Taylor's Series

In Appendix D of the 5th edition of Intermediate Physics for Medicine and Biology, Russ Hobbie and I review Taylor’s Series. Our Figures D.3 and D.4 show better and better approximations to the exponential function, e^x, found by using more and more terms of its Taylor’s series. As we add terms, the approximation improves for small |x| and diverges more slowly for large |x|. Taking additional terms from the Taylor’s series approximates the exponential by higher and higher order polynomials. This is all interesting and useful, but the exponential looks similar to a polynomial, at least for positive x, so it is not too surprising that polynomials do a decent job approximating the exponential.

A more challenging function to fit with a Taylor’s Series would look nothing like a polynomial, which always grows to plus or minus infinity at large |x|. I wonder how the Taylor’s Series does approximating a bounded function; perhaps a function that oscillates back and forth a lot? The natural choice is the sine function.

The Taylor’s Series of sin(x) is

sin(x) = x – x³/6 + x⁵/120 – x⁷/5040 + x⁹/362880 - …

The figure below shows the sine function and its various polynomial approximations.

The sine function and its various polynomial approximations,
from: http://www.peterstone.name/Maplepgs/images/Maclaurin_sine.gif

The red curve is the sine function itself. The simplest approximation is simply sin(x) = x, which gives the yellow straight line. It looks good for |x| less than one, but quickly diverges from sine at large |x|. The green curve is sin(x) = x – x³/6. It rises to a maximum and then falls, much like sin(x), but  heads off to plus or minus infinity relatively quickly. The cyan curve is sin(x) = x – x³/6 + x⁵/120. It captures the first peak of the oscillation well, but then fails. The royal blue curve is sin(x) = x – x³/6 + x⁵/120 – x⁷/5040. It is an excellent approximation of sin(x) out to x = π. The violet curve is sin(x) = x – x³/6 + x⁵/120 – x⁷/5040 + x⁹/362880. It begins to capture the second oscillation, but then diverges. You can see the Taylor’s Series is working hard to represent the sine function, but it is not easy.

Appendix D in IPMB gives a table of values of the exponential and its different Taylor’s Series approximations. Below I create a similar table for the sine. Because the sine and all its series approximations are odd functions, I only consider positive values of x.

A table of values for sin(x) and its various polynomial approximations.

One final thought. Russ and I title Appendix D as “Taylor’s Series” with an apostrophe s. Should we write “Taylor Series” instead, without the apostrophe s? Wikipedia just calls it the “Taylor Series.” I’ve seen it both ways, and I don’t know which is correct. Any opinions?

Progress Toward a Deployable SQUID-Based Ultra-Low Field MRI System for Anatomical Imaging

When surfing the web, I like to visit medicalphysicsweb.org. This site, maintained by the Institute of Physics, always publishes interesting and up-to-date information about physics applied to medicine. Readers of the 5th Edition of Intermediate Physics for Medicine and Biology should visit it regularly.

A recent article discusses a paper by Michelle Espy and her colleagues about “Progress Toward a Deployable SQUID-Based Ultra-Low Field MRI System for Anatomical Imaging” (IEEE Transactions on Applied Superconductivity, Volume 25, Article Number 1601705, June 2015). The abstract is given below.

Magnetic resonance imaging (MRI) is the best method for non-invasive imaging of soft tissue anatomy, saving countless lives each year. But conventional MRI relies on very high fixed strength magnetic fields, ≥ 1.5 T, with parts-per-million homogeneity, requiring large and expensive magnets. This is because in conventional Faraday-coil based systems the signal scales approximately with the square of the magnetic field. Recent demonstrations have shown that MRI can be performed at much lower magnetic fields (∼100 μT, the ULF regime). Through the use of pulsed prepolarization at magnetic fields from ∼10–100 mT and SQUID detection during readout (proton Larmor frequencies on the order of a few kHz), some of the signal loss can be mitigated. Our group and others have shown promising applications of ULF MRI of human anatomy including the brain, enhanced contrast between tissues, and imaging in the presence of (and even through) metal. Although much of the required core technology has been demonstrated, ULF MRI systems still suffer from long imaging times, relatively poor quality images, and remain confined to the R and D laboratory due to the strict requirements for a low noise environment isolated from almost all ambient electromagnetic fields. Our goal in the work presented here is to move ULF MRI from a proof-of-concept in our laboratory to a functional prototype that will exploit the inherent advantages of the approach, and enable increased accessibility. Here we present results from a seven-channel SQUID-based system that achieves pre-polarization field of 100 mT over a 200 cm³ volume, is powered with all magnetic field generation from standard MRI amplifier technology, and uses off the shelf data acquisition. As our ultimate aim is unshielded operation, we also demonstrated a seven-channel system that performs ULF MRI outside of heavy magnetically-shielded enclosure. In this paper we present preliminary images and compare them to a model, and characterize the present and expected performance of this system.

