Friday, April 20, 2018

Listmania! IPMB


Amazon used to have a feature called Listmania! You could make a list of up to 40 books that was visible at Amazon's website. Ten years ago I created a Listmania! list related to Intermediate Physics for Medicine and Biology, reproduced below. Because the list is old, it does not include recent books (such as The Optics of Life) or books that I have discovered recently (such as The First Steps in Seeing). To learn about newer books, search this blog for posts labeled "book review." Amazon has discontinued Listmania!, but you can still find the lists if you look hard. I miss it.

If you are interested in what I read for pleasure, look here.

Enjoy!

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Intermediate Physics for Medicine and Biology

 

 


Bradley J. Roth
The list author says: "Books that are cited by the 4th Edition of Intermediate Physics for Medicine and Biology. These are some of the best biological and medical physics books I know of, and are books that have been useful to me during my career."
Intermediate Physics for Medicine and Biology, 4th Edition (Biological and Medical Physics, Biomedical Engineering)
Intermediate Physics for Medicine and Biology, 4th Edition (Biological and Medical Physics, Biomedical Engineering)
"All the books listed below are cited in the 4th Edition of Intermediate Physics for Medicine and Biology, written by Russ Hobbie and me." 
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Applied Mathematics Series 55]
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables [Applied Mathematics Series 55]
"A math handbook that has everything you'll ever need to know."
The 2nd Law: Energy, Chaos, and Form (Scientific American Library Paperback)
The 2nd Law: Energy, Chaos, and Form (Scientific American Library Paperback)
"I love this coffee table book about the second law of thermodynamics.  A painless way to introduce yourself to the subject."
Introduction to Radiological Physics and Radiation Dosimetry
Introduction to Radiological Physics and Radiation Dosimetry
"Classic in the Medical Physics field."
The Essential Exponential! (For the Future of Our Planet)
The Essential Exponential! (For the Future of Our Planet)
"This book explains why we devoted an entire chapter of Intermediate Physics for Medicine and Biology to the exponential function."
Physics With Illustrative Examples From Medicine and Biology: Mechanics (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Mechanics (Biological and Medical Physics, Biomedical Engineering)
"A classic textbook."
Physics With Illustrative Examples From Medicine and Biology: Electricity and Magnetism (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Electricity and Magnetism (Biological and Medical Physics, Biomedical Engineering)
"The second edition of the book has much the same content as the first, but the quality of the printing and illustrations is vastly improved."
Physics With Illustrative Examples From Medicine and Biology: Statistical Physics (Biological and Medical Physics, Biomedical Engineering)
Physics With Illustrative Examples From Medicine and Biology: Statistical Physics (Biological and Medical Physics, Biomedical Engineering)
"Benedek and Villars were pioneers in biological and medical physics textbooks."
Random Walks in Biology
Random Walks in Biology
"The best book about the role of diffusion in biology that I know of."
Foundations of Medical Imaging
Foundations of Medical Imaging
"Fine book to study imaging algorithms."
Introduction to Membrane Noise
Introduction to Membrane Noise
"Great book on a little-known topic."
Air and Water
Air and Water
"One of my favorites. Written by a physiologist with an interest in physics (as opposed to Hobbie and I, who are physicists interested in physiology)."
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles
"My favorite modern physics textbook."
The Feynman Lectures on Physics (3 Volume Set) (Set v)
The Feynman Lectures on Physics (3 Volume Set) (Set v)
"What physics list could be complete without Feynman?"
From Clocks to Chaos
From Clocks to Chaos
"Excellent book to learn the biological and medical applications of chaos."
The Machinery of Life
The Machinery of Life
"Wonderful picture book.  Great way to visualize the relative sizes of biological objects."
Bioelectricity and Biomagnetism
Bioelectricity and Biomagnetism
"Good, thick tome on bioelectricity."
