Abstract
Xrays are used to obtain diagnostic information and for cancer therapy. We begin by discussing the production of xrays, primarily via bremsstrahlung radiation. We introduce some quantities that are important for measuring how the absorbed photon energy relates to the response of a detector, which might be a film, an ionization chamber, a chemical detector, or a digital detector. Several specialized techniques are described, such as angiography, mammography, fluoroscopy, and computed tomography. Radiation therapy is presented, including an analysis of intensity modulated radiation therapy. Biological mechanisms of radiation damage are outlined, and the chapter ends with a discussion of the risk of radiation and the linear nothreshold model.
Keywords
 Radon Concentration
 Modulation Transfer Function
 Linear Energy Transfer
 Boron Neutron Capture Therapy
 Excess Relative Risk
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Notes
 1.
See Platzman (1961); also Attix (1986, pp. 339–343).
 2.
There is also a problem at high energies because the range of the electrons is large. If they are to come to rest within the chamber, the size of the chamber becomes comparable to the photon attenuation coefficient.
 3.
The fluorescent radiation has a wavelength of about 545 nm (green), and each absorbed highenergy photon has sufficient energy to produce about 14,000 fluorescence photons. However, the efficiency of production is only about 5 % so 700 photons are produced. Some of these escape or are absorbed. Each xray photon produces about 150 photons of visible and ultraviolet light that strike the emulsion—more than enough to blacken the film in the region where the xray photon was absorbed by the screen.
 4.
An argument based on Eq. 2.14 can be used to show that \(\log _{10}x =(1/2.303)\ln x =0.43\ln x\).
 5.
Even though the film may have a linear response over a larger range, doubling the exposure usually makes the film too dense to read.
 6.
Although a digital detector has greater dynamic range, proper exposure is still important. Too low an exposure introduces noise; an excessive exposure increases he dose to the patient unnecessarily.
 7.
The word peak is included because the voltage from power supplies in older machines had considerable “ripple” caused by the alternating voltage from the power lines. Even in modern machines, the voltage pulse applied to the tube may not have a purely rectangular waveform, and kVp may not uniquely determine the xray spectrum during the pulse. Modern kilovolt power supplies are described by Sobol (2002).
 8.
The pointspread function of a detector is easily measured. A point source is created by passing the xrays through a pinhole in a piece of lead placed directly on the detector. The resulting image is the pointspread function. We saw in Chap. 12 how this is related to the modulation transfer function. Standard techniques have been developed for measuring the modulation transfer function (MTF) (ICRU Report 41 1986; ICRU Report 54 1996).
 9.
Examples of parallel subsystems are the two emulsion layers on doublecoated film, and the effect of primary and scattered radiation on the formation of the image.
 10.
 11.
It is sometimes useful to write it as
$$\begin{aligned} C_{\text{noise in}} & \equiv\frac{(f\Phi S)^{1/2}}{f\Phi S}=\frac{1}\% {f^{1/2}S^{1/2}}\frac{\left( \Delta\Phi\right) _{\text{rms}}}{\Phi}=\frac {1}{f^{1/2}S^{1/2}}\frac{A(\Delta X)_{\text{rms}}}{AX}\\ & =\frac{1}{f^{1/2}S^{1/2}}\frac{(\Delta X)_{\text{rms}}}{X}. \end{aligned}$$  12.
An analogous phenomenon is seen when counting individual photons with a radiation detector at a fixed average rate. The number counted in a given time interval fluctuates, with the fractional fluctuation inversely proportional to the square root of the counting time.
 13.
There are statistical fluctuations in the signal as well as the noise. The variance of the difference between signal and noise will be the sum of the variances in the signal and in the noise. This has the effect of increasing the noise by a factor of \(\sqrt {2}\), which can be absorbed in the value of \(K\) that is chosen. See Problem 760.
 14.
The ability to detect the signal accurately is greater when the observer knows the nature of the signal and is only asked whether it is or is not present. That is, the ability of an observer to detect a signal is less in the more realistic situation where the observer does not know what the signal is or where it might be in the radiograph.
 15.