Let’s compare a standard 1.5-Tesla clinical MRI system with Espy et al.’s ultra-low field device. To compare quantities using the same units, I will always express the magnetic field strength in mT; a typical clinical MRI machine has a field of 1500 mT. In Section 18.3 of IPMB, Russ Hobbie and I show that the magnetization depends linearly on the magnetic field strength (Equation 18.9). The static magnetic field of Espy et al.’s machine is 0.2 mT, but for 4000 ms before spin excitation a polarization magnetic field of 100 mT is turned on. Thus, there is a difference of a factor of 15 in the magnetization, with the ultra-low device having less and the clinical machine more. Once the polarization field is turned off, the magnetic field in the ultra-low device reduces to 0.2 mT. This is only four times the earth’s magnetic field, about 0.05 mT. Espy et al. use Hemlholtz coils to cancel the earth’s field.

A magnetic field of 1500 mT is usually produced by a coil that must be kept at low temperatures to maintain the wire as a superconductor. A 100 mT field does not require superconductivity, but the needed current generates enough heat that the 510-turn copper coil needs to be cooled by liquid nitrogen, and even still the current in the coil must be turned off half the time (50% duty cycle) to avoid overheating.

Once the polarization field turns off, the spins precess with the Larmor frequency for a 0.2 mT magnetic field. The gyromagnetic ratio of protons is 42.6 kHz/mT, implying a Larmor frequency of 8.5 kHz, compared with 64,000 kHz for a clinical machine. So, the Larmor frequencies differ by a factor of 7500.

The magnetic resonance signal recorded in a clinical system is large compared to an ultra-low device because the magnetization is larger by a factor of 15 and the Larmor frequency is larger by a factor of 7500, implying a signal over a hundred thousand times larger. Espy et al. get around this problem by measuring the signal with a Superconducting Quantum Interference Device (SQUID) magnetometer, like those used in magnetoencephalography (see Section 8.9 in IPMB).

Preliminary experiments were performed in a heavy and expensive magnetically shielded room (again, like those used when measuring the MEG). However, second-order gradiometer pickup coils reduce the noise sufficiently that the shielded room is unnecessary.

To perform imaging, Espy et al. use magnetic field gradients of about 0.00025 mT/mm, compared with 0.01 mT/mm gradients typical for clinical MRI. For two objects 1 mm apart, a clinical imaging system would therefore produce a fractional frequency shift of 0.01/1500 = 0.0000067 or 6.7 ppm, whereas a low-field device has a fractional shift of 0.00025/0.2 = 0.00125 or 1250 ppm. Therefore, the clinical magnetic field needs to be extremely homogeneous (on the order of parts per million) to avoid artifacts, whereas a low-field device can function with heterogeneities hundreds of times larger.

The relaxation time constants for gray matter in the brain are reported by Espy et al. as about T1 = 600 ms and T2 = 80 ms. In clinical devices, the value of T1 is about half that, and T2 is about the same. Based on Figure 18.12 and Equation 18.35 in IPMB, I’m not surprised that T2 is largely independent of the magnetic field strength. However, in strong fields T1 increases as the square of the magnetic field (or the square of the Larmor frequency), so I was initially expecting a much smaller value of T1 for the low-field device. But once the Larmor frequency drops to values less than the typical correlation time of the spins, T1 becomes independent of the magnetic field strength (Equation 18.34 in IPMB). I assume that is what is happening here, and explains why T1 drops by only a factor of two when the magnetic field is reduced by a factor of 7500.

I find the differences between radio-frequency excitation pulses to be interesting. In clinical imaging, if the excitation pulse has a duration of about 1 ms and the Larmor frequency is 64,000 kHz, there are 64,000 oscillations of the radio-frequency magnetic field in a single π/2 pulse. Espy et al. used a 4 ms duration excitation pulse and a Larmor frequency of 8.5 kHz, implying just 34 oscillations per pulse. I have always worried that illustrations such as Figure 18.23 in IPMB mislead because they show the Larmor frequency as being not too different from the excitation pulse duration. For low-field MRI, however, this picture is realistic.