Textbook of Medical Physiology
Textbook of Medical Physiology
"The classic physiology textbook."
Radiobiology for the Radiologist
Radiobiology for the Radiologist
"Great place to learn about the biological effects of radiation."
Medical Imaging Physics
Medical Imaging Physics
"Standard textbook in medical physics. Hendee is a pioneer in the field."
Ion Channels of Excitable Membranes, Third Edition
Ion Channels of Excitable Membranes, Third Edition
"The bible for information about ion channels."
Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care
Machines in Our Hearts: The Cardiac Pacemaker, the Implantable Defibrillator, and American Health Care
"Learn about the history of pacemakers and defibrillators."
The Physics of Radiation Therapy
The Physics of Radiation Therapy
"The place to go to learn about radiation therapy."
Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields
Bioelectromagnetism: Principles and Applications of Bioelectric and Biomagnetic Fields
"Fine textbook on bioelectricity."
Powers of Ten (Revised) (Scientific American Library Paperback)
Powers of Ten (Revised) (Scientific American Library Paperback)
"Classic work describing how the world looks at different length scales. Required reading by anyone interested in science."
Electric Fields of the Brain: The Neurophysics of EEG,  2nd Edition
Electric Fields of the Brain: The Neurophysics of EEG, 2nd Edition
"Great way to learn about the physics of the electroencephalogram."
Bioelectricity: A Quantitative Approach
Bioelectricity: A Quantitative Approach
"Standard textbook for a class in bioelectricity."
Numerical Recipes 3rd Edition: The Art of Scientific Computing
Numerical Recipes 3rd Edition: The Art of Scientific Computing
"My go-to book on numerical methods."
Electricity and Magnetism (Berkeley Physics Course, Vol. 2)
Electricity and Magnetism (Berkeley Physics Course, Vol. 2)
"Best introduction to electricity and magnetism I know. Part of the great Berkeley Physics Course."
Statistical Physics: Berkeley Physics Course, Vol. 5
Statistical Physics: Berkeley Physics Course, Vol. 5
"Great intuitive introduction to statistical mechanics.  Part of the Berkeley Physics Course."
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth Edition)
Div, Grad, Curl, and All That: An Informal Text on Vector Calculus (Fourth Edition)
"Need a little review of vector calculus? This is the place to find it."
Scaling: Why is Animal Size so Important?
Scaling: Why is Animal Size so Important?
"Great book on biological scaling."
How Animals Work
How Animals Work
"Great physiology book. Quirky, but fun."
Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)
Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in Nonlinearity)
"Best book for a first course in nonlinear dynamics."
Life in Moving Fluids: The Physical Biology of Flow (Princeton Paperbacks)
Life in Moving Fluids: The Physical Biology of Flow (Princeton Paperbacks)
"Best book I know of on biological fluid dynamics. Not too mathematical, but full of insight. I recommend all of Vogel's books."
Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems
Vital Circuits: On Pumps, Pipes, and the Workings of Circulatory Systems
"Great for understanding the fluid dynamics of the circulatory system."
Lady Luck: The Theory of Probability (Dover Books on Mathematics)
Lady Luck: The Theory of Probability (Dover Books on Mathematics)
"I often find probability theory boring, but not this book. An oldie but goodie."
The Geometry of Biological Time (Interdisciplinary Applied Mathematics)
The Geometry of Biological Time (Interdisciplinary Applied Mathematics)
"Classic by Art Winfree, who was a leading mathematical biologists.  Be sure to get the 2nd edition."
When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias
When Time Breaks Down: The Three-Dimensional Dynamics of Electrochemical Waves and Cardiac Arrhythmias
"Winfree's classic on the nonlinear dynamics of the heart."
Cardiac Electrophysiology: From Cell to Bedside, 4e
Cardiac Electrophysiology: From Cell to Bedside, 4e
"Comprehensive reference on cardiac electrophysiology."

Friday, April 13, 2018

Blog to IPMB Mapping

One reason I write this blog is to help instructors who are teaching from Intermediate Physics for Medicine and Biology. The blog, however, is over ten years old, and there are more than 500 posts. Teachers may not be able to find what they need.

