Skip to main content

Medical Uses of X-Rays

  • 128k Accesses

Abstract

X-rays are used to obtain diagnostic information and for cancer therapy. We begin by discussing the production of x-rays, 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 no-threshold 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

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-12682-1_16
  • Chapter length: 42 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   84.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-12682-1
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   109.99
Price excludes VAT (USA)
Hardcover Book
USD   119.99
Price excludes VAT (USA)
Fig. 16.1
Fig. 16.2
Fig. 16.3
Fig. 16.4
Fig. 16.5
Fig. 16.6
Fig. 16.7
Fig. 16.8
Fig. 16.9
Fig. 16.10
Fig. 16.11
Fig. 16.12
Fig. 16.13
Fig. 16.14
Fig. 16.15
Fig. 16.16
Fig. 16.17
Fig. 16.18
Fig. 16.19
Fig. 16.20
Fig. 16.21
Fig. 16.22
Fig. 16.23
Fig. 16.24
Fig. 16.25
Fig. 16.26
Fig. 16.27
Fig. 16.28
Fig. 16.29
Fig. 16.30
Fig. 16.31
Fig. 16.32
Fig. 16.33
Fig. 16.34
Fig. 16.35
Fig. 16.36
Fig. 16.37
Fig. 16.38
Fig. 16.39
Fig. 16.40
Fig. 16.41
Fig. 16.42
Fig. 16.43
Fig. 16.44
Fig. 16.45
Fig. 16.46
Fig. 16.47
Fig. 16.48
Fig. 16.49
Fig. 16.50
Fig. 16.51
Fig. 16.52
Fig. 16.53
Fig. 16.54

Notes

  1. 1.

    See Platzman (1961); also Attix (1986, pp. 339–343).

  2. 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. 3.

    The fluorescent radiation has a wavelength of about 545 nm (green), and each absorbed high-energy 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 x-ray 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 x-ray photon was absorbed by the screen.

  4. 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. 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. 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. 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 x-ray spectrum during the pulse. Modern kilovolt power supplies are described by Sobol (2002).

  8. 8.

    The point-spread function of a detector is easily measured. A point source is created by passing the x-rays through a pinhole in a piece of lead placed directly on the detector. The resulting image is the point-spread 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. 9.

    Examples of parallel subsystems are the two emulsion layers on double-coated film, and the effect of primary and scattered radiation on the formation of the image.

  10. 10.

    This is very similar to the arguments about the fraction of photons absorbed by a visual pigment molecule in Eq. 14.68. Changes in the value of \(f\) in Fig. 14.43 shift the response curve along the axis.

  11. 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. 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. 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. 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. 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 signal-to-noise ratio of the detector output divided by the square of the signal-to-noise ratio of the detector input. The more general definitions of DQE and NEQ are discussed in Wagner (1983) and Wagner (1977).

  16. 16.

    Line pairs (abbreviated lp) are analogous to the period of a square wave.

  17. 17.

    See NCRP Report 100 (1989) for early data; Mettler et al. (2008) for 2008 data.

  18. 18.

    This is a simplification. It is possible for a double strand break to repair properly. See Hall and Giaccia (2012, p. 18).

  19. 19.

    In general, cells exhibit the greatest sensitivity in M and G\(_{2}\).

  20. 20.

    See Attix (1986), Chap. 10ff or Khan (2010), Chap. 8.

  21. 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 high-energy 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. 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. 23.

    An older, related quantity is the effective dose equivalent, \(H_{E}=W_{T}QD\).

  24. 24.

    Values of \(H_{T}\) were provided by C. McCollough.

  25. 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. 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. 27.

    For example, see Lubin (1998a); Cohen (1998); Lubin (1998b); Cohen (1999); Lubin (1999), BEIR VI (1999) and Cohen (2007).

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

    Google Scholar 

  • 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

    Google Scholar 

  • Alberts B et al (2002) Molecular biology of the cell, 4th edn. Garland, New York, p 230

    Google Scholar 

  • Armato SG, van Ginneken B (2008) Anniversary paper: image processing and manipulation through the pages of Medical Physics. Med Phys 35:4488–4500

    Google Scholar 

  • Attix FH (1986) Introduction to radiological physics and radiation dosimetry. Wiley, New York

    Google Scholar 

  • 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

    Google Scholar 

  • 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)

    Google Scholar 

  • 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)

    Google Scholar 

  • BEIR Report VI (1999) Health effects of exposure to radon. National Academy Press, Washington, DC. (Committee on Health Risks of Exposure to Radon)

    Google Scholar 

  • 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)

    Google Scholar 

  • Birch R, Marshall M (1979) Computation of bremsstrahlung X-ray spectra and comparison with spectra measured with a Ge(Li) detector. Phys Med Biol 24:505–517

    Google Scholar 

  • Boice JD Jr (1996) Risk estimates for radiation exposures. In: Hendee WR, Edwards FM (eds). Health effects of exposure to low-level ionizing radiation. Institute of Physics, Bristol.