Does low-field MRI have advantages? You don’t need the heavy, expensive superconducting coil to generate a large static field, but you do need SQUID magnetometers to record the small signal, so you don’t avoid the need for cryogenics. The medicalphysicsweb article weighs the pros and cons. For instance, the power requirements for a low-field device are relatively small, and it is more portable, but the imaging times are long. The safety hazards caused by metal are much less in a low-field system, but the impact of stray magnetic fields is greater. I’m skeptical about the ultimate usefulness of ultra low-field MRI, but it’ll be fun to watch if Espy and her team can prove me wrong.

What My Dogs Forced Me To Learn About Thermal Energy Transfer

I’m a dog lover, so I have to enjoy an American Journal of Physics paper that begins “For many years, I have competed and judged in American Kennel Club obedience trials.” The title of the paper is also delightful: “What my Dogs Forced Me to Learn About Thermal Energy Transfer” (Craig Bohren, American Journal of Physics, Volume 83, Pages 443−446, 2015). Bohren’s hypothesis is that an animal perceives hotness and coldness not directly from an object’s temperature, as one might naively expect, but from the flux density of thermal energy. I could follow his analysis of this idea, but I prefer to use the 5th edition of Intermediate Physics for Medicine and Biology, because Russ Hobbie and I have already worked out almost all the results we need.

Chapter 4 of IPMB analyzes diffusion. We consider the concentration, C, of particles as they diffuse in one dimension. Initially (t = 0), there exists a concentration difference C₀ between the left and right sides of a boundary at x = 0. We solve the diffusion equation in this case, and find the concentration in terms of an error function

  C(x,t) = C₀/2 [ 1 – erf(x/√4Dt) ] ,                  Eq. 4.75

where D is the diffusion constant. A plot of C(x,t) is shown in Fig. 4.22 (we assume C = 0 on the far right, but you could add a constant to the solution without changing the physics, so all that really matters is the concentration difference).

Fig. 4.22 The spread of an initially sharp boundary due to diffusion.

In Homework Problem 19, Russ and I show that the analysis of particle diffusion is equivalent to an analysis of heat conduction, with the thermal diffusivity D given by the thermal conductivity, κ, divided by the specific heat capacity, c, and the density, ρ

D = κ/cρ .

So, by analogy, if you start (t = 0) with a uniform temperature on the left and on the right sides of a boundary at x = 0, with an initial temperature difference ΔT between sides, the temperature distribution T(x,t) is (to within a additive constant temperature) the same as the concentration distribution calculated earlier.

T(x,t) = ΔT /2 [ 1 – erf(x/√4Dt) ] .

The temperature of the interface is always ΔT/2. If a dog responded to simply the temperature at x = 0 (where its thermoreceptors are presumably located), it would react in a way strictly proportional to the temperature difference ΔT. But Bohren’s hypothesis is that thermoreceptors respond to the energy flux density, κ dT/dx.

Now let us look again at Fig. 4.22. The slope of the curve at x = 0 is the key quantity. So, we must differentiate our expression for T(x,t). We get

κ dT/dx = - κ ΔT/2 d/dx [ erf(x/√4Dt) ] .

By the chain rule, this becomes (with u = x/√4Dt)

κ dT/dx  = - κ ΔT / (2 √4Dt ) d(erf(u))/du .

The derivative of the error function is given in IPMB on page 179

d/du (erf(u)) = 2/√π e^-u2 .

At the interface (u = 0), this becomes 2/√π. Therefore

κ dT/dx = - κ ΔT /√4πDt .

Bohren comes to the same result, but by a slightly different argument.

The energy flux density depends on time, with an infinite response initially (we assume an abrupt difference of temperature on the two sides of the boundary x = 0) that falls to zero as the time becomes large. The flux density depends on the material parameters by the quantity κ/√D , which is equivalent to √cρκ and is often called the thermal inertia.

Bohren goes on to analyze the case when the two sides have different properties (for example, the left might be a piece of aluminum, and the right a dog’s tongue), and shows that you get a similar result except the effective thermal inertia is a combination of the thermal inertia on the left and right. He does not solve the entire bioheat equation (Sec. 14.11 in IPMB), including the effect of blood flow. I would guess that blood flow would have little effect initially, but it would play a greater and greater role as time goes by.

