Five New Homework Problems About Diffusion

Diffusion is a central concept in biological physics, but it's seldom taught in physics classes. Russ Hobbie and I cover diffusion in Chapter 4 of Intermediate Physics for Medicine and Biology.

The one-dimensional diffusion equation,

is one of the “big three” partial differential equations. Few analytical solutions to this equation exist. The best known is the decaying Gaussian (Eq. 4.25 in IPMB). Another corresponds to when the concentration is initially constant for negative values of x and is zero for positive values of x (Eq. 4.75). This solution is written in terms of error functions, which are integrals of the Gaussian (Eq. 4.74). I wonder: are there other simple examples illustrating diffusion? Yes!

In this post, my goal is to present several new homework problems that provide a short course in the mathematics of diffusion. Some extend the solutions already included in IPMB, and some illustrate additional solutions. After reading each new problem, stop and try to solve it!

Section 4.13

Problem 48.1. Consider one-dimensional diffusion, starting with an initial concentration of C(x,0)=C_o for x less than 0 and C(x,0)=0 for x greater than 0. The solution is given by Eq. 4.75

where erf is the error function.

(a) Show that for all times the concentration at x=0 is C₀/2.

(b) Derive an expression for the flux density, j = -D ∂C/∂x at x = 0. Plot j as a function of time. Interpret what this equation is saying physically. Note: 

Problem 48.2. Consider one-dimensional diffusion starting with an initial concentration of C(x,0)=C_o for |x| less than L and 0 for |x| greater than L.

(a) Plot C(x,0), analogous to Fig. 4.20.

(b) Show that the solution

obeys both the diffusion equation and the initial condition.

(c) Sketch a plot of C(x,t) versus x for several times, analogous to Fig. 4.22.

(d) Derive an expression for how the concentration at the center changes with time, C(0,t). Plot it.

Problem 48.3. Consider one-dimensional diffusion in the region of x between -L and L. The concentration is zero at the ends, C(±L,t)=0.

(a) If the initial concentration is constant, C(x,0)=C_o, this problem cannot be solved in closed form and requires Fourier series introduced in Chapter 11. However, often such a problem can be simplified using dimensionless variables. Define X = x/L, T = t/(L²/D) and Y = C/C_o. Write the diffusion equation, initial condition, and boundary conditions in terms of these dimensionless variables.

(b) Using these dimensionless variables, consider a different initial concentration Y(X,0)=cos(Xπ/2). This problem has an analytical solution (see Problem 25). Show that Y(X,T)=cos(Xπ/2) e^-π2T/4 obeys the diffusion equation as well as the boundary and initial conditions.

Problem 48.4. In spherical coordinates, the diffusion equation (when the concentration depends only on the radial coordinate r) is (Appendix L)

Let C(r,t) = u(r,t)/r. Determine a partial differential equation governing u(r,t). Explain how you can find solutions in spherical coordinates from solutions of analogous one-dimensional problems in Cartesian coordinates.

Problem 48.5. Consider diffusion in one-dimension from x = 0 to ∞. At the origin the concentration oscillates with angular frequency ω, C(0,t) = C_o sin(ωt).

(a) Determine the value of λ that ensures the expression

obeys the diffusion equation.

(b) Show that the solution in part (a) obeys the boundary condition at x = 0.

(c) Use a trigonometric identity to write the solution as the product of a decaying exponential and a traveling wave (see Section 13.2). Determine the wave speed.

(d) Plot C(x,t) as a function of x at times t = 0, π/2ω, π/ω, 3π/2ω, and 2π/ω.

(e) Describe in words how this solution behaves. How does it change as you increase the frequency?

Of the five problems, my favorite is the last one; be sure to try it. But all the problems provide valuable insight. That’s why we include problems in IPMB, and why you should do them. I have included the solutions to these problems at the bottom of this post (upside down, making it more difficult to check my solutions without you trying to solve the problems first).

Random Walks in Biology,
by Howard Berg.

Interested in learning more about diffusion? I suggest starting with Howard Berg’s book Random Walks in Biology. It is at a level similar to Intermediate Physics for Medicine and Biology.

After you have mastered it, move on to the classic texts by Crank (The Mathematics of Diffusion) and Carslaw and Jaeger (Conduction of Heat in Solids). These books are technical and contain little or no biology. Mathephobes may not care for them. But if you’re trying to solve a tricky diffusion problem, they are the place to go.

Enjoy!

The Mathematics of Diffusion.

The Conduction of Heat in Solids.

