Basic Physics Concepts Firas Mourtada, Ph.D. D. ABR Associate Professor MD Anderson Cancer Center Radioactive Decay dN/dt = -lN dN/N = -ldt N(t) = N0e-lt N0 = initial number N(t) = number at time t l = decay constant Half - life N(t)/N0 = 0.5 = e-lt ln(0.5) = -lt t = T1/2 = 0.693/l half-life of 198Au half life 2.7 days 1.000 0.900 fraction of remaining activity 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100 0.000 0 5 10 15 days 20 25 30 Decay of Au198 Half-life = 2.7 days Days 1.000 1.000 0 0 5 5 10 10 15 15 Fraction of Initial Activity Fraction of Initial Activity 0.100 0.100 0.010 0.010 0.001 0.001 0.000 0.000 Days 20 20 25 Decay 25 of Au198 Half-life = 2.7 days Decay of Au198 Half-life = 2.7 days 30 30 Half - Lives Isotope 226Ra 137Cs 198Au 192Ir 125I 103Pd Half - Life 1620 years 30 years 2.7 days 73.83 days 59.4 days 16.97 days Mean life N(t)/N0 = e-1 = e-lt lt = 1 t = Tav =1/l Tav = T 1/2/.693 = 1.44 T 1/2 Au198 half-life vs mean life Half-life = 2.7 days 1.0 Fraction of Initial Activity 0.9 half-life mean life 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.0 0 5 10 15 Days 20 25 30 Brachytherapy Source Strength Specification Sealed photon source Encapsulated so radioactive material can not be lost from physical or chemical stress under foreseeable circumstances. Usually double metal wall encapsulation, prevents escape of radioactive material and absorbs unwanted betas. All brachytherapy sources are sealed sources except 192Ir wire or hairpins, which have core exposed when cut. ISO considers 192Ir wire or hairpins to be a closed source. Brachytherapy Source Strength Specification Mass- Early 20th century Activity- Early 20th century Apparent Activity-Mid 20th century Air Kerma Strength-Late 20th century Mass Radium – Mme. Curie prepared first 226Ra standards, quantified amount by expressing mass of sample in g or mg. Activity 226Ra alpha decays to 222Rn, all photons are emitted by radon or radon daughter products. 222Radon seeds produced by collecting radon gas from decay of radium and encapsulating in gold tubing. Method needed to permit correlation of 222Rn to 226Ra clinical experience. Activity Defined 1 Curie (Ci) to be the amount of radon in equilibrium with 1 g of radium. A 1 Ci radon seed has same activity as 1 g of radium. Early experiments indicated 1 Ci of radon emitted 3.7 * 1010 alpha per second. 1 Ci defined as 3.7*1010 disintegrations per second (d.p.s.). Activity Later experiments established amount of radon in equilibrium with 1 g of radium gives 3.61 * 1010 d.p.s. Curie definition remains 3.7 *1010 d.p.s. milliCurie (mCi) is 3.7 * 107 d.p.s. Roughly, conversion is 27 mCi per 1 GBq Apparent Activity Apparent activity - activity of a bare source that produces the same exposure rate at calibration distance as the specified source. Expressed in mCi for brachytherapy. Particularly useful for low energy photon sources, e.g., 125I, 103Pd mgRaeq Post WWII, other reactor produced isotopes began to be used as radium substitutes in radiotherapy. Source strength was expressed in mCi, but also needed a method to take advantage of clinical experience