Help is here! I have prepared a mapping of the sections in IPMB to the weekly blog posts (see an excerpt below). You can find it here, or through the book website, or download the pdf (but the links might not work). Now an instructor teaching, say, Section 1.1 (Distances and Sizes) can find eight related posts. I will keep the file up-to-date as new posts appear.


Some posts, including many of my favorites, are not associated with a particular section; I did not include those. A few posts fit with two or three sections, and appear several times. The majority relate to a single section.

What do I write about most? Four sections in IPMB have ten or more related posts.
  • Section 9.10, Possible Effects of Weak External Electric and Magnetic Fields, 11 posts. Many of these posts debunk myths about the dangers of weak low-frequency fields.
  • Section 17.7, Radiopharmaceuticals and Tracers, 11 posts. Several posts discuss potential shortages of technetium.
  • Section 16.2, The Risk of Radiation, 19 posts. These posts are about radiation accidents, the “risk” of very low doses of radiation, and the linear-no-threshold model.
  • Section 7.10, Electrical Stimulation, 20 posts. This section reflects my research interests, with multiple posts describing pacemakers, defibrillators, and neural stimulation.
Which chapters have the most posts? In first place are Chapters 8 (Biomagnetism) and 16 (Medical Uses of X-Rays), each with 39. Tied for last are Chapters 3 (Systems of Many Particles) and 5 (Transport Through Neutral Membranes), each with only 11. I guess I don’t like to post about thermodynamics.

I hope this mapping from IPMB to the blog helps instructors use the textbook. Enjoy!

Friday, April 6, 2018

Radiobiology for the Radiologist

In Section 16.9 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the biological effects of radiation. We write
“This section provides a brief introduction to radiobiology, but it ignores many important details. For these details see Hall and Giaccia (2012).”
Radiobiology for the Radiologist, by Eric Hall and Amato Giaccia, is a leading graduate textbook in radiology and medical physics. It analyzes the “four Rs" of radiobiology:
  1. Repair: “If only one strand [of DNA] is broken, there are efficient mechanisms that repair it over the course of a few hours using the other strand as a template” (p. 481, IPMB).
  2. Reassortment: “Even though radiation damage can occur at any time in the cell cycle (albeit with different sensitivity), one looks for chromosome damage during the next M [cell division, or mitosis] phase” (p. 481-482, IPMB).
  3. Reoxygenation: “A number of chemicals enhance or inhibit the radiation damage…One of the most important chemicals is oxygen, which promotes the formation of free radicals and hence cell damage. Cells with a poor oxygen supply are more resistant to radiation than those with a normal supply” (p. 482, IPMB).
  4. Repopulation: IPMB doesn’t address this last “R” specifically, but it is the most obvious of the four: After a dose of radiation, surviving cells grow and divide, repopulating the tumor.
Hall and Giaccia’s Figure 6.13 summarizes the reoxygenation process (below I show a similar, open-access figure by Padhani et al., European Radiology, 17:861-872, 2007).


Reoxygenation is one reason why radiation is divided into small daily fractions rather than one large dose. Hall and Giaccia write
“A modest dose of x-rays to a mixed population of aerated and hypoxic cells results in significant killing of aerated cells but little killing of hypoxic cells. Consequently, the viable cell population immediately after irradiation is dominated by hypoxic cells. If sufficient time is allowed before the next radiation dose, the process of reoxygenation restores the proportion of hypoxic cells to about 15%. If this process is repeated many times, the tumor cell population is depleted, despite the intransigence to killing by x-rays of the cells deficient in oxygen.”
They later use the four Rs to summarize why fractions are important during radiotherapy.
“The basis of fractionation in radiotherapy can be understood in simple terms. Dividing a dose into several fractions spares normal tissues because of repair of sublethal damage between dose fractions and repopulation of cells if the overall time is sufficiently long. At the same time, dividing a dose into several fractions increases damage to the tumor because of reoxygenation and reassortment of cells into radiosensitive phases of the cycle between dose fractions.