    Google Scholar 

  • Brenner DJ, Elliston CD (2004) Estimated radiation risks potentially associated with full-body CT screening. Radiology 232:735–738

    Google Scholar 

  • Broad WJ (1980) Riddle of the Nobel debate. Science 207:37–38

    Google Scholar 

  • 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

    Google Scholar 

  • Brooks RA, DiChiro G (1976a) Principles of computer assisted tomography (CAT) in radiographic and radioisotope imaging. Phys Med Biol 21:689–732

    Google Scholar 

  • Brooks RA, DiChiro G (1976b) Statistical limitations in x-ray reconstructive tomography. Med Phys 3:237–240

    Google Scholar 

  • Cohen BL (1995) Test of the linear—no threshold theory of radiation carcinogenesis for inhaled radon decay products. Health Phys 68(2):157–174

    Google Scholar 

  • Cohen BL (1998) Response to Lubin’s proposed explanations of our discrepancy. Health Phys 75(1):18–22

    Google Scholar 

  • Cohen BL (1999) Response to the Lubin rejoinder. Health Phys 76(4):437–439

    Google Scholar 

  • Cohen, B. L. (2002). Cancer risk from low-level radiation. [see comment]. AJR Am J Roentgenology 179(5):1137–1143

    Google Scholar 

  • Cohen BL (2007) The cancer risk from low-level radiation (Chapter 3). In: Tack D, Gevenois PA (eds) Radiation dose from adult and pediatric multidetector computed tomography. Springer, Berlin

    Google Scholar 

  • Cormack AM (1980) Nobel award address: early two-dimensional reconstruction and recent topics stemming from it. Med Phys 7(4):277–282

    Google Scholar 

  • Cowen AR, Davies AG, Sivananthan MU (2008a) The design and imaging characteristics of dynamic, solid-state, flat-panel x-ray image detectors for digital fluoroscopy and fluorography. Clin Radiol 63:1073–1085

    Google Scholar 

  • Cowen AR, Kengyelics SM, Davies AG (2008b) Solid-state, flat-panel, digital radiography detectors and their physical imaging characteristics. Clin Radiol 63:487–498

    Google Scholar 

  • Delaney TF, Kooy HM (2008) Proton and charged particle radiotherapy. Williams and Wilkins, Philadelphia

    Google Scholar 

  • DeVita VT (2003) Hodgkin’s disease—Clinical trials and travails. N Engl J Med 348(24):2375–2376

    Google Scholar 

  • DiChiro G, Brooks RA (1979) The 1979 Nobel prize in physiology or medicine. Science 206:1060–1062

    Google Scholar 

  • 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

    Google Scholar 

  • Doss M, Little MP, Orton C (2014) Point/Counterpoint: low-dose radiation is beneficial, not harmful. Med Phys 41:070601. doi:http://dx.doi.org/10.1118/1.4881045

  • Douglas JG, Koh WJ, Austin-Seymour M, Laramore GE (2003) Treatment of salivary gland neoplasms with fast neutron radiotherapy. Arch Otolaryngol Head Neck Surg 129(9):944–948

    Google Scholar 

  • 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 fast-neutron therapy)

    Google Scholar 

  • Goitein M (2008) Radiation oncology: a physicist’s eye view. Springer, New York

    Google Scholar 

  • Hall EJ (2000) Radiobiology for the radiologist, 5th edn. Lippincott Williams & Wilkins, Philadelphia

    Google Scholar 

  • Hall EJ (2002) Helical CT and cancer risk: introduction to session I. Pediatr Radiol 32:225–227

    Google Scholar 

  • Hall EJ (2003) The bystander effect. Health Phys 85(1):31–35

    Google Scholar 

  • Hall EJ, Giaccia AJ (2012) Radiobiology for the radiologist, 7th edn. Lippincott Williams & Wilkins, Philadelphia

    Google Scholar 

  • Harris CC, Hamblen DP, Francis JE Jr (1969) Basic Principles of Scintillation Counting for Medical Investigators. ORNL-2808

    Google Scholar 

  • Health Physics Society (2010) Radiation risk in perspective. Position statement of the health physics society. http://hps.org/ hpspublications/positionstatements.html./risk ps010-2.pdf