I told you these books are technical! (Page 45 of Crank)

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The Radial Isochron Clock

Section 10.8 of Intermediate Physics for Medicine and Biology describes the radial isochron clock, a toy model for electrical stimulation of nerve or muscle. Russ Hobbie and I write:

Many of the important features of nonlinear systems do not occur with one degree of freedom. We can make a very simple model system that displays the properties of systems with two degrees of freedom by combining the logistic equation for variable r with an angle variable θ that increases at a constant rate:

We can interpret (r,θ) as the polar coordinates of a point in the xy plane. When [time] t has increased from 0 to 1 the angle has increased from 0 to 2π, which is equivalent to starting again with θ = 0. This model system has been used by many authors. Glass and Mackey (1988) have proposed that it be called the radial isochron clock.

Fig. 10.19 of IPMB.

We use this model to analyze phase resetting. Let the clock run until it settles into a stable limit cycle, in which case the signal x(t) is a sinusoidal oscillation. Then apply a stimulus that suddenly increases x by an amount b (see Fig. 10.19 in IPMB) and observe the resulting dynamics. The system returns to its limit cycle, but with a different phase. The first plot in the figure below shows the period T/T₀ of the oscillator just after a stimulus is applied at T_S/T₀; it's the same illustration as in Fig 10.20b of IPMB. Something dramatic appears to be happening at T_S/T₀ = 0.5. What's going on?

The radial isochron clock for different stimulus times and strengths.

The problem with a plot of T/T₀ versus T_S/T₀ is that I have difficulty relating it to the behavior of the signal as a function of time, x(t). Above I plot x versus t for four cases:

• T_S/T₀ = 0.25, b = 0.95 (blue dot). In this case, the stimulus is applied soon after the peak when the signal is decreasing. The sudden jump in x increases the signal so it has further to fall (it must recover lost ground), delaying its descent. As a result is the signal is a behind (is shifted to the right of) the signal that would have been produced had there been no stimulus (red dashed). The figure for b = 1.05 is almost indistinguishable from b = 0.95, so I won’t show it.
• T_S/T₀ = 0.75, b = 0.95 (red dot). The stimulus is applied after the trough when the signal is increasing. The stimulus helps it rise, so it reaches its peak earlier (is shifted to the left) compared to the signal with no stimulus. Again, the figure for b = 1.05 is similar.
• T_S/T₀ = 0.50, b = 0.95 (green dot). When we apply the stimulus near the bottom of the trough, the behavior depends sensitively on stimulus strength b. If b were exactly one and it was applied precisely at the minimum, the result would be x = 0 forever. This would be an unstable equilibrium, like balancing a pencil on its tip. If b was not exactly one, then the key issue is if the signal starts slightly negative (in phase with the unperturbed signal) or slightly positive (out of phase). For b = 0.95, the signal moves to a slightly negative value that corresponds to a trough, meaning that the resulting signal is in phase with the unperturbed signal.
• T_S/T₀ = 0.50, b = 1.05 (yellow dot). If b was a little stronger, then the stimulus moves x to a slightly positive value corresponding to a peak, meaning that the resulting signal is out of phase with the unperturbed signal. Because T/T₀ = 1.5 is equivalent to T/T₀ = 0.5 (the phase just wraps around), the jump of T/T₀ in the top frame does not correspond to a discontinuous physical change.

The drama at T_S/T₀ = 0.5 and b = 1 arises because the stimulus nearly zeros out the signal. The phase of the signal changes from zero to 180 degrees as b changes from less than one to greater than one, but the amplitude of the signal r goes to zero, so the variables x and y change in a continuous way. Some of the homework problems for Section 10.8 in IPMB ask you to explore this on your own. Try them.

The moral of the story is that an abstract illustration—such as Fig. 10.20b in Intermediate Physics for Medicine and Biology—summarizes the behavior of a nonlinear system, but it can’t replace intuition about how the system behaves as a function of time. You need to understand your system “in your gut.” This isn’t true just for the radial isochron clock; it's true for any system. Forget this lesson at your peril!

Anisotropy in Bioelectricity and Biomechanics

Structures: Or Why Things Don't Fall Down,
by James Gordon.

In this third and final post about James Gordon’s book Structures: Or Why Things Don’t Fall Down, I analyze shear.

If tension is about pulling and compression is about pushing, then shear is about sliding. In other words, a shear stress measures the tendency for one part of a solid to slide past the next bit: the sort of thing which happens when you throw a pack of cards on the table or jerk the rug from under someone’s feet. It also nearly always occurs when anything is twisted, such as one’s ankle or the driving shaft of a car…

In Intermediate Physics for Medicine and Biology, Russ Hobbie and I introduce the shear stress, shear strain, and shear modulus, but we don’t do much with them. After Gordon defines these quantities, however, he launches into to a fascinating discussion about shear and anisotropy: different properties in different directions.

