An Advanced Undergraduate Laboratory in Living State Physics

One weakness of Intermediate Physics for Medicine and Biology is that it doesn’t have an associated laboratory. Students need to learn how to perform experiments and use instruments.

An Advanced Undergraduate Laboratory
In Living State Physics,
by Wikswo, Vickery, and Venable.

Fortunately, instructors wanting to develop a lab don’t need to start from scratch. My PhD advisor, John Wikswo, and his colleagues Barbara Vickery and John Venable created An Advanced Undergraduate Laboratory in Living State Physics at Vanderbilt University around 1980. I didn’t take this lab class, but my wife Shirley did (she obtained a masters degree in physics from Vanderbilt), and she still has the lab manual. 

Wikswo obtained a grant from the National Science Foundation to support the development of the lab. He collaborated with John Venable, a biologist on the Vanderbilt faculty. When I was a graduate student, Venable was the Associate Dean of the College of Arts and Sciences. Barbara Vickery was a Vanderbilt undergraduate biomedical engineering major.

The lab wasn’t designed for any particular textbook, but Wikswo was an early adopter of Russ Hobbie’s Intermediate Physics for Medicine and Biology, and I think I can see its influence. I don’t have an electronic copy of the 250-page lab manual; you would have to contact Wikswo for that. Below I quote parts of it.

1.1 An Introduction to the Living State Physics Laboratory

The undergraduate physics curriculum at a typical university might include an introductory class in biophysics or medical physics in addition to the more traditional curriculum of mechanics, electricity and magnetism, light and sound, thermodynamics, and modern physics. While introductory and advanced laboratory classes cover these standard fields of physics, generally there has been little opportunity for an undergraduate student to gain laboratory experience in biophysics or medical physics. The need for such experience is particularly acute today for preprofessional and scientifically oriented students. Of these students, physics majors are not being exposed to an important area of experimental physics, and pre-medical students and majors in other departments such as Molecular Biology, Chemistry, and Biomedical Engineering are presently receiving only a minimal exposure to modern biophysical techniques and instrumentation. Thus by introducing an advanced undergraduate laboratory in physics applied to living systems, we expect to broaden the experience in experimental physics for physics majors and non-majors alike.

Several options were available to us in designing this laboratory. We could, for example, have structured the laboratory to emphasize applications of physics to certain living systems such as the nervous system, the cardiovascular system, and the special senses. Rather than take this system-oriented approach, we have chosen to organize the course by areas of physics. The course will draw on techniques and ideas from the whole breadth of physics (mechanics, electricity, thermodynamics, optics, etc.) and apply these to topics of biophysical interest [the same approach as IPMB]. Since we will study intact living systems such as people and frogs, as well as isolated living preparations and inanimate molecules and models, this laboratory will use physics to study topics conventionally identified with both biophysics and medical physics, as well as with electrophysiology, physical chemistry, biomedical engineering and molecular biology. Because of the intended breadth of the planned experiments and their organization by area of physics rather than by biological system, we have chosen to title this laboratory “An Advanced Undergraduate Laboratory in Living State Physics”. The generality of the term “Living State Physics” is intended to parallel the generality of the term “solid state physics”, which as an experimental discipline utilizes the complete spectrum of physical concepts and techniques...

1.2 Summary of Experiments

a. Introduction to Bioelectric Phenomena. The first of the three experiments in this section is an exercise with an oscilloscope and an electronic stimulator which will allow the student to obtain a familiarity with the use of these instruments. In the second and third experiments, the Thornton Modular Plug-In System is used to provide familiarity with the basic physics describing the electromyogram and the electroencephalogram…

b. The Heart Experiments. This section should enable the student to gain an understanding of the basic principles of cardiac physiology. In the laboratory, the student will measure the frog and the human electrocardiogram…

c. Nerve Action Potential… [Students perform an] in-depth study of the properties of nerve propagation in the isolated sciatic nerve of a frog. In both experiments, from extracellular recordings of the nerve action potential it will be possible to demonstrate the graded response of the nerve bundle, the strength-duration relationship of stimuli producing a threshold response, bi-directional conduction, and the monophasic response…

d. Nerve Modeling. In the first experiment, the passive cable properties of the nerve are studied by using a resistor-capacitor network that represents a section of a nerve axon… The active properties of the nerve are investigated in the second experiment. An electronic nerve model which has a design based on a system of equations similar to those developed by Hodgkin and Huxley is used…

