Friday, October 26, 2018

Earl Bakken (1924-2018)

Earl Bakken (1924-2018)
Earl Bakken. From the Bakken Museum, via wikipedia.
Earl Bakken, cofounder of the medical device company Medtronic, died Sunday at the age of 94. In 1957 he developed the first external, battery-operated artificial pacemaker. Russ Hobbie and I don’t mention Bakken by name in Intermediate Physics for Medicine and Biology, but we do discuss pacemakers.
Cardiac pacemakers are a useful treatment for certain heart diseases. The most frequent are an abnormally slow pulse rate (bradycardia) associated with symptoms such as dizziness, fainting (syncope), or heart failure. These may arise from a problem with the SA node (sick sinus syndrome) or with the conduction system (heart block). One of the first uses of pacemakers was to treat complete or third degree heart block. The SA node and the atria fire at a normal rate but the wave front cannot pass through the conduction system. The AV node or some other part of the conduction system then begins firing and driving the ventricles at its own, pathologically slower rate…. A pacemaker stimulating the ventricles can be used to restore a normal ventricular rate.
In the 1950s, famed cardiac surgeon Dr. Walt Lillehei was at the University of Minnesota, where he met Bakken. Kirk Jeffrey tells the story of their collaboration in Machines In Our Hearts.
Machines In Our Hearts, by Kirk Jeffrey
Machines In Our Hearts, by Kirk Jeffrey
The myocardial pacing wire was the first electrical device ever to be implanted in the human body and left there for a period of time. Surgeons at Minnesota were now able to pace children for days or weeks after heart surgery. By October 1957, they had used the technique with 18 patients. But Lillehei now grew uneasy about the Grass stimulator because it was bulky and plugged into the electrical system. The surgeon wanted to get his heart patients out of bed and moving around, but the stimulator had to accompany them on a wheeled cart. The electrical cord was a further nuisance. ‘‘Many of these [patients] were kids. They wanted to wander around and get active. Well, they were active. They couldn’t go any further than the cord. We had to string wires down the hall. . . . And then, if they needed an X ray or something that couldn’t be done in the room, you couldn’t get on the elevator so you had to string them down the stairwells. It seemed that almost everything you wanted was on a different floor. We needed something battery-operated.’’

From Machines In Our Hearts.
The plug-in stimulator was more than an inconvenience, for by introducing the myocardial pacing wire, Lillehei and his associates had connected the hearts of their surgical patients to the 110-volt electrical system of the hospital. Everyone in the program knew that an electrical surge might send patients into ventricular fibrillation or that a power outage could leave them without pacemaker support. On October 31, 1957, an equipment failure at a large Twin Cities power plant caused an outage lasting nearly three hours in Minneapolis. The University hospital had auxiliary power in its surgical suites and recovery area, but not in patients’ rooms. None of his heart patients died—but Lillehei viewed the event as a warning. Lillehei…turned to Earl Bakken…a young engineer who owned a small medical electronics business called Medtronic and repaired and serviced equipment for the Department of Surgery.

Bakken…realized that he could simply build a stimulator that used transistors and small batteries. ‘‘It was kind of an interesting point in history,’’ he recalled—‘‘a joining of several technologies.’’ In constructing the external pulse generator, Bakken borrowed a circuit design for a metronome that he had noticed a few months earlier in an electronics magazine for hobbyists. It included two transistors. Invented a decade earlier, the transistor was just beginning to spread into general use in the mid-1950s. Hardly anyone had explored its applications in medical devices. Bakken used a nine-volt battery, housed the assemblage in an aluminum circuit box, and provided an on-off switch and control knobs for stimulus rate and amplitude.

