Consider a system into which a small amount of heatQ flows. In many cases the temperature of the system rises [by an amount ΔT]… The heat capacityC of the system is defined as
C = Q/ΔT .
(3.39)
Heat capacity has units of J K-1. It depends on the size of the object and the substance it is made of. The specific heat capacity, c, is the heat capacity per unit mass (J K−1 kg−1).
The specific heat of air is 1006 J kg−1 K−1, a typical value for a gas… Water, however, … has a very high specific heat for a liquid… about 4200 J kg−1 K−1… It thus takes about four times as much heat to raise the temperature of a kilogram of water one degree as it does to raise the temperature of an equal mass of air.
A factor of four is significant, but frankly I would have expected an even bigger difference. The reason for the somewhat similar values of the specific heat capacity for air and water is that we are comparing heat capacity per unit mass. It may be more intuitive to compare heat capacity per unit volume. To convert from per kilogram to per cubic meter you must multiply the specific heat capacity per unit mass by the mass per unit volume, which is just the density, ρ. The density of air and water are very different. Air has a density of about 1.2 kg m−3, whereas water has a density of 1000 kg m−3.
We can express the specific heat capacity per unit volume as the product ctimes ρ. For air cρ is 1207 J K−1 m−3 but for water cρ is 4,200,000 J K−1 m−3. So water has a vastly higher specific heat capacity compared to air when expressed as per unit volume.
The relative similarity of specific heat between air and water can be misleading, however, because specific heat is measured on per-mass basis. One cubic meter of air weighs only about 1.2 kg, while a cubic meter of water weighs 1000 kg. It thus takes about 3500 times as much heat to raise the temperature of a given volume of water one degree as it does to raise the temperature of the same volume of air.
For similar volumes of air and water in thermal equilibrium, the heat stored in the air is negligible compared to that stored in the water. Biological tissue is mostly water, so that means air holds a lot less heat than tissue, per unit volume. This has implications for biological processes, such as heat exchange between air and tissue in the lungs.
The forces in the hip joint can be several times a person’s weight, and the use of a cane can be very effective in reducing them.
Indeed, a cane is very useful, as Russ and I show in Section 1.8 of IPMB. But what if you don’t have a cane handy, or if you prefer not to use one? You limp. In this post, we examine the biomechanics of limping.
When you limp, you lean toward the injured side to reduce the forces on the hip. The reader can analyze limping in this new homework problem.
Section 1.8
Problem 11 ½. The left side of the illustration below analyzes normal walking and reproduces Figures 1.11 and 1.12. The right side shows what happens when you walk with a limp. By leaning toward the injured side you reduce the distance between the hip joint and the body’s center of gravity, and your leg is more vertical than in the normal case.
Pertinent features of the anatomy of the leg: normal (left) and limping (right).
(a) Reproduce the analysis of Section 1.7 to calculate of the forces on the hip during normal walking using the illustration on the left. Begin by making a free-body diagram of the forces acting on the leg like in Figure 1.13. Then solve the three equations for equilibrium: one for the vertical forces, one for the horizontal forces, and one for the torques. Verify that the magnitude of the force on the hip joint is 2.4 times the weight of the body.
(b) Reanalyze the forces on the hip when limping. Use the geometry and data shown in the illustration on the right. Assume that any information missing from the diagram is the same as for the case of normal walking; For example, the abductor muscle makes an angle of 70° with the horizontal for both the normal and limping cases. Draw a free-body diagram and determine the magnitude of the force on the hip joint in terms of the weight of the body.
I won’t solve the entire problem for you, but I’ll tell you this: limping reduces the force on the hip from 2.4 times the body weight in the normal case to 1.2 times the body weight in the case of limping. No wonder we limp!
The main reason for the lower force when limping is the smaller moment arm. If we calculate torques about the head of the femur, then in the normal case the moment arm for the force that the ground exerts on the foot is 18 – 7 = 11 cm. When limping, this moment arm reduces to 9 – 7 = 2 cm. The moment arm for the abductor muscles (the gluteus minimus and gluteus medius) is the same in the two cases. Therefore, rotational equilibrium can be satisfied with a small muscle force when limping, although a large muscle force is required normally. The torque is a critical concept for understanding biomechanics.
