BIOMEDICAL PHYSICS

11.1. Physics and medicine

- overview, general intro, what goals (understand, diagnose, treat) 

11.2. Scaling

Area scales

Let us use a quantity L = the "linear dimension", some measure of how big an object (or an animal, or some part of it) is. We do not now care about exactly what shape the object has, nor whether we are measuring the length, width, height, diameter, radius or other such quantity of it. We will here focus on issues independent of that.

By an area scale A we mean the relation between some area of two objects of the same shape but different linear dimension L. For example if the the side of a square is L then its area is A = L² so if one square has twice the side length of another, it will have four times its are: A₁/A₂ = L₁²/L₂². But the same would be true for a circle A with twice the radius of another; if one has the radius L₁ and the other L₂ then

• A₁ = pL₁² and the other A₂ = pL₂² so A₁/A₂ = pL₁²/pL₂² = L₁²/L₂²

and if their diameters had been L₁ and L₂ we would also have

• A₁/A₂ = p(L₁/2)²/p(L₁/2)² = L₁²/L₂²

Volume scales

In a similar way, the volume of any three-dimensional geometrical body is proportional to its linear scale cubed, e.g.

• for two cubes, we have V₁/V₂ = L₁³/L₂³where L = the side length

• for two spheres, V₁/V₂ = (4pL₁³/3)/(4pL₂³/3) = V₁/V₂

Physical properties which depend on A

• the rate of heat loss, which is a power (amount of energy per time, unit: 1 watt).

This is related to the general formula for heat transport by conduction through a material, DQ/Dt = -kADT/Dx where DQ = amount of conducted thermal energy, Dt = time, DT = temperature difference between the hot and cold end or surface of the material, Dx = the length of the object through which heat is conducted or the thickness of the surface through which it moves, k = thermal conductivity (a material constant, low for good thermal insulators) and A = the area of the surface or a cross section of the object. (This formula is no longer required in the IB's Thermal physics. The dependency on the area can also be related to the L = sAT⁴ formula in Astrophysics which is generally valid for radiation of heat as well as for light: L = the power in watts, A = the area of the radiating surface, s = the Stefan-Boltzmann constant. For other than "black" bodies the formula can be completed by multiplication with a unitless constant, emissivity, which is 1 for a black body and smaller for others.

• muscle and bone strength (force): the force depends on microscopic forces between cells and/or molecules; the force per such is about constant, their number proportional to the area of a cross section of the muscle or bone

• pressure P = F/A where F = the force a gas or liquid exerts on the surface A, unit 1 pascal = 1Nm^-2.

• stress = force/area (sometimes the force is called load).

Similar to pressure, but relevant to solid objects.

[The stress can be tensile stress, if a force is pulling the object from its ends (as the force of tension in a rope), compressive stress (the object is being compressed) or shear stress (the force is acting parallel to the chosen cross section A).]

Physical properties which depend on V

• mass, for objects which have a roughly constant density d where d = m/V gives m = dV. Many living beings are made mostly of water, where this is true.

• weight = force of gravity, depends on mass and therefore volume as above. F_G = mg where g = the gravity constant.

Absolute and relative quantities

An absolute quantity is one which has a direct dependency on a the linear dimension L (ex. surface area) while a relative quantity is the ratio or product of two or more absolute quantities (surface area per unit body mass, heat loss rate per unit mass) or some other quantity related to one dependent on L (oxygen absorption rate per unit mass).

Consequences for animals

This explains why an elephant does not look like a scaled-up version of a mosquito. The mass of an animal is, assuming that animal tissue is mostly made up of water with about the same density, proportional to volume, and therefore to L³. Bone strength on the other hand, is proportional to bone cross-section area and therefore to L². So the elephants legs must be much thicker compared to the overall size of the animal than the mosquito's.

Fig: Elephant and mosquito.

Different types of forces may also be important on different scales. Surface tension (caused by forces between water molecules) is important for insects (they may walk on water or be trapped inside a drop of water) while they are of little importance to larger animals.

11.3. Biomedical mechanics

Centre of mass (or gravity)

The center of mass is a point in a body such that all the forces of gravity acting on the atoms of it can be assumed to act in that one point for the purposes of translational and rotational mechanical problems. For homogenous, simple objects the CM- or CG-point is in the geometric center.

[If the body is not large enough for the gravity constant g to be different in different parts of it, then CM and CG would be the same. For living beings this comlplication can be ignored; the situation is different for astronomical bodies where a tidal effect can be observed]

Experimentally, the CG of a rigid body can be found by hanging it in a pivot point around which it can rotate freely and drawing a vertical line downwards from it, and then repeat this from another pivot point. Where the lines intersect the CG point is.

