“Often brachytherapy is performed by implanting a source of radiation formed as a line. Below is a new homework problem for calculating the dose of radiation assuming a small line source. You will do the calculation with and without a shield surrounding the source.Brachytherapy(brachymeans short) involves implanting directly in a tumor sources for which the radiation falls off rapidly with distance because of attenuation, short range, or 1/r^{2}. Originally the radioactive sources (seeds) were implanted surgically, resulting in high doses to the operating room personnel. In theafterloadingtechnique, developed in the 1960s, hollow catheters are implanted surgically and the sources inserted after the surgery. Remote afterloading, developed in the 1980s, places the sources by remote control, so that only the patient receives a radiation dose.”

The Sievert integral is analyzed and tabulated in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables by Abramowitz and Stegun. It can be generalized to include end effects. The integral is named after Rolf Sievert, the Swedish medical physicist who is honored by the SI unit for equivalent dose: the sievert.Section 17.11

Problem 56 ½. Brachytherapy is often performed using a radioactive source shaped as a line of lengthLhaving a total cumulated activityÃand a mean energy emitted per unit cumulated activityΔ. Assume Eq. 17.50 describes the specific absorbed fractionΦin the surrounding tissue having an energy absorption coefficientμ_{en}and densityρ.

(a) Calculate the doseDa distancehaway from the center of the line source (assumehis much less than bothLand 1/μ_{en}). Letxindicate the position along the source, and setx= 0 at the center, sor^{2}=x^{2}+h^{2}. The total dose is an integral over the length of the source, which has a cumulated activity per unit lengthÃ/L. Evaluate this integral using the substitutionx=htanθ. In the limits of integration, ignore end effects by lettingLextend to infinity. You may need the trigonometric relationships d(tanθ)/dθ= sec^{2}θand 1 + tan^{2}θ= sec^{2}θ.

(b) Repeat the calculation in part (a), except add a coaxial cylindrical shield of thicknessbsurrounding the line source, made of a material having an absorption coefficientμ_{atten}. The dose from a small section of the source is now attenuated by an additional factor of exp(-μ_{atten}bsecθ). Justify the factor of secθin the exponential. Show that the dose can now be written as the result from part (a) times 2/πtimes a definite integral, called the Sievert integral. Derive an expression for the Sievert integral.

(c) Make a drawing that indicates the physical meaning ofh,b,x,r,L, andθ. Explain why the dose is inversely proportional toL.

## No comments:

## Post a Comment