What do you do if both hips are injured? When walking, you first lean to one side and then the other; you waddle. This reduces the forces on the hip, but results in a lot of swinging from side to side as you walk.
If you are having trouble solving this new homework problem, contact me and I’ll send you the solution.
Problem 5 ½. About 1 in every 2500 people is born with cystic fibrosis, an autosomal recessive disorder. What is the probability of the gene responsible for cystic fibrosis in the population? What fraction of the population are carriers of the disease?
To answer these questions, first we must know that an “autosomal recessive disorder” is one in which you only get the disease if you have two copies of a recessive gene. To a first approximation, there are often two variants (or alleles) of a gene governing a particular protein: dominant (A) and recessive (a). In order to have cystic fibrosis, you must have two copies of the recessive allele (aa). If you have only one copy (Aa), you are healthy but are a carrier for the disease: your children could potentially get the disease if your mate is also a carrier. If you have no copies of the recessive allele (AA) then you’re healthy and your children will also be healthy.
Let’s assume the probability of the dominant allele is p, and the probability of the recessive allele is q. Since we assume there are only two possibilities, we know that p + q = 1. Our goal is to find q, the probability of the gene responsible for cystic fibrosis in the population.
When two people mate, they each pass on to their offspring one of their two copies of the gene. The probability that both parents are dominant (AA), so the child is normal, is p2. The probability that both parents are recessive (aa), so the child has the disease, is q2. There are two ways for the child to be a carrier: A from dad and a from mom, or a from dad and A from mom. So, the probability of a child being a carrier (Aa) is 2pq. There are only three possibilities or genotypes: AA, Aa, and aa. The sum of their probabilities must equal one: p2 + 2pq + q2 = 1. But this expression is equivalent to (p + q)2 = 1, and we already knew that p + q = 1, so the result isn’t surprising.
The only people that suffer from cystic fibrosis have the genotype aa, so q2 is equal to the fraction of people with the disease. The problem states that this fraction is 1/2500 (0.04%). So, q is the square root of 1/2500, or 1/50 (2%; wasn’t that nice of me to make the fraction be the reciprocal of a perfect square?). One out of every fifty copies of the gene governing cystic fibrosis is defective (that is, it is the recessive version that can potentially lead to the disease). If q is 1/50, then p is 49/50 (98%). The fraction of carriers is 2pq, or 3.92%. The only reason this result is not exactly 4% is that we don’t count someone with the disease (aa) as a carrier, even though they couldpass the disease to their children (a carrier by definition has the genotype Aa).
If we are rounding off our result to the nearest percent, then 1 out of every 25 people (4% of the population) are carriers.
This calculation is based on several assumptions: no natural selection, no inbreeding, and no selection of embryos based on genetic testing. Cystic fibrosis is such a severe disease that often victims don’t survive long enough to have children (modern medicine is making this less true). The untreated disease is so lethal that one wonders why natural selection didn’t eliminate it from our gene pool long ago. One possible reason is that carriers of cystic fibrosis might be better able to resist other diseases—such as cholera, typhoid fever, or tuberculosis—than are normal people.
The most famous of medieval scientists was born in Somerset about 1214. We know that he lived till 1292, and that in 1267 he called himself an old man. He studied at Oxford under Grosseteste, and caught from the great polymath a fascination for science; already in that circle of Oxford Franciscans the English spirit of empiricism and utilitarianism was taking form. He went to Paris about 1240, but did not find there the stimulation that Oxford had given him…
Bacon is known for his support of the role of experiment in science. So much of medieval thought was based on religion and mysticism, and an emphasis on science and experiment is refreshing.
We must not think of him [Bacon] as a lone originator, a scientific voice crying out in the scholastic wilderness. In every field he was indebted to his predecessors, and his originality was the forceful summation of a long development. Alexander Neckham, Bartholomew the Englishman, Robert Grosseteste, and Adam Marsh had established a scientific tradition at Oxford; Bacon inherited it, and proclaimed it to the world. He acknowledged his indebtedness, and gave his predecessors unmeasured praise. He recognized also his debt—and the debt of Christendom—to Islamic science and philosophy, and through these to the Greeks…
Like Russ Hobbie and I, Bacon appreciated the role of math in science. Durant summarized Bacon’s view as “though science must use experiment as its method, it does not become fully scientific until it can reduce its conclusions to mathematical form.”