[Quantitatively, the x-coordinate of CM and in practice the CG can be found in any chosen x-dimension from the formula

x_CM = (x₁ + x₂ + x₃ + ....)/(m₁ + m₂ +m₃ +....) = (x₁ + x₂ + x₃ + ....)/m_tot

where x_i and m_i are the x-coordinates and masses of the particles of the body. Corresponding formulas give the y- and z-coordinates]

Forces and torques in the human body

Recall from mechanics the mechanical equilibrium conditions:

• translational equilibrium: the resultant force is zero in any dimension
• rotational equilibrium: the resultant torque is zero around any pivot point

In the elbow joint, the triceps muscle with its tendons acts with a force F at an attachment point at some distance r from the joint around which the lower arm can rotate. The resulting torque t = Fr and is therefore greater the bigger r is. For humans r is rather small, wherefore we are weaker than some monkeys with less muscle mass but a higher r in the elbow joint.

Fig: Force and torque in an elbow joint

When lifting a heavy object in a forward-bending position, the upper body rotates around the pelvic joint which acts as a pivot point. The force of gravity ("load") on the lifted object acts downwards at the shoulders. The back muscles are attached very near the pivot point giving them a lower r-value than the load force. To keep the clockwise and anticlockwise torques constant (which for slow lifting is approximately the case) there must be a much larger force in the back muscles. This can lead to injuries, and it is better to lift with the back in a more vertical position (with bent legs) since that decreases the r for the load and thereforce the force in the back.

Fig: Lifting objects and force in the muscles of the back

Lever systems in the human body

Recall from Mechanics that torque is

t = Fr sin q

for the force F acting at the distance r from a pivot point. If the angle between the force and the line from where it acts and the pivot is 90^o, we can write

t = Fr

The torque supplied by a given force is therefore larger the longer "arm" it acts on. It is therefore possible to lift a heavy stone with an iron bar if the pivot is arranged so that the force of gravity on the stone gets a shorter r than the lifting force. This method of lifting is called a lever system. There are different types of levers, where the pivot may be placed in different places relative to the lifting force (the "effort") and the force of gravity on the object to be lifted, the "load".

Fig. Heavy stone lifted with bar, pivot near the heavy stone.

The ratio between the forces called load (F_load) and effort (F_effort) is defined as the

Mechanical Advantage = load/effort [DB p. 11]

From Mechanics we have that

• translational equilibrium (forces balance out in any dimension) => constant velocity, that is: an object remains at rest or in uniform motion
• rotational equilibrium (torques balance out around any pivot) => constant angular velocity, that is: an object remains at rest or in rotational motion at a constant angular velocity.

If the object is lifted at a constant (angular) velocity, we can approximately write

• t_load = t_effort so
• F_loadr_load = F_effortr_effort which gives
• Mechanical Advantage = F_load/F_effort = r_effort/ r_load

The distance moved in a circular path by the points where the forces act is the arc of a circular sector:

Fig: Circular sector, radius r, arc length s, angle j

The distance s moved is related to the r as s = 2prj/360^o or if a is given in radians, s = jr. The angle j must be the same for both load and effort if the lever is not broken, so the ration between the distances moved will be:

• s_effort/ s_load = jr_effort/ jr_load = r_effort/ r_load = the Mechanical Advantage

Dividing this by an arbitrary time t gives a speed ratio

• s_effort/ s_load = (s_effort/t)/(s_load/t) = v_effort / v_load

which here is called the velocity ratio (for a short t this makes no difference, instantaeneous speed and velocity are the same):

Velocity Ratio = distance moved by effort/ distance moved by load [DB p. 11]

From above it is evident that the Velocity Ratio (VR) = the Mechanical Advantage (MA).

Applications in the human body: throwing and lifting

The triceps muscle on the back of the upper arm is attached to a point near the elbow joint, while the force of gravity on an object in the hand will be further from this pivot. This makes that lifting things more difficult (MA = r_effort/r_load < 1 since r_load > r_effort), but the gain is that at the same time VR > 1, and since MA = VR = v_effort / v_load we also have v_load > v_effort. The result is that humans can throw things (e.g. a stone or a spear) or hit with an object used as a weapon with a high velocity v_load, even if the speed at which the triceps muscle can contract is limited for biological reasons.

Fig. Elbow joints and arms on human and monkey.

