The Bystander Effect and a Supralinear Dose-Response Curve

When discussing the biological effects of radiation in Intermediate Physics for Medicine and Biology, Russ Hobbie and I describe the bystander effect. 

Ionization damage is not the entire story. The bystander effect in radiobiology refers to the “induction of biological effects in cells that are not directly traversed by a charged particle, but are in close proximity to cells that are” (Hall 2003; Hall and Giaccia 2012).

Hall (2003) “The Bystander Effect,”
Health Physics, 85:31–35.

I sometimes reread the references we cite, looking for interesting tidbits to share in this blog. Below is the abstract to the 2003 article by Eric Hall about the bystander effect (Health Physics, Vol. 85, Pages 31–35).

The bystander effect refers to the induction of biological effects in cells that are not directly traversed by a charged particle. The data available concerning the bystander effect fall into two quite separate categories, and it is not certain that the two groups of experiments are addressing the same phenomenon. First, there are experiments involving the transfer of medium from irradiated cells, which results in a biological effect in unirradiated cells. Second, there is the use of sophisticated single particle microbeams, which allow specific cells to be irradiated and biological effects studied in their neighbors; in this case communication is by gap junction. Medium transfer experiments have shown a bystander effect for cell lethality, chromosomal aberrations and cell cycle delay. The type of cell, epithelial vs. fibroblast, appears to be important. Experiments suggest that the effect is due to a molecule secreted by irradiated cells, which is capable of transferring damage to distant cells. Use of a single microbeam has allowed the demonstration of a bystander effect for chromosomal aberrations, cell lethality, mutation, and oncogenic transformation. When cells are in close contact, allowing gap junction communication, the bystander effect is a much larger magnitude than the phenomenon demonstrated in medium transfer experiments. A bystander effect has been demonstrated for both high- and low-LET radiations but it is usually larger for densely ionizing radiation such as alpha particles. Experiments have not yet been devised to demonstrate a comparable bystander effect on a three-dimensional normal tissue. Bystander studies imply that the target for the biological effects of radiation is larger than the cell and this could make a simple linear extrapolation of radiation risks from high to low doses of questionable validity.

Our discussion of the bystander effect in IPMB closely parallels that given by Hall. But in his article Hall wrote this

This bystander effect can be induced by radiation doses as low as 0.25 mGy and is not significantly increased up to doses of 10 Gy

and this

When 10% of the cells on a dish are exposed to two or more alpha particles, the resulting frequency of induced oncogenic transformation is indistinguishable from that when all the cells on the dish are exposed to the same number of alpha particles.

What?!? The bystander effect is not increased when the dose increases by a factor of forty thousand? You can fire three alpha particles per cell at only one out of every ten cells and the response is the same as if you fire three alpha particles per cell at every cell? I don’t understand. 

Another surprising feature of the data is that all these changes are different than they are for zero dose. That means the dose-response curve must start at zero, jump up to a significant level, and then be nearly flat. Such a dose-response behavior is different than that predicted by the linear no-threshold model (linearly extrapolating from what is known about radiation risk at high doses down to low doses). Indeed, that is what Hall is hinting at in the last sentence of his abstract.

Below is a slightly modified version of Figure 16.51 from IPMB. It shows different assumptions for how tissue responds to a dose of radiation. Data exists (the data points with error bars) for moderate doses, but what happens at very low doses? The standard dogma is the linear no-threshold model (LNT), which is a linear extrapolation from the data at moderate doses down to zero. Some believe there is a threshold below which low doses of radiation have no effect, and a few researchers even claim that very low doses can be beneficial (hormesis). Hall’s hypothesis is that the bystander effect would have a larger impact at low doses than predicted by the linear no-threshold model. It would be a supralinear effect. Based on Fig. 6 of Hall’s article, the effect would be dramatic, like the red bystander curve I added to our figure below. 

Possible responses of tissue to various doses of radiation. The two lowest-dose
measurements are shown. With zero dose there is no excess effect.
Adapted from Fig. 16.51 of Intermediate Physics for Medicine and Biology.

Previously in this blog, I have expressed skepticism of the linear no-threshold model, leaning more toward a threshold model in which very low doses have little or no effect. Hall’s claim implies the opposite: very low doses would have a bigger effect than expected from the linear no-threshold model. What do I make of this? First, let me say that I’m speculating in a field that’s outside my area of expertise; I’m not a radiation biologist. But to me, it seems odd to say that zapping 10% of the cells with alpha particles will have the same effect as zapping 100% of the cells with alpha particles. And it sounds strange to say that the response is not significantly affected by increasing the dose by a factor of 40,000. I don't usually ask for assistance from my readers, but if anyone out there has an explanation for how this dramatic supralinear effect works, I would appreciate hearing it. 