  • Hempelmann LH (1949) Potential dangers in the uncontrolled use of shoe fitting fluoroscopes. N Engl J Med 241:335–336

    Google Scholar 

  • Hendee WR, Ritenour ER (2002) Medical imaging physics, 4th edn. Wiley-Liss, New York

    Google Scholar 

  • Higson DJ (2004) The bell tolls for LNT. Health Phys 87(Supplement 2):S47–S50

    Google Scholar 

  • Hogstrom KR, Almond PR (2006) Review of electron beam therapy physics. Phys Med Biol 51:R455–489

    Google Scholar 

  • Hounsfield GN (1980) Nobel award address: computed medical imaging. Med Phys 7(4):283–290

    Google Scholar 

  • Hubbell JH, Seltzer SM (1996) Tables of x-ray mass attenuation coefficients and mass energy-absorption 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) X-ray imaging with amorphous selenium: x-ray to charge conversion gain and avalanche multiplication gain. Med Phys 29(11):2464–2471

    Google Scholar 

  • ICRP (1991) The 1990 recommendations of the international commission on radiological protection. Ann ICRP 21:1–3

    Google Scholar 

  • ICRP (2007) The 2007 recommendations of the international commission on radiation protection. Ann ICRP Publication No. 103. Elsevier, New York

    Google Scholar 

  • ICRU Report 31 (1979) Average energy required to produce an ion pair. International Commission on Radiation Units and Measurements, Bethesda

    Google Scholar 

  • ICRU Report 33 (1980, Reprinted 1992). Radiation Quantities and Units Bethesda, MD, International Commission on Radiation Units and Measurements.

    Google Scholar 

  • ICRU Report 37 (1984) Stopping powers for electrons and positrons. International Commission on Radiation Units and Measurements, Bethesda

    Google Scholar 

  • ICRU Report 39 (1985) Determination of dose equivalents resulting from external radiation sources. International Commission on Radiation Units and Measurements, Bethesda

    Google Scholar 

  • ICRU Report 41 (1986) Modulation transfer function of screen-film systems. International Commission on Radiation Units and Measurements, Bethesda

    Google Scholar 

  • ICRU Report 54 (1996) Medical imaging—the assessment of image quality. International Commission on Radiation Units and Measurements, Bethesda

    Google Scholar 

  • ICRU Report 74 (2005) Patient dosimetry for x rays used in medical imaging. J ICRU 5(2):1–113

    Google Scholar 

  • Kalender WA (2011) Computed tomography: fundamentals, system technology, image quality and applications, 3rd edn. Publicis, Erlangen

    Google Scholar 

  • Kassis AI (2004) The amazing world of Auger electrons. Rad Biol 80(11–12):789–803

    Google Scholar 

  • Kathren RL (1996) Pathway to a paradigm: the linear nonthreshold dose-response model in historical context: the American Academy of Health Physics 1995 Radiology Centennial Hartman Oration. Health Phys 70(5):621–635

    Google Scholar 

  • 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

    Google Scholar 

  • Khan FM (2003) The physics of radiation therapy, 3rd edn. Philadelphia, Lippincott Williams & Wilkins, p 163, 204, 210, 215

    Google Scholar 

  • Khan FM (2010) The physics of radiation therapy, 4th edn. Lippincott Williams & Wilkins, Philadelphia

    Google Scholar 

  • Kondo S (1993) Health effects of low-level radiation. Kinki University Press, Osaka. English translation: Medical Physics, Madison

    Google Scholar 

  • Körner M, Weber CH, Wirth S, Pfeifer K-J, Reiser MF, Treitl M (2007) Advances in digital radiography: physical principles and system overview. Radiographics 27:675–686

    Google Scholar 

  • 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

    Google Scholar 

  • 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

    Google Scholar 

  • Lubin JH (1999) Response to Cohen’s comments on the Lubin rejoinder. Health Phys 77(3):330–332

    Google Scholar 

  • Lutz G (1999) Semiconductor radiation detectors. Springer, New York

    Google Scholar 

  • Macovski A (1983) Medical imaging systems. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  • McCollough CH, Schueler BA (2000) Educational treatise: calculation of effective dose. Med Phys 27(5):828–837

    Google Scholar 

  • 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

    Google Scholar 

  • Mettler FA, Huda W, Yoshizumi TT, Mahesh M (2008) Effective doses in radiology and diagnostic nuclear medicine: a catalog. Radiology 248:254–263

    Google Scholar 

  • Metz CE, Doi K (1979) Transfer function analysis of radiographic imaging systems. Phys Med Biol 24(6):1079–1106