Cloth is one of the commonest of all artificial materials and it is highly anisotropic….If you take a square of ordinary cloth in your hands—a handkerchief might do—it is easy to see that the way in which it deforms under a tensile load depends markedly upon the direction in which you pull it. If you pull, fairly precisely, along either the warp or the weft threads, the cloth will extend very little; in other words, it is stiff in tension. Furthermore, in this case, if one looks carefully, one can see that there is not much lateral contraction as a result of the pull…Thus the Poisson’s ratio…is low.

However, if you now pull the cloth at 45° to the direction of the threads—as a dressmaker would say, ‘in the bias direction’—it is much more extensible; that is to say, Young’s modulus in tension is low. This time, though, there is a large lateral contraction, so that, in this direction, the Poisson’s ratio is high.

This analysis led me to ruminate about the different role of anisotropy in bioelectricity versus biomechanics. The mechanical behavior Gordon describes is different than the electrical conductivity of a similar material. As explained in Section 7.9 of IPMB, the current density and electric field in an anisotropic material are related by a conductivity tensor (Eq. 7.39). A cloth-like material would have the same conductivity parallel and perpendicular to the threads, and the off-diagonal terms in the tensor would be zero. Therefore, the conductivity tensor would be proportional to the identity matrix. Homework Problem 26 in Chapter 4 of IPMB shows how to write the tensor in a coordinate system rotated by 45°. The result is that the conductivity is the same in the 45° direction as it is along and across the fibers. As far as its electrical properties are concerned, cloth is isotropic!

I spent much of my career analyzing anisotropy in cardiac muscle, and I was astonished when I realized how different anisotropy appears in mechanics compared to electricity. Gordon’s genius was to analyze a material, such as cloth, that has identical properties in two perpendicular directions, yet is nevertheless mechanically anisotropic. If you study muscle, which has different mechanical and electrical properties along versus across the fibers, the difference between mechanical and electrical anisotropy is not as obvious.

This difference got me thinking: is the electrical conductivity of a cloth-like material really isotropic? Well, yes, it must be when analyzed in terms of the conductivity tensor. But suppose we look at the material microscopically. The figure below shows a square grid of resistors that represents the electrical behavior of tissue. Each resistor is the same, having resistance R. To determine its macroscopic resistance, we apply a voltage difference V and determine the total current I through the grid. The current must pass through N vertical resistors one after the other, so the total resistance through one vertical line is NR. However, there are N parallel lines, reducing the total resistance by a factor of 1/N. The net result: the resistance of the entire grid is the resistance of a single resistor, R.

The electrical behavior of tissue represented by a grid of resistors.

Now rotate the grid by 45°. In this case, the current takes a tortuous path through the tissue, with the vertical path length increasing by the square root of two. However, more vertical lines are present per unit length in the horizontal direction (count ’em). How many more? The square root of two more! So, the grid has a resistance R. From a microscopic point of view, the conductivity is indeed isotropic.

The electrical behavior of tissue represented by a rotated grid of resistors.

Next, replace the resistors by springs. When you pull upwards, the vertical springs stretch with a spring constant k. Using a similar analysis as performed above, the net spring constant of the grid is also k.

The mechanical behavior of tissue represented by a grid of springs.

Now analyze the grid after it's been rotated by 45°. Even if the spring constant were huge (that is, if the springs were very stiff), the grid would stretch by shearing the rotated squares into diamonds. The tissue would have almost no Young’s modulus in the 45° direction and the Poisson's ratio would be about one; the grid would contract horizontally as it expanded vertically (even if the springs themselves didn't stretch at all). This arises because the springs act as if they're connected by hinges. It reminds me of those gates my wife and I installed to prevent our young daughters from falling down the steps. You would need horizontal struts or vertical ties to prevent such shearing.

The mechanical behavior of tissue represented by a rotated grid of springs.

In conclusion, you can't represent the mechanical behavior of an isotropic tissue using a square grid of springs without struts or ties. Such a microscopic structure corresponds to cloth, which is anisotropic. A square grid fails to capture properly the shearing of the tissue. You can, however, represent the electrical behavior of an isotropic tissue using a square grid of resistors without “electrical struts or ties.”

Gordon elaborated on the anisotropic mechanical properties of cloth in his own engaging way.

In 1922 a dressmaker called Mlle Vionnet set up shop in Paris and proceeded to invent the “bias cut.” Mlle Vionnet had probably never heard of her distinguished compatriot S. D. Poisson—still less of his ratio—but she realized intuitively that there are more ways of getting a fit than by pulling on strings…if the cloth is disposed at 45°…one can exploit the resulting large lateral contraction so as to get a clinging effect.