One of the most important questions raised in IPMB is: What is the true risk from low doses of radiation. The bystander effect is one factor that goes into answering this question. We need to understand it better.

The career of Eric Hall. https://www.youtube.com/watch?v=eh6erz_Ds0M

 

Klein-Nishina Formula for Polarized Photons

In Chapter 15 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss Compton Scattering. An incident photon scatters off a free electron, producing a scattered photon and a recoiling electron. We write

The inclusion of dynamics, which allows us to determine the relative number of photons scattered at each angle, is fairly complicated. The quantum-mechanical result is known as the Klein-Nishina formula (Attix 1986). The result depends on the polarization of the photons. For unpolarized photons, the cross section per unit solid angle for a photon to be scattered at angle θ is

 

where

is the classical radius of the electron. [The variable x is the ratio of the incident photon’s energy to the rest energy of the electron.]

What happens for polarized photons? In that case, the scattering may depend on the angle φ with respect to the direction of the electric field. The resulting scattering formula is

Unpolarized light means that you average over all angles φ, implying that factors of cos²φ become ½. A bit of algebra should convince you that when the expression above is averaged over φ it’s equivalent to Eq. 15.16 in IPMB.

In order to analyze polarized photons, we must consider the two polarization states, φ = 0 and φ = 90°.

φ = 0

The incident and scattered photon directions define a plane. Assume the electric field associated with the incident photon lies in this plane, as shown in the drawing below. From a classical point of view, the electric field will cause the electron to oscillate, resulting in dipole radiation (a process called Thomson scattering). A dipole radiates perpendicular to its direction of oscillation, but not parallel to it. Therefore, you get scattering for θ = 0 and 180°, but not for θ = 90°.

A schematic diagram of Compton scattering for polarized light with φ = 0.

A quantum-mechanical analysis of this behavior (Compton scattering) accounts for the momentum of the incident photon and the recoil of the electron. In the quantum case, some scattering occurs at θ = 90°, but it is suppressed unless the energy of the incident photon is much greater than the rest mass of the electron (x >> 1).

φ = 90°

For Thomson scattering, if the electric field oscillates perpendicular to the scattering plane (shown below) then all angles θ are perpendicular to the dipole and therefore should radiate equally. This effect is also evident in a quantum analysis unless x >> 1.

A schematic diagram of Compton scattering for polarized light with φ = 90°.

The figure below is similar to Fig. 15.6 in IPMB. The thick, solid lines indicate the amount of scattering (the differential cross section) for unpolarized light, as functions of θ. The thin dashed curves show the scattering for φ = 0 and the thin dash-dot curves show it for φ = 90°. The red curves are for a 10 keV photon, whose energy is much less than the 511 keV rest energy of an electron (x << 1). The behavior is close to that of Thomson scattering. The light blue curves are for a 1 GeV photon (1,000,000 keV). For such a high energy (x >> 1) almost all the energy goes to the recoiling electron, with little to the scattered photon. The dashed and dash-dot curves are present, but they overlap with the solid curve and are not distinguishable from it. Polarization makes little difference at high energies. 

The differential cross section for Compton scattering. The incident photon energy for each curve is shown on the right. The solid curves are for unpolarized light, the dashed curves are for light with φ = 0, and the dash-dot curves are for φ = 90°. Adapted from Fig. 15.6 in IPMB.

Why is there so little backscattering (θ = 180°) for high energy photons? It’s because the photon has too much momentum to have its direction reversed by a light electron. It would be like a truck colliding with a mosquito, and after the collision the truck recoils backwards. That’s extraordinarily unlikely. We all know what will happen: the truck will barrel on through with little change to its direction. Any scattering occurs at small angles. 

 

Notice that Thomson scattering treats light as a wave and predicts what an oscillating electric field will do to an electron. Compton scattering treats light as a photon having energy and momentum, which interacts with an electron like two colliding billiard balls. That is wave-particle duality, and is at the heart of a quantum view of the world. Who says IPMB doesn’t do quantum mechanics?

Is Shot Noise Also White Noise?

In Chapters 9 and 11 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss shot noise.

9.8.1 Shot Noise

The first (and smallest) limitation [on our ability to measure current] is called shot noise. It is due to the fact that the charge is transported by ions that move randomly and independently through the channels....

11.16.2 Shot Noise

Chapter 9 also mentioned shot noise, which occurs because the charge carriers have a finite charge, so the number of them passing a given point in a circuit in a given time fluctuates about an average value. One can show that shot noise is also white noise [my italics].