This simple equality exists only because we are using a model with Poisson statistics. The DQE is defined more generally as the square of the signaltonoise ratio of the detector output divided by the square of the signaltonoise ratio of the detector input. The more general definitions of DQE and NEQ are discussed in Wagner (1983) and Wagner (1977).
 16.
Line pairs (abbreviated lp) are analogous to the period of a square wave.
 17.
 18.
This is a simplification. It is possible for a double strand break to repair properly. See Hall and Giaccia (2012, p. 18).
 19.
In general, cells exhibit the greatest sensitivity in M and G\(_{2}\).
 20.
 21.
The nomenclature here is quite confusing. ICRP used to define the dose equivalent, also denoted by \(H\), as \(QD\), where \(q\) was called the quality factorof the radiation. The radiation weighting factor is very similar, and essentially numerically equivalent, to the earlier quality factor, \(q\). Values of \(q\) recommended by Nuclear Regulatory Commission (NRC) are 1 for photons and electrons, 10 for neutrons of unknown energy and highenergy protons, and 20 for \(\alpha \) particles, multiply charged ions, fission fragments, and heavy particles of unknown charge. The ICRP has its own recommendations, that differ slightly for protons and neutrons. See McCollough and Schueler (2000).
 22.
Both the sievert and the gray are J kg\(^{1}.\) Different names are used to emphasize the fact that they are quite different quantities. One is physical, and the other includes biological effects. An older unit for \(H\) is the rem. 100 rem = 1 Sv.
 23.
An older, related quantity is the effective dose equivalent, \(H_{E}=W_{T}QD\).
 24.
Values of \(H_{T}\) were provided by C. McCollough.
 25.
Radon is chemically inert gas that escapes from the earth. Since it is chemically inert, we breathe it in and out. When it decays in the air (the decay scheme is described in Sect. 17.12), the decay products attach themselves to dust particles in the air. When we breathe these dust particles, some become attached to the lining of the lungs, irradiating adjacent cells as they undergo further decay.
 26.
The dose to the lungs from radon progeny is about 1 mGy yr\(^{1}\). This is multiplied by \(W_{r}=20\) and \(W_{T}=0.12\) (lungs) to arrive at an effective dose of 2.4 mSv yr\(^{1}\).
 27.
References
AAPM Report 87 (2005) Diode in vivo dosimetry for patients receiving external beam radiation therapy. American Association of Physicists in Medicine, College Park. Report of Task Group 62 of the Radiation Therapy Committee
AAPM Report 96 (2007) The measurement, reporting and management of radiation dose in CT. American Association of Physicists in Medicine, College Park. Report of Task Group 23 of the Diagnostic Imaging Council CT Committee
Alberts B et al (2002) Molecular biology of the cell, 4th edn. Garland, New York, p 230
Armato SG, van Ginneken B (2008) Anniversary paper: image processing and manipulation through the pages of Medical Physics. Med Phys 35:4488–4500
Attix FH (1986) Introduction to radiological physics and radiation dosimetry. Wiley, New York
Ayotte P, Lévesque B, Gauvin D, McGregor RG, Martel R, Gingras S, Walker WB, Létourneau E G (1998) Indoor exposure to \(^{222}\)Rn: a public health perspective. Health Phys 75(3):297–302
Barth RF (2003) A critical assessment of boron neutron capture therapy: an overview. J Neurooncol 62:1–5. (The entire issue of the journal is devoted to a review of BNCT)
BEIR Report V (1990) Health effects of exposure to low levels of ionizing radiation. National Academy Press, Washington, DC. (Committee on the Biological Effects of Ionizing Radiation)
BEIR Report VI (1999) Health effects of exposure to radon. National Academy Press, Washington, DC. (Committee on Health Risks of Exposure to Radon)
BEIR Report VII (2005) Health risks from exposure to low levels of ionizing radiation. National Academy Press, Washington, DC. (Committee to Assess Health Risks from Exposure to Low Levels of Ionizing Radiation)
Birch R, Marshall M (1979) Computation of bremsstrahlung Xray spectra and comparison with spectra measured with a Ge(Li) detector. Phys Med Biol 24:505–517
Boice JD Jr (1996) Risk estimates for radiation exposures. In: Hendee WR, Edwards FM (eds). Health effects of exposure to lowlevel ionizing radiation. Institute of Physics, Bristol.
Brenner DJ, Elliston CD (2004) Estimated radiation risks potentially associated with fullbody CT screening. Radiology 232:735–738
Broad WJ (1980) Riddle of the Nobel debate. Science 207:37–38