with radium. Used mgRaeq. mgRaeq mgRaeq yields same exposure rate at calibration distance as 1 mg Ra encapsulated by 0.5mm Pt. The exposure rate at 1 cm from 1 mg Ra(0.5mm) is 8.25R/hr. Exposure Rate constant (G) is G = 8.25 [(R-cm2)/(mg-hr)] - Ra(0.5mm Pt) G = 7.71 [(R-cm2)/(mg-hr)] - Ra(1.0mm Pt) mg-hours or mgRaeq-hours Number of mg or mgRaeq in implant times the duration of the implant in hours Source Specification Activity - mCi - 3.7 x 107 d.p.s. Apparent activity - activity of a bare point source that produces same exposure rate at calibration distance as the specified source. mg Ra eq - amount of 226Ra that produces same exposure rate at calibration distance as specified isotope Exposure Rate Constants Isotope 226Ra (0.5mmPt) 137Cs 192Ir 198Au 125I 103Pd Gd(R-cm2/mCi-hr) 8.25 3.3 4.69 2.38 1.51 1.48 Conversion - mCi to mg Ra eq # of mg Ra eq = (Gx/GRa) * # of mCix Conversion - mCi to mg Ra eq Examples 137Cs # of mg Ra eq = (3.3/8.25) * # of mCi137Cs = 0.4 * # of mCi137Cs 192Ir # of mg Ra eq = (4.69/8.25) * # of mCi192Ir = 0.569* # of mCi192Ir 1 mCi of 137Cs -dN / dt = 3.7 * 107 dps = lN N = 3.7*107 dps / l l = 0.693 / T1/2 = 0.693 / (30 y * p * 107 s/y) l = 7.36 *10-10 s-1 N = 5.03 * 1016 atoms of 137Cs A0 = 6.023 * 1023 atoms per 137 g (1 mole) of 137Cs Mass of 1mCi = (5.03 * 1016 / 6.02 * 1023)*137 = 10 mg Dose Rate Calculation - Seeds dD/dt = A G f Btiss Bw Bs fan /d2 A = activity G = exposure rate constant f = f-factor - R to cGy conversion factor Btiss = attenuation and scattering in tissue Bw, Bs = attenuation and scattering for source encapsulation and self attenuation fan= anisotropy factor d = distance to calculation point Attenuation and Scattering Functions Tissue Btiss = 1+ka(md)kb Btiss = a + bd + gd2 +Dd3 Wall Bw = exp(-mwtw) Source Bs = exp(-msts) Meisberger Coefficients Ratio of in-water exposure to in-air exposure - Tissue attenuation/scattering Meisberger, L.L., Keller, R., Shalek, R.J., The effective attenuation in water of gold-198, iridium-192, cesium-137, radium-226, and cobalt-60, Radiology 90, 953, 1968. Meisberger ratio = A + Br + Cr^2 + Dr^3, r distance in cm from source to point of calculation A B C D r(cm) 0.50000 1.00000 1.50000 2.00000 2.50000 3.00000 3.50000 4.00000 4.50000 5.00000 Cs137 1.009100000 -0.009015000 -0.000345900 -0.000028170 Cs137 1.00450 0.99971 0.99470 0.98946 0.98396 0.97818 0.97210 0.96570 0.95896 0.95186 Ir192 Au198 Ra226 1.012800000 1.036000000 1.000500000 0.005019000 -0.008134000 -0.004423000 -0.001178000 0.001111000 -0.001707000 -0.000020080 -0.000159700 0.000074480 Ir192 Au198 Ra226 1.01501 1.03219 0.99787 1.01662 1.02882 0.99444 1.01761 1.02576 0.99028 1.01797 1.02290 0.98542 1.01767 1.02011 0.97994 1.01671 1.01729 0.97388 1.01508 1.01429 0.96730 1.01274 1.01102 0.96026 1.00970 1.00734 0.95282 1.00594 1.00314 0.94502 Meisberger Ratio Cs137 1.04 Ir192 Au198 1.03 Ra226 Dose in water/Dose in air 1.02 1.01 1.00 0.99 0.98 0.97 0.96 0.95 0.94 0.00 1.00 2.00 3.00 distance from source(cm ) 4.00 5.00 6.00 Dose Rate Calculation - Linear Sources Quantization Method - divide source into multiple point sources Quantization