    Google Scholar 

  • Miralbell R, Lomax A, Cella L, Schneider U (2002) Potential reduction of the incidence of radiation-induced second cancers by using proton beams in the treatment of pediatric tumors. Int J Radiat Oncol Biol Phys 54(3):824–829

    Google Scholar 

  • Mossman KL (2001) Deconstructing radiation hormesis. Health Phys 80(3):263–269

    Google Scholar 

  • 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

    Google Scholar 

  • NCRP Report 100 (1989) Exposure of the U.S. population from diagnostic medical radiation. National Council of Radiation Protection and Measurements, Bethesda

    Google Scholar 

  • NCRP Report 136 (2001) Evaluation of the linear-nonthreshold dose-response model for ionizing radiation. National Council of Radiation Protection and Measurements, Bethesda

    Google Scholar 

  • NCRP Report 160 (2009) Ionizing radiation exposure of the population of the United States. National Council on Radiation Protection and Measurements, Bethesda

    Google Scholar 

  • 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

    Google Scholar 

  • Pisano ED, Yaffe MJ (2005) Digital mammography. Radiology 234:353–362

    Google Scholar 

  • Platzman RL (1961) Total ionization in gases by high-energy particles: an appraisal of our understanding. Intl J Appl Radiat Is 10:116–127

    Google Scholar 

  • Ratliff ST (2009) Resource letter MPRT-1: medical physics in radiation therapy. Am J Phys 77:774–782

    Google Scholar 

  • Rowlands JA (2002) The physics of computed radiography. Phys Med Biol 47: R123–R126

    Google Scholar 

  • 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 multi-energy photon-counting K-edge imaging in pre-clinical computed tomography. Phys Med Biol 53:4031–4047

    Google Scholar 

  • Schulz RJ, Kagan AR (2002) On the role of intensity-modulated radiation therapy in radiation oncology. Med Phys 29(7):1473–1482

    Google Scholar 

  • Shani G (1991) Radiation dosimetry: instrumentation and methods. CRC, Boca Raton

    Google Scholar 

  • Shani G (2001) Radiation dosimetry: instrumentation and methods, 2nd edn. CRC, Boca Raton

    Google Scholar 

  • Smith AR (2009) Vision 20/20: proton therapy. Med Phys 36:556–568

    Google Scholar 

  • Sobol WT (2002) High frequency x-ray generator basics. Med Phys 29(2):132–144

    Google Scholar 

  • Steel GG (1996) From targets to genes: a brief history of radiosensitivity. Phys Med Biol 41(2):205–222

    Google Scholar 

  • 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

    Google Scholar 

  • Suit H, Urie M (1992) Proton beams in radiation therapy. J Natl Cancer Inst 84(3):155–164

    Google Scholar 

  • Tack D, Gevenois PA (eds) (2007) Radiation dose from adult and pediatric multidetector computed tomography. Springer, Berlin

    Google Scholar 

  • Tubiana M, Feinendegen LE, Yang C, Kaminski JM (2009) The linear no-threshold relationship is inconsistent with radiation biologic and experimental data. Radiology 251:13–22

    Google Scholar 

  • Uffmann M, Schaefer-Prokop C (2009) Digital radiography: the balance between image quality and required radiation dose. Eur J Radiol 72:202–208

    Google Scholar 

  • Upton AC (2003) The state of the art in the 1990’s: NCRP report no. 136 on the scientific bases for linearity in the dose-response relationship for ionizing radiation. Health Phys 85(1):15–22

    Google Scholar 

  • van Eijk CWE (2002) Inorganic scintillators in medical imaging. Phys Med Biol 47:R85–R106

    Google Scholar 

  • Wagner HN Jr (ed) (1968) Principles of nuclear medicine. Elsevier, p 153, 162

    Google Scholar 

  • Wagner RF (1977) Toward a unified view of radiological imaging systems. Part II: Noisy images. Med Phys 4(4):279–296

    Google Scholar 

  • Wagner RF (1983) Low contrast sensitivity of radiologic, CT, nuclear medicine, and ultrasound medical imaging systems. IEEE Trans Med Imaging MI-2(3):105–121

    Google Scholar 

  • 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

    Google Scholar 

  • Williams CR (1949) Radiation exposures from the use of shoe-fitting fluoroscopes. N Engl J Med 241:333–335

    Google Scholar 

  • William TM, James DC (1989) Radiation Oncology, 6th edn. Mosby, St. Louis

    Google Scholar 

  • Xu XG, Bednarz B, Paganetti H (2008) A review of dosimetry studies on external-beam radiation treatment with respect to second cancer induction. Phys Med Biol 53:R193–R241

    Google Scholar 

  • Yu CX, Amies CJ, Svatos M (2008) Planning and delivery of intensity-modulated radiation therapy. Med Phys 35:5233–5241

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Russell K. Hobbie .