Wikipedia adds:

Vionnet's bias cut clothes dominated haute couture in the 1930s, setting trends with her sensual gowns worn by such internationally known actresses as Marlene Dietrich, Katharine Hepburn, Joan Crawford and Greta Garbo. Vionnet’s vision of the female form revolutionized modern clothing, and the success of her unique cuts assured her reputation.

Structures: Or Why Things Don't Fall Down, by J. E. Gordon.

The Pitfalls of Using Handbooks and Formulae

Structures: Or Why Things Don't Fall Down, by J. E. Gordon.

Last week I discussed James Gordon’s book Structures: Or Why Things Don’t Fall Down. The book contains several appendices. The first appendix is ostensibly about using handbooks and formulas to make structural calculations.

Over the last 150 years the theoretical elasticians have analysed the stresses and deflections of structures of almost every conceivable shape when subjected to all sorts and conditions of loads…Fortunately a great deal of this information has been reduced to a set of standard cases or examples the answers to which can be expressed in the form of quite simple formulae.

Then, to my surprise, Gordon changes tack and warns about pitfalls when using these formulas. His counsel, however, applies to all calculations, not just mechanical ones. In fact, his advice is invaluable for any young scientist or engineer. Below, I quote parts of this appendix. Read carefully, and whenever you encounter a word specific to mechanics substitute a general one, or one related to your own field.

[Formulae] must be used with caution.

Appendix 1 of Structures.

1. Make sure that you really understand what the formula is about.
2. Make sure that it really does apply to your particular case.
3. Remember, remember, remember, that these formulae take no account of stress concentrations or other special local conditions.

After this, plug the appropriate loads and dimensions into the formula—making sure that the units are consistent and that the noughts are right. [I’m not sure what “noughts” are, but I think the Englishman Gordon is saying to make sure the decimal point is in the right place.] Then do a little elementary arithmetic and out will drop a figure representing a stress or a deflection.

Now look at this figure with a nasty suspicious eye and think if it looks and feels right. In any case you had better check your arithmetic; are you sure that you haven’t dropped a two?...

If the structure you propose to have made is an important one, the next thing to do, and a very right and proper thing, is to worry about it like blazes. When I was concerned with the introduction of plastic components into aircraft I used to lie awake night after night worrying about them, and I attribute the fact that none of these components ever gave trouble almost entirely to the beneficent effects of worry. It is confidence that causes accidents and worry which prevents them. So go over your sums not once or twice but again and again and again.

Structures: Or Why Things Don't Fall Down.

This is the attitude I try to instill in my students when teaching from Intermediate Physics for Medicine and Biology. I implore them to think before they calculate, and then think again to judge if their answer makes sense. Students sometimes submit an answer to a homework problem (almost always given to five or six significant figures) that is absurd because they didn't look at their answer with a “nasty suspicious eye.” I insist they "remember, remember, remember" the assumptions and limitations of a mathematical model and its resulting formulas. Maybe Gordon goes a little overboard with his “night after night” of lost sleep, but at least he cares enough about his calculation to wonder “again and again and again” if it is correct. A little worry is indeed a “right and proper thing.”

Who would of expected such wisdom tucked away in an appendix about handbooks and formulae?

Structures: Or Why Things Don't Fall Down

Structures: Or Why Things Don't Fall Down,
by James Gordon.

When I was in graduate school, I read a fascinating book by James Gordon titled Structures: Or Why Things Don’t Fall Down. It showed me to how engineers think about mechanics. Recently, I reread Structures and read for the first time its sequel The New Science of Strong Materials: Or Why You Don’t Fall Through the Floor. I enjoyed both books thoroughly.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss two mechanical properties of a material: stiffness and strength. Stiffness describes how much a material lengthens when pulled (that is, strains when stressed), and is quantified by its Young’s modulus. Strength measures how much stress a material can withstand before failing. Gordon summarizes these ideas succinctly.

A biscuit is stiff but weak, steel is stiff and strong, nylon is flexible and strong, raspberry jelly is flexible and weak. The two properties together describe a solid about as well as you can reasonably expect two figures to do.

Just two figures, however, are not sufficient to characterize a material, especially when it's used to build a structure.

The worst sin in an engineering material is not lack of strength or lack of stiffness, desirable as these properties are, but lack of toughness, that is to say, lack of resistance to the propagation of cracks.

Toughness is opposite to brittleness, and is related to but not identical to ductility. It is quantified by the work of fracture—the energy needed to produce a new surface by propagation of a crack through the material—a concept introduced by Alan Griffith during his research on fracture mechanics.