The advantages of prolongation of treatment are to spare early reactions and to allow adequate reoxygenation in tumors. Excessive prolongation, however, allows surviving tumor cells to proliferate during treatment.”
I like the colorful figures in Hall and Giaccia's book. For instance, is that French ram wearing a beret?


For those planning to buy a copy of the 7th edition of Radiobiology for the Radiologist, I have news. The 8th edition will be published later this year! Hang on for a few more months, and then purchase the new edition.

Friday, March 30, 2018

The Radiation Dose from Radon: A Back-of-the-Envelope Estimation

I like Fermi problems: those back-of-the-envelope order-of-magnitude estimates that don’t aim for accuracy, but highlight underlying principles. I also enjoy devising new homework exercises for the readers of this blog. Finally, I am fascinated by radon, that radioactive gas that contributes so much to the natural background radiation. Ergo, I decided to write a new homework problem about estimating the radiation dose from breathing radon.

What a mistake. The behavior of radon is complex, and the literature is complicated and confusing. Part of me regrets starting down this path. But rather than give up, I plan to forge ahead and to drag you--dear reader--along with me.
Section 17.12
Problem 57 1/2. Estimate the annual effective dose (in Sv yr-1) if the air contains a trace of radon. Use the data in Fig. 17.27, and assume the concentration of radon corresponds to an activity of 150 Bq m-3, which is the action level at which the Environmental Protection Agency suggests you start to take precautions. Make reasonable guesses for any parameters your need.
Here is my solution (stop reading now if you first want to solve the problem yourself). In order to be accessible to a wide audience, I avoid jargon and unfamiliar units.
One bequerel is a decay per second, and a cubic meter is 1000 liters, so we start with 0.15 decays per second per liter. The volume of air in your lungs is about 6 liters, implying that approximately one atom of radon decays in your lungs every second.

Radon decays by emitting an alpha particle. You don't, however, get just one. Radon-222 (the most common isotope of radon) alpha-decays to polonium-218, which alpha-decays to lead-214, which beta-decays twice to polonium-214, which alpha-decays to lead-210 (see Fig 17.27 in Intermediate Physics for Medicine and Biology). The half-life of lead-210 is so long (22 years) that we can treat it as stable. Each decay of radon therefore results in three alpha particles. An alpha particle is ejected with an energy of about 6 MeV. Therefore, roughly 18 MeV is deposited into your lungs each second. If we convert to SI units (1 MeV = 1.6 × 10-13 joule), we get about 3 × 10-12 joules per second.

Absorbed dose is expressed in grays, and one gray is a joule per kilogram. The mass of the lungs is about 1 kilogram. So, the dose rate for the lungs is 3 × 10-12 grays per second. To find the annual dose, multiply this dose rate by one year, or 3.2 × 107 seconds. The result is about 10-4 gray, or a tenth of a milligray per year.

