Fundamental Limits of Spatial Resolution in PET

In Chapter 17 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss positron emission tomography.

17.10 Positron Emission Tomography

If a positron emitter is used as the radionuclide, the positron comes to rest and annihilates an electron, emitting two annihilation photons back to back. In positron emission tomography (PET) these are detected in coincidence.

Anyone who has looked at PET images will be struck by their low spatial resolution. They provide valuable functional information, but little anatomical detail. Why?

“Fundamental Limits of
Spatial Resolution in PET,”
by William Moses.

In a 2011 article in Nuclear Instruments and Methods in Physics Research A (“Fundamental Limits of Spatial Resolution in PET,” Volume 648, Supplement 1, Pages S236–S240), William Moses analyses what factors contribute to PET spatial resolution.

Abstract: The fundamental limits of spatial resolution in positron emission tomography (PET) have been understood for many years. The physical size of the detector element usually plays the dominant role in determining resolution, but the combined contributions from acollinearity, positron range, penetration into the detector ring, and decoding errors in the detector modules often combine to be of similar size. In addition, the sampling geometry and statistical noise further degrade the effective resolution. This paper quantitatively describes these effects, discusses potential methods for reducing the magnitude of these effects, and computes the ultimately achievable spatial resolution for clinical and pre-clinical PET cameras.

Detector size

The most obvious limitation of spatial resolution comes from the detector size. Usually detection occurs in a scintillator crystal that converts a gamma ray to visible light, which is detected by a photomultiplier. The width of the scintillator limits the spatial solution of the image. A typical detector size is about 4 mm.

Positron range

A positron is emitted with an energy of about a million electronvolts. It then travels through tissue until it slows enough to capture an electron and and give off two 0.511 MeV photons. The range of the positron sets a limit to the spatial resolution. Different isotopes emit positrons with different energies. One of the most widely used isotopes for functional studies is ¹⁸F, which has a range of about half a millimeter. Most other common isotopes used in PET have longer ranges.

Acollinearity

If a positron and electron are at rest when they annihilate, they emit two 0.511 MeV photons. To conserve momentum, these photons must travel in opposite directions. If, however, the positron-electron pair has some kinetic energy when they annihilate, the photons are not emitted exactly in opposite directions. Usually they deviate from 180° by up to 0.25°. This translates into about one to two millimeters of blurring in typical detector rings.

Decoding

Decoding is complicated. Many PET devices have more scintillators than photomultipliers, so the photomultipliers take turns recording from different scintillators (a process called multiplexing). The PET scanner must then decode all this information, and this decoding process is not perfect. Moses estimates that decoding introduces an uncertainty of about a third of the detector width, or around a couple millimeters in spatial resolution.

Penetration

The 0.511MeV photons can penetrate into the ring of detectors used in a PET device, causing blurring. In the illustration below, if the source (green) contains an isotope that emits two photons, then for some angles those photons (red) are detected by a single detector, but for other angles (blue) they are detected by multiple detectors. 

An illustration based on Fig. 2 of
“Fundamental Limits of Spatial Resolution in PET,”
by William Moses, showing how penetration of a
photon into different detectors causes blurring.

Sampling Error

A detector ring is more sensitive to sources at some positions compared to others (see Moses’s Fig. 3 for an explanation). This effect tends to degrade by spatial resolution by about 25%.

 

If you add all these uncertainties in quadrature, you get a spatial resolution of about 6 mm. This is worse resolution that you would have for magnetic resonance imaging or computed tomography, which is why PET images look so blurry. They are often overlaid on an MRI (see Fig. 17.25 in IPMB).

If you decided to build a PET system with the best possible spatial resolution (regardless of complexity or cost), you could eliminate all of the sources of uncertainty except positron range and acollinearity, implying a spatial resolution of about 2 mm (worse for isotopes other than ¹⁸F). PET is never going to image small-scale anatomical detail.

Qualifying Exams

First page of the 2020 Physics
Qualifying Exam.

I have discussed the Oakland University Medical Physics PhD Qualifying Exam previously in this blog. It’s a series of three written exams over math, physics, and biology, plus an oral exam.

I’ve collected the written qualifying exams over the last ten years (2011-2020) in a single file that you’re welcome to download. These exams are broad but not deep. They cover material at a level similar to, or somewhat higher than, Intermediate Physics for Medicine and Biology. Anyone working at the intersection of physics with biology should master these topics. The exams allow you to hone your problem solving skills. If you’re a student interested in applying physics to physiology, or math to medicine, but are stuck at home and not able to attend classes because of the covid-19 pandemic, you might want to try solving the 300 problems in this collection. (Slightly less than 300, because there wasn’t a math exam in 2014 and because we sometimes repeated questions from previous years.) If you want to try some of the exams from 2010 or earlier, you can find them at https://sites.google.com/view/bradroth/home/medical-physics-graduate-program/qualifying-exams?authuser=0.