On a monkey (chimpanzee?), the triceps is attached a bit further from the elbow joint, which makes it "stronger" than a human, but not as good at throwing things. The same phenomenon can be further developed with tools that increase the rload even more: clubs for hitting, slings for throwing stones and spear-throwing tools like an atlatl.

Another application is the known fact that lifting objects with a bent back puts more stress and a higher risk for injuries on the muscles in the back than lifting with bent legs and a straight back:

Fig: Lifting with bent and straight back

The back muscles are attached close to the hip joint acting as a pivot, while the arms are attached to the shoulder much further away. We cannot avoid a situation where rload > reffort, but we can decrease the needed Feffort by affecting the angle q_load :

• t_load = t_effort now becomes
• F_loadr_loadsinq_load = F_effortr_effortsinq_effort so
• F_effort = F_loadr_loadsinq_load / r_effortsinq_effort

When lifting with a back bent forward, q_load is close to 90^o, but when bending the legs q_load is much smaller.

11.4. Biomedical thermal physics

Metabolism

All the time food is being digested in the stomach and other organs and in addition to various nutrients being utilised, its chemical energy is turned into thermal energy which varies from the basal metabolic rate (when sleeping or unconscious) to higher metabolic rates, for example at physical activity when a lot more thermal power is generated in the muscles.

Temperature regulation

Humans like most mammals are keeping a rather constant body temperature, which means that depending on the metabolic rate and the external circumstances (temperature and others) there may sometimes be an excess and sometimes a deficiency of thermal energy. Heat may flow into or out of the body in the same ways as earlier in thermal energy:

• conduction: whenever materials are in touch with each other heat will be conducted via molecular collisions; heat is also transported through any material in this way. The transportation is more or less effective depending on the material. E.g. metals conduct heat very well and may cause burns.

• convection: heat is transported when a material at a higher temperature than 0 K is moving; blood can transport heat in a body; the flowing air around a body can do so more or less effectively depending on the amount of clothing and its speed (hence the "wind chill factor" which takes into account not only air temperature but also speed).

• radiation: this transportation method is not dependent on any medium; heat can be lost more effectively from a larger area .....

• evaporation: this is when a wet shirt cools us, since water is vaporised even below its boiling point. The kinetic energies of the molecules are distributed as in the Maxwell-Boltzmann curve; those which have a very high energy may break free from the liquid which leads to a lower average kinetic energy in the remaining ones. This evaporative cooling differs from the 3 other "proper" ways of thermal energy transportation in that the body cannot be heated in this way.

Energy and efficiency

We take in energy in the form of food and expend it to work done (e.g. lifting objects) and waste heat. In this sense the human body works like an engine, see the Thermal physics topic. E.g. the efficiency is, as in Mechanics,

e or h = E_out/E_in or P_out/P_in [not in DB but a similar definition is given in thermal physics, DB p.6]

11.5. Biomedical waves : Sound and hearing

Intensity

Sound intensity I is defined as

I = P/A [not in DB]

where P = the power transported by a wave and A the area through which wavefronts (of e.g. sound) progress. (This quantity is also used in Astrophysics for the light emitted by a star).

The energy of an oscillating particle is periodically changes from kinetic to elastic potential energy. For an oscillation of a mass m on a spring with the amplitude A, the energy will be E = ½kA², where k = the spring constant. For these oscillations we have (here given without proof) that

• T = 2pÖ(k/m) which with f = 1/T gives
• k = 4p²mf²

which gives

• E = ½(4p²mf²)A² and P = E/t = 2p²mf²A²

wherefore the power and also intensity of a sound wave are proportional to the squares of the frequency and amplitude.

The decibel scale

The ability of the human ear to detect sound (its loudness) depends on its frequency and the intensity level. The ear is most sensitive around a frequency of a few thousand Hz, where the lowest detectable frequency - the "treshold of hearing" is about I₀ = 10^-12 Wm^-2.

A logarithmic scale (similar to the pH-scale in chemistry and the magnitude scale in Astrophysics) has been constructed, such that

b = 10 log ( I / I₀ ) where I₀ = 10^-12 Wm^-2 [DB p. 11]

where the sound intensity level in the dimensionless unit "bel" is log (I/I₀) and b = the intensity level in decibels, dB.

The ear

This organ consists of and outer, middle and inner ear. The middle ear transforms sound pressure variations to larger ones in the fluid in an organ called the cochlea, from which they are converted to nerve signals sent to the brain.