A strained material contains strain energy which would like to be released just as a raised weight contains potential energy and would like to fall…The relief of strain energy …. [is] proportional to the square of the crack length…On the other side of the account book is the surface energy…needed to form the new surfaces and clearly increases as only the first power of the depth of the crack…When the crack is shallow it is consuming more energy as surface energy than it is releasing as relaxed strain energy and therefore conditions are unfavorable for it to propagate. As the crack gets longer however these conditions are reversed and beyond the ‘critical Griffith length’ l_g the crack is producing more energy than it is consuming, so it may start to run away in an explosive manner.

In heterogeneous materials, internal interfaces act as crack stoppers. This makes wood exceptionally tough; Its cellular, fibrous structure prevents a crack from propagating. Toughness is important in biological materials that must undergo large strains without breaking. Wood is not dense (compared to, say, steel), so you get lots of toughness for little weight, which is one reason wood is so popular as a building material. On the other hand, wood isn’t very stiff, and it swells, burns, and rots.

Gordon provides deep insight into the behavior of structures and materials. Consider the stress in the wall of a cylindrical pressure vessel (a long cylinder with spherical end caps). The circumferential stress in the cylinder's wall is given by the Law of Laplace (see IPMB, Chapter 1, Problem 18). The longitudinal stress is equal to the stress in the end caps (the stress in a sphere is two times that in a cylinder, see Problem 19). Thus

the circumferential stress in the wall of a cylindrical pressure vessel is twice the longitudinal stress...One consequence of this must have been observed by everyone who has ever fried a sausage. When the filling inside the sausage swells and the skin bursts, the split is almost always longitudinal.

Then Gordon develops this theme.

Figure 5 from Structures.

If we make a tube or cylinder from such a material [as rubber] and then inflate it, by means of an internal pressure, so as to involve a circumferential strain of 50 per cent or more, then the inflation or swelling process will become unstable, and the tube will bulge out...into a spherical protrusion which a doctor would describe as an “aneurism”....Since veins and arteries do, in fact, generally operate at strains around 50 per cent, and since, as any doctor will tell you, one of the conditions it is most desirable to avoid in blood-vessels is the production of aneurisms, any sort of rubbery elasticity is quite unsuitable....The only sort of elasticity which is completely stable under fluid pressures at high strains is that which is represented by Figure 5 [showing the stress increasing exponentially with the strain]. With minor variations, this shape of stress-strain curve is very common indeed for animal tissue....Materials with this [exponential] type of stress-strain curve are extremely difficult to tear. One reason is, perhaps, that the strain energy stored under such a curve--and therefore available to propagate fracture...is minimized."

He continues

Perhaps partly for these reasons the molecular structure of animal tissue does not often resemble that of rubber or artificial plastics. Most of these natural materials are highly complex, and in many cases they are of a composite nature, with at least two components; that is to say, they have a continuous phase or matrix which is reinforced by means of strong fibres of filaments of another substance. In a good many animals this continuous phase or matrix contains a material called 'elastin', which has a very low modulus and a [flat] stress-strain curve...The elastin is, however, reinforced by an arrangement of bent and zig-zagged fibres of collagen...a protein, very much the same as tendon, which has a high modulus...Because the reinforcing fibres are so much convoluted, when the material is in its resting or low-strain condition they contribute very little to its resistance to extension, and the initial elastic behavior is pretty well that of elastin. However, as the composite tissue stretches the collagen fibres begin to come taut; thus in the extended state the modulus of the material is that of the collagen, which more or less accounts for Figure 5.

As you probably can tell, Gordon writes wonderfully and explains mechanics so it's understandable to a layman. His writing is a model of clarity.

Structures: Or Why Things Don't Fall Down.

Structures was my first exposure to continuum mechanics, but certainly not my last. I was a member of the Mechanical Engineering Section when I worked at the National Institutes of Health, so I was surrounded by outstanding mechanical engineers. My friend Peter Basser—himself a mechanical engineer—would lend me his books, and I recall reading classics such as Love’s A Treatise on the Mathematical Theory of Elasticity and Schlichting’s Boundary Layer Theory. I was impressed by Basser’s model of infusion-induced swelling in the brain and Richard Chadwick’s studies of cardiac biomechanics (Richard was another member of our Mechanical Engineering Section). In many ways, NIH provided a liberal education in physics applied to biology and medicine.

Throughout my career, most of my research has focused on bioelectricity and biomagnetism. Recently, however, I have been working on problems in biomechanics. But that is another story.

Computerized Transverse Axial Scanning (Tomography)

“Computerized Transverse Axial
Scanning (Tomography).”

Section 16.8 of Intermediate Physics for Medicine and Biology discusses computed tomography. Russ Hobbie and I describe the history of this technique.