e. Skeletal Muscle. The first of the two experiments in this section is an introduction to the active and passive mechanical properties of skeletal muscle using the frog gastrocnemius muscle. The experiment includes measurement of the muscle twitch, the ability of the muscle to do work, and the maximum tension developed by the muscle at different lengths, as well as demonstration of the phenomena of temporal summation and the graded response of muscle. The second experiment involves characterization of the mechanical properties of muscle in its resting and contractile states…

f. Diffusion. In this experiment, a Cenco model is used for qualitative demonstration of the transport phenomenon of diffusion, showing the exponential approach to equilibrium and how the relative sizes of molecules and pores affect diffusion rates.

g. Compartmental Modeling. The usefulness of compartmental modeling in analysis of some systems is demonstrated by constructing one- and two-compartment models for several open and closed thermal systems. The theoretical models are analyzed mathematically…

h. The Physical Aspects of Vision. The minimum number of photons that the human eye can detect in a single detectable flash is the minimum number of photons whose absorption by photoreceptor cells in the eye leads to the firing of an impulse in the brain. This threshold value is determined by recording the fraction of detected flashes as a function of relative intensity of the flashes… by utilizing Poisson statistics.

i. Ultrasound… The experiments introduce the physics of mechanical waves by using ultrasound transducers, a two-dimensional ultrasound target, and an existing ultrasound scanner and transient analyzer to demonstrate wave propagation, attenuation, reflection, refraction, pulse-echo principles, piezoelectric crystals and the concepts of cross-section and spatial resolution.

The first time I ever saw my wife was when she was in Wikswo's office asking a question about one of the lab exercises. I needed to talk to him about some very important issue related to my research, and she was in the way! Well, one thing led to another and....

I recall how Shirley and my friend Ranjith Wijesinghe were lab partners doing the vision experiment. It required sitting in a small, dark enclosure for about half an hour while their eyes became adapted to the dark. I had only recently met Shirley, and I recall being jealous of Ranjith for getting to spend such a private time with her! 

One of the most memorable parts of the lab was the pithing of the frog. None of the students liked doing that. Wikswo had a fun way of demonstrating the fight-of-flight response during the electrocardiogram lab. He would measure the ECG on one of the students, and then take out a giant syringe and say something like “now watch what happens to her heart rate when I inject her with this adrenaline.” Of course no one ever got injected, but the student was always so startled that her heart rate would jump dramatically.

If you are considering developing you own laboratory for Intermediate Physics for Medicine and Biology, you could start with Wikswo’s lab, and then add some of the experiments discussed in these American Journal of Physics papers. Good luck!

J. D. Prentice and K. G. McNeill (1962) “Measurement of the Beta Spectrum of I¹²⁸ in an Undergraduate Laboratory,” American Journal of Physics, Volume 30, Pages 66–67.  

Peter J. Limon and Robert H. Webb (1964) “A Magnetic Resonance Experiment for the Undergraduate Laboratory,” American Journal of Physics, Volume 32, Pages 361–364.    

L. J. Bruner (1979) “Cardiovascular Simulator for the Undergraduate Physics Laboratory,” American Journal of Physics, Volume 47, Pages 608–611.  

H. W. White, P. E. Chumbley, R. L. Berney, and V. H. Barredo (1982) “Undergraduate Laboratory Experiment to Measure the Threshold of Vision,” American Journal of Physics, Volume 50, Pages 448–450. 

Colin Delaney and Juan Rodriguez (2002) “A Simple Medical Physics Experiment Based on a Laser Pointer,” American Journal of Physics, Volume 70, Pages 1068–1070. 

Danny G. Miles Jr. and David W. Bushman (2005) “Protein Gel Electrophoresis in the Undergraduate Physics Laboratory,” American Journal of Physics, Volume 73, Pages 1186–1189. 

Luis Peralta (2006) “A Simple Electron-Positron Pair Production Experiment,” American Journal of Physics, Volume 74, Pages 457–461.  

Joseph Peidle, Chris Stokes, Robert Hart, Melissa Franklin, Ronald Newburgh, Joon Pahk, Wolfgang Rueckner, and Aravi Samuel (2009) “Inexpensive Microscopy for Introductory Laboratory Courses,” American Journal of Physics, Volume 77, Pages 931–938. 