[More details about the functioning of the ear are found in many textbooks and here omitted in this version of the compendium]

Audible sounds

The character of the sounds we can hear is determined by their "pitch" (frequency), their "loudness" (perceived intensity) and their "timbre" (their characteristic wave shape, which for most real sounds is not purely sinusoidal but the result of a superposition of several tones and overtones. This is what makes the same note played with a flute and a clarinet sound different).

The range of audible frequencies for the human ear is ca 20 - 20000 Hz, although the upper limit decreases with age. The threshold of hearing and the perceived loudness depend on the frequency:

Fig. intensity-frequency diagram

Hearing tests and hearing losses

Still to write are these parts of the IB syllabus:

D.2.10. hearing tests, type and extent of hearing loss

D.2.11. audiograms with bone and air conductions curves 

D.2.12. conductive and sensory hearing losses, possible origins

D.2.13. selective freq losses, loss of speech discrimination

D.2.14. hearing aids

Resistance (impedance) matching in the ear

Let us review some electric circuit theory: Say that we have a V = 4.5V battery connected to a R₁ = 10W resistor. Let this constitute part 1 of the circuit. If we then connect another resistor R₂ in series with R₁, this resistor will be part 2. The first resistor and the voltage we assume to be constant; the second can be varied. The question is now: what value should we give R₁ so that the power dissipated in it will be maximal?

Try first with R₂ = R₁ = 10W, which gives R_tot = R₁ + R₂ = 20W. So the current (which in a serial connection is the same in both resistors) is given by R_tot = V/I => I = V/R_tot = 4.5 V/ 20W = 0.225A. The power dissipated in R₂ is then P₂ = R₂I² = 10W*(0.225A)² = 0.50625 W. Is this the maximal power?

Try instead with R₂ = 5W. Now R_tot = 15W and I = 4.5V/15W = 0.3A. So P₂ = (5W*0.3A)² = 0.45W, which is less than above.

What about trying with R₂ = 15W? Then R_tot = 25W and I = 4.5V/25W = 0.18A. And then P₂ = 15W*(0.18A)² = 0.486W. Also less than the first attempt.

Trying other values will reveal that R₁ = R₂ will maximize the power in the second part of the circuit. This is called resistance matching (or, for AC circuits with capacitors and solenoids where ordinary resistance is replaced by a similar quantity, impedance Z = V/I, impedance matching). This can be shown in various ways; we may combine P₂ = R₂I² with I = V/(R₁ +R₂) to get P₂ = V²R₂/(R₁+R₂)² and make a graph of P₂ as a function of R₂ for some constant R₁; the graph will have a maximum at R₂ = R₁. It can also be shown with calculus as below.

[Calculus-based proof: We have the function y = ax/(b + x)² where a = V² and b = R₁ are constants, and x = R₂ the variable. We find the maximum of y(x) by differentiating it and solving y'(x) = 0. In this we will use the rule that the derivative of f/g is (f'g-g'f)/g², here f(x) = ax and g(x) = (b+x)² = b² + 2bx + x² :

• y'(x) = [a*(b+x)² - (2b + 2x)*ax]/(b+x)⁴ which excludes x = -b and is zero if
• [a*(b+x)² - (2b + 2x)*ax] = 0 giving ab² + 2abx +ax² - (2abx + 2ax²) = 0 and then ab² - ax² = 0 which since a = V² is not 0 gives b² - x² = 0 and then
• x² = b²; where the only valid solution here is x = b, so R₁ = R₂ ]

Practical applications of this is e.g. building loudspeaker systems, where maximal power is transmitted to the next part of the system if its impedance is the same as that of the previous.

But what does all this electric circuit theory have to do with the functioning of the ear? Well, its all about transferring waves, that is oscillations or pressure maxima (compressions) in air in the middle ear to water in the inner ear. This leads to the field of fluid (= liquid or gas) mechanics. In that field, many theories are very similar to those in electricity.

The equation R = V/I can be replaced by a similar one where

• instead of voltage or potential difference V there is a pressure difference between the ends of a tube where a fluid is to flow
• instead of the current I = amount of charge moved/time there is a fluid flow, volume of fluid moved per time

• instead of electric resistance we have fluid resistance, which like electric resistance in a wire depends on its length and cross-section area of the tube, but also on how "thick-flowing" or viscuous the fluid is (like water, oil or honey). This is described by the quantity viscosity.

It will also be true that the total fluid resistance for two tubes in series or parallel will follow the same R_tot = R₁ + R₂ and 1/R_tot = 1/R₁ + 1/R₂ formulas as in electricity. And as in electricity, the fluid resistance can be replaced by a similar quantity, fluid impedance, and to effectively transmit the power in the sound waves from middle to inner ear we need the same impedance matching as above. And this will be a big problem, since the fluid impedance of air and water are very different.