Appendices

Symbols Used in Chapter 16

Table 8

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 \(K\) x-ray 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 x-ray 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 energy-level 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).

  1. (a)

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

  2. (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.

  3. (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.

  1. (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. \(h\nu \).

  2. (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.

  3. (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.

  4. (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 0.08-MeV 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\(^{-2}\).

  1. (a)

    What is the skin dose where the beam enters the body?

  2. (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.

  3. (c)

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

  4. (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 x-ray 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 x-rays upward through the feet and right at the reproductive organs of the children! A typical unit had an x-ray tube operating at 50 kVp with a current of 5 mA.

  1. (a)

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

  2. (b)

    Assume that the x-rays are radiated uniformly in all directions (this is not a good assumption) and that the x-rays 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 x-ray 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 cross-sections for the materials in a gadolinium oxysulfide screen for 50-keV photons are

$$ \begin{tabular} [c]{lll}Element & Cross-section per atom (m$^{2}$) & A\\ & & \\[-10pt]Gd & \multicolumn{1}{c}{$1.00\times10^{-25}$} & $157$\\ S & \multicolumn{1}{c}{$3.11\times10^{-27}$} & $32$\\ O & \multicolumn{1}{c}{$5.66\times10^{-28}$} & $16$ \end{tabular} \ \ $$
  1. (a)

    What is the cross-section per target molecule of GdO\(_{2}\)S?

  2. (b)

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

  3. (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 x-ray 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 50-keV 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 50-keV 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 100-keV photons have to continuously and uniformly strike a crystal of 1-cm\(^{2}\) 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.

  1. (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.

  2. (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.

  3. (c)

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

  4. (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 cross-section energy dependence, Eq. 15.8.

  1. (a)

    What are the units of \(B\)?

  2. (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.

  3. (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 0.1-cm filtration curve is equal to 1.

Problem 20.

X-ray 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 half-value layer HVL–the thickness of a specified absorber (often Cu or Al) that reduces the intensity of the beam to one-half.

  1. (a)

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

  2. (b)

    For a monoenergetic beam, how does the quarter-value layer QVL relate to HVL?

  3. (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 x-ray beam. It is the thickness of a specified absorber that attenuates the beam to one-half 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 x-ray beam does this correspond to?

Problem 22.

Assume an antiscatter grid is made of lead sheets 3-mm 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 \(s=y-n\).

  1. (a)

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

  2. (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.

  3. (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 x-ray tubes for mammography. The electron energy levels in Mo are as follows:

$$ \begin{tabular} [c]{llllll}$K$ & 20\thinspace000~eV & $L_{\text{I}}$ & 2886~eV & $M_{\text{I}}$ & 505~eV\\ & & $L_{\text{II}}$ & 2625~eV & $M_{\text{II}}$ & 410~eV\\ & & $L_{\text{III}}$ & 2520~eV & $M_{\text{III}}$ & 392~eV\\ & & & & $M_{\text{IV}}$ & 230~eV\\ & & & & $M_{\text{V}}$ & 227~eV \end{tabular} \ \ \ $$

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 fat-water combination and in 1 mm of glandular tissue for 50-keV photons? For 30-keV 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: \(\alpha =w\alpha _{1}+(1-w)\alpha _{2}\).

  1. (a)

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

  2. (b)

    Averaging over a \(360\operatorname {{{}^\circ }}\) scan involves integrating uniformly over all weights:

    $$ \sigma^{2}=\int_{0}^{1}\left[ w^{2}\sigma_{0}^{2}+(1-w)^{2}\sigma_{0}^{2}\right] \,dw. $$

    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

$$ (\text{arterial enhancement})=\frac{5.15\theta_{\text{Xe}}}{(\mu/\rho )_{w}/(\mu/\rho)_{\text{Xe}}}C_{\text{Xe}}(t), $$

where the arterial enhancement is in Hounsfield units, \(C_{\text {Xe}}\) is the concentration of xenon in the lungs (end tidal volume), and

$$ \theta_{\text{Xe}}=(0.011)(\text{Hct})+0.10. $$

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 range-energy relationship:

$$ R=AE^{p}. $$

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 \(S = C/T\).

  1. (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.

  2. (b)

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

  3. (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.

  4. (d)

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

  5. (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 10-MeV 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 0.5-MeV 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

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Hobbie, R., Roth, B. (2015). Medical Uses of X-Rays. In: Intermediate Physics for Medicine and Biology. Springer, Cham. https://doi.org/10.1007/978-3-319-12682-1_16

Download citation