Sorry, but I don’t have written solutions for these exams. You can always email me (roth@oakland.edu) if you’re stuck.

Enjoy!

Boron Neutron Capture Therapy

In Chapter 16 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss boron neutron capture therapy.

Boron neutron capture therapy (BNCT) is based on a nuclear reaction which occurs when the stable isotope ¹⁰B is irradiated with neutrons, leading to the nuclear reaction (in the notation of Chap. 17)

Both the alpha particle and lithium are heavily ionizing and travel only about one cell diameter. BNCT has been tried since the 1950s; success requires boron-containing drugs that accumulate in the tumor. The field has been reviewed by Barth (2003).

Barth (2003)
J Neurooncol 62:1–5.

The citation is to an article by Rolf Barth of Ohio State University.

Barth, RF (2003) A Critical Assessment of Boron Neutron Capture Therapy: An Overview. Journal of Neuro-Oncology, Volume 62, Pages 1–5.

The abstract of this seventeen-year-old review states

Boron neutron capture therapy (BNCT) is based on the nuclear reaction that occurs when boron-10 is irradiated with neutrons of the appropriate energy to produce high-energy alpha particles and recoiling lithium-7 nuclei. BNCT has been used clinically to treat patients with high-grade gliomas, and a much smaller number with primary and metastatic melanoma. The purpose of this special issue of the Journal of Neuro-Oncology is to provide a critical and realistic assessment of various aspects of basic and clinical BNCT research in order to better understand its present status and future potential. Topics that are covered include neutron sources, tumor-targeted boron delivery agents, brain tumor models to assess therapeutic efficacy, computational dosimetry and treatment planning, results of clinical trails in the United States, Japan and Europe, pharmacokinetic studies of sodium borocaptate and boronophenylalanine (BPA), positron emission tomography imaging of BPA for treatment planning, and finally an overview of the challenges and problems that must be faced if BNCT is to become a useful treatment modality for brain tumors. Clinical studies have demonstrated the safety of BNCT. The next challenge is an unequivocal demonstration of therapeutic efficacy in one or more of the clinical trails that either are in progress or are planned over the next few years.

I was wondering what’s happening in this field lately, so I searched the Physics World website and found a fascinating and recent article by Tami Freeman.

Boron neutron capture therapy (BNCT), a technique that deposits highly targeted radiation into tumour cells, was first investigated as a cancer treatment back in the 1950s. But the field remains small, with only 1700 to 1800 patients treated to date worldwide. This may be about to change.

“The field of BNCT seems to be progressing rapidly at the moment,” said Stuart Green, director of medical physics at University Hospital Birmingham. “The big difference compared with five or ten years ago is that the commercial interest from a variety of companies is strong now and this is driving the field…”

Speaking at the Medical Physics & Engineering Conference (MPEC), Green updated on the status of BNCT programmes worldwide, noting that clinical experience is continually increasing. The US Food and Drug Administration has now approved two boron drugs for clinical use. But by far the majority of treatments, over 1150 to date, have taken place in Japan, initially using the Kyoto University Reactor in the early 2000s, and more recently using three Sumitomo accelerator systems in Kyoto, Fukushima and Osaka.

“Very importantly, earlier this year we had the first ever medical device approval for BNCT, for treatment in Japan of recurrent head-and-neck cancer,” said Green. “This is a significant marker for the entire field....”

“For the first time, there’s a substantial and sustained effort in the commercial sector to drive this field forward,” Green concluded. “We should keep an eye on BNCT over the next few years, there’s a lot happening, and hopefully our community can play a key role.”

Why the renewed interest in this technique? First, the original clinical applications of BNCT used neutrons from a nuclear reactor. Now accelerator-based neutron sources are available that can be installed in a hospital. Second, researchers are working hard on boron-containing drugs. Currently, boronophenylalanine and sodium borocaptate are the most common drugs used clinically. Improving the delivery of these drugs, or designing entirely new drugs, could increase the usefulness of BNCT. 

We live in exciting times.

Boron Neutron Capture Therapy Animation. 

https://www.youtube.com/watch?v=cQrbfBiQpfk

Generation of Unidirectionally Propagating Action Potentials Using a Monopolar Electrode Cuff

Ungar et al. (1986) Ann.
Biomed. Engin., 14:437–450.

In Chapter 7 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss electrical stimulation of nerves. One of the end-of-the-chapter homework problems asks the students to

Design a stimulator that will result in one-way propagation… For an application of such [a device] during functional electrical stimulation, see Ungar et al. (1986).

The reference is

Ungar IJ, Mortimer JT, Sweeney JD (1986) “Generation of Unidirectionally Propagating Action Potentials Using a Monopolar Electrode Cuff,” Annals of Biomedical Engineering, Volume 14, Pages 437–450.

