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.

Application of a spring-dashpot system to clinical lung tumor motion data

“…(Note that these were incorrectly drawn in the Wilson and Meyer paper 35 as Maxwell units.) Asymptotic analysis 35 shows that when the SI motion of the tumor is dominant the motion is well-modeled by a single Voigt unit acting in the SI direction, with the tumor attached at one end and the abdominal system providing input at the other. It is this psuedo-one dimensional class of tumor motion which we consider in this paper.…”

“…This forcing input signal, x * (t), is therefore related to the measured data by x* = δf, with the scale factor δ, having units s −2 , being determined under optimization. This approach reduces the dimension of the parameter space of Wilson and Meyer 35 by one, and also allows a wider range of parameter values to be studied, both of which enable the model to be more readily optimized. Following Wilson and Meyer 35 but with the above notation, the tumor motion is given by…”

“…The forcing term of the original Wilson and Meyer 35 model was modeled on the mechanics of a rigid rod whose undetermined length was to be optimized, and which connected the diaphragm to the location of the external markers. The forcing has been simplified here by the term δf just described.…”

“…A more physical approach to the problem models lung motion as a contact problem of elasticity theory by describing the physiology of breathing using elastic constants to directly model the lung tissue. 34 A different approach was explored by Wilson and Meyer 35,36 who presented a comprehensive physical 3D system of springs and dashpots to model the correlation of an external abdominal respiratory signal and the lung tumor motion, rather than directly modeling the actual lung tissue. 24 Wilson and Meyer 35 showed mathematically that it is possible to formally simplify the three-dimensional model to a one-dimensional model when SI motion of the tumor dominates.…”

“…To overcome some of the physical intrinsic limitations for the practical application of the above approach, the first aim of this paper was to refine the Wilson and Meyer 35 model in the one-dimensional realm. The relationship between lung tumor and external abdominal marker movement is considered for the primary dimension of motion, namely, the SI direction for the tumor and the AP direction for the abdominal signal.…”