Data Provided: A formula sheet and table of physical constants is attached to this paper. MEDICAL PHYSICS: Aspects of Medical Imaging and Technology

Transcription

1 Dt Provided: A formul sheet nd tble of physicl constnts is ttched to this pper. Ancillry Mteril: None DEPARTMENT OF PHYSICS AND ASTRONOMY Autumn semester exm (Acdemic yer ) MEDICAL PHYSICS: Aspects of Medicl Imging nd Technology 2 HOURS The pper is divided into two sections: A nd B The student should nswer ll questions in Section A. One sentence nswers re sufficient for ll questions in this section. The student should nswer two questions from section B. 1 TURN OVER

2 SECTION A (Answer ll questions in this section: 2 mrks ech) 1. Wht is the reltionship between convolution nd the Fourier Trnsform? 2. A Gussin might be used s representtion of n imging point spred function. Write down the generl mthemticl formul for Gussin nd sketch the curve. 3. Write down the definition of the definite integrl of the function f(x) between the limits nd b. 4. Wht is the Fourier Trnsform of delt-function? 5. A 10-bit nlogue to digitl converter is used to digitise n electricl signl whose vlues vry between 0 nd 1V. Wht re the expected chrcteristics of the digitised signl? 6. Two chrged point-entities of 0.3µC nd 0.1µC re seprted by 1cm in vcuum. Clculte the force between them. 7. Write down the mthemticl eqution tht describes the Biot-Svrt Lw. 8. Nme two properties tht re used to chrcterise the performnce of trnsducer. 9. In the context of digitistion hrdwre, wht do the letters DAC stnd for? 10. Mxwell sttes tht n ccelerting chrge rdites. Wht is the nture of this rdition nd t wht speed does it propgte in free spce? 11. Plce the following four terms in order of incresing wvelength: visible rdition, microwves, gmm-rys, ultrviolet. 12. Briefly clrify how ttenution differs from bsorption. 13. The energy flux density of n X-ry bem is better known s which quntity? With which units is it ssocited? 2 CONTINUED

3 14. Sketch the coil/windings rrngement tht might be used to generte grdient B- field in n MRI scnner. 15. An unstble nucleus is excessively lrge becuse of the presence of too mny neutrons nd protons in the nucleus. By wht mechnism is it likely to decy? 16. In the context of ionising rdition, wht do the letters HVL refer to? 17. Wht is the threshold energy bove which pir production is possible? 18. By which units is the Liner Energy Trnsfer coefficient chrcterised? 19. Wht do the letters TLD stnd for in the context of ionising rdition dosimetry? 20. Wht is the benefit of using fluorescent screen in screened-film rdiogrphy? 3 TURN OVER

4 SECTION B (Answer two questions from this section: 30 mrks ech) B1. Describe n experiment to demonstrte exponentil bsorption, using rdioctive source, n ionising detector, stopwtch nd collection of luminium pltes. [10] b. The tble below shows dt obtined by the experiment. Wht mgnitude of error is ssocited with ech count vlue if Poisson sttistics is ssumed nd errors re quoted s ± 1 stndrd devition? Explicitly enter vlue for ech row in the tble. Counts in 1 minute Plte thickness (mm) Count Error (1SD) ? ? ? ? ? ? ? [5] c. Sketch plot of the results nd use this to estimte the liner ttenution coefficient of Aluminium in this experiment. [5] d. The dt offers improved interprettion if the grph cn be linerised. Linerise the dt nd produce stright line grph. [5] e. Clculte the grdient. Is it wht you expect it to be? [2] f. In wht context is the experiment tht you hve described relevnt to dignostic imging prctise? [3] 4 CONTINUED

5 B2. With the help of digrms briefly describe the principl components nd opertion of CT scnner for production of dignostic imge. [10] b. The following mtrix is representtive of n imge mtrix in CT. Compute the verticl nd horizontl projection sums [2] c. Strting with null mtrix (ll zeroes), use n itertive bck projection technique to reconstruct the vlues in the imge mtrix. Comment on you nswer. [10] d. Express the reconstruction method you used in lgebric form. [6] e. CT is more sophisticted dignostic tool thn plnr X-ry imging. However, ptient doses re significntly different in ech cse. Quote representtive vlue for ech. [2] 5 TURN OVER

6 B3. Write down the formul tht describes the Fourier expnsion of the function f(x), including the integrls responsible for determining the coefficients. [6] b. Consider tht the sin nd cos coefficients cn be mpped to 2D spce, which is lso conveniently represented by the complex plne. With the help of digrm nd some explntory mthemtics, demonstrte how the sin, cos coefficients cn be combined to represent Fourier Coefficients in exponentil form. [6] c. Extension of this concept to the continuous domin yields the Fourier Trnsform. Write down the two integrls tht describe the Fourier Trnsform of the function f(x) nd its inverse. [2] d. Discuss how Gussin, nd the frequency spectrum of Gussin relte to ech other. [5] e. Mgnetic Resonnce Imging mkes extensive use of the Fourier Trnsform. With the help of digrms, explin the role of the ltter for production of n imge from grdient fields, using frequency encoding s specific exmple of the wy in which the trnsform is used for signl loclistion. [9] f. Wht do the letters SAR stnd for in respect of mgnetic resonnce imging? [2] END OF EXAMINATION PAPER 6 CONTINUED