The abstract to their article states

Unidirectionally propagating action potentials, which can be used to implement transmission failure on peripheral nerve through “collision block,” have been generated electrically on cat myelinated peripheral nerve using a monopolar electrode cuff with the conductor positioned closest to the “arrest” end of the cuff. A single cathode located at least 5 mm from the arrest end resulted in unidirectional propagation with minimal current and charge injection. The range of stimulus current values that produced unidirectional propagation increased with increases in longitudinal asymmetry of cathode placement over the range of asymmetries tested (1.7:1 to 7:1). The stimulus current pulse that minimized charge injection was quasitrapezoidal in shape with a plateau pulse width of approximately 350 μsec and an exponential trailing phase having a fall time (90%–10%) of approximately 600 μsec. These parameters were found to be independent of cuff geometry. Arrest efficiency was not degraded using a cuff of sufficient internal diameter to prevent nerve compression in chronic implantation. The critical current density within the extracellular space of the electrode cuff required to produce conduction failure at the arrest end was estimated to be 0.47 ± 0.08 mA/mm².

I’ll explain how to make a one-way stimulator using the illustration below, adapted from Ungar et al.’s Figure 1.

The design of a one-way neural stimulator.
Adapted from Fig. 1 of Ungar et al. (1986).

The nerve (blue) is threaded through a cylindrical insulating cuff (red), which resembles a short segment of a plastic drinking straw. The cathode (green) is inside the cuff; it stimulates the action potential. The anode (not shown) is far away. Current (purple curves) comes out of the cathode and enters the nerve axons, depolarizing them. Once it reaches the end of the cuff the current spreads out as it returns to the distance anode. Current there leaves the axons, hyperpolarizing them, and lowering their transmembrane potential. The locations where the axons are hyperpolarized are labeled the virtual anodes.

The key to making a one-way stimulator is to place the cathode off-center in the cuff. Most of the current leaves the cuff through the end nearest the cathode (right end), so the current density is stronger there (the purple current lines are crowded together). Only a small fraction of the current leaves the cuff through the end farther from the cathode (left end), so the virtual anode is weaker there.

Depolarization under a cathode excites an action potential, which then propagates outward in both directions (left and right). If, however, the stimulus strength is strong enough, the hyperpolarization at a virtual anode can block propagation. If the current has the correct strength, the stronger virtual anode on the right will block propagation, while the weaker virtual anode on the left won’t. In that case, an action potential will propagate to the left (the escape end of the stimulator) but will not propagate to the right (the arrest end).

Ira Ungar and his collaborators were able to test their stimulator for different cathodal current strengths. For a very weak stimulus, the cathode is below threshold and no action potential is excited. For a moderately weak stimulus, the cathode excites an action potential that then propagates to both the left and the right; both virtual anodes are too weak to block propagation. For a moderately strong stimulus, the right virtual anode is strong enough to block the action potential and you have one-way propagation. For a very strong stimulus, both ends block propagation, and no action potential leaves the stimulator.

Why construct a one-way stimulator? Suppose you have a nerve that’s constantly firing action potentials, causing unwanted muscle contraction (spasticity). You could stop the downstream propagation of those action potentials by electrically stimulating action potentials further downstream. The stimulated action potentials propagating upstream will collide with the original action potentials propagating downstream, annihilating them (colliding action potentials can’t pass through each other because they’re each followed by a region of refracotoriness). This works great unless your stimulator sends its own volley of action potentials propagating downstream to excite the muscle. To avoid this problem, you need a one-way stimulator, so you only excite action potentials propagating upstream to block those causing the trouble, but none propagating downstream.

The senior author on this article was J. Thomas Mortimer, now emeritus professor in the Neural Engineering Center at Case Western Reserve University. He earned his PhD from Case and then spent his entire career there. He has developed an online Applied Neural Control Toolkit to teach how nerves work. I heard Mortimer speak during one of the Neural Prosthesis Workshops at the National Institutes of Health; he was inspirational.

Mortimer and his team’s development of a one-way stimulator is a classic example of how physics and engineering can contribute to medicine and biology.

The Death and Life of the Great Lakes

Many readers of this blog live far from me—sometimes in countries on the other side of the world—and may not be interested in local topics germane to Michigan. Yet the Great Lakes are important to everyone; they hold 20% of the earth’s liquid freshwater. The challenges faced by the Great Lakes aren’t unique; they’re relevant to other watersheds worldwide.

For those who aren’t Michiganders, a drawing of the five Great Lakes looks like a map of Michigan. 

The Great Lakes
From Phizzy at Wikipedia.

