In this work, we present a methodology to analytically determine microdosimetric quantities in radioimmunotherapy and targeted radiotherapy with alpha particles. Monte Carlo simulations using the Geant4-DNA toolkit, which provides interaction models at the microscopic level, are performed for monoenergetic alpha particles traversing spherical sites with diameters of 1, 5 and 10 μm. An analytical function is fitted against the data in each case to model the energy imparted by monoenergetic particles to the site, as well as the variance of the distribution of energy imparted. Those models allow us to obtain the mean and dose-mean values of specific energy (⁠${\rm{z}}$⁠) and lineal energy (⁠${\rm{y}}$⁠) for polyenergetic arrangements of alpha particles. The energetic spectrum is estimated by considering the distance that each particle needs to travel to reach the sensitive target. We apply this methodology to a simple case in radioimmunotherapy: a spherical cell that has its membrane uniformly covered by 211At, an alpha emitter, with a spherical target representing the nucleus, placed at the center of the cell. We compare the results of our analytical method with calculations using Geant4-DNA of this specific setup for three nucleus sizes corresponding to our three functions. For nuclei with diameter of 1 μm and 5 μm, all mean and dose-mean quantities for ${\rm{y}}$ and ${\rm{z}}$ were in an agreement within 4% to Geant4-DNA calculations. This agreement improves to approximately 1% for dose-mean lineal energy and dose-mean specific energy. For the 10-μm-diameter case, discrepancies scale to approximately 9% for mean values and 3% for dose-mean values. Dose-mean values are within Geant4-DNA uncertainties in all cases. Our method provides accurate analytical calculations of dose-mean quantities that may be further employed to characterize radiobiological effectiveness of targeted radiotherapy. The spatial distributions of sources and targets are required to calculate microdosimetric-relevant quantities.

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