11.6. Ultrasound

Ultrasounds and infrasounds

By ultrasounds we mean sounds with a higher frequency than the ca 20000 Hz that a (young) human can hear. These sounds can be heard or sometimes even produced by some animals, such as dogs and bats. Sounds with a lower frequency than we can hear (ca 20 Hz) are called infrasounds and used by elephants to communicate over long distances.

Ultrasounds are produced with piezoelectric crystals, which change shape when an electric potential difference is applied to them, and

therefore can vibrate and produce sound waves when a high-frequency AC voltage is used. Ultrasounds in medical use typically have frequencies about 1-10 MHz.

The SONAR principle (radar with sound)

A radar (radio detecting and ranging) sends out a pulse of radio waves and measures the time it takes for it to be reflected back to the radar antenna. Knowing the speed of the wave, it is then possible to calculate how far away the target - a ship or an airplane - is. The SONAR ("sound navigation ranging") or pulse-echo technique is similar for sounds waves, and has been used since WW2 to find submarines under water. In doing this, the sound pulse may be reflected or refracted not only by a target submarine or the ocean floor, but also by layers of water with different temperature and/or salinity.

In medical use, the ultrasound pulse is (partially) reflected when reaching the boundary between tissues where sound travels at slightly different speeds, around the typical value ca 1540 ms^-1 for water (which human bodies mostly consist of). In air this speed is ca 340 ms^-1 and in bone ca 4000 ms^-1 which gives a very effective reflection, but makes it difficult to "see" organs behind lungs or bones.

In order to avoid a strong reflection when the pulse enters from the transducer into the body (= water), a water-based gel can be used to ensure that the pulse never travels through air.

A-scans and B-scans

The A-scan produces a graph of the received echo intensity as a function of time since the pulse was sent out. The following pulse is not sent until the first has gone through the body, and the duration of a pulse is short compared to the separation in time between them. Pulse echoes (intensity peaks) represent the passing of the pulse through tissue boundaries

Fig. b06a: Tissues and A-scan graph and B-scan points (G372)

In a B-scan, the peaks are represented by points that by a computer are made more intense, larger, differently coloured or otherwise to represent the height of the peak in the A-scan. By moving the transducer and/or having several of them a set of B-scans give a two-dimensional image of the organs inside the body.

Fig b06b: set of B-scans and image of inner organ

Factors affecting the choice of frequency

· resolution: the higher the frequency, the shorter the wavelength, and the smaller details can be observed; e.g. if f = 10 MHz and v = 1500 m/s then limit is given by:

v = fλ => λ = v / f = 1500 ms-1 / 1000 000 Hz = 0.0015 m = 1.5 mm

· penetration: the higher the frequency, the more of the wave is absorbed or scattered by the water, and this makes it difficult to see organs deeper in the body (compare this to the scattering of light in the atmosphere: blue light with a high frequency is scattered more than red with a low frequency, therefore the sky looks blue in the day, and the sunset red).

11.7. EEG and ECG (not required in the IB)

[ECG = electrocardiogram: Small electrical voltages are used to detect the action of muscle cells, especially the heart muscle. Different stages in the operation of the heart muscle produce different characteristics on a graph of voltage as a function of time. Malfunctions in the heart are detected as distortions of these graphs from the normal type

EEG = electroencefaphalogram: same for brain ]

11.8. Biomedical atomic physics : X-rays

X-ray imaging

Production and properties of X-rays

From Atomic physics, recall that when electrons in a vacuum tube are accelerated by a p.d or voltage of ca 10000V and then hit a metal target, where some (ca 1 %) of their kinetic energy is released as X-rays are produced, and the rest turns into thermal energy. These X-rays are high-frequency and high-energy (E = hf) photons, with an f higher than UV-rays but lower than gamma rays. The target material is often tungsten (wolfram) like the filament in a light bulb, since it has a high melting point. To avoid overheating of the spot where the stream of accelerated electrons hit, the target is usually rotated. The X-ray spectrum (graph of intensity as a function of X-ray frequency or wavelength) has 3 important features:

• I.The continuous curve, caused by the release of X-rays as Bremsstrahlung

• II. The cutoff minimum wavelength (or maximum frequency) which is found when an electron gives all its energy to the photon: qV = hf = hc/l

• III. The characteristic peaks, which arise when the incoming electrons strike out an inner-shell electron in the target atom, and an X-ray photon is emitted when a free electron falls down to replace the one stricken out.