If you want the equivalent dose in sieverts, multiply the absorbed dose in grays by 20, which is the radiation weighting factor for alpha particles. To get the effective dose, multiply by the tissue weighting factor for the lungs, 0.12. The final result is 0.24 mSv per year.
This all seems nice and logical, except the result is a factor of ten too low! It is probably even worse than that, because my initial radon concentration was higher than average and in Table 16.6 of IPMB Russ Hobbie and I report a value of 2.28 mSv for the average annual effective dose. My calculation here is an estimate, so I don't expect the answer to be exact. But when I saw such a low value I was worried and started to read some of the literature about radon dose calculations. Here is what I learned:
  1. The distribution of radon progeny (such as 214Po) is complicated. These short-lived isotopes are charged and behave differently than an unreactive noble gas like radon. They stick to particles in the air. Your dose depends on how dusty the air is.
  2. How these particles interact with our lungs is even more difficult to understand. Some large particles are filtered out by the upper respiratory track
  3. The range of a 6-MeV alpha particle is only about 50 microns, so some of the energy is deposited harmlessly into the gooey mucus layer lining the airways (see https://www.ncbi.nlm.nih.gov/books/NBK234233). Ironically, if you get bronchitis your mucus layer thickens, protecting you from radon-induced lung cancer.  
  4. The progeny and their dust particles stick to the bronchi walls like flies to flypaper, increasing their concentration.
  5. Filtering out dust and secreting a mucus layer reduces the dose to the lungs, while attaching the progeny to the airway lining increases it. My impression from the literature is that the flypaper effect dominants, and explains why my estimate is so low.
  6. The uranium-238 decay chain shown in Fig. 17.27 is the source of radon-222, but other isotopes arise from other decay chains. The thorium-232 decay chain leads to radon-220, called thoron, which also contributes to the dose.
  7. I am not confident about my value for the mass. The lungs are a bloody organ; about half of their mass is blood. I don't know whether or not the blood is included in the reported 1 kg mass. The radon literature is oddly silent about the lung mass, and I don't know how these authors calculate the dose without it. 
  8. I ignored the energy released when progeny beta-decay, which would cause a significant error if my aim was to calculate the absorbed dose in grays. But if I want the effective dose in sieverts I should be alright, because the radiation weighting factor for electrons is 1 compared to 20 for alpha particles. 
  9. The radon literature is difficult to follow in part because of strange units, such as picocuries per liter and working level months (see https://www.ncbi.nlm.nih.gov/books/NBK234224).
  10. Radon can get into the water as well as the air. If you drink the water, your stomach gets a dose. With a half-life of days, the radon in this elixir has time to enter your blood and irradiate your entire body.
  11. Does the dose from radon lead to lung cancer? That depends on the accuracy of the linear no-threshold model. If there is a threshold, then such a small dose may not represent a risk.
  12. If you want to learn more about radon, read NCRP Report 160, ICRP Publication 103, or BEIR VI. Of course, you should start by reading Section 17.12 in IPMB.
What do I take away from this estimation exercise? First, radon dosimetry is complicated. Second, biology problems are messy, and while order-of-magnitude estimates are still valuable, your results need large error bars.

Friday, March 23, 2018

95gTc and 96gTc as Alternatives to Medical Radioisotope 99mTc

In Chapter 17 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the most widely used radioisotope in nuclear medicine: 99mTc (technetium-99m). Previously in this blog (here, here, and here) I described the looming shortage of 99mTc. In a recent paper in the open access journal Heliyon (Volume 4, Article Number e00497, 2018), Hayakawa et al. review “95gTc and 96gTc as alternatives to medical radioisotope 99mTc.” I don’t know enough nuclear medicine to judge if 95gTc and 96gTc are realistic alternatives to 99mTc, but the idea is intriguing. Below I reproduce an abridged and annotated version of the introduction to this interesting paper (my comments are in italics and enclosed in brackets []). Enjoy!
“Various radioisotopes, such as 99mTc (half-life 6.02 h [hours]), 201Tl [thallium-201] (half-life 3.04 d [days]), and 133Xe [xenon-133] (half-life 5.27 d), are used for single-photon emission computed tomography (SPECT) in medical diagnostic scans. In particular, 99mTc has become the most important medical radioisotope at present… Over 30 commonly used radiopharmaceuticals are based on 99mTc [for example, 99mTc–sestamibi, 99mTc–tetrofosmin, and 99mTc-exametazime]... The 99mTc radioisotopes are supplied by 99Mo/99mTc generators, which continuously generate 99mTc through the β-decay of the parent nucleus 99Mo [molybdenum-99 is trapped in alumina (Al2O3) where it decays to pertechnetate (TcO4-); eluting solution flowing through the alumina collects the 99mTc]... This supply method provides two excellent advantages. First, it is possible to transport 99Mo/99mTc generators from a production facility to any place in the world because the half-life of 99Mo is... 2.75 d. Second, when a 99Mo/99mTc generator is transported to a hospital, 99mTc can be produced fresh for up to 2 weeks by daily milking/elution from this 99Mo/99mTc generator. At present, the parent nucleus 99Mo is produced in nuclear reactors by the neutron-induced fission of 235U [uranium-235] in highly enriched uranium (HEU) targets, in which the fraction of 235U is approximately 90%. However, some nuclear reactors that have supplied 99Mo require major repairs or shutdown [for example, the Chalk River reactor in Ontario, Canada], which may lead to a 99mTc shortage. Thus, many alternative methods to produce 99Mo or 99mTc [such as in a cyclotron] without HEU have been proposed…
The September 11th terrorist attacks in Washington D.C. [these attacks actually took place in New York City, at the Pentagon in Arlington County Virginia, and near Shanksville, Pennsylvania] in 2001 also affected medical radioisotope production from the viewpoint of the safeguards of nuclear materials. The control of fissionable nuclides such as 235U and 239Pu [plutonium-239] is important for the safeguards of nuclear materials… The International Atomic Energy Agency (IAEA) hopes to discontinue 99mTc production using HEU targets, which can be transmuted into nuclear weapons... In the near future, 99mTc will be supplied by nuclear reactors using LEU [low-enriched uranium] targets in addition to HEU. The Nuclear Energy Agency (NEA) reported the prediction that the 99Mo/99mTc supply will be larger than the world demand when the scheduled nuclear reactors using LEU start 99Mo production…