Bacon’s work on optics and vision overlaps with topics in IPMB. Durant notes that “one result of these studies in optics [performed by Bacon and others] was the invention of spectacles.” I can hardly think of a better example of physics interacting with physiology than eyeglasses. Durant concludes:
Experimenting with lenses and mirrors, Bacon sought to formulate the laws of refraction, reflection, magnification, and microscopy. Recalling the power of a convex lens to concentrate many rays of the sun at one burning point, and to spread the rays beyond that point to form a magnified image, he wrote:
We can so shape transparent bodies [lenses], and arrange them in such a way with respect to our sight and the objects of vision, that the rays will be refracted and bent in any direction we desire; and under any angle we wish we shall see the object near or at a distance. Thus from an incredible distance we might read the smallest letters…
These are brilliant passages. Almost every element in their theory can be found before Bacon, and above all in al-Haitham [an Arab scientist also known as Alhazen]; but the material was brought together in a practical and revolutionary vision that in time transformed the world. It was these passages that led Leonard Digges (d. c. 1571) to formulate the theory of which the telescope was invented.
I enjoy reading the Durants’ books. They contain not only the usual political and military history of the world, but also the history of science, art history, music history, comparative religion, linguistics, the history of medicine, philosophy, and literature. While The Story of Civilization may not be the definitive source on any of these topics, it is the best integration of all of them into one work that I am aware of. Had the Durants lived longer, future volumes (which they tentatively titled The Age of Darwin and The Age of Einstein) might have focused even more on the role of science in civilization.
I won’t finish The Story of Civilization anytime soon; I still have seven volumes to go. The series runs to over ten thousand pages, single-spaced, small font (I had to buy more powerful reading glasses for this project). I’ll continue to search for discussions of medical physics and biological physics throughout.
The Story of Civilization. 1. Our Oriental Heritage, 2. The Life of Greece, 3. Caesar and Christ, 4. The Age of Faith, 5. The Renaissance, 6. The Reformation, 7. The Age of Reason Begins, 8. The Age of Louis XIV, 9. The Age of Voltaire, 10. Rousseau and Revolution, and 11. The Age of Napoleon.
Just as early radiologists did not understand the dangers of high radiation doses, today we are naive to imaging’s carbon footprint and its implications for public health. The world’s temperature has already risen more than 1 °C from preindustrial levels. We see the effects of climate change across the world, from extreme wildfires and stronger storms to rising sea levels and ocean acidification. If we continue with “business as usual,” children born today will experience a planet that is 4 °C warmer than in preindustrial times and the associated health consequences. These consequences are disproportionately felt by children, the elderly, those with preexisting conditions, and outdoor workers. As our climate crisis worsens, radiologists must urgently consider our role in climate change.
According to Schoen et al., the health care system may be responsible for nearly ten percent of American’s greenhouse gas emissions. TEN PERCENT! Yikes. They suggest that radiology departments are “likely a major contributor to energy use within hospital systems.” They identify four ways to address the energy use in radiology.
Imaging Exams
Schoen et al. claim that “over a year, the energy use of one CT [computed tomography] scanner was comparable with that of 5 four-person households, and the energy use of one MR [magnetic resonance] scanner was close to that of 26 four-person households.” I always thought MRI was the ideal imaging method, but it turns out it’s an energy hog, contributing significantly to radiology’s carbon footprint. There are few easy ways to reduce energy use; perhaps use ultrasound more when appropriate and adopt new technologies that shorten imaging time.
Scanners in the Off State
Imaging systems use a lot of energy even in standby mode. You must keep the superconducting coil of a MRI scanner cold all the time, not just when it’s imaging. Solutions are not simple. Schoen et al. suggest using scanners 24 hours a day (patients won’t like that) and working with manufacturers to find ways of reducing energy use when a scanner is not operating.
Wasteful Habits
We have to cut the waste in radiology departments. Simple improvements would be to turn off computers and picture archiving and communication systems (PACSs) at night or when not in use, and reducing excess packaging. I support these easy changes, but wonder if they’ll have a major impact on our carbon footprint.