The area under the graph represents the total intensity of the X-rays produced.

X-ray quality

The term "X-ray quality" refers to the intensity of the rays at different wavelengths, and affects the intensity that actually reaches an organ inside the patient and then the detecting device (film or electronic detector).

Factors affecting X-ray quality and their results for features I-III

A. Increasing the potential difference V

I. When more kinetic energy is carried by the electrons, the broad peak of the continuous curve shifts towards higher f / lower l.

II. The maximum f increases/ minimum l decreases

III. Depending on where the energy levels of the shells in the target atom are, new characteristic peaks may appear (but all peaks are at their constant places in the spectrum).

B. Increasing the electron current

More electrons per time are sent through the X-ray tube (e.g. by increasing the current heating the cathode thus making thermionic emission of electrons more efficient). The drawback is the heating of the target material also increases.

I. No change in the shape of the characteristic curve, but the overall intensity increases

II. No change in the max f / min. l.

III. No change in the position of the characteristic peaks or in which of them are found, but the general increase in intensity is noted in them as well.

C. Increasing the atomic number Z of the target material

For the target atoms, a higher Z makes the collision between an accelerated electron and target atom more likely to result in the emission of an X-ray photon than in a heating of the target. The problem lies in balancing this against the need for target materials with a high melting point. Common materials are tungsten (wolfram) with Z = 74 and a melting point = 3370 ^oC, and platinum with Z = 78 but a melting point of 1770 ^oC). More effective cooling (by rotation or otherwise) of the target opens possibilities for using a higher Z.

I. No change in the shape of the continuous curve, but the overall intensity values increase.

II. No change in max f. / min l.

III. The characteristic peaks for a different target atom are different - usually the ones in materials with a higher Z are found at a higher f/ lower l.

D. Energy selective filters

When EM waves pass through suitable materials, some wavelengths are absorbed more than others. For visible light, this may lead to white light changing to coloured light when passing a transparent but "coloured" material, e.g. coloured glass. There are materials which have a similar effect on X-rays, absorbing selectively at lower f / higher l and thus improving the resulting X-ray "quality". For X-rays as for any EM radiation, the term "harder" refers to a spectrum where higher frequencies are more dominating. Blue light is "harder" than red or white light.

I. The shape of the continuous curve changes so that its broad peak is shifted towards higher f / lower l. The overall intensity level and total area under the curve decreases.

II. No change in max f. / min l.

III. No change in the position of the characteristic peaks or in which of them are found, but their relative height may change since they are unevenly affected by the decrease in intensity.

X-ray attenuation

Attenuation means the damping or decrease in intensity for the X-rays when they pass through the material in the patient's body. Compare this to light becoming fainter when it passes deeper and deeper into the ocean.

A. Simple coherent scattering (and a few words on Compton scattering)

If the X-ray photon an energy E = hf that does not fit any of the energy level differences in the atom in the patient's tissue, it may change its direction without losing energy. This will, however, mean that the intensity of radiation passing through the body in a given direction (from the X-ray tube to the film or other detecting device) decreases. This mostly happens in soft tissue but to some extent also in bones.

[Compton scattering occurs when the incoming photon collides with an electron in such a way that it loses part of its momentum p = E/c and therefore energy (without losing either speed, which is always c, or mass, which it does not have!. This process adds to the decrease in intensity of photons with a given wavelength since the photon must lose energy in that way]

B. Photoelectric effect

Recall from Atomic physics that a photon hitting a target atom may strike out an electron losing all (in proper PE effect) or part of its energy. The stricken electron may be replaced by another under emission of a new photon, but mostly with a different, lower, energy. The medical relevance of this is that since bone tissue contains a different distribution of elements than surrounding soft tissue (muscles) it is possible to design the X-ray so that its peak in the intensity curve matches the energy needed for photoelectric effect in bones. There will then be a relatively greater difference in attenuation in bones vs. other tissue, and a sharper image of the skeleton can be produced.

Attenuation coefficient and intensity

The change (decrease) in intensity ΔI when radiation passes the distance Δx through materia depends on the intensity I before passing Δx as:

ΔI = -μIΔx

where μ = a linear attenuation coefficient in the unit m^-1. If we turn this into a differential equation

dI = -μIdx

we will get the solution:

I = I₀e^-μx DB p. 11

where I₀ = the intensity before hitting the material and I the intensity at a depth x in it.

