My Final Question

Suppose I died tomorrow. When my soul came before that great scientist in the sky, she might say “because of your contributions to Intermediate Physics for Medicine and Biology, I’ll answer one question for you. What is your question?” I can think of many things I would like to know, but the question I’d ask is: “is the linear no-threshold model appropriate at low doses?”

What’s the linear no-threshold model, and why’s it so important? Russ Hobbie and I explain it in Chapter 16 of IPMB.

In dealing with radiation to the population at large, or to populations of radiation workers, the policy of the various regulatory agencies has been to adopt the linear no-threshold (LNT) model to extrapolate from what is known about the excess risk of cancer at moderately high doses and high dose rates to low doses, including those below natural background.

If the excess probability of acquiring a particular disease is αH in a population N [where H is the equivalent dose per person in sieverts and α is a proportionality constant], the average number of extra persons with the disease is

                 m = α N H.         (16.42)

The product NH, expressed in person Sv, is called the collective dose. It is widely used in radiation protection, but it is meaningful only if the LNT assumption is correct. Small doses to very large populations can give fairly large values of m, assuming that the value of α determined at large doses is valid at small doses.”

Let me give you an example of why this question is so consequential. Should our society spend its time and money trying to reduce radon exposure in people’s homes? Radon is a radioactive gas that is produced in the decay chain of uranium. This noble gas can seep into basements, where it may be breathed into the lungs. The decay of radon and its progeny can cause lung cancer. However, the typical yearly dose from radon is very low, about 2 mSv. For an individual the resulting cancer risk is tiny, but if the linear no-threshold model is correct then when multiplied by the population of the United States (over 300,000,000) there can be tens of thousands of cancer deaths each year caused by radon. On the other hand, if a threshold exists below which there is no risk of cancer, then radon probably causes few if any cancer deaths. So, from a public health perspective, the answer to my question about the validity of the linear no-threshold model is crucial.

Other examples are

• The severity of low-dose, widespread exposure to radiation caused by a terrorist attack, such as a small amount of radioactivity dissolved in the water supply of a major city, 
• The hazard caused by low-dose x-ray backscatter scanners used at airports for security, 
• The danger from the release into the Pacific Ocean of minuscule amounts of radioactivity in water leftover from the Fukushima nuclear accident, or 
• The risks associated with storing radioactive waste from nuclear power plants in underground storage facilities.

All these situations have one thing in common: small individual doses to a large number of people. Public health officials need to know whether or not the collective dose is significant, and is it something we should spend our scarce resources trying to minimize.

On that fateful day I fear that even if she answers my last question, I won’t be able share it with you, dear readers (I don’t think they allow blogging down there). We’ll just have to examine the evidence available today.

Quackwatch

One goal of Intermediate Physics for Medicine and Biology is to provide readers with an understanding of the physics underlying biomedicine, so they can recognize and refute pseudoscientific ideas. For instance, in Chapter 9 of IPMB Russ Hobbie and I discuss the physics behind the discredited claim that weak, low frequency electromagnetic fields (ranging in frequency from 60 Hz powerline fields to cell phone radiowaves) are dangerous.

These days, with so much pseudoscience parading as fact, and with the United State’s Secretary of the Department of Health and Human Services being a leading proponent of anti-science nonsense, what we need is something to point out and refute all this quackery. What we need is Quackwatch.org. According Quackwatch’s mission statement, 

Quackwatch is a network of Web sites and mailing lists developed by Stephen Barrett, M.D. and maintained by the Center for Inquiry (CFI). Their primary focus is on quackery-related information that is difficult or impossible to get elsewhere. Dr. Barrett’s activities include:

• Investigating questionable claims

• Answering inquiries about products and services

• Advising quackery victims

• Distributing reliable publications

• Debunking pseudoscientific claims

• Reporting illegal marketing

• Improving the quality of health information on the Internet

• Attacking misleading advertising on the Internet 

For those of you who prefer social media, you can follow Quackwatch on Facebook and Twitter.

Quackwatch was established by Stephen Barrett, a retired medical doctor. These days he’s in his 90’s and deserves a rest after a lifetime of defending science. Unfortunately, there’s no rest for the weary; we need him now more than ever.

Five years ago, Quackwatch become part of the Center for Inquiry, another group that opposes pseudoscience malarkey and that publishes the Skeptical Inquirer magazine. I mention both Quackwatch and the Center for Inquiry in my book Are Electromagnetic Fields Making Me Ill?

Thank you Stephen Barrett for giving the world your wonderful website Quackwatch.org. I wish we all deserved it. 

 

Quackery: A History of Fake Medicine and Cure-alls. CBS Sunday Morning.

https://www.youtube.com/watch?v=G_3K0lFuvHQ

My 50+ Years of Antiquackery Activity with Stephen Barrett and William M. London. Center for Inquiry.

https://www.youtube.com/watch?v=EjYIjBae0wM

Sine and Cosine Integrals and the Delta Function

Trigger warning: This post is for mature audiences only; it may contain Fourier transforms and Dirac delta functions. 