Brooks AL (2003) Developing a scientific basis for radiation risk estimates: goal of the DOE low dose research program. Health Phys 85(1):85–93
Brooks RA, DiChiro G (1976a) Principles of computer assisted tomography (CAT) in radiographic and radioisotope imaging. Phys Med Biol 21:689–732
Brooks RA, DiChiro G (1976b) Statistical limitations in xray reconstructive tomography. Med Phys 3:237–240
Cohen BL (1995) Test of the linear—no threshold theory of radiation carcinogenesis for inhaled radon decay products. Health Phys 68(2):157–174
Cohen BL (1998) Response to Lubin’s proposed explanations of our discrepancy. Health Phys 75(1):18–22
Cohen BL (1999) Response to the Lubin rejoinder. Health Phys 76(4):437–439
Cohen, B. L. (2002). Cancer risk from lowlevel radiation. [see comment]. AJR Am J Roentgenology 179(5):1137–1143
Cohen BL (2007) The cancer risk from lowlevel radiation (Chapter 3). In: Tack D, Gevenois PA (eds) Radiation dose from adult and pediatric multidetector computed tomography. Springer, Berlin
Cormack AM (1980) Nobel award address: early twodimensional reconstruction and recent topics stemming from it. Med Phys 7(4):277–282
Cowen AR, Davies AG, Sivananthan MU (2008a) The design and imaging characteristics of dynamic, solidstate, flatpanel xray image detectors for digital fluoroscopy and fluorography. Clin Radiol 63:1073–1085
Cowen AR, Kengyelics SM, Davies AG (2008b) Solidstate, flatpanel, digital radiography detectors and their physical imaging characteristics. Clin Radiol 63:487–498
Delaney TF, Kooy HM (2008) Proton and charged particle radiotherapy. Williams and Wilkins, Philadelphia
DeVita VT (2003) Hodgkin’s disease—Clinical trials and travails. N Engl J Med 348(24):2375–2376
DiChiro G, Brooks RA (1979) The 1979 Nobel prize in physiology or medicine. Science 206:1060–1062
Doi K (2006) Diagnostic imaging over the last 50 years: research and development in medical imaging science and technology. Phys Med Biol 51:R5–R27
Doss M, Little MP, Orton C (2014) Point/Counterpoint: lowdose radiation is beneficial, not harmful. Med Phys 41:070601. doi:http://dx.doi.org/10.1118/1.4881045
Douglas JG, Koh WJ, AustinSeymour M, Laramore GE (2003) Treatment of salivary gland neoplasms with fast neutron radiotherapy. Arch Otolaryngol Head Neck Surg 129(9):944–948
Duncan W (1994) An evaluation of the results of neutron therapy trials. Acta Oncol 33(3):299–306. (This issue of the journal is devoted to fastneutron therapy)
Goitein M (2008) Radiation oncology: a physicist’s eye view. Springer, New York
Hall EJ (2000) Radiobiology for the radiologist, 5th edn. Lippincott Williams & Wilkins, Philadelphia
Hall EJ (2002) Helical CT and cancer risk: introduction to session I. Pediatr Radiol 32:225–227
Hall EJ (2003) The bystander effect. Health Phys 85(1):31–35
Hall EJ, Giaccia AJ (2012) Radiobiology for the radiologist, 7th edn. Lippincott Williams & Wilkins, Philadelphia
Harris CC, Hamblen DP, Francis JE Jr (1969) Basic Principles of Scintillation Counting for Medical Investigators. ORNL2808
Health Physics Society (2010) Radiation risk in perspective. Position statement of the health physics society. http://hps.org/ hpspublications/positionstatements.html./risk ps0102.pdf
Hempelmann LH (1949) Potential dangers in the uncontrolled use of shoe fitting fluoroscopes. N Engl J Med 241:335–336
Hendee WR, Ritenour ER (2002) Medical imaging physics, 4th edn. WileyLiss, New York
Higson DJ (2004) The bell tolls for LNT. Health Phys 87(Supplement 2):S47–S50
Hogstrom KR, Almond PR (2006) Review of electron beam therapy physics. Phys Med Biol 51:R455–489
Hounsfield GN (1980) Nobel award address: computed medical imaging. Med Phys 7(4):283–290
Hubbell JH, Seltzer SM (1996) Tables of xray mass attenuation coefficients and mass energyabsorption coefficients from 1 keV to 20 MeV for elements Z=1 to 92 and 48 additional substances of dosimetric interest. National Institute of Standards and Technology Report NISTIR 5632. http: //www.nist.gov/pml/data/xraycoef
Hunt DC, Kirby SS, Rowlands JA (2002) Xray imaging with amorphous selenium: xray to charge conversion gain and avalanche multiplication gain. Med Phys 29(11):2464–2471
ICRP (1991) The 1990 recommendations of the international commission on radiological protection. Ann ICRP 21:1–3
ICRP (2007) The 2007 recommendations of the international commission on radiation protection. Ann ICRP Publication No. 103. Elsevier, New York
ICRU Report 31 (1979) Average energy required to produce an ion pair. International Commission on Radiation Units and Measurements, Bethesda