[Compare this to radioactive decay in Atomic physics where the decay probability constant λ in the unit s^-1 and the differential equation

dN = - λNdt

gave

N = N₀e^-lt [DB p.8]

The half-value thickness

For similar mathematical reasons we will also get something corresponding to the half-life

T_½ = ln2 / λ [DB p. 8]

namely the half-value thickness x_½ = the depth at which the intensity has been halved:

x_½ = ln2 / μ DB p.11

The attenuation coefficient depends on:

• the wavelength or frequency (harder X-rays with higher f may penetrate better thus having a lower attenuation coefficient and a higher half-value thickness
• the type of tissue material penetrated

X-ray detection

Basic detection: films and electronics

The X-rays can be detected with certain photographic films sensitive to them or secondary radiation from intensifying screens. The developed films are studied agains an illuminated background. The X-rays can also be detected and recorded by electronic components similar to those in digital cameras, and displayed on a computer monitor.

X-ray image enhancement

To improve the quality of the images one can:

· let the rays pass collimating grid of lead plates, which will suppress rays not moving in the desired direction

· move the collimating grid (or the source and the film) in such a way that the images of the grid are blurred and those of the patient sharp

· use contrast-enhancing substances which make medically interesting organs more visible when introduced into the patient (barium meal for the intestines, injected iodine for the blood system)

· use intensifying screens, that is materials which when hit by X-rays release lower-frequency radiation which is more easily registered by the photographich film/ electronic detector

Computer tomography (CAT scan)

In an ordinary X-ray image, the rays are sent through a whole part of the patient's body at one time and the intensity of the rays afterwards redorded with a film or otherwise. In computerised axial tomography, one could in principle use just one very narrow ray and one small detector and then move them in a circle around the patient, although in practice several rays and an arc-shaped array of detectors are used to produce an image more quickly:

Fig. b08---: Patient (P) with X-ray source and detector array, ring

The intensity of one narrow ray - "axis" - that has passed through the patient is only one number; to produce a two dimensional image of the patient the computer calculates what the X-ray absorption must be in different parts of the "slice" of the patient. This problem can be compared to finding what numbers must be placed in a matrix when the sums of rows, columns and diagonals are known:

Fig b08----: Box 3 x 3 with numbers 1,1,1; 2,1,7; 1,1,2 and sums including diagonals below 3,4,3,10,4 and on the right side upwards 4,10,3

To produce a 3-dimensional image many such thin "slice-images" are made by moving the source and detector array in a direction perpendicular to the picture above.

11.9. Other imaging techniques

MRI or NMR (so-called "magnetic X-ray")

The nuclei of atoms are rotating, "spinning", and for those with an odd mass number (sum of protons and neutrons) such as 1H this makes them act as small magnets. If the atoms are placed in a strong magnetic field B, then they can be aligned either in the same or in the opposite direction as the field B.

To make them switch between these alternatives ("spin up" and "spin down") requires or releases energy, which is proportional to the strength of the field B. If the B-field is directed "up", then energy is released when the nucleus switches from the opposite direction to one aligned with B:

Fig b09a: B-field , E-levels (G996)

This energy can be absorbed or emitted photons with the energy E = hf, where the frequency is typically in the radio frequency (RF) part of the electromagnetic spectrum. The frequency must be such that the energy of the photon matches the energy absorbed or released in the switching of the nucleus; this phenomenon is called nuclear magnetic resonance or NMR.

Since the needed energy and therefore the radio photon frequency depends on the magnetic field B, one can "excite" (switch to the higher energy state) nuclei on only one slice of the body by using a magnetic gradient, that is a B field that grows stronger the further along a chosen direction we go. By applying another B-field, also with a gradient, perpendicular to that, we can find out where is this slice an emitted RF photon comes from - that will be revealed by its frequency. Add to this that the apparatus can be rotated around the patient and a computer used to find where in the patient we have a lot of emissions/absorptions, then we can get a very detailed 3-dimensional image of where in the patient there is more or fewer H-atoms.

This does, however not give very good medical information about the patient, since these atoms are present in large numbers everywhere, in water which makes up most of the body and in all typical organic molecules. What is done then is to measure how fast the excited H-atoms "decay" back into the lower energy state; this is to some extent affected by the neighbouring atoms, and therefore the type of molecule, and consequently the type of tissue we have. This is called a spin-echo technique.

Of other atoms present in organic molecules, ¹²C and ¹⁶O are unsuitable (not having odd nucleon numbers), but ³¹P has been used in studies of metabolism (where phosphorus in ADP and ATP molecules are essential).