The Nobel Prize in Physiology or Medicine was shared in 1979 by a physicist, Allan Cormack, and an engineer, Godfrey Hounsfield…. Hounsfield, working independently [of Cormack], built the first clinical [computed tomography] machine, which was installed in 1971. It was described in 1973 in the British Journal of Radiology. The Nobel Prize acceptance speeches (Cormack 1980; Hounsfield 1980) are interesting to read. A neurologist, William Oldendorf, had been working independently on the problem but did not share in the Nobel Prize…

Oddly, Russ and I did not include Hounsfield’s 1973 paper in our list of references. I decided to dig it up and have a look. The reference and abstract is:

Hounsfield GN (1973) Computerized transverse axial scanning (tomography): Part I. Description of the system. Br J Radiol 46:1016-1022

This article describes a technique in which X-ray transmission readings are taken through the head at a multitude of angles: from these data, absorption values of the material contained within the head are calculated on a computer and presented as a series of pictures of slices of the cranium. The system is approximately 100 times more sensitive than conventional X-ray systems to such an extent that variations in soft tissues of nearly similar density can be displayed.

A dozen comments:

1. This is Hounsfield’s most highly cited paper, with 4667 citations according to Google Scholar. That's a respectable number (ten times more than any of my papers have), yet seems curiously small for a Nobel Prize-winning advance.
2. Hounsfield’s paper is the first of a trilogy. Hounsfield is not a coauthor on the other two; they report clinical studies using the new technique.
3. Hounsfield lists his institution as “Central Research Laboratories of EMI Limited.” EMI is famous in the music industry; it is the recording label responsible for the early hits of the Beatles.
4. Hounsfield’s paper has only three references: two to his own preliminary reports and one to an article by Oldendorf. He didn’t cite Cormack’s papers.
5. Hounsfield sounds most impressed not by recreating three-dimensional images from two-dimensional projections (which to me is the big advance) but instead by the increased sensitivity of the technique to small differences in x-ray absorption coefficient.
6. Figure 3, illustrating the scanning device and sequence, is similar to Fig. 16.25 in IPMB.

   

   Fig. 16.25 of IPMB.

7. Hounsfield measured 160 points in each translation and performed 180 rotations. Each two-dimensional image was represented by an 80 × 80 grid of pixels.
8. The reconstruction method was different from the two Russ and I analyze in Chapter 12 of IPMB: i) Fourier transform reconstruction and ii) filtered back-projection. Instead, Hounsfield just fit his data using the least squares method (see Section 11.1 of IPMB). Hounsfield writes “Each beam path [in the CT scan], therefore, forms one of a series of 28,800 simultaneous equations, in which there are 6,400 variables and, providing that there are more equations than variables, the values of each [pixel] …. can be solved.”
9. The Hounsfield unit was introduced in Fig. 9, but he did not, of course, call it that. Interestingly, his definition is different than what is used today. Equation 16.25 in IPMB gives the Hounsfield unit as
   
   
   where μ_tissue and μ_water are x-ray attenuation coefficients. In his paper, Hounsfield defines the unit the same way, except he replaces 1000 by 500.
10. The article describes preliminary experiments using an iodine-containing contrast agent and digital subtraction, analogous to Fig. 16.23 in IPMB.
11. The computer equipment pictured in Hounsfield’s paper look big and clunky today. I can only guess what paltry computer power he had available for these first reconstructions.
12. I love the British Journal of Radiology, also known as BJR. (What journal did you think that Bradley John Roth would like?)

I’ll conclude with Hounsfield’s final paragraph. To my ear, it sounds like classic British understatement.

It is possible that this technique may open up a new chapter in X-ray diagnosis. Previously, various tissues could only be distinguished from one another if they differed appreciably in density. In this procedure absolute values of the absorption coefficient of the tissues are obtained. The increased sensitivity of computerized X-ray section scanning thus enables tissues of similar density to be separated and a picture of the soft tissue structure within the cranium to be built up.

Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography

Page 394 of IPMB.

Section 14.7 of Intermediate Physics for Medicine and Biology discuses optical coherence tomography (OCT). Russ Hobbie and I write

Optical range measurements using the time delay of reflected or backscattered light from pulses of a few femtosecond (10^-15 s) duration can be used to produce images similar to those of ultrasound…Since it is difficult to measure time intervals that short, most measurements are done using interference properties of the light. Optical coherence tomography is conceptually similar to range measurements but uses interference measurements…It is widely used in ophthalmology….

Fig. 14.15 of IPMB.