ICRU Report 33 (1980, Reprinted 1992). Radiation Quantities and Units Bethesda, MD, International Commission on Radiation Units and Measurements.
ICRU Report 37 (1984) Stopping powers for electrons and positrons. International Commission on Radiation Units and Measurements, Bethesda
ICRU Report 39 (1985) Determination of dose equivalents resulting from external radiation sources. International Commission on Radiation Units and Measurements, Bethesda
ICRU Report 41 (1986) Modulation transfer function of screenfilm systems. International Commission on Radiation Units and Measurements, Bethesda
ICRU Report 54 (1996) Medical imaging—the assessment of image quality. International Commission on Radiation Units and Measurements, Bethesda
ICRU Report 74 (2005) Patient dosimetry for x rays used in medical imaging. J ICRU 5(2):1–113
Kalender WA (2011) Computed tomography: fundamentals, system technology, image quality and applications, 3rd edn. Publicis, Erlangen
Kassis AI (2004) The amazing world of Auger electrons. Rad Biol 80(11–12):789–803
Kathren RL (1996) Pathway to a paradigm: the linear nonthreshold doseresponse model in historical context: the American Academy of Health Physics 1995 Radiology Centennial Hartman Oration. Health Phys 70(5):621–635
Khan FM (1986) Clinical electron beam dosimetry. In Keriakes JG, Elson HR, Born CG (eds) Radiation oncology physics. American Association of Physicists in Medicine, College Park
Khan FM (2003) The physics of radiation therapy, 3rd edn. Philadelphia, Lippincott Williams & Wilkins, p 163, 204, 210, 215
Khan FM (2010) The physics of radiation therapy, 4th edn. Lippincott Williams & Wilkins, Philadelphia
Kondo S (1993) Health effects of lowlevel radiation. Kinki University Press, Osaka. English translation: Medical Physics, Madison
Körner M, Weber CH, Wirth S, Pfeifer KJ, Reiser MF, Treitl M (2007) Advances in digital radiography: physical principles and system overview. Radiographics 27:675–686
Lubin JH (1998a) On the discrepancy between epidemiologic studies in individuals of lung cancer and residential radon and Cohen’s ecologic regression. Health Phys 75(1):4–10
Lubin JH (1998b) Rejoinder: Cohen’s response to “On the discrepancy between epidemiologic studies in individuals of lung cancer and residential radon and Cohen’s ecologic regression”. Health Phys 75(1):29–30
Lubin JH (1999) Response to Cohen’s comments on the Lubin rejoinder. Health Phys 77(3):330–332
Lutz G (1999) Semiconductor radiation detectors. Springer, New York
Macovski A (1983) Medical imaging systems. PrenticeHall, Englewood Cliffs
McCollough CH, Schueler BA (2000) Educational treatise: calculation of effective dose. Med Phys 27(5):828–837
McCollough CH, Chen GH, Kalender W, Leng S, Samei E, Taguchi K, Wang G, Yu L, Pettigrew RI (2012) Achieving routine submillisievert CT scanning: report from the summit on management of radiation dose in CT. Radiology 264(2):567–580
Mettler FA, Huda W, Yoshizumi TT, Mahesh M (2008) Effective doses in radiology and diagnostic nuclear medicine: a catalog. Radiology 248:254–263
Metz CE, Doi K (1979) Transfer function analysis of radiographic imaging systems. Phys Med Biol 24(6):1079–1106
Miralbell R, Lomax A, Cella L, Schneider U (2002) Potential reduction of the incidence of radiationinduced second cancers by using proton beams in the treatment of pediatric tumors. Int J Radiat Oncol Biol Phys 54(3):824–829
Mossman KL (2001) Deconstructing radiation hormesis. Health Phys 80(3):263–269
NCRP Report 94 (1987) Exposure of the population in the United States and Canada from natural background radiation. National Council of Radiation Protection and Measurements, Bethesda
NCRP Report 100 (1989) Exposure of the U.S. population from diagnostic medical radiation. National Council of Radiation Protection and Measurements, Bethesda
NCRP Report 136 (2001) Evaluation of the linearnonthreshold doseresponse model for ionizing radiation. National Council of Radiation Protection and Measurements, Bethesda
NCRP Report 160 (2009) Ionizing radiation exposure of the population of the United States. National Council on Radiation Protection and Measurements, Bethesda
Orton C (1997) Fractionation: radiobiological principles and clinical practice (Chapter 21). In: Khan FM, Gerbi B (eds) Treatment planning in radiation oncology. Williams and Wilkins, Baltimore
Pisano ED, Yaffe MJ (2005) Digital mammography. Radiology 234:353–362
Platzman RL (1961) Total ionization in gases by highenergy particles: an appraisal of our understanding. Intl J Appl Radiat Is 10:116–127