In Chapter 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I examine some properties of Fourier transforms. In particular, we consider three integrals of sines and cosines. After some analysis, we conclude that these integrals are related to the Dirac delta function, δ(ω – ω’), equal to infinity at ω = ω’ and zero everywhere else (it’s a strange function consisting of one really tall, thin spike).

Are these equations correct? I now believe that they’re almost right, but not entirely. I propose that instead they should be 

You’re probably thinking “what a pity, the second three equations looks more complicated than the first three.” I agree. But let me explain why I think they’re better. Hang on, it’s a long story.

Let’s go back to our definition of the Fourier transform in Eq. 11.57 of IPMB

The first thing to note is that y(t) consists of two parts. The first depends on cos(ωt), which is an even function, meaning cos(–ωt) = cos(ωt). There’s an integral over ω, implying that many different frequencies contribute to y(t), weighted by the function C(ω). But one thing we know for sure is that when you add up the contributions from all these many frequencies, the result must be an even function (the sum of even functions is an even function). The second part depends on sin(ωt), which is an odd function, sin(–ωt) = – sin(ωt). Again, when you add up all the contributions from these many frequencies weighted by S(ω), you must get an odd function. So, we can say that we’re writing y(t) as the sum of an even part, y_even(t), and an odd part, y_odd(t). In that case, we can rewrite our Fourier transform expressions as

We should be able to take our expression for y_even(t), put our expression for C(ω) into it, and then—if all works as it should—get back y_even(t). Let’s try it and see if it works. To start I’ll just rewrite the first of the four equations listed above

Now for C(ω) I’ll use the third of the four equations listed above. In that expression, there is an integral over t, but t is a dummy variable (it’s an “internal” variable; after you do the integral, the result does not depend on t), so to keep things from getting confusing we’ll call the dummy variable by another name, t'

Next switch the order of the integrals, so the integral over t' is on the outside and the integral over ω is on the inside

Ha! There, inside the bracket, is one of those integrals were’re talking about. Okay, the variables ω and t are swapped, but otherwise it’s the same. So, let’s put in our new expression for the integral

The 2π’s cancel, and a factor of one half comes out. An integral containing a delta function just picks out the value where the argument of the delta function is zero. We get

But, we know that y_even(t) is an even function, meaning y_even(–t) equals y_even(t). So finally

It works! We go “around the loop” and get back our original function.

You could perform another calculation just like this one but for y_odd(t). Stop reading and do it, to convince yourself that again you get back to where you started from, y_odd(t) = y_odd(t).

Now, you folks who are really on the ball might realize that if you had used the old delta function relationships given in IPMB (the first three equations in this post), they would also work. (Again, try it and see.) So why use my fancy new formulas? Well, if you have an integral that adds up a bunch of cos(ωt), you know you’re gonna get an even function. There’s no way it can be equal to δ(ω – ω’), because that function is neither even nor odd. So, it just doesn’t make sense to say that summing up a bunch of even functions gives you something that isn’t even. In my new formula, that sum of two delta functions is an even function. The same argument holds for the integral with sin(ωt), which must be odd.

Finally (and this is what got me started down this rabbit hole), you often see the delta function written as

Jackson even gives this equation, so it MUST be correct. (For those of who aren’t physicists, John David Jackson wrote the highly regarded graduate textbook Classical Electrodynamics, known by physics graduate students simply as “Jackson.”)

In Jackson’s equation, i is the square root of minus one. So, this representation of the delta function uses complex numbers. You won’t see it in IPMB because Russ and I avoid complex numbers (I hate them).

Let’s use the Euler formula e^iθ = cosθ + i sinθ to change the integral in Jackson’s delta function expression to

Now use a couple trig identities, cos(A–B) = cosA cosB + sinA sinB and sin(A–B) = sinA cosB –cosA sinB, to get

This is really four integrals,

Then, using the relations between these integrals and the delta function given in IPMB (the first three equations at the top of this post), you get that the sum of these integrals is equal to

which is obviously wrong; we started with 2πδ and ended up with 4πδ. Even worse, do the same calculation for δ(ω + ω') with a plus instead of a minus in front of the ω'. I’ll leave it to you to work out the details, but you’ll get zero! Again, nonsense. However, if you use the integral relations I propose above (second set of three integrals at the top of this blog), everything works just fine (try it).

Gene Surdutovich, my new coauthor for the sixth edition of IPMB, and I are still deciding how to discuss this issue in the new edition (which we are hard at work on but I doubt will be out within the next year). I don’t want to get bogged down in mathematical minutia that isn’t essential to our book’s goals, but I want our discussion to be correct. Once the sixth edition is published, you can see how we handle it.