At the electronics repair shop that he had founded with his brother-in-law in 1949, Bakken had customized many instruments for researchers at the University of Minnesota Medical School…When Bakken delivered the battery-powered external pulse generator to Walt Lillehei in January 1958, it seemed to the inventor another special order, nothing more. The pulse generator was hardly an aesthetic triumph, but it was small enough to hold in the hand and severed all connection between the patient’s heart and the hospital power system. Bakken’s business had no animal-testing facility, so he assumed that the surgeons would test the device by pacing laboratory dogs. They did ‘‘a few dogs,’’ then Lillehei put the pacemaker into clinical use. When Bakken next visited the university, he was surprised to find that his crude prototype was managing the heartbeat of a child recovering from open-heart surgery.
The Bakken Museum. From wikipedia.
Russ Hobbie is retired from the University of Minnesota, and still lives in the area, so he is particularly familiar with Earl Bakken. He served on the board of the Bakken Museum, which is devoted to bioelectricity (my kind of museum). Russ say he ‘‘was impressed by Bakken's vision, energy, and desire to help people. The Bakken Museum has an extensive outreach program which does a lot of good things.’’ They recently posted a statement honoring their founder.

Medtronic is one of the largest medical device companies. I had a job interview there years ago, but I didn’t get the position. I came away impressed by the company, and wish I had bought stock.

Bakken is a member of the dwindling greatest generation; he was an airborne radar maintenance instructor during World War II. He had a long, full life, and we will miss him.

Below are a couple of videos about Earl Bakken. Enjoy!

Friday, October 19, 2018

The Bond Number

In graduate school, I did experiments on nerves submerged in saline. When dissecting or manipulating the tissue, I was amazed by the effect of surface tension compared to gravity; bubbles would stick to my forceps, and when I raised my scalpel out of the saline all kinds of stuff came along with it.

A dimensionless number describes this effect: the Bond number (sometimes called the Eötvös number). Several dimensionless numbers—such as the Reynolds number, the Peclet number, and the Lewis number—are mentioned in Intermediate Physics for Medicine and Biology. They indicate the relative importance of two physical mechanisms. Because they are dimensionless, their value does not depend on the units used. I find them useful.

The Bond number, Bo, is the ratio of the gravitational force mg (where m is the mass and g is the acceleration of gravity) to the force due to surface tension γL (where γ is the surface tension and L is some characteristic length such as the radius for a sphere), so Bo = mg/γL. If the Bond number is much greater than one, gravity dominates and surface tension is relatively insignificant. If the Bond number is much less than one, surface tension dominates and gravity hardly matters (like during my nerve experiments). When it’s close to one, both forces are similar.

The Bond number can help answer questions such as: Can animals stand on water? Insects can but people can’t. Why? Consider how the Bond number scales with size. An animal’s mass is equal to its density ρ times its volume, which varies as L3. In that case, the Bond number has an L3 in the numerator and an L in the denominator, so it scales as L2.

The Bond number
If an animal is large enough, gravity wins and it sinks.

Let's estimate the size of an animal when the Bond number is one.
The distance is equal to the square root of the surface tension divided by the density and the acceleration of gravity
Take g = 9.8 m/s2, ρ = 1000 kg/m3, and γ = 0.07 N/m (for water on earth). We find that L = 3 mm. For an insect, the good news is that it can stand on water. The bad news is that surface tension holds it in place so it can hardly move. As Steven Vogel says in Life in Moving Fluids: “Staying up is easy, but getting around is awkward.”

The Bond number is also useful for studying capillary action. Water will climb up the walls of a tube because of cohesion between the wall and the fluid. Technically, this effect depends on the nature of the surface as measured by its contact angle; you can coat a surface with a substance that repels water so that water climbs down the tube, but we will ignore all that and assume the contact angle is so small that the water “wets” the surface and cohesion is the same as surface tension. The gravitational force of the climbing water is gρπR2H, where R is the tube radius and H is the height that the water climbs. The surface tension force is γ times the circumference of the tube, 2πR. The Bond number becomes gρRH/2γ. We expect water to rise until gravity and surface tension are in balance, or until about Bo = 1. The height is then H = 2γ/gρR. For a 1 mm radius tube, the height is 14 mm, or roughly half an inch.

Can capillary action explain the ascent of sap in trees? The tubes in a tree's xylem have a radius of about 20 microns. In that case, water will rise about 0.6 meters. That’s about two feet, which doesn’t explain how water gets up a 100-m giant redwood. Suffice to say, the trunk of a redwood tree does not act as a giant wick, sucking water up its trunk by capillary action.

Wilfrid Noel Bond
Wilfrid Noel Bond, from Wikipedia.
The Bond number is named after the English physicist Wilfrid Noel Bond (1897-1937). He studied fluid dynamics, including viscosity and surface tension. He died tragically early at age 39.