Energy Sources
Alternative energy sources—including ones like wind, solar, and nuclear—will reduce greenhouse gas emissions. This is something individual radiologists, or even radiology departments, have little control over, but if major health care systems demand cleaner energy sources they might be able to influence regional utilities and politicians.
Conclusion
Schoen, McGinty, and Quirk discuss an important issue, and I thank them for raising it. Their call to action must be addressed by radiologists in collaboration with hospital administrators, academic researchers, and medical device companies. All of us—including the past, present, and future patients needing radiological services—must advocate for reducing our impact on the climate.
I’ll give Schoen et al. the last word by quoting the eloquent final paragraph of their publication.
Radiology faces many challenges, from improving diversity to changes in reimbursement in a budget-neutral system. Addressing climate change is an opportunity to protect vulnerable populations and increase our value in the health care system. Initiatives to address climate change align with the ACR’s [American College of Radiology’s] core purpose of serving both patients and society. Our field has made great strides in patient safety by decreasing radiation doses. Similarly, through our technological expertise and awareness, we can decrease our carbon footprint, with the ultimate goal of mitigating climate change and preventing a looming public health crisis.
Listen to a podcast of Julia Schoen discussing sustainability and radiology.
In 2015 I described the mechanical bidomain model in a chapter of Cardiomyocytes: Methods and Protocols. This book was part of the series Methods in Molecular Biology, and each chapter had a unusual format. The research was outlined, with the details relegated to an extensive collection of endnotes. A second edition of the book was proposed, and I dutifully submitted an updated chapter. However, the new edition never come to pass. Rather than see my chapter go to waste, I offer it to you, dear reader. You can download a draft of my chapter for the second edition here. For those of you who have time only for a summary, below is the abstract.
The mechanical bidomain model provides a macroscopic description of cardiac tissue
biomechanics, and also predicts the microscopic coupling between the extracellular matrix and the intracellular cytoskeleton of cardiomyocytes. The goal of this chapter is
to introduce the mechanical bidomain model, to describe the mathematical methods
required for solving the model equations, to predict where the membrane forces acting
on integrin proteins coupling the intracellular and extracellular spaces are large, and to
suggest experiments to test the model predictions.
The main difference between the chapter in the first edition and the one submitted for the second was a new section called “Experiments to Test the Mechanical Bidomain Model.” There I describe how the model can reproduce data obtained when studying colonies of embryonic stem cells, sheets of engineered heart tissue, and border zones between normal and ischemic regions in the heart. The chapter ends with this observation:
I particularly like a new figure in the second edition. It’s a revision of a figure created by Xavier Trepat and Jeffrey Fredberg that compares mechanobiology to a game of tug-of-war. I added the elastic properties of the extracellular space (the green arrows), saying “It is as if the game of tug-of-war is played on a flexible surface, such as a flat elastic sheet.” In other words, tug-of-war on a trampoline.
Enjoy!
The “tug-of-war” model of tissue biomechanics, adapted from an illustrationby Trepat and Fredberg. Top: the intracellular (yellow), extracellular (green) and
integrin (blue) forces acting on a monolayer of cells. Middle: The analogous forces
among the players of a game of tug-of-war. This figure is extended beyond that of
Trepat and Fredberg by allowing both the intracellular and extracellular spaces to
move. Bottom: Representation of the mechanical bidomain model by a ladder of
springs.
Next is a more rigorous simulation of an aquaporin
A simulation of a water channel in a cell membrane, performed by The Theoretical and Biophysics Group at the NIH Center for Macromolecular Modeling and Bioinformatics. https://www.youtube.com/watch?v=GSi5-y6NHjY
Russ and I cite the paper by Murara et al. (2000). The full citation is
Human red cell AQP1 is the first functionally defined member of the aquaporin family of membrane water channels. Here we describe an atomic model of AQP1 at 3.8 Å resolution from electron crystallographic data. Multiple highly conserved amino-acid residues stabilize the novel fold of AQP1. The aqueous pathway is lined with conserved hydrophobic residues that permit rapid water transport, whereas the water selectivity is due to a constriction of the pore diameter to about 3 Å over a span of one residue. The atomic model provides a possible molecular explanation to a longstanding puzzle in physiology—how membranes can be freely permeable to water but impermeable to protons.