Between its upper and lower peninsulas, Michigan borders four of the five lakes. I live half way between the southern tip of Lake Huron and the western edge of Lake Erie. I’ve visited all five lakes, and I’m concerned about their welfare.

The Death and Life of the Great Lakes,
by Dan Egan.

 

I recently read The Death and Life of the Great Lakes (2017), by Dan Egan. (This book was recommended to me by Congresswoman Elissa Slotkin, who’s now running for reelection in my home district). I’ve lived in Michigan for 22 years, but much of this story was new to me. In his introduction, Egan writes

Iconic disasters have a history of prompting government action. Three years after the Cuyahoga River fire of 1969, Congress passed the Clean Water Act. Two decades later, when the Exxon Valdez ran aground and dumped 10.8 million gallons of crude oil into Alaska’s Prince William Sound, images of cleanup crews using paper towels to cleanse tarred birds helped press Congress into doing something it should have done years earlier. It mandated double-hulled oil tankers.

But the disaster unfolding today on the Great Lakes didn’t ignite like a polluted river or gush like oil from a cracked hull, and so far there is no galvanizing image of this slow-motion catastrophe, though a few come to my mind. One is the bow of an overseas ship easing its way into the first navigation lock on the St. Lawrence Seaway, the Great Lakes’ “front door” to fresh waves of biological pollution. Another is a satellite photo of a green-as-paint toxic algae slick smothering as much as 2,000 square miles of Lake Erie.

Yet another is the grotesque mug of an Asian carp, a monster-sized carp imported to the United States in the 1960s and used in government experiments to gobble up excrement in Arkansas sewage lagoons. The fish, which can grow to 70 pounds and eat up to 20 percent of their weight in plankton per day, escaped into the Mississippi River basin decades ago and have been migrating north ever since. There are now mustering at the Great Lakes’ “back door”—the Chicago canal system that created a manmade connection between the previously isolated Great Lakes and the Mississippi basin, which covers about 40 percent of the continental United States. The only thing blocking the fish’s swim through downtown Chicago and into Lake Michigan is an electrical barrier in the canal—one that has a history of unexpected shutdowns.

The Chicago canal has also turned the Great Lakes’ ballast water problem [ships from the Atlantic Ocean having exotic species in their ballast water and then dumping it into the Great Lakes] into a national one, because there are dozens of invasive species poised to ride its waters out of the lakes and into the rivers and water bodies throughout the heart of the continent. Species like the spiny water flea, the threespine stickleback, the bloody red shrimp and the fishhook water flea. All organisms you probably haven’t heard of. Yet.

Few out West, after all, had ever heard of quagga mussels—until they tumbled down the Chicago canal and metastasized across the Mississippi basin and, eventually, into the arid West, likely as hitchhikers aboard recreational boats towed over the Rocky Mountains. The mussels have since unleashed havoc on hydroelectric dams, drinking water systems and irrigation networks in Utah, Nevada and California and the federal government estimates that if the mussels make their way into the Northwest’s Columbia River hydroelectric dam system they could do a half billion dollars of damage—per year.

Much of Egan’s book is about how invasive species interact with native animals. In Chapter 2 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the predator-prey problem modeled by the Lotka-Volterra equations. It is a useful toy model for understanding the interplay of two species, but it doesn’t explain the workings of a large and complex ecosystem.

The Great Lakes (at least the four that border Michigan) were once isolated from the Atlantic Ocean by Niagara Falls, a barrier than even the most determined fish could not surmount. The opening of the Saint Lawrence Seaway allowed a series of nonnative animals to enter the lakes. First came the sickening sea lamprey that nearly caused the native lake trout to go extinct. Next was an explosion of alewife (a river herring). Scientists introduced coho and Chinook salmon to prey upon the alewives. In each case, the ramifications of a new species were difficult to predict. Then zebra mussels and the quagga muscles arrived, and finally the round goby. It’s a horror story, and I suspect an invasion of the Asian carp is inevitable.

Besides these exotic species, the Great Lakes (especially Lake Erie) suffer from algae blooms triggered by fertilizer runoff. Add in climate change, which is causing the water level in the lakes to fluctuate, and you have an unstable and unhealthy ecosystem. 

I’ll give Egan the last word.

If we can close these doors to future invasions [of invasive species], we may give the lakes, and the rest of the country, time to reach a new equilibrium, a balance between what is left of the natural inhabitants and all the newcomers... And if we do this, then we can focus on the major problems that still plague the lakes, which include the overapplication of farm fertilizer that is helping to trigger the massive toxic algae outbreaks, the impact a warming globe is having on the lake’s increasingly unstable water levels and the need to protect lake waters from outsiders seeking to drain them for their own profit.

Dan Egan on The Death and Life of the Great Lakes at the 2018 L.A. Times Festival of Books. https://www.youtube.com/watch?v=jtq_cN1kPkg