I haven’t seen my new delta function/Fourier integral relationships in any other textbook or math handbook. This makes me nervous. Are they correct? Moreover, Intermediate Physics for Medicine and Biology does not typically contain new mathematical results. Maybe I haven’t looked hard enough to see if someone else published these equations (if you’ve seen them before, let me know where…please!). Maybe I’ll find these results in Morse and Feshbach (another one of those textbooks known to all physics graduate students) or some other mathematical tome. I need to make a trip to the Oakland University library to look through their book collection, but right now its too cold and snowy (we got about four to five inches of the white stuff in the last 48 hours).

The Air They Breathe

I’m used to thinking about climate change from a physics perspective: what technologies can we use to reduce the amount of carbon dioxide and methane going into the atmosphere. Even when I consider the health effects of climate change, I tend to focus on the technical aspects (as you might expect from an author of a book titled Intermediate Physics for Medicine and Biology). Moreover, I often consider the long-term risks of climate change, and how it will harm future generations.

The Air They Breathe,
by Debra Hendrickson.

In her wonderful new book The Air They Breathe: A Pediatrician on the Front Lines of Climate Change, Debra Hendrickson has a different perspective. She explains how climate change is harming her young patients today. Specifically, she highlights four ways they are in danger.

1. Bad air from burning fossil fuels and from forest fires caused by climate change hurts children, particularly those with breathing problems like asthma. Here in Michigan, sometimes climate change seems a distant threat. But I remember the summer of 2023, when the air in the Detroit area was filled with smoke from fires in Canada. Hendrickson often makes her points by examples of specific children, such as a young girl named Anna, whose asthma was worsened by a forest fire burning near her home in Reno, Nevada in 2013. In The Air They Breathe, Hendrickson writes

Since Anna’s visit to my clinic that afternoon, thousands of other wildfires have raged through California, just a few miles to our west. They have grown bigger and more explosive, devouring not just forests, but towns. Every summer and fall now, waves of smoke pass through my city, and more of my young patients cough and wheeze. In 2018, the Mendocino Complex wildfire would become the largest California had ever seen, darkening the skies for weeks. Only two years later, in 2020, the August Complex fire would shatter that record, becoming the first to burn more than a million acres. And in 2021 we spent not just days breathing smoke, as we did in 2013, but months, as both the Dixie and Caldor fires raged a few miles away.

When I look back today, I see that the Rim Fire was not an isolated event, as it seemed to us then; it was the beginning of a trend. It was a sample of the world we are creating for our children.

2. Excessive heat can cause heatstroke in children, particularly in infants left in hot cars and high school football players who practice in the extreme heat. Children are especially sensitive to overheating. Heat waves can kill. Hendrickson tells the story of Joey Azuela, a child who almost died when hiking on a hot summer day near Phoenix, Arizona, saved only after being rushed to a hospital where he was covered with ice and injected with cold saline. She writes

Heatstroke is treated with extreme urgency; minutes make the difference between life and death. Joey Azuela is alive because he was cooled so quickly. Yet as the world watches temperatures climb, we drift, and delay; we risk pushing the planet to tipping points of rapid and uncontrollable changes, from which we cannot recover. The speed of our response is everything. It will determine not just the type of future our children have, but whether they have a future, at all.

3. Trauma and post traumatic stress disorder can occur in children who experience disasters caused by climate change, such as a hurricane, flood, or forest fire. Hendrickson examines in particular how in 2017 hurricane Harvey dumped as much as 40 inches of rain on Houston, Texas. One boy, Lucus, had to escape the rising water with his mother and siblings from their neighbor’s roof, saved by a passing boat. She writes

Natural disasters have always plagued us; the events themselves are nothing new. But a warming world is turning up their dial, and with it, the potential for trauma. Though some years are better than others, weather-related catastrophes are clearly trending worse over time: becoming more frequent, more powerful, and more destructive. Globally, natural disasters have increased fivefold over the last half century. Extreme weather events—the worst examples of these disasters, like 100-year floods and Category 4 hurricanes—are growing steadily more severe, and more common.

4. Infectious diseases, such as an increase in malaria caused by a greater range for mosquitoes, are becoming more common with global warming. Hendrickson tells us about Darah, an infant born in New Jersey who got the Zika virus from her mother while in the womb, and who suffered from microcephaly: an underdeveloped brain. She explains

To understand the connection between climate change and Darah’s case, we have to zoom out from her small New Jersey apartment and see that she shares this planet with trillions of other living things. That her body is linked to the Earth not just by water and air, but by a rich sea of organisms, friend and foe, living within and around her. Many of them are being affected by rising temperatures and shifting rains; by changes in habitats and seasons.

One point this book makes clear is the health care and climate change are not separate issues. The two are intertwined. Another point is that this is not merely a problem that we will all face in the coming decades. It’s happening now, as described by the horrific stories of these children. I found this book to be a call to action. It motivates me to make an even greater effort to address global warming, because—as Hendrickson warns us—“The only heroes our children have are us.”