HW3 (solution)

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University of Illinois, Urbana Champaign *

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380

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Chemistry

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Dec 6, 2023

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ECE/BIOE 380 - Homework 3 (Solution) 1. Circle the best answer (20 pts) a. (True or False) The emitted x-ray spectrum (both bremsstrahlung and characteristic) depends on both the target material and the applied voltage. (2pts) b. (True or False) The main reason why a vacuum is used in the x-ray tube is to absorb low- energy x-rays that are not useful for imaging. (2pts) c. (True or False) Materials with high density are preferred for being used as anode target because the increased number of atoms/nucleus in a unit volume facilitates bremsstrahlung radiation. (2pts) d. (True or False) Both bremsstrahlung and characteristic radiations produce electrons that are ejected from the shells of the atom. (2pts) e. (True or False) Some low-energy electrons can still eject electrons from outer shells of an atom but the resulting characteristic x-ray energy is too low to be differentiated from bremsstrahlung x-ray. (2pts) f. (True of False) High energy x-ray sources are always preferred because they provide the best penetration of tissue. (2pts) g. (True or False) Materials such as molybdenum and rhodium are preferred over tungsten for mammography (i.e., breast x-ray) because they provide x-ray with lower energy, which provides higher contrast of tissue. (2pts) h. (True of False) Collimators are used in x-ray sources to filter the x-ray spectrum and remove low-energy x-rays. (2pts) i. (Check all that apply) Which factors below always contribute to better x-ray imaging quality (SNR, contrast, spatial resolution): (4pts) i. Obese patients ii. Higher x-ray energy iii. Thick intensifying screen for the detector iv. Compton scattering v. Beam hardening vi. Administration of contrast 2. A region of the body has a 10 cm thickness of muscle with 3 cm of bone imbedded as shown in the figure below. Assuming parallel x-ray beams are used to image the body. ( 30 pts ) a. Calculate the x-ray transmission along paths through muscle alone and through muscle and bone for photon energies of 30 and 100 keV using the attenuation parameters provided in Table I. Assume monoenergetic x-ray photons. The intensity of the x-rays at the source is I 0 . Calculate the intensity of x-rays I d at the detector as a function of I 0 . (5 pts for each case, 20 pts total)

b. Determine the contrast between the background (muscle only) and muscle plus bone. Which energy of x-ray has the better contrast? (Hint: contrast is given by the ratio of intensity difference over background intensity). (4pts for each contrast measurement and 2pts for the comparison, total 10 pts)

3. The contrast of an x-ray image is defined by the following equation, where I t is the intensity of the target tissue signal (e.g., bone), and I b is the intensity of the background (e.g., muscle). (10 pts) ࠵? = ࠵? ! − ࠵? " ࠵? " Compton scattering adds an additional intensity ࠵? # to both the background and tissue signals. The goal is to maximize the contrast between the tissue (e.g., bone) and background (e.g., muscle). a. What is the new contrast ࠵?′ in the presence of Compton scattering? ( show your work ) (5 pts) (a) ࠵? $ = % &’ ( ! ( " ) (b) ࠵? $ = % & ’ ( " ( ! ) (c) ࠵? $ = % ( # ’ ( " ( ! ) (d) ࠵? $ = &’ ( " ( ! ) % b. Identify the plot that most accurately displays the contrast as a function of increased Compton scattering. Explain your answer . (5 pts) ࠵?′ arbitrary units ࠵? ! ࠵? " $ Plot (a)

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According to the equation, when Compton scattering is zero, C’ = C; and when Compton scattering is infinity, C’ approaches 0. Therefore, plot (a) is the correct plot. 4. Sodium iodide is a typical material used in nuclear medicine imaging for detecting gamma rays emitted from radioactive isotopes such as technetium-99m. Assume that the half value layer (HVL) of sodium iodide is 0.3 cm. Question: what percentage of gamma rays will pass through a 1.5 cm-thick sodium iodide crystal? ( 15 pts ) 5. Photon energy (20 pts) a. What voltage should be applied to accelerate an electron from zero velocity in order that it acquire the same energy as a photon of wavelength λ=600nm? (Hint: Planck constant h = 6.63 x 10 -34 J·s) (5 pts) ࠵?′ arbitrary units ࠵? ! ࠵? " $ Plot (b)

b. 100 trillion x-ray photons of wavelength 1 nm are produced for an incident x-ray. The x-ray was emitted on a patient body with a surface area of 100 cm 2 and exposure time of 2 seconds. i. What is the total x-ray energy in unit of Joule? (5 pts) ii. What is the power of the x-ray in unit of Watt? (5 pts) iii. What is the intensity of the x-ray in unit of Watt/m 2 ? (5 pts)