The basic apparatus [for OCT] is shown in Fig. 14.15….The light pulse travels over an optical fiber to a 50/50 beam splitter. Part travels to the sample, where it is reflected back to the 50/50 coupler and then to the detector. The other half of the light goes to the reference mirror, where it is also reflected back to the detector. Changing the position of the reference mirror changes the depth of the image plane in the sample….

Fig. 14.17 of IPMB.

It is possible to make many kinds of images. Fig. 14.17 shows the parabolic velocity profile of blood flowing in a retinal blood vessel 176 μm diameter. It was obtained by measuring the Doppler shift in light scattered from moving blood cells.

Yazdanfar et al. (2000).

Figure 14.17 is from the paper “Imaging and Velocimetry of the Human Retinal Circulation with Color Doppler Optical Coherence Tomography,” by Siavash Yazdanfar, Andrew Rollins, and Joseph Izatt (Optical Letters, Volume 25, Pages 1448–1450, 2000). [For some reason Russ and I did not include the year in our citation—another item for the errata].

Abstract: Noninvasive monitoring of blood flow in retinal microcirculation may elucidate the progression and treatment of ocular disorders, including diabetic retinopathy, age-related macular degeneration, and glaucoma. Color Doppler optical coherence tomography (CDOCT) is a technique that allows simultaneous micrometer-scale resolution cross-sectional imaging of tissue microstructure and blood flow in living tissues. CDOCT is demonstrated for the first time in living human subjects for bidirectional blood-flow mapping of retinal vasculature.

I like Fig 14.17 because it combines ideas from Chapters 1, 11, 13, and 14 of IPMB. It also highlights the excellent spatial resolution you can obtain with OCT.

The illustration below shows the geometry associated with Fig. 14.17. The light is reflected by blood cells moving at speed v, causing a Doppler shift in its frequency. By adjusting the reference mirror, different depths are selected. The vessel makes an angle θ relative to the incident light. As the depth is scanned across the vessel, the Doppler shift determines the blood flow profile.

The geometry associated with IPMB Fig. 14.17.

To help the reader learn more about the physics of OCT and Fig. 14.17, I have written two new homework problems. The solutions are included at the bottom of the post (upside down, to encourage readers to solve the problems themselves first). Enjoy!

Section 14.7

Problem 24 ⅓. This problem and the next explore the physics behind Fig. 14.17, which shows the velocity profile in a blood vessel measured using color Doppler optical coherence tomography. The data is based on an article by Yazdanfar et al. (2000). For this problem ignore the index of refraction of the tissue and assume θ = 60°.

(a) If the wavelength, λ, of the incident light is 832 nm and the wavelength bandwidth, Δλ, is 15 nm, determine the frequency, f, and the frequency bandwidth, Δf, in THz.

(b) The coherence time, τ_coh, is equal to 1/(πΔf). Calculate τ_coh in fs, and the coherence length, x₂ – x₁, in microns. The coherence length determines the spatial resolution of the measurement.

(c) Use Eq. 13.42 to derive an expression for the speed of blood flow in the direction of the light, v', in terms of the Doppler frequency shift, df. Assume that the speed of light, c, is much greater than v'. Calculate v' if df = 4 kHz. The Doppler technique measures the component of motion in the direction of the light. Determine the speed v along the vessel.

Problem 24 ⅔. The Doppler shift, df, of OCT data as a function of depth z across a blood vessel is given below. For viscous flow in a tube (Sec. 1.17), the blood speed varies parabolically across the vessel cross section (Eq. 1.37). Fit a parabola to this data of the form df = Az² + Bz + C, and determine constants A, B, and C. Use these constants to find the peak value of df in this vessel, the location of the center of the vessel, and the vessel diameter (the width of the parabola when df = 0). The measured diameter corresponds to an oblique section at θ = 60°. Correct this result to get the true diameter.

z (mm)	df (kHz)
0.15	3.26
0.20	5.20
0.25	6.12
0.30	6.02
0.35	4.89
0.40	2.75

I will give the final word to Yazdanfar, Rollins, and Izatt, who conclude

In summary, CDOCT has been applied for what is believed to be the first time to retinal blood-flow mapping in the human eye. Depth-resolved quantification of retinal hemodynamics may prove helpful in understanding the pathogenesis of several ocular diseases. Unlike fluorescein angiography, CDOCT is entirely noninvasive and does not require dilation of the pupil. Furthermore, CDOCT operates at longer wavelengths than does laser Doppler velocimetry, so light exposure times can be safely increased. CDOCT is believed to be the first technique for determining, with micrometer-scale resolution, the depth, diameter, and flow rate of blood vessels within the living retina.

Solution to new Homework Problem 24 ⅓

Solution to new Homework Problem 24 ⅔.