Because the Tc [technitium] chemistry is the same, all the radiopharmaceuticals based on 99mTc can, in principle, be applied to other Tc isotopes. There are five Tc isotopes with half-lives in the range from hours to days: 94mTc (half-life 52 m [minutes]), 94gTc (half-life 4.88 h), 95mTc (half-life 60 d), 95gTc (half-life 20 h), and 96gTc (half-life 4.28 d) [superscript "g" stands for ground state, whereas superscript "m" stands for metastable excited state]… The half-life of 96gTc (4.28 d) is long enough for worldwide delivery from a production facility and lengthy use of up to 2 weeks in hospitals. 95gTc (20 h) can also be transported to a wide area and used for 3–5 days in hospitals. Thus, 95gTc and 96gTc are candidates for alternative γ-ray emitters. However, the decay rates of 95gTc and 96gTc are lower than that of 99mTc by a factor of 3.3 and 17, respectively, because the decay rate of a radioisotope is inversely proportional to its half-life. This fact leads to the question of whether these isotopes can work as 99mTc medical radioisotopes.

In the current study, we present the relative γ-ray flux of these isotopes with simple assumptions. We also estimate the patient radiation does [dose] per Tc-labeled tracer using… PHITS [Particle and Heavy Ion Transport code System, a general purpose Monte Carlo particle transport simulation code]... Various nuclear reactions that are production methods of Tc isotopes, such as (p, n) reactions [in which a proton enters the nucleus and a neutron leaves it]…, deuteron [hydrogen-2]-induced reactions…, and 96Ru [ruthenium-96] (n, p)96gTc reactions…, were studied. We consider the production by the (p, n) reaction on an enriched Mo isotope. We also calculate the production rate using a typical PET [positron emission tomography] medical cyclotron [If you must make 96gTc in a cyclotron, why not make 99mTc in the cyclotron instead?]. Because the energies of decay γ-rays of these Tc isotopes are typically higher than 200 keV, they are not suitable for the traditional SPECT cameras. Thus, we also discuss the property of possible ETCC [Electron-Tracking Compton Camera] for high energy γ-rays.”

Friday, March 16, 2018

Another Analytical Example of Filtered Back Projection

Some people collect stamps, aged bottles of wine, or fine art. I collect analytical examples of tomographic reconstruction: determining a function f(x,y) from its projections F(θ,x’). Today I'll share my latest acquisition: an example of filtered back projection. I discussed a similar example in a previous post, but you can't have too many of these. This analysis illustrates the process that Russ Hobbie and I describe in Section 12.5 of Intermediate Physics for Medicine and Biology.

The method has two steps: filtering the projection (that is, taking its Fourier transform, multiplying by a filter function, and doing an inverse Fourier transform) and then back projecting. We start with a projection F(θ,x'), which in the clinic would be the output of your tomography machine.
This projection is independent of the angle θ, implying that the function f(x,y) looks the same in all directions. A plot of F(θ,x') as a function of x’ is shown below.
I suggest you pause for a minute and guess f(x,y) (after all, our goal is to build intuition). Once you make your guess, continue reading.