Perhaps I will try Bohren’s experiment: I’ll give Auggie (my daughter Kathy’s lovable foxhound) a cylinder of aluminum and a cylinder of stainless steel, and see if he can distinguish between the two. My prediction is that, rather than either metal, he prefers rawhide.

The blog is dead. Long live the blog!

Intermediate Physics for
Medicine and Biology.

All good things must come to an end. After nearly eight years of posting an entry to this blog every Friday morning, I must say goodbye. This is the last entry to my blog dedicated to the 4th edition of Intermediate Physics for Medicine and Biology. I hope you have found it useful.

Next week will be the first entry to the blog for the FIFTH edition of Intermediate Physics for Medicine and Biology! You can now purchase the 5th edition at the Springer website. Amazon does not have a page for the book yet, but it should be coming soon.

What’s new in the 5th edition? The preface states

The Fifth Edition does not add any new chapters, but almost every page has been improved and updated. Again, we fought the temptation to expand the book and deleted material when possible. Some of the deleted material is available at the book’s website: [https://sites.google.com/view/hobbieroth]. The Fifth Edition has 12% more end-of-chapter problems than the Fourth Edition; most highlight biological applications of the physical principles. Many of the problems extend the material in the text. A solutions manual is available to those teaching the course. Instructors can use it as a reference or provide selected solutions to their students. The solutions manual makes it much easier for an instructor to guide an independent-study student. Information about the solutions manual is available at the book’s website.

The 5th edition is only 13 pages longer than the 4th. By deleting or condensing obsolete or less important topics, we made room for some new sections, including

• 1.2 Models (about toy models and their different roles in biology and physics)
• 1.15 Diving (SCUBA diving and the bends)
• 2.6 The Chemostat
• 8.1.2 The Cyclotron
• 8.8.4 Magnetic nanoparticles
• 9.10.5 Microwaves, Mobile Phones, and Wi-Fi
• 10.7 Proportional, Derivative, and Integral Control
• 13.1.3 Shear waves
• 13.7.4 Elastography (using ultrasound imaging)
• 13.7.5 Safety (of ultrasound imaging)
• 14.2 Electron Waves and Particles: The Electron Microscope
• 14.9.2 Photodynamic Therapy
• 14.15 Color Vision
• 18.14 Hyperpolarized MRI of the Lung

There is no errata yet, but despite our best efforts to find and remove all mistakes I suspect we will start finding errors in the 5th edition soon, and we’ll tell you about them if we do. As always, if YOU find errors in the book, please let us know.

Enjoy!

Churchill’s moral

The Second World War,
by Winston Churchill.

Winston Churchill is one of my heroes. After completing my PhD dissertation, I rewarded myself by reading Churchill’s history The Second World War; all six volumes. I am amazed how someone could make so much history leading England through World War Two, and then write that history so well. I love his language, and the way he uses his memos and letters to illustrate his thoughts at the time the events happened, rather than relying on his memories of those events years later. His life story is fascinating, and is told eloquently by the late William Manchester in his biography of Churchill, The Last Lion.

One unique feature of Churchill’s history is that he gave it a moral:

Moral of the Work

In War: Resolution

In Defeat: Defiance

In Victory: Magnanimity

In Peace: Goodwill

All books should have a moral, which sums up its key message in just a handful of carefully chosen words, and highlights the crucial lessons readers should learn. A moral is not an abstract, meant to summarize the book. Rather than explaining what the book is about, a moral tells why you should bother reading the book at all.

What would be the moral of the 4th edition of Intermediate Physics for Medicine and Biology? I cannot capture the essence of IPMB as well as Churchill did his history, but I must try. The moral of IPMB should express how physics explains and constrains biology and medicine, how you cannot truly understand biology until you can describe it in the language of mathematics, how so much of what is learned in introductory physics has direct applications to modern medicine, how toy models provide a way to reduce complex biological processes to their fundamental mechanisms, how the goal of expressing phenomena in mathematics is not merely to calculate numbers but to tell a physical story, and how solving homework problems is the most important way (more important that reading the book!) to learn crucial modeling skills. So, with apologies to Sir Winston, I present the moral of Intermediate Physics for Medicine and Biology:

Moral of the Work

 In Physics: Physiology

In Math: Medicine

In Models: Comprehension

In Equations: Insight