Introduction to Membrane Noise,
by Louis DeFelice

How does one show that shot noise is white noise (independent of frequency)? I’m going to follow Lou DeFelice’s explanation in his book Introduction to Membrane Noise (cited in IPMB). I won’t give a rigorous proof. Instead, I’ll first state Campbell’s theorem (without proving it), and then show that the whiteness of shot noise is a consequence of that theorem.

Campbell’s Theorem

To start, I’ll quote DeFelice, but I will change the names of a few variables.

Suppose N impulses i(t) arrive randomly in the time interval T. The sum of these will result in a random noise signal I(t). This is shown qualitatively in Figure 78.1.

Below is my version of Fig. 78.1.

A diagram illustrating the sum of N impulses, i(t), each shown in red, arriving randomly in the time interval T. The blue curve represents their sum, I(t), and the green dashed line represents the average, <I(t)>. Adapted from Fig. 78.1 in Introduction to Membrane Noise by Louis DeFelice.

DeFelice shows that the average of I(t), which I’ll denote <I(t)>, is

Here he lets T and N both be large, but their ratio (the average rate that the impulses arrived) remains finite.

He then shows that the variance of I(t), called σ_I², is

Finally, he writes

In order to calculate the spectral density of I(t) from i(t) we need Rayleigh’s theorem [also known as Parseval’s theorem]…

where î(f) is the Fourier transform of i(t) [and f is the frequency].

He concludes that the spectral density S_I(f) is given by

These three results (for the average, the variance, and the spectral density) constitute Campbell’s theorem.

Shot Noise

Now, let’s analyze shot noise by using Campbell’s theorem assuming the impulse is a delta function (zero everywhere except at t = 0 where it’s infinite). Set i(t) = q δ(t), where q is the charge of each discrete charge carrier.

First, the average <I(t)> is simply Nq/T, or the total charge divided by the total time. 

Second, the variance is the integral of the delta function squared. When any function is multiplied by a delta function and then integrated over time, you get that function evaluated at time zero. So, the integral of the square of the delta function gives the delta function itself evaluated at zero, which is infinity. Yikes! The variance of shot noise is infinite.

Third, to get the spectral density of shot noise we need the Fourier transform of the delta function. 

The key point is that S_I(f) is independent of frequency; it’s white.

DeFelice ends with

This [the expression for the spectral density] is the formula for shot noise first derived by Schottky (1918, pp. 541-567) in 1918. Evidently, the variance defined as

is again infinite; this is a consequence of the infinitely small width of the delta function.

As DeFelice reminds us, shot noise is white because the delta function is infinitely narrow. As soon as you assume i(t) has some width (perhaps the time it takes for a charge to cross the membrane), the spectrum will fall off at high frequencies, the variance won’t be infinite (thank goodness!), and the noise won’t be white. The bottom line is that shot noise is white because the Fourier transform of a delta function is a constant.

Conclusion

Perhaps you’re thinking I haven’t helped you all that much. I merely changed your question from “why is shot noise white” to “how do I prove Campbell’s theorem.” You have a point. Maybe proving Campbell’s theorem can be the story of another post.

I met Lou DeFelice in 1984, when I was a graduate student at Vanderbilt University and he came to give a talk. In the summer of 1986, my PhD advisor John Wikswo and I traveled to Emory University to visit DeFelice and Robert DeHaan. During that trip, Wikswo and I were walking across the Emory campus when Wikswo decided he knew a short cut (he didn’t). He left the sidewalk and entered a forest, with me following behind him. After what seemed like half an hour of wandering through a thicket, we emerged from the woods at a back entrance to the Yerkes Primate Research Center. We’re lucky we weren’t arrested.

DeFelice joined the faculty at Vanderbilt in 1995, and we both worked there in the late 1990s. He was a physicist by training, but spent most of his career studying electrophysiology. Sadly, in 2016 he passed away.

The Unit of Vascular Resistance: A Naming Opportunity

The metric system is based on three fundamental units: the kilogram (kg, mass), the meter (m, distance), and the second (s, time). Often a combination of these three is given a name (called a derived unit), usually honoring a famous scientist. For example, a newton, the unit of force named after the English physicist and mathematician Isaac Newton (1642 – 1727), is a kg m s⁻²; a joule, the unit of energy named for English physicist James Joule (1818 – 1889), is a kg m² s⁻²; a pascal, the unit of pressure named for French mathematician and physicist Blaise Pascal (1623 – 1662), is a kg m⁻¹ s⁻²; a watt, the unit of power named for Scottish engineer James Watt (1736 – 1819), is a kg m² s⁻³; and a rayl, the unit of acoustic impedance named for English physicist John Strutt (1842 – 1919) who is also known as Lord Rayleigh, is a kg m⁻² s⁻¹.