7 PHYSICAL CONSTANTS & MATHEMATICAL FORMULAE Physicl Constnts electron chrge e = C electron mss m e = kg = MeV c 2 proton mss m p = kg = MeV c 2 neutron mss m n = kg = MeV c 2 Plnck s constnt h = J s Dirc s constnt ( = h/2π) = J s Boltzmnn s constnt k B = J K 1 = ev K 1 speed of light in free spce c = m s m s 1 permittivity of free spce ε 0 = F m 1 permebility of free spce µ 0 = 4π 10 7 H m 1 Avogdro s constnt N A = mol 1 gs constnt R = J mol 1 K 1 idel gs volume (STP) V 0 = 22.4 l mol 1 grvittionl constnt G = N m 2 kg 2 Rydberg constnt R = m 1 Rydberg energy of hydrogen R H = 13.6 ev Bohr rdius 0 = m Bohr mgneton µ B = J T 1 fine structure constnt α 1/137 Wien displcement lw constnt b = m K Stefn s constnt σ = W m 2 K 4 rdition density constnt = J m 3 K 4 mss of the Sun M = kg rdius of the Sun R = m luminosity of the Sun L = W mss of the Erth M = kg rdius of the Erth R = m Conversion Fctors 1 u (tomic mss unit) = kg = MeV c 2 1 Å (ngstrom) = m 1 stronomicl unit = m 1 g (grvity) = 9.81 m s 2 1 ev = J 1 prsec = m 1 tmosphere = P 1 yer = s

8 Polr Coordintes x = r cos θ y = r sin θ da = r dr dθ 2 = 1 ( r ) + 1r 2 r r r 2 θ 2 Sphericl Coordintes Clculus x = r sin θ cos φ y = r sin θ sin φ z = r cos θ dv = r 2 sin θ dr dθ dφ 2 = 1 ( r 2 ) + 1 r 2 r r r 2 sin θ ( sin θ ) + θ θ 1 r 2 sin 2 θ 2 φ 2 f(x) f (x) f(x) f (x) x n nx n 1 tn x sec 2 x e x e x sin ( ) 1 x ln x = log e x 1 x cos 1 ( x sin x cos x tn ( 1 x cos x sin x sinh ( ) 1 x cosh x sinh x cosh ( ) 1 x sinh x cosh x tnh ( ) 1 x ) ) 1 2 x x 2 2 +x 2 1 x x x 2 cosec x cosec x cot x uv u v + uv sec x sec x tn x u/v u v uv v 2 Definite Integrls x n e x dx = n! (n 0 nd > 0) n+1 π e x2 dx = π x 2 e x2 dx = 1 2 Integrtion by Prts: 3 b u(x) dv(x) dx dx = u(x)v(x) b b du(x) v(x) dx dx

9 Series Expnsions (x ) Tylor series: f(x) = f() + f () + 1! n Binomil expnsion: (x + y) n = (1 + x) n = 1 + nx + k=0 ( ) n x n k y k k n(n 1) x 2 + ( x < 1) 2! (x )2 f () + 2! nd (x )3 f () + 3! ( ) n n! = k (n k)!k! e x = 1+x+ x2 2! + x3 x3 +, sin x = x 3! 3! + x5 x2 nd cos x = 1 5! 2! + x4 4! ln(1 + x) = log e (1 + x) = x x2 2 + x3 3 n Geometric series: r k = 1 rn+1 1 r k=0 ( x < 1) Stirling s formul: log e N! = N log e N N or ln N! = N ln N N Trigonometry sin( ± b) = sin cos b ± cos sin b cos( ± b) = cos cos b sin sin b tn ± tn b tn( ± b) = 1 tn tn b sin 2 = 2 sin cos cos 2 = cos 2 sin 2 = 2 cos 2 1 = 1 2 sin 2 sin + sin b = 2 sin 1( + b) cos 1 ( b) 2 2 sin sin b = 2 cos 1( + b) sin 1 ( b) 2 2 cos + cos b = 2 cos 1( + b) cos 1 ( b) 2 2 cos cos b = 2 sin 1( + b) sin 1 ( b) 2 2 e iθ = cos θ + i sin θ cos θ = 1 ( e iθ + e iθ) 2 nd sin θ = 1 ( e iθ e iθ) 2i cosh θ = 1 ( e θ + e θ) 2 nd sinh θ = 1 ( e θ e θ) 2 Sphericl geometry: sin sin A = sin b sin B = sin c sin C nd cos = cos b cos c+sin b sin c cos A

10 Vector Clculus A B = A x B x + A y B y + A z B z = A j B j A B = (A y B z A z B y ) î + (A zb x A x B z ) ĵ + (A xb y A y B x ) ˆk = ɛ ijk A j B k A (B C) = (A C)B (A B)C A (B C) = B (C A) = C (A B) grd φ = φ = j φ = φ x î + φ y ĵ + φ z ˆk div A = A = j A j = A x x + A y y + A z z ) curl A = A = ɛ ijk j A k = ( Az y A y z φ = 2 φ = 2 φ x + 2 φ 2 y + 2 φ 2 z 2 ( φ) = 0 nd ( A) = 0 ( A) = ( A) 2 A ( Ax î + z A ) ( z Ay ĵ + x x A ) x y ˆk