Ratliff ST (2009) Resource letter MPRT1: medical physics in radiation therapy. Am J Phys 77:774–782
Rowlands JA (2002) The physics of computed radiography. Phys Med Biol 47: R123–R126
Schlomka JP, Roessl E, Dorscheid R, Dill S, Martens G, Istel T, Bäumer C, Herrmann C, Steadman R, Zeitler G, Livne A, Proksa R (2008) Experimental feasibility of multienergy photoncounting Kedge imaging in preclinical computed tomography. Phys Med Biol 53:4031–4047
Schulz RJ, Kagan AR (2002) On the role of intensitymodulated radiation therapy in radiation oncology. Med Phys 29(7):1473–1482
Shani G (1991) Radiation dosimetry: instrumentation and methods. CRC, Boca Raton
Shani G (2001) Radiation dosimetry: instrumentation and methods, 2nd edn. CRC, Boca Raton
Smith AR (2009) Vision 20/20: proton therapy. Med Phys 36:556–568
Sobol WT (2002) High frequency xray generator basics. Med Phys 29(2):132–144
Steel GG (1996) From targets to genes: a brief history of radiosensitivity. Phys Med Biol 41(2):205–222
Suess C, Polacin A, Kalender WA (1995) Theory of xenon/computed tomography cerebral blood flow methodology. In: Tomonaga M, Tanaka A, Yonas H (eds). Quantitative cerebral blood flow measurements using stable xenon/CT: clinical applications. Futura, Armonk
Suit H, Urie M (1992) Proton beams in radiation therapy. J Natl Cancer Inst 84(3):155–164
Tack D, Gevenois PA (eds) (2007) Radiation dose from adult and pediatric multidetector computed tomography. Springer, Berlin
Tubiana M, Feinendegen LE, Yang C, Kaminski JM (2009) The linear nothreshold relationship is inconsistent with radiation biologic and experimental data. Radiology 251:13–22
Uffmann M, SchaeferProkop C (2009) Digital radiography: the balance between image quality and required radiation dose. Eur J Radiol 72:202–208
Upton AC (2003) The state of the art in the 1990’s: NCRP report no. 136 on the scientific bases for linearity in the doseresponse relationship for ionizing radiation. Health Phys 85(1):15–22
van Eijk CWE (2002) Inorganic scintillators in medical imaging. Phys Med Biol 47:R85–R106
Wagner HN Jr (ed) (1968) Principles of nuclear medicine. Elsevier, p 153, 162
Wagner RF (1977) Toward a unified view of radiological imaging systems. Part II: Noisy images. Med Phys 4(4):279–296
Wagner RF (1983) Low contrast sensitivity of radiologic, CT, nuclear medicine, and ultrasound medical imaging systems. IEEE Trans Med Imaging MI2(3):105–121
Wagner RF, Weaver KE, Denny EW, Bostrom RG (1974) Toward a unified view of radiological imaging systems. Part I: Noiseless images. Med Phys 1(1):11–24
Williams CR (1949) Radiation exposures from the use of shoefitting fluoroscopes. N Engl J Med 241:333–335
William TM, James DC (1989) Radiation Oncology, 6th edn. Mosby, St. Louis
Xu XG, Bednarz B, Paganetti H (2008) A review of dosimetry studies on externalbeam radiation treatment with respect to second cancer induction. Phys Med Biol 53:R193–R241
Yu CX, Amies CJ, Svatos M (2008) Planning and delivery of intensitymodulated radiation therapy. Med Phys 35:5233–5241
Author information
Authors and Affiliations
Corresponding author
Appendices
Symbols Used in Chapter 16
Problems
16.2.1 Section 16.1
Problem 1.
Use Eqs. 15.2 and 16.2 to answer the following questions. Then compare your answers to values given in tables, such as those in the Handbook of Chemistry and Physics. What is the minimum energy of electrons striking a copper target that will cause the \(K\) xray lines to appear? What is the approximate energy of the \(K_{\alpha }\) line? Repeat for iodine, molybdenum, and tungsten.
Problem 2.
When tungsten is used for the anode of an xray tube, the characteristic tungsten \(K_{\alpha }\) line has a wavelength of \(2.1\times 10^{11}\) m. Yet a voltage of \(69,525\) V must be applied to the tube before the line appears. Explain the discrepancy in terms of an energylevel diagram for tungsten.
Problem 3.
Henry Moseley first assigned atomic numbers to elements by discovering that the square root of the frequency of the \(K_{\alpha }\) photon is linearly related to \(z\). Solve Eq. 16.2 for \(z\) and show that this is true. Plot \(z\) vs. the square root of the frequency and compare it to data you look up.
Problem 4.
Equation 16.3b, indicating the number of photons of energy \(h\nu \) produced by bremsstrahlung, is known as Kramer’s law, and is plotted as crosses in Fig. 16.5 (except for the drop at low energies caused by attenuation that is not included in Kramer’s law).