Timothy A. Stiles (2014) “Ultrasound Imaging as an Undergraduate Physics Laboratory Exercise,” American Journal of Physics, Volume 82, Pages 490–501.  

Elliot Mylotta, Ellynne Kutschera, and Ralf Widenhorn (2014) “Bioelectrical Impedance Analysis as a Laboratory Activity: At the Interface of Physics and the Body,” American Journal of Physics, Volume 82, Pages 521–528.    

Alexander Hydea and Oleg Batishchevb (2015) “Undergraduate Physics Laboratory: Electrophoresis in Chromatography Paper,” American Journal of Physics, Volume 83, Pages 1003–1011.

Owen Paetkau, Zachary Parsons, and Mark Paetkau (2017) “Computerized Tomography Platform Using Beta Rays,” American Journal of Physics, Volume 85, Pages 896–900. 

Heaps of Precessing Protons

Spin Dynamics,
by Malcolm Levitt.

Last week’s post quoted from Spin Dynamics: Basics of Nuclear Magnetic Resonance, by Malcolm Levitt. This week I’ll talk more about this excellent textbook. Russ Hobbie and I cite Spin Dynamics in Intermediate Physics for Medicine and Biology when relating the proton relaxation time constants T₁ and T₂ to the correlation time τ_c. Our Fig. 18.12 shows this relationship in a log-log plot.

Fig. 18.12  Plot of T₁ and T₂ vs correlation time of the fluctuating magnetic field at the nucleus. The dashed lines are for a Larmor frequency of 29 MHz; the solid lines are for 10 MHz. Experimental points are shown for water (open dot) and ice (solid dots).

What do we mean by the “correlation time”? Levitt explains.

The parameter τ_c is called the correlation time of the fluctuations. Rapid fluctuations have a small value of τ_c, while slow fluctuations have a large value of τ_c. For rotating molecules in a liquid, τ_c is in the range of tens of picoseconds to several nanoseconds.

Qualitatively, the correlation time indicates how long it takes before the random field changes sign.

In practice, the correlation time depends on the physical parameters of the system, such as the temperature. Generally, correlation times are decreased by warming the sample, since an increase in temperature corresponds to more rapid molecular motion. Conversely, correlation times are increased by cooling the sample.

Levitt presents a plot similar to Fig. 18.12 in IPMB, except on linear-linear rather than log-log axes. 

Adapted from Fig. 16.16 of Spin Dynamics. The T₁ relaxation time as a function of the correlation time for random field fluctuations.

His curve is calculated for a static magnetic field of 11.74 T, which corresponds to a Larmor frequency, f_Larmor, of 500 MHz (a considerably stronger magnetic field than in our Fig. 18.12). The minimum of the curve is when τ_c equals the reciprocal of 2πf_Larmor, or about 0.32 ns. Levitt writes

It is a fortuitous circumstance that the most common experimental situation in solution NMR, namely medium-size molecules in non-viscous solutions near room temperature, falls close to the T₁ minimum. The small values of T₁ permit more rapid averaging of NMR signals, and hence a relatively high signal-to-noise ratio within a given experimental time. 

Think of the correlation time as a measure of the molecule’s rotation or tumbling time, characteristic of the molecular environment. One reason magnetic resonance imaging provides such excellent soft tissue contrast is because the relaxation times T₁ and T₂ are so sensitive to their surroundings. Relaxation happens most quickly when the tumbling time is similar to the period of precession, just as spin flipping is most effective when the radiofrequency field is in resonance with the precessing protons.

I like Spin Dynamics, in part because it has its own sound track. Russ and I have a lot of auxiliary stuff associated with Intermediate Physics for Medicine and Biology, but we don’t have a sound track. I’ll have to work on that.

To close, I quote from Levitt’s lyrical introduction to Spin Dynamics. Enjoy!

“Commonplace as such experiments have become in our laboratories, I have not yet lost that sense of wonder, and of delight, that this delicate motion should reside in all ordinary things around us, revealing itself only to him who looks for it.”

E. M. Purcell, Nobel Lecture, 1952

In December 1945, Purcell, Torrey and Pound detected weak radiofrequency signals generated by the nuclei of atoms in ordinary matter (in fact, about 1 kg of paraffin wax). Almost simultaneously, Bloch, Hansen and Packard independently performed a different experiment in which they observed radio signals from the atomic nuclei in water. There two experiments were the birth of the field we now know as Nuclear Magnetic Resonance (NMR).