Below is a illustration of the aquaporin molecule. The view is perpendicular to the membrane, and the little hole in the middle is the pore.
This book is about electric and magnetic fields, and their effect on your body. We
will examine the use of magnets for pain relief, the risk of power line magnetic
fields, the safety of cell phones, and the possibility that microwave weapons are
responsible for the Havana syndrome. Many medical treatments are based on electromagnetism,
including well established ones like heart pacemakers and neural
prostheses, and more questionable ones such as bone healing, transcutaneous electrical
nerve stimulation, and transcranial direct current stimulation. Innumerable
books and articles have been written about each of these topics; my goal in this book
is to examine them together, to get the big picture.
This book is not a research monograph. It presents no original discoveries and
makes no attempt to be comprehensive. Moreover, it omits numerous details and
technicalities that experts often argue about. It does, however, try to offer an overall
view of the field that is accurate.
My target readers are nonscientists: journalists, politicians, teachers, students,
and anyone who has heard about electric and magnetic fields interacting with biological
tissue and wants to learn more. I use no mathematics, avoid jargon, and
employ abbreviations only when repeating the same mouthful of words over and
over again becomes tedious. I tried my best to make the book understandable to a
wide audience….
Sometimes the effect of electric and magnetic fields is controversial. For any
debate, I have tried to present both sides. Nevertheless, readers will soon catch on
that I’m a skeptic. Each chapter title is a question, of which my answer is usually
“probably not” or “no.”
Here is the Table of Contents.
Introduction
Can Magnets Cure All Your Ills?
Can a 9-Volt Battery Make You Smarter?
Do Power Lines Cause Cancer?
Will Electrical Stimulation Help Your Aching Back?
Is Your Cell Phone Killing You?
Did 5G Cell Phone Radiation Cause Covid-19?
Did Cuba Attack America with Microwaves?
Is That Airport Security Scanner Dangerous?
Conclusion
Although Russ Hobbie is not a coauthor on my new book, readers familiar with IPMB will see his influence on each page. In one of our last email exchanges before he passed away, I sent Russ an early draft of the book and he claimed to love it (that may have been Russ being kind, as he always was).
Enjoy!
Listen to me read the final chapter of Are Electromagnetic Fields Making Me Ill?
To examine this behavior in more detail, let’s turn to Mark Denny’s book Air and Water: The Biology and Physics of Life’s Media. Denny has an entire section on the consequences of the attenuation of light. Below I present a modified version of his Figure 11.13B, plotting the attenuation coefficient as a function of wavelength. The most important point is that the attenuation of air is much less than that of water. The difference doesn’t look too striking in this figure, because the attenuation is plotted on a logarithmic scale, but the attenuation of water is at least a hundred times greater than the attenuation of air, and for large wavelengths (red light) the difference is far greater. On the right I added a scale for the penetration depth, which is just the reciprocal of the attenuation coefficient. For air, the penetration depth is at least ten kilometers, and often much more. This is good, because we definitely want sunlight to pass through the atmosphere and reach the earth’s surface.
For water, the attenuation coefficient has a minimum around 470 nm, which is in the blue part of the visible spectrum. It then rises as the wavelength increases into the green, yellow, and red parts of the spectrum. Again, don’t let the logarithmic plot fool you. Between blue and red the attenuation coefficient increases by a factor of a hundred. Red light can only penetrate a few meters into water, but blue light reaches depths of hundreds of meters. Except very near the surface, aquatic animals live in a blue world.
No sunlight reaches the bottom of the ocean, ten kilometers down.
I’ll let Denny describe more ramifications of the strong dependence of attenuation on color.