(a)
Sketch a plot of \(d\Phi /dE\) versus energy (\(0 <h\nu <h\nu _0\)) using Eq. 16.3b.

(b)
Use Eq. 16.3b, integrate \(d\Phi /dE\) over energy from 0 to \(h\nu _0\), and show that Kramer’s law predicts that the number of photons goes to infinity if attenuation is not taken into account.

(c)
Integrate Eq. 16.3a from 0 to \(h\nu _0\) and show that the energy of the bremsstrahlung radiation predicted by Kramer’s law is finite, even if the number of photons is infinite. Explain how this is possible. Derive an expression for the total bremsstrahlung energy.
Problem 5.

(a)
The energy fluence spectrum for a thin target \(d\Psi /d(h\nu )\) in Fig. 16.3 is constant (call it \(C^{\prime }\)) for \(h\nu <h\nu _0\) and zero for higher energies. Calculate the photon particle fluence rate \(d\Phi /d(h\nu )\) and plot it vs. \(h\nu \).

(b)
Use the chain rule to express the photon particle fluence rate \(d\Phi /d\lambda \) for a thin target as a function of wavelength \(\lambda \) and plot it.

(c)
Express Eq. 16.3a, giving the energy fluence rate \(d\Psi /d(h\nu )\) for a thick target as a function of photon frequency \(h\nu \), as an equation for \(d\Psi /d\lambda \) as a function of wavelength \(\lambda ,\) and plot it.

(d)
Repeat the analysis in part (c) for Eq. 16.3b, giving the photon fluence rate \(d\Phi /d\lambda \) for a thick target. Plot it.
16.2.2 Section 16.2
Problem 6.
A beam of 0.08MeV photons passes through a body of thickness \(L\). Assume that the body is all muscle with \(\rho =1.0\times 10^{3}\) kg m\(^{3}\). The energy fluence of the beam is \(\Psi \) J m\(^{2}\).

(a)
What is the skin dose where the beam enters the body?

(b)
Assume the beam is attenuated by an amount \(e^{\mu L}\) as it passes through the body. Calculate the average dose as a function of the fluence, the body thickness, and μ.

(c)
What is the limiting value of the average dose as \(\mu L\rightarrow 0\)?

(d)
What is the limiting value of the average dose as \(\mu L\rightarrow \infty \)? Does the result make sense? Is it useful?
Problem 7.
The obsolete unit, the roentgen (R), is defined as \(2.08\times 10^{9}\) ion pairs produced in \(0.001293\) g of dry air. (This is 1 cm\(^{3}\) of dry air at standard temperature and pressure.) Show that if the average energy required to produce an ion pair in air is 33.7 eV (an old value), then 1 R corresponds to an absorbed dose of \(8.69\times 10^{3}\) Gy and that 1 R is equivalent to \(2.58\times 10^{4}\) C kg\(^{1}\).
Problem 8.
During the 1930s and 1940s it was popular to have an xray fluoroscope unit in shoe stores to show children and their parents that shoes were properly fit. These marvellous units were operated by people who had no concept of radiation safety and aimed a beam of xrays upward through the feet and right at the reproductive organs of the children! A typical unit had an xray tube operating at 50 kVp with a current of 5 mA.

(a)
What is the radiation yield for 50keV electrons on tungsten? How much photon energy is produced with a 5mA beam in a 30s exposure?

(b)
Assume that the xrays are radiated uniformly in all directions (this is not a good assumption) and that the xrays are all at an energy of 30 keV. (This is a very poor assumption.) Use the appropriate values for striated muscle to estimate the dose to the gonads if they are at a distance of 50 cm from the xray tube. Your answer will be an overestimate. Actual doses to the feet were typically 0.014–0.16 Gy. Doses to the gonads would be less because of \(1/r^{2}\). Two of the early articles pointing out the danger are Hempelmann (1949) and Williams (1949).
16.2.3 Section 16.3
Problem 9.
Rewrite Eq. 16.9 in terms of exponential decay of the viewing light and relate the optical density to the attenuation coefficient and thickness of the emulsion.
Problem 10.
Derive the useful rule of thumb \(\Delta (\)OD\()=0.43\gamma \,\Delta X/X\).
Problem 11.
The atomic crosssections for the materials in a gadolinium oxysulfide screen for 50keV photons are

(a)
What is the crosssection per target molecule of GdO\(_{2}\)S?

(b)
How many target molecules per unit area are there in a thickness \(\rho dx\) of material?

(c)
What is the probability that a photon interacts in traversing 1.2 kg m\(^{2}\) of GdO\(_{2}\)S?
Problem 12.
The film speed is often defined as the reciprocal of the exposure (in roentgens) required to give an optical density that is 1 greater than the base density. Assume that in Fig. 16.6 a relative exposure of 1 corresponds to \(10^{5}\) C kg\(^{1}\). Calculate the film speed.
Problem 13.
A dose of \(1.74\times 10^{4}\) Gy was estimated for part of the body just in front of an unscreened xray film. Suppose that a screen permits the dose to be reduced by a factor of 20. Calculate the skin dose on the other side of the body (the entrance skin dose) assuming 50keV photons and a body thickness of 0.2 m. Ignore buildup, and assume that only unattenuated photons are detected.
Problem 14.
Find an expression for photon fluence per unit absorbed dose in a beam of monoenergetic photons. Then find the photon fluence for 50keV photons that causes a dose of 10\(^{5}\) Gy in muscle.
Problem 15.
A dose of 100 Gy might cause noticeable radiation damage in a sodium iodide crystal. How long would a beam of 100keV photons have to continuously and uniformly strike a crystal of 1cm\(^{2}\) area at the rate of \(10^{4}\) photon s\(^{1}\), in order to produce this absorbed dose? For NaI, \(\mu _{\text {en}}/\rho =0.1158\) m\(^{2}\) kg\(^{1}\).
Problem 16.
Another method to measure the absorbed dose is by calorimetry. Show that if all the energy imparted warms the sample, the temperature rise is \(2.39\times 10^{4}\operatorname {{}^{\circ }\textrm {C}}\) per Gy.
16.2.4 Section 16.4
Problem 17.
Plot μ for lead, iodine, and barium from 10 to 200 keV.
Problem 18.
Use a spreadsheet to make the following calculations. Consider a photon beam with 100 kVp.