Before then, physicists knew a lot about atomic nuclei, but only through experiments on exotic states of matter, such as those found in particle beams, or through energetic collisions in accelerators. How amazing to detect atomic nuclei using nothing more sophisticated than a few army surplus electronic components, a rather strong magnet, and a block of wax!

In his Nobel prize address, Purcell was moved to the poetic description of his feeling of wonder, cited above. He went on to describe how

“in the winter of our first experiments… looking on snow with new eyes. There the snow lay around my doorstep—great heaps of protons quietly precessing in the Earth’s magnetic field. To see the world for a moment as something rich and strange is the private reward for many a discovery…”

In this book, I want to provide the basic theoretical and conceptual equipment for understanding these amazing experiments. At the same time, I want to reinforce Purcell’s beautiful vision—the heaps of snow, concealing innumerable nuclear magnets, in constant precessional motion. The years since 1945 have shown us that Purcell was right. Matter really is like that. My aim in this book is to communicate the rigorous theory of NMR, which is necessary for really understanding NMR expeirments, but without losing sight of Purcell’s heaps of precessing protons.

Can T2 Be Longer Than T1?

In Chapter 18 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss magnetic resonance imaging. A key process in MRI is when the magnetization vector M is rotated away from the static magnetic field and is then allowed to relax back to equilibrium. To be specific, let’s assume that the static field is in the z direction, and the magnetization is rotated into the x-y plane. The magnetization M_z along the static field returns to its equilibrium value M₀ exponentially with time constant T₁. The M_x and M_y components relax to zero with time constant T₂. Russ and I write

The transverse relaxation time [T₂] is always shorter than T₁. Here is why. A change of M_z requires an exchange of energy with the [thermal] reservoir. This is not necessary for changes confined to the xy plane... M_x and M_y can change as M_z changes, but they can also change by other mechanisms, such as when individual spins precess at slightly different frequencies, a process known as dephasing.

Is T₂ always less than T₁? Let me start by giving you the bottom line: T₂ is usually less than T₁, and for most purposes we can assume T₂ < T₁. But Russ and I wrote “always,” meaning no exceptions. It’s not always true that T₂ < T₁.

“Relaxation: Can T₂ Be Longer Than T₁?”
by Daniel Traficante.

To see why, look at the 1991 article by Daniel Traficante in the journal Concepts in Magnetic Resonance (Volume 3, Pages 171–177), “Relaxation: Can T₂ Be Longer Than T₁?” Traficante begins by analyzing the relaxation equations introduced in Section 18.4 of IPMB,

      dM_x/dt = − M_x/T₂     dM_y/dt = − M_y/T₂    dM_z/dt = (M₀ − M_z)/T₁ .

If we start at t = 0 with M_x = M₀ and M_y = M_z = 0 (the situation after a 90° radiofrequency pulse), the magnetization is

       M_x = M₀ e^−t/T₂          M_y = 0                    M_z = M₀ (1 − e^−t/T₁) . 

(For the experts, this is correct in the frame of reference rotating with the Larmor frequency.) We are particularly interested in how the magnitude of the magnetization vector |M| changes (or, to avoid taking a square root, how the square of the magnetization changes, M² = M_x² + M_y² + M_z²). In our example, we find

                M²/M₀² = e^−2t/T₂ + (1 − e^−t/T₁)².

Traficante claims that many researchers mistakenly believe that |M| is equal to M₀ at all times; the vector simply rotates in the x-z plane, with its tip following the blue dashed arc in each figure below. Figure 18.5 in IPMB proves that Russ and I did not make that mistake. For the usual case when T₂ << T₁, the x-component decays quickly, while the z-component grows slowly, so |M| starts at M₀, quickly shrinks to a small value, and then slowly rises back to M₀. In the x-z plane, the tip of M follows the red path shown below. Clearly |M| is always less than M₀ (the red curve is well under the blue arc).

The path of the tip of M, for T₂ << T₁. 

 

If T₂ equals T₁, Traficante shows that in the x-z plane the tip of M follows a straight line, and again |M| is less than M₀.

The path of the tip of M, for T₂ = T₁.