If you want to learn more about the Bond number and other dimensionless numbers, see Vogel’s 1998 article in Physics Today about “Exposing Life’s Limits with Dimensionless Numbers.”

Friday, October 12, 2018

A Trick to Generate Exam Problems

Intermediate Physics for Medicine and Biology.
Intermediate Physics for Medicine and Biology.
When teaching a class based on Intermediate Physics for Medicine and Biology, instructors need to write problems for their exams. My goal in this post is to explain a trick for creating good exam problems. 

One of my favorite homework problems in IPMB is from Chapter 4.
Problem 37. The goal of this problem is to estimate how large a cell living in an oxygenated medium can be before it is limited by oxygen transport. Assume the extracellular space is well stirred with uniform oxygen concentration C0. The cell is a sphere of radius R. Inside the cell oxygen is consumed at a rate Q molecule m−3 s−1. The diffusion constant for oxygen in the cell is D.
(a) Calculate the concentration of oxygen in the cell in the steady state.
(b) Assume that if the cell is to survive the oxygen concentration at the center of the cell cannot become negative. Use this constraint to estimate the maximum size of the cell.
(c) Calculate the maximum size of a cell for C0 = 8 mol m−3, D = 2 x 10−9 m2 s−1, Q = 0.1 mol m−3 s−1. (This value of Q is typical of protozoa; the value of C0 is for air and is roughly the same as the oxygen concentration in blood.)
Homework problems for Chapter 4 in Intermediate Physics for Medicine and Biology.
Homework problems for Chapter 4 in
Intermediate Physics for Medicine and Biology.
I usually work this problem in class. Not only do students practice solving the steady-state diffusion equation, but also they estimate the maximum size of a cell from some basic properties of oxygen. In the Solution Manual—available to instructors only (email us)—we explain the purpose of each problem in a preamble. Here is what the solution manual says about Problem 37:
This important “toy model” considers the maximum size of a spherical cell before its core dies from lack of oxygen. One goal of biological physics is to show how physics constrains evolution. In this case, the physics of diffusion limits how large an animal can be before needing a circulatory system to move oxygen around.
How do you create an exam problem on this subject? Here's the trick: Do Problem 37 in class and then put a question on the exam identical to Problem 37 except “sphere” is replaced by “cylinder”. The problem is only slightly changed; just enough to determine if the student is solving the problem from first principles or merely memorizing. In addition, nerve and muscle fibers are cylindrical, so the revised problem may provide an even better model for those cells. Depending on the mathematical abilities of your students, you may need to provide students with the Laplacian in cylindrical coordinates. (If the exam is open book then they can find the Laplacian in Appendix L).

Here’s a second example: Chapter 1 considers viscous flow in a tube; Poiseuille flow. On the exam, ask the student to analyze viscous flow between two stationary plates.
Section 1.17
Problem 36 ½. Consider fluid flow between two stationary plates driven by a pressure gradient. The pressure varies in the x direction with constant gradient dp/dx, the plates are located at y = +L and y = -L, and the system is uniform in the z direction with width H. The fluid has viscosity η.

(a) Draw a picture the geometry.
(b) Consider a rectangular box of fluid centered at the origin and derive a differential equation like Eq. 1.35 governing the velocity vx(y).

(c) Solve this differential equation to determine vx(y), analogous to Eq. 1.37. Assume a no-slip boundary condition at the surface of each plate. Plot vx(y) versus y.

(d) Integrate the volume fluence and find the total flow i. How does i depend on the plate separation, 2L? How does this compare with the case of flow in a tube?
An interesting feature of this example is that i depends on the third power of L, whereas for a tube it depends on the fourth power of the radius. Encourage the student to wonder why.

Third example: A problem in Chapter 7 compares three different functions describing the strength-duration curve for electrical stimulation. On your exam, have the students analyze a fourth case.
Section 7.10

Problem 46 ½. Problem 46 analyzes three possible functions that could describe the strength-duration curve, relating the threshold current strength required for neural excitation, i, to the stimulus pulse duration, t. Consider the function i = A/tan-1(t/B). Derive expressions for the rheobase iR and chronaxie tC in terms of A and B. Write the function in the form used in Problem 46. Plot i versus t.
And still more: Problem 32 in Chapter 8 examines magnetic stimulation of a nerve axon using an applied electric field Ei(x) = E0 a2/(x2 + a2). Give a similar problem on your exam but use a different electric field, such as Ei(x) = E0 exp(-x2/a2).