(a)
Use Eq. 16.3b to calculate the photon fluence from a thick target at 1, 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100 keV.

(b)
The specific gravity of aluminum is 2.7. Make a table of the photon fluence at these energies emerging from 2 and 3 mm of aluminum. Compare the features of this table to Fig. 16.15.

(c)
Use trapezoidal integration to show that the average photon energy is 44 keV after 2mm filtration and 47 keV after 3mm filtration.

(d)
Repeat for 120 kVp and show that the average energies after the same filtrations are 52 and 55 keV.
Problem 19.
To get a qualitative understanding of Fig. 16.15, assume the photon particle fluence is given by Eq. 16.3b multiplied by a factor \(exp(BL/(h\nu )^3)\), where \(B\) is a constant, \(L\) is the thickness of the aluminum filtration (in cm) and the \(1/(h\nu )^3\) dependence on the photon energy (in keV) arises from the photoelectric crosssection energy dependence, Eq. 15.8.

(a)
What are the units of \(B\)?

(b)
Use some simple numerical method to estimate \(B\) from Fig. 16.15. One method might be to calculate the maximum of the photon fluence curve and adjust \(B\) so the maximum occurs at the correct photon energy.

(c)
For the value of \(B\) you found in part (b), plot the three relative photon fluence curves as a function of photon energy, as shown in Fig. 16.15. Normalize the curves so the peak of the 0.1cm filtration curve is equal to 1.
Problem 20.
Xray beams have a spectrum of photon energies. It would be very laborious to measure the spectrum every time we want to check the quality of the beam. In addition to kVp, one simple measurement that is used to check beam quality (related to the energy spectrum) is the halfvalue layer HVL–the thickness of a specified absorber (often Cu or Al) that reduces the intensity of the beam to onehalf.

(a)
For a monoenergetic beam, relate HVL to the attenuation coefficient. What is the HVL if the attenuation coefficient is \(0.46\operatorname {mm}^{1}\)?

(b)
For a monoenergetic beam, how does the quartervalue layer QVL relate to HVL?

(c)
Suppose a beam has equal numbers of photons at two different energies. The attenuation coefficients at these energies are \(0.46\operatorname {mm}^{1}\) and \(0.6\operatorname {mm}^{1}\). Find the HVL and QVL for this beam. You may need to plot a graph or use a computer algebra program.
Problem 21.
The half value layer (HVL) is often used to characterize an xray beam. It is the thickness of a specified absorber that attenuates the beam to onehalf the original value. Figure 16.41 refers to a beam with a 3.0 mm Cu HVL. What is the value of the attenuation coefficient? What monoenergetic xray beam does this correspond to?
Problem 22.
Assume an antiscatter grid is made of lead sheets 3mm long with a spacing between sheets of 0.3 mm. Ignore the thickness of the sheets. If all photons hitting the sheets are absorbed, what is the largest angle from the incident beam direction that a photon can be scattered and still emerge?
16.2.5 Section 16.5
Problem 23.
Suppose that two measurements are made: one of the combination of signal and noise, \(y=s+n\), and one of just the noise \(n\). One wishes to determine \(s=yn\).

(a)
Find \(s\overline {s}\) in terms of \(y\), \(\overline {y}\), \(n\), and \(\overline {n}\).

(b)
Show that if \(y\) and \(n\) are uncorrelated, \(\overline {(s\overline {s})^{2}}=\overline {(y\overline {y})^{2}}+\overline {(n\overline {n})^{2}}\) and state the mathematical condition for being uncorrelated.

(c)
If \(y\) and \(n\) are Poisson distributed, under what conditions is the \(\sqrt {2}\) factor of Footnote 13 needed?
16.2.6 Section 16.7
Problem 24.
A molybdenum target is used in special xray tubes for mammography. The electron energy levels in Mo are as follows:
What is the energy of the \(K_{\alpha }\) line(s)? The \(K_{\beta }\) line(s) (defined in Fig. 16.2)?
Problem 25.
As a simple model for mammography, consider two different tissues: a mixture of 2/3 fat and 1/3 water, with a composition by weight of 12 % hydrogen, 52 % carbon and 36 % oxygen; and glandular tissue, composed of 11 % hydrogen, 33 % carbon, and 56 % oxygen. The density of the fat and water combination is 940 kg m\(^{3}\), and the density of glandular tissue is 1020 kg m\(^{3}\). What is the attenuation in 1 mm of the fatwater combination and in 1 mm of glandular tissue for 50keV photons? For 30keV photons?
16.2.7 Section 16.8
Problem 26.
It is often said that the number of photons that must be detected in order to measure a difference in fluence with a certain resolution can be calculated from \(N=(\Delta \Phi /\Phi )^{2}\). (For example, if we want to detect a change in \(\Phi \) of 1 % we would need to count \(10^{4}\) photons.) Use Eq. 16.20 to make this statement more quantitative. Discuss the accuracy of the statement.
Problem 27.
Spiral CT uses interpolation to calculate the projections at a fixed value of \(z\) before reconstruction. This has an effect on the noise. Let \(\sigma _{0}\) be the noise standard deviation in the raw projection data and \(\sigma \) be the noise in the interpolated data. The interpolated signal, \(\alpha \), is the weighted sum of two values: \(\alpha =w\alpha _{1}+(1w)\alpha _{2}\).

