A method is proposed that incorporates the effects of intratreatment organ motion due to breathing on the dose calculations for the treatment of liver disease. Our method is based on the convolution of a static dose distribution with a probability distribution function (PDF) which describes the nature of the motion. The organ motion due to breathing is assumed here to be one-dimensional (in the superior-inferior direction), and is modeled using a periodic but asymmetric function (more time spent at exhale versus inhale). The dose distribution calculated using convolution-based methods is compared to the static dose distribution using dose difference displays and the effective volume (Veff) of the uninvolved liver, as per a liver dose escalation protocol in use at our institution. The convolution-based calculation is also compared to direct simulations that model individual fractions of a treatment. Analysis shows that incorporation of the organ motion could lead to changes in the dose prescribed for a treatment based on the Veff of the uninvolved liver. Comparison of convolution-based calculations and direct simulation of various worst-case scenarios indicates that a single convolution-based calculation is sufficient to predict the dose distribution for the example treatment plan given.

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A method is presented for solving initial and boundary value problems for the energy dependent and one speed neutron transport equations. It consists in constructing an asymptotic expansion of the neutron density ψ(r, v, τ) with respect to a small parameter ε, which is the ratio of a typical mean free path of a neutron to a typical dimension of the domain under consideration. The density ψ is expressed as the sum of an interior part ψi, a boundary layer part ψb, and an initial layer part ψ0. Then ψi is sought as a power series in ε, while ψb decays exponentially with distance from a boundary or interface at a rate proportional to ε−1. Similarly ψ0 decays at a rate proportional to ε−1 with time after the initial time. For a near critical reactor, the leading term in ψi is determined by a diffusion equation. The leading term in ψb is determined by a half-space problem with a plane boundary. The initial and boundary conditions for the diffusion equation are obtained by requiring ψ0 and ψb to decay away from the initial instant and from the boundary, respectively. The results are illustrated by specializing them to the one speed case. The method may make it possible to treat more realistic and more complex problems than can be handled by other methods.