What if T₂ >> T₁? Then M_z would rapidly rise to its equilibrium value M₀ while M_x would slowly fall to zero. 

The path of the tip of M, for T₂ >> T₁.

In this case, |M| would become larger than M₀ (the red curve passes outside of the blue arc). Traficante argues that an increase in |M| above M₀ would be unphysical (I suspect it would violate one of the laws of thermodynamics), so T₂ cannot be much larger than T₁.

Can T₂ be just a little larger than T₁? The straight-line plot for T₂ = T₁ suggests that |M| stays less than M₀ with room to spare. I tried to make a new homework problem asking you to find the relation between T₁ and T₂ that would prevent |M| from ever rising above M₀. The analysis was more complicated than I expected, so I skipped the homework problem. Below is my hand-waving argument to find the largest allowed value of T₂.

You can use a Taylor series analysis to show that |M| is less than M₀ for small times (corresponding to the lower right corner of the plots above), regardless of the values of T₁ and T₂. For longer times, I’ll suppose that |M| might become larger than M₀, but it can’t oscillate back-and-forth, going from smaller to larger to smaller and so on (I haven’t proven this, hence the hand waving). So, what we need to focus on is how |M| (or, equivalently, M²) behaves as t goes to infinity (corresponding to the upper left corner of the plots). If M² is less than M₀² at large times, then it should be less than M₀² at all times and we have not violated any laws of physics. If M² is greater than M₀² at large times, then we have a problem.

A little algebra applied to our previous equation gives

                       M²/M₀² = 1 + e^–2t/T₂  + e^–2t/T₁ – 2e^–t/T₁ .

At long times, the term with –2t/T₁ in the exponent must be smaller than the term with –t/T₁, so we can ignore it. That leaves two terms to compete, a positive term with –2t/T₂ in the exponent and a negative one with –t/T₁. The term with the smaller decay constant will ultimately win, so M² will never become greater than M₀² if T₂ < 2T₁.

I admit, my argument is complicated. If you see an easier way to prove this, let me know.

Traficante concludes

It is a common misconception that after a pulse, the net magnetization vector simply tips backwards toward the z axis, while maintaining a constant length. Instead, under the normal conditions when T₂* [for now, let’s ignore the difference between T₂ and T₂*] is less than T₁, the resultant first shrinks, and then grows back toward its initial value as it tips back toward the z axis. This behavior is clearly shown by examining the basic equations that describe both the decay of the magnetization in the xy plane and its growth up along the z axis. From these equations, the magnitudes of the xy and z components, as well as their [vector] sums, can be calculated as a function of time. This same behavior is demonstrated even when T₂* is equal to T₁—the resultant still does not maintain a constant value of 1.0 as it tips back. 

The resultant does not exceed 1.0 at any time during the relaxation if the T₂/T₁ ratio does not exceed 2. However, experimental evidence has been obtained that shows that the ratio can be greater than 1.

Spin Dynamics,
by Malcom Levitt

Malcolm Levitt, in his book Spin Dynamics: Basics of Nuclear Magnetic Resonance, comes to the same conclusion.

The following relationship holds absolutely

        T₂ < 2 T₁ (theoretical limit).

In most cases, however, it is usually found that T₂ is less than, or equal to, T₁:

        T₂ < T₁ (usual practical limit).

The case where 2T₁ > T₂ > T₁ is possible but rarely encountered.

  In a footnote, Levitt expands on this idea.

The case where T₂ > T₁ is encountered when the spin relaxation is caused by fluctuating microscopic fields which are predominately transverse rather than longitudinal.

I would like to thank Steven Morgan for calling this issue to my attention. Russ and I now address it in the errata. In general, we appreciate readers finding mistakes in Intermediate Physics for Medicine and Biology. If you find something in our book that looks wrong, please let us know.

The SI Logo

Intermediate Physics for Medicine and Biology uses the metric system. On page 1, Russ Hobbie and I write

“The metric system is officially called the SI system (systeme internationale). It used to be called the MKS (meter kilogram second) system.”

In 2018, the International Bureau of Weights and Measures changed how the seven SI base units are defined. They are now based on seven defining constants. This change is summarized in the SI logo.

The SI logo, produced by the
International Bureau of Weights and Measures.