Radioactive tracers

Some elements or chemical compounds for biochemical reasons tend to accumulate in certain organs. For example iodine does so in the thyroid gland (Sw. sköldkörteln, Fi. kilpirauhanen). If there are radioactive isotopes of iodine in the environment which may be the case after a nuclear accident, then one can take an excess of non-radioactive iodine so that the body cannot take up more of it for some time. Then the accumulation of radioactive iodine is avoided and the risk of thyroid cancer decreased.

A similar phenomenon can be used for medical treatment with radioactive tracers. This means that isotopes of elements which typically accumulate in some organ but have a short half-life so that they do not contaminate the body for a longer time are injected, and the uptake in some organ studied. If the uptake is smaller or bigger than usual it indicates that this organ is not functioning properly.

The detection is done either by checking blood or other fluid samples with a Geiger-Muller-detector (or similar) for alpha and beta emitters which send out radiation that does not easily get out from the body. For gamma emitters the detection can be done with an external "gamma camera", a detector for gamma rays. Since gamma rays are electrically neutral they cannot be found with a GM-tube, but have to be studied indirectly. A "scintillation detector" uses certain compounds (sodium iodide with small amounts of tellurium added) where gamma rays can be absorbed and the subsequently emitted photons of visual light observed.

PET = positron-emission tomography with coincidence measurements

This is a special case of diagnosis with radioactive tracers. Recall that there are three types of beta decay: emission of electrons, of their antiparticle positrons, and electron capture. The second of these types is used, and when then the positrons are annihilated by ordinary electrons in the body, two gamma photons are emitted.

Even if a photon does not have a mass, it does according to relativity theory have a momentum, and to conserve the momentum in the annihilation event, the photons move out in opposite directions. (The emitted positron has a very small momentum and kinetic energy compared to that of the photons, since all the mass in the positron and electron is converted to energy as E = m_totc²).

The beta emitter is inserted into the patient (e.g. as radioactive oxygen ¹⁵O inhaled) and a ring of gamma detectors around e.g. the head used to detect the gamma photons. To distinguish the relevant photons from others a computer only counts the photon hitting a detector if another hits the detector exactly opposite to it at the same time (coincidence). In addition the same imaging techniques as in CAT scans are used to produce a three-dimensional image of the brain.

11.10. Biomedical nuclear physics

Types of ionising radiation

D.5.1. biol effects of ionizing radiation

"Dose" units

- mention Bq also

Absorbed dose = Absorbed Energy / mass [DB p. 11]

Exposure = total charge / mass [DB p. 11]

Dose equivalent = quality factor x Absorbed dose [DB p. 11]

D.5.2. exposure, absorbed dose, quality factor, dose equivalent in dosimetry

Precautions for ionising radiation

There are 3 main ways of reducing the dose of ionising radiation one gets.

1. Time : the source causes a certain equivalent dose per time unit, e.g. in microsieverts/hour, and limiting the time of exposure decreases the total dose received.

2. Shielding : to place material which absorbs a significant part of the radiation between oneself and the source (Here one must be careful to avoid secondary radiation emitted by the shielding material when the primary radiation collides with its atoms, or "Bremsstrahlung", the electromagnetic radiation emitted whenever a moving charge is accelerated or decelerated.

3. Distance : for any point source of radiation the "intensity" of radiation hitting a given area is inversely proportional to the square of the distance from it

(In Finnish : remember ASE = aika, suoja, etäisyys)

Radiation therapy

D.5.7. radiation therapy for cancer basics ??

D.5.8. probs: choice of isotope for given diagnostic or therap.???

Physical, biological and effective half-life

Recall from nuclear physics that the number of decayed atoms of the N present in a sample in a given time Dt is

DN = - lNDt

where l = the decay constant, related to the half-life of the nuclide as T_½ = ln 2 / l [DB p. 8]. This half-life will here be called the physical half-life T_R (and the decay constant l_R) of the radioactive nuclide in question. But some of the radioactive atoms in the patient's body may also leave it before they have decayed with exhaled air, urine, feces, vomit, semen or other ways of losing materia from the body. These processes do not precisely follow any simple mathematical formula, but a reasonable approximation is that they are proportional to the number of radioactive nuclei in the body and the time given, and to some biological decay constant l_B. We will then have the total or effective decay constant

• l_E = l_B + l_R which since any l = ln2/ T gives
• ln2/T_E = ln2/T_B + ln2/T_R or cancelling ln2 then:

1 / T_E = 1/T_B + 1 / T_R [DB p. 11]