Venkataranan Ramakrishna, Biological Physicist

Failed physicist?

I was reading the November issue of APS News (A publication of the American Physical Society) when I noticed an article titled “Failed Physicist? From Biologist Turned Nobel Laureate to Author.” The article was about Venkataraman Ramakrishnan, winner of the 2009 Nobel Prize in Chemistry for “studies of the structure and function of the ribosome.” He received a PhD in physics before switching to biology. In the article, he calls himself a “failed physicist.”

Many readers of Intermediate Physics for Medicine and Biology may be in a similar position of having been trained in physics but now learning biology. Ramakrishnan provides an interesting case study in how to make such a transition. I looked up his biographical statement on the Nobel Prize website, and I reproduce excerpts below. Changing fields is not easy, but it is possible, and can ultimately lead to groundbreaking research.

Choosing Basic Science

[When Ramakrishna was growing up in India, he was looking for a university for his undergraduate studies.] A faculty member in the physics department in [the University of] Baroda, S.K. Shah, told me about a brand new curriculum they were introducing for their undergraduate course. It began with the Berkeley Physics Course, and was supplemented by the Feynman Lectures on Physics before moving on to more specialized areas. I therefore decided to enroll in the B.Sc. course in physics in Baroda, my hometown. Since I was only 16 when I began this course, I sensed that my parents, especially my father, were relieved that I was not leaving home….

I found myself tremendously interested by the articles in biology in Scientific American, to which I have subscribed off and on through the years. It appeared that hardly a month went by without a major breakthrough in the life sciences, whereas physics was having a hard time making any fundamental progress. Certainly I felt that if I continued in physics, I would be doing boring and tedious calculations rather than making really interesting advances. The result was that I felt so frustrated that I withdrew from my thesis work and spent an inordinate amount of time on extracurricular activities….

[Ramakrishna eventually finished his BS in physics, and then obtained a PhD in Physics at Ohio University]…. By that time I had already decided I was going to switch to biology.

Transition to Biology

Since I hardly knew any biology, I felt I needed formal training of some sort. I could go to graduate school again, with the option of getting a second Ph.D. or go to medical school, which was ironic since I had turned down the opportunity to do precisely that when I was younger. I took the MCAT …. but despite scoring in the 99th percentile in all the subjects, I only got one interview (at Yale) because I was not a U.S. citizen or even a permanent resident at that point…. However, I had also written to a number of graduate programs. Many of them said that they would not accept someone who already had a Ph.D. The chairman of the Molecular Biophysics and Biochemistry ... department at Yale, Franklin Hutchinson, wrote to me saying that while they could not take me as a graduate student, he would circulate my CV to faculty members for a potential postdoctoral position. Two of them responded: One was Don Engelman, and the other, ironically, was Tom Steitz, with whom I shared the Nobel Prize. Although I found their work very interesting, I thought doing a postdoc directly from a degree in physics would leave me with too narrow a background in biology to be an effective scientist. So when three schools accepted me into their graduate program, I chose to go to the University of California, San Diego (UCSD)…. During the first year, I did several lab rotations in biology and took as many undergraduate courses as I could possibly manage, including genetics, taught by Dan Lindsley, a well-known Drosophila geneticist, and biochemistry, where I was inspired by the brilliant and enthusiastic lectures of Paul Price.

In my second year [at UCSD], I settled down to do research in Mauricio Montal's lab. Mauricio had developed an ingenious method of incorporating conducting channels into lipid bilayers formed by bringing together two defined monolayers, and was thus doing single molecule biophysics at a time when nobody called it that. Around this time, however, I read an article in Scientific American by Don Engelman and Peter Moore about their ribosome work, and became interested in it. It also struck me that there was no longer any reason to continue on to obtain a second Ph.D. because I felt I had acquired the background I needed. I therefore wrote to Don Engelman, one of the two people from Yale who had responded to me earlier. Don was interested in membrane proteins, a subject I was already working on in Mauricio's lab. Don wrote back and said that he and Peter had a position open on their ribosome project, and I could always work on membrane-related projects once I got there. Peter arranged to meet me in San Diego in early 1978 and offered me a postdoctoral position soon afterwards. Thus began my lifelong interest in ribosomes….

In the APS News article, Ramakrishnan said something that could be the motto for IPMB.

Physics is a great training for every science because it teaches you quantitative and mathematical thinking, and that way of approaching problems is becoming increasingly important in every field, including biology.

Want to learn more? Ramakrishna has a new book out: Gene Machine: The Race to Dicipher the Secrets of the Ribosome. It's on my list of books to read. Below are a couple videos in which you can hear from Ramakrishna himself. Enjoy!