The attenuation coefficient of water varies with wavelength… Attenuation is high in the UV [ultraviolet] and IR [infrared], and is minimal for light at visible wavelengths. Given that life initially evolved in an aqueous medium, it may not be a coincidence that “visible” light corresponds to those wavelengths for which water is most transparent. The same argument can be applied to the pigments used by plants to capture light for photosynthesis. All of the major photosynthetic pigments (chlorophylls, carotenoids, and phycobilins) absorb light in the range of 400 to 700 nm, the range at which water is least attenuating…
Even within the visible range, the attenuation coefficient of water varies substantially (fig. 11.13B); red light is attenuated much more strongly than blue light. Again this effect is well known to SCUBA divers, who note that the apparent color of objects changes rapidly with depth. For instance, a camera case that is bright red at the surface appears gray at a depth of only a few meters because all of the available red light has been absorbed by water above… The rapid absorption of red light has had evolutionary consequences for plants. Because chlorophyll a (the most common photosynthetic pigment) absorbs strongly at a wavelength of 680 nm (red light), it is a relatively ineffective means for gathering light at depth. However, plants which live deep beneath the water’s surface have accessory pigments (carotenoids and phycobilins) that absorb at shorter wavelengths.
I’ve already hinted at the view of nature… this book expands upon, which I identify as biophysical. The term implies a unification of biology and physics. It encapsulates the notion that the substances, shapes, and actions that constitute life are governed and constrained by the universal laws of physics, and that illuminating the connections between physical rules and biological manifestations reveals a framework upon which the dazzling variety of life is built. The notion of universality is central to the utility of physics, and to its appeal… Biophysics extends to the living world the quest for unity that lies at the heart of physics.
So, what are these four principles that Raghu says shape our living world?
Self-Assembly: “the idea that the instructions for building with biological components—whether molecules, cells, or tissues—are encoded in the physical characteristics of the components themselves.”
Regulatory Circuits: “The wet, squishy building blocks of life assemble into machines that can sense their environment, perform calculations, and make logical decisions.”
Predictable Randomness: “The physical processes underlying the machinery of life are fundamentally random but, paradoxically, their average outcomes are reliably predictable.”
Scaling: “Physical forces depend on the size and shape in ways that determine the forms accessible to living, growing, and evolving organisms.”
So Simple a Beginning is a very different book than Intermediate Physics for Medicine and Biology. SSaB is an introductory book for the general public; IPMB is an intermediate textbook for upper-level undergraduates in the sciences. SSaB examines life from the molecular scale to organs and organisms; IPMB focuses more on tissue-scale physiology and up, with only passing mention of molecular biology and biochemistry. SSaB has no math; IPMB has equations on nearly every page. SSaB has no end-of-chapter homework problems; one of IPMB’s strengths is its large collection of exercises for the reader. SSaB is elegantly and beautifully written; IPMB’s prose is workmanlike, nothing too graceful but adequate for the job. Finally, SSaB contains dozens of Raghu’s charming drawings and paintings; IPMB’s figures tend to be competent but not artistic. I’m gonna send out a strongly worded letter to whoever’s in charge of distributing talent. No one should have the ability to write well and draw skillfully. Raghu does both. That’s cheating.
I’ll end with the final paragraph of Raghu’s introduction. He quotes Darwin’s famous last paragraph of On the Origin of Species. He probably wanted to title his book This View of Life but Stephen Jay Gould already claimed that phrase for his series of essays about evolution. Instead, Raghu took So Simple a Beginning. Raghu’s writing reminds me of Gould, one of my favorite authors. He writes like Gould would have written had Gould been a physicist.
As interesting as these topics and examples may be, their cumulative effect is greater than the sum of their parts. Biophysics transforms the way we look at the world. At the end of On the Origin of Species, Darwin writes:
There is grandeur in this view of life, with its several powers, having been originally breathed into a few forms or into one; and that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being, evolved.
I hope to convince you that Nature has a grandeur even deeper than what Darwin discerned. Rather than a contrast between the fixed, clockwork laws of physics and the generation of endless and beautiful forms, the two are inextricably linked. We can identify the crucial “simple beginning” not as the origin of life, nor the formation of our planet, but as the primeval emergence of the physical laws that characterize our universe. The influence of these laws on life didn’t end billions of years ago, but rather shaped and continue to shape all the wonderful forms around us and within us. To discern simplicity amid complexity and to draw connections between life’s diverse phenomena and universal physical concepts gives us a deeper appreciation of ourselves, our fellow living creatures, and the natural world that we inhabit. I hope you’ll agree.
I am an emeritus professor of physics at Oakland University, and coauthor of the textbook Intermediate Physics for Medicine and Biology. The purpose of this blog is specifically to support and promote my textbook, and in general to illustrate applications of physics to medicine and biology.