First let’s see where the seven base units appear in IPMB. Then we’ll examine the seven defining constants.

kilogram

The most basic units of the SI system are so familiar that Russ and I don’t bother defining them. The kilogram (mass, kg) appears throughout IPMB, but especially in Chapter 1, where density plays a major role in our analysis of fluid dynamics.

meter

We define the meter (distance, m) in Chapter 1 when discussing distances and scales: “The basic unit of length in the metric system is the meter (m): about the height of a 3-year-old child.” Both the meter and the kilogram are critical when discussing scaling in Chapter 2.

second

The second (time, s) is another unit that’s so basic Russ and I take it for granted. It plays a particularly large role in Chapter 10 when discussing nonlinear dynamics.

ampere

The SI system becomes more complicated when you add electrical units. IPMB defines the ampere (electrical current, A) in Section 6.8 about current and Ohm’s law: “The units of the current are C s⁻¹ [C is the unit of charge, a coulomb] or amperes (A) (sometimes called amps).”

kelvin

The unit for absolute temperature—the kelvin (temperature, K)—plays a central role in Chapter 3 of IPMB, when describing thermodynamics.

mole

The mole (number of molecules, mol) appears in Chapter 3 when relating microscopic quantities (Boltzmann’s constant, elementary charge) to macroscopic quantities (the gas constant, the Faraday). John Wikswo and I have introduced a name for a mole of differential equations (the leibniz), but the International Bureau of Weights and Measures inexplicably did not add it to their logo.

candela

Russ and I introduce the candela (luminous intensity, cd) in Section 14.12 of IPMB, when comparing radiometry to photometry: “The number of lumens per steradian is the luminous intensity, in lm sr⁻¹. The lumen per steradian is also called the candela.” The steradian (the unit of solid angle) used to play a more central role in the SI system, but appears to have been demoted.

Now we examine the seven constants that define these units.

Planck’s constant

In IPMB, the main role of Planck’s constant (h, 6.626 × 10⁻³⁴ J s) is to relate the frequency and energy of a photon. Quantum mechanics doesn’t play a major role in IPMB, so Planck’s constant appears less often than you might expect.

speed of light

Like quantum mechanics, relativity does not take center stage in IPMB, so the speed of light (c, 2.998 × 10⁸ m s⁻¹) appears rarely. We use it in Chapter 14 when relating the frequency of light to its wavelength, and in Chapter 17 when relating the mass of an elementary particle to its energy.

cesium hyperfine frequency

The cesium hyperfine frequency (Δν, 9.192 × 10⁹ Hz) defines the second. It never appears in IPMB. Why cesium? Why this particular atomic transition? I don’t know.

elementary charge

The elementary charge (e, 1.602 × 10⁻¹⁹ C) is used throughout IPMB, but is particularly important in Chapter 6 about bioelectricity.

Boltzmann’s constant

Boltzmann’s constant (k_B, 1.381 × 10⁻²³ J K⁻¹) appears primarily in Chapter 3 of IPMB, but also anytime Russ and I mention the Boltzmann factor.

Avogadro’s number

Like Boltzmann’s constant, Avogadro’s number (N_A, 6.022 × 10²³ mol⁻¹) shows up first in Chapter 3.

luminous efficacy

The luminous efficacy (K_cd, 683 lm W⁻¹) appears in Chapter 14 of IPMB: “The ratio P_v/P at 555 nm is the luminous efficacy for photopic vision, K_m = 683 lm W⁻¹.” I find this constant to be different from all the others. It’s a prime number specified to only three digits. Suppose a society of intelligent beings evolved on another planet. Their physicists would probably measure a set of constants similar to ours, and once we figured out how to convert units we would get the same values for six of the constants. The luminous efficacy, however, would depend on the physiology of their eyes (assuming they even have eyes). Perhaps I make too much about this. Perhaps the luminous efficacy merely defines the candela, just as Avogardo’s number defines the mole and Boltzmann’s constant defines the kelvin. Still, to me it has a different feel.

You can learn more about the SI units and constants in the International Bureau of Weights and Measures’ SI brochure. I’m fond of the SI logo, which reminds me of the circle of fifths. If you’re new to the metric systems, you might want to paste the logo into your copy of Intermediate Physics for Medicine and Biology; I suggest placing it in the white space on page 1, just above Table 1.1.

Page 1 of Intermediate Physics for Medicine and Biology,
with the SI Logo added at the top.