A spring–dashpot system for modelling lung tumour motion in radiotherapy

Abstract

A 3D system of springs and dashpots is presented to model the motion of a lung tumour during respiration. The main guiding factor in configuring the system is the spatial relationship between abdominal and lung tumour motion. A coupled, non-dimensional triple of ordinary differential equations models the tumour motion when driven by a 3D breathing signal. Asymptotic analysis is used to reduce the system to a single equation driven by a 3D signal, in the limit of small lateral and transverse tumour motions. A numerical scheme is introduced to solve this equation, and tested over wide parameter ranges. Real clinical data is used as input to the model, and the tumour motion output is in excellent agreement with that obtained by a prototype tumour tracking system, with model parameters obtained by optimization. The fully 3D model has the potential to accurately model the motion of any lung tumour given an abdominal signal as input, with model parameters obtained from an internal optimization routine.

Keywords:

• modelling
• asymptotic analysis
• numerical analysis
• optimization
• lung
• tumour
• respiration
• radiotherapy

AMS Subject Classification::

• 92C05
• 92C10
• 92C50

Acknowledgements

The authors would like to thank Dr Juergen Wilbert and Kurt Baier from the University of Wuerzburg, Germany, for providing clinical data sets, and the referee for helpful comments.

Notes

^1. Note that t∼1 assumes that there are no short or long time scales; no rapid changes, and no longer-term changes are considered.

^2. Recall that the dimensionless group ω _Y is proportional to spring stiffness and inversely proportional to tumour mass m ^*. The ‘weak spring’ case is equivalent to a ‘heavy tumour’ case, and likewise for the remaining cases of this section.

^3. Recall that the dimensionless group λ _Y is proportional to friction coefficient and inversely proportional to the tumour mass m ^*.