Step 1a: Fourier transform

The Fourier transform of the projection F(θ,x') is
with Sf(θ, k) = 0 (Russ and I divide the Fourier transform into a cosine part C and a sine part S, see Eq. 11.57 in IPMB). Cf(θ, k) is a function of k, the wave number (also known as the spatial frequency). I won’t show all calculations; to gain the most from this post, fill in the missing steps. Cf(θ, k) is plotted below as a function of k.

Step 1b: Filter

To filter, multiply Cf(θ, k) by |k|/2π (Eq. 12.39 in IPMB) to get the Fourier transform of the filtered projection Cg(θ, k)
with Sg(θ, k) = 0. The result is shown below. Notice that filtering removes the dc contribution at k = 0 and causes the function to fall off more slowly at large |k| (it's a high-pass filter).



Step 1c: Inverse Fourier transform

Next use Eq. 11.57 in IPMB to calculate the inverse Fourier transform of Cg(θ, k). Initially I didn’t think the required integral could be solved analytically. I even checked the best integral table in the world, with no luck. However, when I typed the integral into the WolframAlpha website, it gave me an answer.
The above screenshot above contains the cosine integral, Ci(z). After evaluating it at the integral's end points, using limiting expressions for Ci(z) at small z, and expending much effort, I derived the filtered projection G(θ,x')
A plot of G(θ,x') is shown below.

Step 2: Back projection

The back projection (Eq. 12.30 in IPMB) of G(θ,x') requires some care. Because f(x,y) does not depend on direction, you can evaluate the back projection along any radial line. I chose y = 0, which means the rotated coordinate x’ = x cosθ + y sinθ in the back projection is simply x’= x cosθ. You must examine two cases.

Case 1: |x| less than a

In this case, integrate using only the section of G(θ,x') for |x’| less than a.

Instead of integrating from θ = 0 to π, use symmetry, multiply by two, and integrate from 0 to π/2.
At this point, I was sure I could not integrate such a complicated function analytically, but again WolframAlpha came to the rescue.

(Beware: Wolfram assumes you integrate over x, so in the above screenshot x is my θ and b is my x.) The solution contains inverse hyperbolic tangent functions, but once you evaluate them at the end points you get a simple expression

Case 2: |x| greater than a

When |x| is greater than a, angles around θ = 0 use the section of G(θ,x') for |x’| greater than a and angles around θ = π/2 use the section for |x’| less than a.
The angle where you switch from one to the other is θ = cos-1(a/x). (To see why, analyze the right triangle in the above figure with side of length a and hypotenuse of length x.) The back projection integral becomes
If you evaluate this integral, you get zero.


Final Result

Putting all this together, and remembering that f(x,y) doesn’t depend on direction so you can replace x by the radial distance, you find
We did it! We solved each step of filtered back projection analytically, and found f(x,y).

I’ll end with a few observations.
  1. Most clinical tomography devices use discrete data and computer computation. You rarely see the analysis done analytically. Yet, I think the analytical process builds insight. Plus, it's fun.
  2. Want to check our result? Calculate the projection of f(x,y). You get the function F(θ,x') that we started with. To learn how to project, click here.
  3. Regular readers of this blog might remember that I analyzed this function in a previous post, where you can see what you get if you don't filter before back projecting. 
  4. I have a newfound respect for WolframAlpha. It solved integrals analytically that I thought were hopeless. In addition, it's online, free, and open to all.
  5. Most filtered back projection algorithms don’t filter using Fourier transforms. Instead, they use a convolution. I think Fourier analysis provides more insight, but that may be a matter of taste. 
  6. My bucket list includes finding an analytical example of filtered back projection when f(x,y) depends on direction. Wouldn’t that be cool! 
  7. Remember, there is another method for doing tomography: the Fourier method (see Section 12.4 in IPMB). Homework Problems 26 and 27 in Chapter 12 provide analytical examples of that technique.