The Radiation Dose Accumulated by Blood during Extracorporeal Irradiation

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Abstract
The probability distribution of the number of times that a blood cell passes through a shunt is given by the formula [image] where the x''s and y''s represent successive stays in the body and in the shunt. When n is small, this expression may be calculated directly; when n is large, this probability may be estimated by using the fact that N(t) is approximately normally distributed with mean t/(mx + my) and variance t( x + a y2 )/(mx + my)3 , where mx, my and [sigma] x y [sigma] y are the means and variances of the random stays of a blood cell in the body and in the shunt. When transit time through the shunt is constant, my = c and [sigma] y2 = 0. In the particular case when there is good mixing and the random stay of a blood cell in the body may be taken as exponentially distributed, P (x < a) = 1 [long dash] e-[lambda] a, then [image] and N(t) is approximately normal with mean t[lambda](l+[lambda] c) and variance t [lambda]/ (1 + [lambda]c)3. For practical purposes, any one of the methods for computing the distribution of radiation dose[long dash]the exact probability method, the method based on differential equations, or the normal approximation[long dash] is likely to be more precise than the other data associated with the shunting procedure, particularly estimates of total blood volume, flow through and transit time of the shunt. The advantage of the normal approximation lies in the speed and simplicity of the calculations; a little arithmetic to find the mean and standard deviation will serve in most situations to describe adequately the distribution of the radiation dose by means of the normal curve. The advantage of the probability approach, in general, is its adaptability to modifications of the initial assumptions. The normal approximation is relatively insensitive to the actual distribution of the stay of a blood cell in the main circulation, depending only on the mean and variance. It is not difficult to modify the theory to take into account other situations, for example, when treatments are intermittent, spread over enough time that changes in the cell population become significant, or when assuming more than one body compartment, with specified initial distributions of cells.

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