And yet another: Chapter 10 examines the onset of cardiac fibrillation and chaos. The action potential duration APD is related to the diastolic interval (time from the end of the previous action potential to the start of the following one) DI by the restitution curve. Have the student repeat Problem 41 but using a different restitution curve: APDi+1 = 300 DIi/(DIi + 100).

Final example: Problem 36 in Chapter 9 asks the student to calculate the electrical potential inside and outside a spherical cell in the presence of a uniform electric field (Figure 9.19). On your exam, make the sphere into a cylinder.

I think you get the point. On an exam, repeat one the of homework problems in IPMB, but with a twist. Change the problem slightly, using a new function or a modified geometry. You will be able to test the knowledge and understanding of the student without springing any big surprises on the exam. Many problems in IPMB that could be modified in this way.

Warning: This trick doesn’t always work. For instance, in Chapter 1 if you try to analyze fluid flow perpendicular to a stationary object, you run into difficulties when you change the sphere of Problem 46 into a cylinder. The cylindrical version of this problem has no solution! The lack of a solution for low Reynolds number flow around a cylinder is known as Stokes’ Paradox. In that case, you’re just going to have to think up your own exam question.

Friday, October 5, 2018

A Bidomain Model for the Extracellular Potential and Magnetic Field of Cardiac Tissue

My research notebooks from graduate school.
My research notebooks from graduate school.
In graduate school, I worked with John Wikswo measuring the magnetic field of a nerve axon. We isolated a crayfish axon and threaded it through a wire-wound ferrite-core toroid immersed in saline. As the action currents propagated by, they produced a changing magnetic field that induced a signal in the toroid by Faraday induction. Ampere’s law tells us that the signal is proportional to the net current through the toroid, which is the sum of the intracellular current and the fraction of the current in the saline that passes through the toroid, called the return current.

For my PhD dissertation, Wikswo had me make similar measurements on strands of cardiac tissue, such as a papillary muscle. The instrumentation was the same as for the nerve, but the interpretation was different. Now the signal had three sources: the intracellular current, the return current in the saline, and extracellular current passing through the interstitial space within the muscle called the “interstitial return current.” Initially neither Wikswo nor I knew how to calculate the interstitial return current, so we were not sure how to interpret our results. As I planned these experiments, I recorded my thoughts in my research notebook. The July 26, 1984 entry stressed that “Understanding this point [the role of interstitial return currents] will be central to my research and deserves much thought.”

Excerpt from the July 26, 1984 entry in my Notebook 8, page 64.
When working on nerves, I had studied articles by Robert Plonsey and John Clark, in which they calculated the extracellular potential in the saline from the measured voltage across the axon’s membrane: the transmembrane potential. I was impressed by this calculation, which involved Fourier transforms and Bessel functions (see Chapter 7, Problem 30, in Intermediate Physics for Medicine and Biology). I had used their result to calculate the magnetic field around an axon (see Chapter 8, Problem 16 in IPMB), so I set out to extend their analysis again to include interstitial return currents.

Page 1 of Notebook 9, from Sept 13, 1984.
Page 1 of Notebook 9, from Sept 13, 1984.
The key was to use the then-new bidomain model, which accounts for currents in both the intracellular and interstitial spaces. The crucial advance came in September 1984 after I read a copy of Les Tung’s PhD dissertation that Wikswo had loaned me. After four days of intense work, I had solved the problem. My results looked a lot like those of Clark and Plonsey, except for a few strategically placed additional factors and extra terms.
Excerpt from Notebook 9, page 13, the Sept 16, 1984 entry.
First page of Roth and Wikswo (1986) IEEE Trans. Biomed. Eng., 33:467–469.
First page of Roth and Wikswo (1986).
Wikswo and I published these results as a brief communication in the IEEE Transactions on Biomedical Engineering.
Roth, B. J. and J. P. Wikswo, Jr., 1986, A bidomain model for the extracellular potential and magnetic field of cardiac tissue. IEEE Trans. Biomed. Eng., 33:467-469.

