PET fixes direction with a clever trick. A positron emitted by the tracer (FDG, say) annihilates with a nearby electron and sends out two 511 keV photons in exactly opposite directions. If two detectors on the ring register photons within a very short time window (a coincidence), the annihilation lies on the line connecting them — the line of response (LOR). No collimator is needed to select direction, which gives PET its high sensitivity. SPECT detects single photons, so it selects direction mechanically with a lead collimator (at the cost of sensitivity).

Arrange the LOR counts by angle and position and you have a sinogram — the structure from Chapter 2 returns unchanged.

MLEM in its true form

Emission counts are inherently Poisson-distributed: the count in each LOR fluctuates, Poisson-like, around the value expected from the activity distribution. Finding the activity that maximizes this likelihood is MLEM (Maximum Likelihood Expectation Maximization). Chapter 8 introduced MLEM as a "simplified model applied to transmission CT," but MLEM was in fact born for emission tomography (Shepp & Vardi 1982). Here the multiplicative update is not an approximation but exactly the correct likelihood maximization.

$$x_j \;\leftarrow\; \frac{x_j}{\sum_i a_{ij}} \sum_i a_{ij}\, \frac{p_i}{\sum_k a_{ik} x_k}$$

Attenuation correction is achieved by folding the per-LOR attenuation factor $\exp(-\int \mu\, dl)$ into the system matrix $a_{ij}$.

Simulation: emission scan and attenuation correction

From an activity phantom (uniform background uptake plus two hot spots), an attenuated sinogram is formed and reconstructed with MLEM. Turn off "attenuation correction" and iterate: a cupping artifact appears because photons from deep locations are lost to attenuation. Turn it on and the uniform uptake flattens, restoring correct hot-spot quantification. This is why PET/CT uses the CT μ map.

Time of flight (TOF)

Time-of-flight measurement pushes PET resolution further. Measuring the difference $\Delta t$ in arrival times of the two photons reveals how far the annihilation lay from the center of the LOR. The position uncertainty is $\Delta x = c\,\Delta t / 2$ ($c$ is the speed of light); better timing resolution confines it to a narrower segment. Counts that non-TOF PET could only backproject uniformly along the whole LOR are concentrated into the correct segment, raising effective sensitivity (image quality).

Resolution limits, and the textbook's close

PET has physical resolution limits. The positron travels a few millimeters before annihilating (positron range), and the photon pair is not exactly 180° apart (non-collinearity); these remain no matter how good the hardware. Yet only PET/SPECT can quantitatively image functional information such as metabolism and receptor distribution.

This textbook started from CT's line integrals (the Radon transform) and traveled through Fourier reconstruction, iterative methods, compressed sensing, and deep learning, then to MRI's Fourier measurement, and finally to the emission counting of PET/SPECT. The physical quantity measured differs from device to device, but the skeleton of the inverse problem — recovering an image from limited measurements — and the tools that solve it (the Fourier transform, iteration, regularization, statistical models) are strikingly shared. Tomographic imaging is one piece of mathematics wearing many faces.

References

• Shepp LA, Vardi Y. Maximum Likelihood Reconstruction for Emission Tomography. IEEE Transactions on Medical Imaging 1, 113–122 (1982).
• Hudson HM, Larkin RS. Accelerated image reconstruction using ordered subsets of projection data. IEEE Transactions on Medical Imaging 13, 601–609 (1994).
• Ollinger JM, Fessler JA. Positron-emission tomography. IEEE Signal Processing Magazine 14, 43–55 (1997).
• Cherry SR, Sorenson JA, Phelps ME. Physics in Nuclear Medicine, 4th ed. Elsevier (2012).