(a)
Show that the variance in \(\alpha \) is \(\sigma ^{2}=w^{2}\sigma _{0}^{2}+(1w)^{2}\sigma _{0}^{2}\). Plot \(\sigma /\sigma _{0}\) vs. \(W\).

(b)
Averaging over a \(360\operatorname {{{}^\circ }}\) scan involves integrating uniformly over all weights:
$$ \sigma^{2}=\int_{0}^{1}\left[ w^{2}\sigma_{0}^{2}+(1w)^{2}\sigma_{0}^{2}\right] \,dw. $$Find the ratio \(\sigma /\sigma _{0}\).
Problem 28.
An experimental technique to measure cerebral blood perfusion is to have the patient inhale xenon, a noble gas with \(Z=54\), \(A=131\) (Suess et al. 1995). The solubility of xenon is different in red cells than in plasma. The equation used is
where the arterial enhancement is in Hounsfield units, \(C_{\text {Xe}}\) is the concentration of xenon in the lungs (end tidal volume), and
Hct is the hematocrit: the fraction of the blood volume occupied by red cells. Discuss why the equation has this form.
16.2.8 Section 16.9
Problem 29.
Use Equations 16.30 and 16.31 to derive an expression for the probability of eradicating a tumor (no surviving tumor cells) as a function of dose for tumors containing different numbers of cells. Verify that your expression reproduces Fig. 16.38.
16.2.9 Section 30.
Problem 767
Geiger’s rule is an approximation to the rangeenergy relationship:
For protons in water \(A=0.0022\) when \(R\) is in cm and \(E\) is in MeV. The exponent \(p=1.77.\) This is a good approximation for \(E<200\) MeV. Use Geiger’s approximation to find \(dE/dx\) as a function of \(R\) for 100 MeV protons. Make a plot to show the Bragg peak when straggling is ignored.
Problem 31.
Assume the stopping power of a particle, \(S =dT/dx\), as a function of kinetic energy, \(T\), is \(S = C/T\).

(a)
What are the units of \(C\)? From Fig. 15.17, estimate for protons the range of kinetic energies over which \(S=C/T\) is appropriate.

(b)
If the initial kinetic energy at \(x = 0\) is \(T_0\), find \(T(x)\).

(c)
Determine the range \(R\) of the particle as a function of \(C\) and \(T_0\). For protons in water, estimate \(C\) from Fig. 15.26.

(d)
Plot \(S(x)\) vs. \(X\). Compare the shape of the curve to Fig. 16.47. Does this plot contain a Bragg peak?

(e)
Discuss the implications of the shape of \(S(x)\) for radiation treatment using this particle.
16.2.10 Section 16.11
Problem 32.
Calculate \((\overline {S}_{e}/\rho )_{g}^{w}\) in argon for 0.1, 1.0 and 10MeV electrons. The values of \(S_{e}/\rho \) for argon at these energies are 2.918, 1.376, and 1.678 cm\(^{2}\) g\(^{1}\).
Problem 33.
An ion chamber contains 10 cm\(^{3}\) of air at standard temperature and pressure. Find \(q\) vs. \(D\) for 0.5MeV electrons.
16.2.11 Section 16.12
Problem 34.
Suppose that the probability \(p\) per year of some event (death, mutations, cancer, etc.) consists of a spontaneous component \(S\) and a component proportional to the dose of something else, \(D\): \(p=S+AD\). The dose may be radiation, chemicals, sunlight, etc. Investigations of women given mammograms showed that if \(p\) is the probability of acquiring breast cancer, \(S=1.91\times 10^{3}\) and \(A=4\times 10^{4}\) Gy\(^{1}\). How many women had to be studied to distinguish between \(A=0\) and the value above if \(D=2\) Gy? If \(D=10^{2}\) Gy?
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Hobbie, R., Roth, B. (2015). Medical Uses of XRays. In: Intermediate Physics for Medicine and Biology. Springer, Cham. https://doi.org/10.1007/9783319126821_16
Download citation
DOI: https://doi.org/10.1007/9783319126821_16
Published:
Publisher Name: Springer, Cham
Print ISBN: 9783319126814
Online ISBN: 9783319126821
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)