In Chapter 1 of Intermediate Physics for Medicine and Biology, Russ Hobbie and I discuss the human circulatory system. We talk about blood pressure, p, which is usually expressed in mmHg or torr, but in the metric system is given in pascals. We also analyze blood flow or cardiac output, i, sometimes expressed in milliliters per minute, but properly should be m³ s⁻¹. Then Russ and I introduce is the vascular resistance.

We define the vascular resistance R in a pipe or a segment of the circulatory system as the ratio of pressure difference across the pipe or segment to the flow through it:

R = Δp/i .                  (1.58)

The units are Pa m⁻³ s. Physiologists use the peripheral resistance unit (PRU), which is torr ml⁻¹ min.

What name is given to the Pa m⁻³ s, or equivalently the kg m⁻⁴ s⁻¹? Sometimes it’s called the “acoustic ohm,” stressing its analogy to the electrical unit of the ohm (a volt per amp). If the unit for electrical resistance can honor a scientist, German physicist Georg Ohm (1789 – 1854), why can’t the unit for mechanical resistance do the same? Let’s name the unit for vascular resistance!

I know what you’re thinking: we already have a name, the peripheral resistance unit. True, but I see three disadvantages with the PRU. First, it’s based on oddball units (pressure in torr? time in minutes?), so it’s not standard metric. Second, sometimes it’s defined using the second rather than the minute, so it’s confusing and you always must be on your toes to avoid making a mistake. Third, it wastes the chance to honor a scientist. We can do better.

My first inclination was to name this unit after the French physician Jean Poiseuille (1797 – 1869). He is the hero of Sec. 1.17 in IPMB. His equation relating the pressure drop and flow through a tube—often called the Poiseuille law—explains much about blood flow. However, Poiseuille already has a unit. The coefficient of dynamic viscosity has units of kg m⁻¹ s⁻¹, which is sometimes called a poiseuille. It’s not used much, but it would be confusing to adopt it for vascular resistance in addition to viscosity. Moreover, the old cgs unit for viscosity, g cm⁻¹ s⁻¹, is also named for Poiseuille; it’s called the poise, and it is commonly used. With two units already, Poiseuille is out.

Henry Darcy (1803 – 1858) was a French engineer who made important contributions to hydraulics, including Darcy’s law for flow in a porous medium. However, an older unit of hydraulic permeability is the darcy. Having another unit named after Darcy (even if it’s an mks unit instead of an oddball obsolete unit) would complicate things. So, no to Mr. Darcy.

The Irish physicist and mathematician George Stokes (1819 – 1903) helped develop the theoretical justification for the Poiseuille law. I’m a big fan of Stokes. He seems like a perfect candidate. However, the cgs unit of kinematic viscosity, the cm² s⁻¹, is called the stokes. He’s taken.

The Poiseuille law is sometimes called the Hagen-Poiseuille law, after the German scientist Gotthilf Hagen (1797 – 1884). He would be a good candidate for the unit, and some might choose to call a kg m⁻⁴ s⁻¹ a hagen. Why am I not satisfied with this choice? Hagen appears to be more of a hydraulic engineer than a biomedical scientist, and one theme in IPMB is to celebrate researchers who work at the interface between physics and physiology. Nope.

A portrait of William Harvey,
downloaded from wikipedia
(public domain).

My vote goes to William Harvey (1578 – 1657), the English physician who first discovered the circulation of blood. I can find no units already named for Harvey. He doesn’t have a physics education, but he did make quantitative estimates of blood flow to help establish his hypothesis of blood circulation (such numerical calculations were uncommon in his day, but are in the spirit of IPMB). Harvey is a lot easier to pronounce than Poiseuille. Moreover, my favorite movie is Harvey.

We can name the kg m⁻⁴ s⁻¹ as the harvey (Ha), and 1 Ha = 1.25 × 10^-10 PRU (we may end up using gigaharveys when analyzing the peripheral resistance in people). One final advantage of the harvey: for those of you who disagree with me, you can claim that “Ha" actually stands for Hagen.

Ceaseless Motion: William Harvey’s Experiments in Circulation.

https://www.youtube.com/watch?v=w6q50_qNMoA

 

The trailer from the 1950 film Harvey,

starring James Stewart as Elwood P. Dowd.

https://www.youtube.com/watch?v=VvfXvW2wsuQ