Abstract—In this brief communication, a bidomain volume conductor model is developed to represent cardiac muscle strands, enabling the magnetic field and extracellular potential to be calculated from the cardiac transmembrane potential. The model accounts for all action currents, including the interstitial current between the cardiac cells, and thereby allows quantitative interpretation of magnetic measurements of cardiac muscle.
Rather than explain the calculation in all its gory detail, I will ask you to solve it in a new homework problem.
Section 7.9
Problem 31½. A cylindrical strand of cardiac tissue, of radius a, is immersed in a saline bath. Cardiac tissue is a bidomain with anisotropic intracellular and interstitial conductivities σir, σiz, σor, and σoz, and saline is a monodomain volume conductor with isotropic conductivity σe. The intracellular and interstitial potentials are Vi and Vo, and the saline potential is Ve.
a) Write the bidomain equations, Eqs. 7.44a and 7.44b, for Vi and Vo in cylindrical coordinates (r,z). Add the two equations.
b) Assume Vi = σiz/(σiz+σoz) [A I0(kλr) + (σoz/σiz) Vm] sin(kz) and Vo = σiz/(σiz+σoz) [A I0(kλr) - Vm] sin(kz) where λ2 = (σiz+σoz)/(σir+σor). Verify that Vi - Vo equals the transmembrane potential Vm sin(kz). Show that Vi and Vo obey the equation derived in part a). I0(x) is the modified Bessel function of the first kind of order zero, which obeys the modified Bessel equation d2y/dx2 + (1/x) dy/dx = y. Assume Vm is independent of r.
c) Write Laplace’s equation (Eq. 7.38) in cylindrical coordinates. Assume Ve = B K0(kr) sin(kz). Show that Ve satisfies Laplace’s equation. K0(x) is the modified Bessel function of the second kind of order zero, which obeys the modified Bessel equation.
d) At r = a, the boundary conditions are Vo = Ve and σirVi/∂r + σorVo/∂r = σeVe/∂r. Determine A and B. You may need the Bessel function identities dI0/dx = I1 and dK0/dx = - K1, where I1(x) and K1(x) are modified Bessel functions of order one.
In the problem above, I assumed the potentials vary sinusoidally with z, but any waveform can be expressed as a superposition of sines and cosines (Fourier analysis) so this is not as restrictive as it seems.

In part b), I assumed the transmembrane potential was not a function of r. There is little data supporting this assumption, but I was stuck without it. Another assumption I could have used was equal anisotropy ratios, but I didn’t want to do that (and initially I didn’t realize it provided an alternative path to the solution).

The calculation of the magnetic field is not included in the new problem; it requires differentiating Vi, Vo, and Ve to find the current density, and then integrating the current to find the magnetic field via Ampere’s law. You can find the details in our IEEE TBME communication.

Some of you might be thinking “this is a nice homework problem, but how did you get those weird expressions for the intracellular and interstitial potentials used in part b)”? Our article gives some insight, and my notebook provides more. I started with Clark and Plonsey’s result, used ideas from Tung’s dissertation, and then played with the math (trial and error) until I had a solution that obeyed the bidomain equations. Some might call that a strange way to do science, but it worked for me.

I was very proud of this calculation (and still am). It played a role in the development of the bidomain model, which is now considered the state-of-the-art model for simulating the heart during defibrillation.

Wikswo and I carried out experiments on guinea pig papillary muscles to test the calculation, but the cardiac data was not as clean and definitive as our nerve data. We published it as a chapter in Cell Interactions and Gap Junctions (1989). The cardiac work made up most of my PhD dissertation. Vanderbilt let me include our three-page IEEE TBME communication as an appendix. It’s the most important three pages in the dissertation.

Coda: While browsing through my old research notebooks, I found this gem passed from Prof. Wikswo to his earnest but naive graduate student, who dutifully wrote it down in his research notebook for posterity.

Excerpt from Notebook 10, Page 62, January 3, 1985.
Excerpt from Notebook 10, Page 62.