Method Quantization Method Dose Rate Calculation - Linear Sources Sievert Integral q2 dD ( x , y ) / dt = {( A G B tiss B s ) / Ly } exp( m t ) sec q d q q1 L = active length y = perpendicular distance from source to calculation point m = effective attenuation coefficient of wall q= as defined in diagram Sievert Integral Source Geometry 140.00 Com parison of linear source to point source Ra filtered by 0.5 m m Pt linear source - Shalek & Stovall 120.00 cGy/mg-hr 100.00 1.5 cm AL 80.00 point 60.00 40.00 20.00 0.00 0.00 0.50 1.00 1.50 2.00 2.50 distance(cm ) 3.00 3.50 4.00 4.50 5.00 % difference between 1.5cm Ra source and point Ra source 60.00% 50.00% 40.00% 30.00% 20.00% 10.00% 0.00% 0.00 1.00 2.00 3.00 distance(cm) 4.00 5.00 6.00 cGy/mg-hr dist(cm) 1.5 cm AL point 0.25 50.67 126.23 0.50 20.26 31.51 0.75 10.84 13.98 1.00 6.67 7.85 1.50 3.20 3.47 2.00 1.85 1.95 2.50 1.20 1.24 3.00 0.83 0.85 3.50 0.61 0.62 4.00 0.47 0.47 4.50 0.37 0.37 5.00 0.30 0.30 % diff 59.86% 35.71% 22.48% 15.05% 7.91% 4.89% 3.06% 2.85% 2.16% 0.81% 0.40% -0.52% Away & Along Tables Young - Batho Shalek - Stovall Krishnaswamy Young and Batho, British Journal of Radiology, 37, 38, 1962. Young-Batho Source Geometry Shalek and Stovall, American Journal of Roentgenology, Radium Therapy and Nuclear Medicine, CII, 662, 1968. Shalek & Stovall Example Krishnaswamy, Radiology 105, 181, 1972. ACR Standards www.acr.org ACR Standard on Brachy Physics Manually-Loaded Temporary Implants Section IV.C.3. An additional and independent method should be used to validate the dose calculation results of the computerized planning systems. This validation should be consistent with the written prescription and completed before 50% of the dose is delivered. End of Lecture 1 Implant Doses Permanent Implant – D = (dD0/dt )Tav Temporary Implant with T1/2>>T – D = (dD0/dt ) T Temporary Implant with T1/2 not >> T – D = (dD0/dt ) Tav[1 - exp(-T/Tav)] – “milliCuries destroyed” Temporary Implant T1/2 not >>T t D = ( dD 0 / dt ) exp( l t ) dt 0 D = (dD0/dt)[-{exp(-lt)}/{l}]0t D = (dD0/dt)[-{exp(-lT)/l} +{1/l}] D = ((dD0/dt) /l)[1-exp(-lT)] D = (dD0/dt) Tav[1-exp(-lT)] D = (dD0/dt) Tav[1-exp(-T/Tav)] milliCuries destroyed D = (dD0/dt) Tav[1-exp(-lT)] D = (dD0/dt) Tav- (dD0/dt) exp(-lT) Tav D = (dD0/dt) Tav - (dDT/dt) Tav Radiation Protection Time Distance Shielding Time Dose is proportional to exposure time Half the time equals half the dose Distance For radiation protection purposes can assume dose follows inverse square law. Dose at 1 cm = 4 * Dose @ 2cm = 0.25 Dose @ 0.5cm Photon Energies Isotope 226 Ra 137Cs 192Ir 198Au 125I 103Pd Energies(MeV) 0.047 - 2.45 (ave 0.83) 0.662 0.136 - 1.06 (ave 0.38) 0.412 0.0274 - 0.0355 (ave 0.028) 0.0201, 0.023 (ave 0.021) Half -Value Layers Isotope 226Ra 137Cs 192Ir 198Au 125I 103Pd HVL(mm of Pb) 8.0 5.5 2.5 2.5 0.025 0.008 Radiation Protection Example 125I Calculation Assume 125I prostate implant 10 cm to patient surface, 50 mCi total activity, tissue attenuates 95% of dose at 10 cm (5% transmission). (dXsurface /dt) = 1.51 * 50 * (1/10)2 * 0.05 =38 mR/hr (dX1m/dt) = 38 mR/hr * (10/100)2 = 0.4 mR/hr