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We give a novel approach for obtaining an intensity-modulated radiation therapy (IMRT) optimization solution based on the idea of continuous dynamical methods. The proposed method, which is an iterative algorithm derived from the discretization of a continuous-time dynamical system, can handle not only dose-volume but also mean-dose constraints directly in IMRT treatment planning. A theoretical proof for the convergence to an equilibrium corresponding to the desired IMRT planning is given by using the Lyapunov stability theorem. By introducing the concept of “acceptable,” which means the existence of a nonempty set of beam weights satisfying the given dose-volume and mean-dose constraints, and by using the proposed method for an acceptable IMRT planning, one can resolve the issue that the objective and evaluation are different in the conventional planning process. Moreover, in the case where the target planning is totally unacceptable and partly acceptable except for one group of dose constraints, we give a procedure that enables us to obtain a nearly optimal solution close to the desired solution for unacceptable planning. The performance of the proposed approach for an acceptable or unacceptable planning is confirmed through numerical experiments simulating a clinical setup.

Intensity-modulated radiation therapy (IMRT) [

The calculation of nonuniform intensities based on the dose prescription in the planning target volume (PTV) and the neighboring critical or sensitive organs, called organs at risk (OARs), is called inverse planning. IMRT inverse treatment planning uses optimization techniques, with the objective function measuring the quality of a treatment plan, to form the dose distribution with the ability to generate concave dose distributions and provide a specific spacing for the sensitive normal structures. The dose-volume constraint (DVC) expressed as an inequality condition with a set of a volume percentage and prescription dose applied to a PTV or OAR is evaluated on specific, predefined subvolumes of the organ and restricts the relative volume of a structure that receives more or less than a particular threshold. The use of DVCs is a normal way to determine the objective and has been the standard way of evaluating the treatment in practice; however, they are generally handled in limited ways in current optimization algorithms. Namely, because the inequality with percentages is difficult to treat as an objective function in optimization, the conventional methods are mostly constructed by using gradient techniques such as Newton’s method and the conjugate gradient method of minimizing dose-based and biology-based objective functions [

In this paper, we present an optimization method to handle not only dose-volume but also mean-dose constraints

Now, we can get an acceptable solution for any totally acceptable IMRT treatment planning; however, we cannot obtain a nearly optimal solution close to the desired solution for every unacceptable planning. Therefore, as a strategy to search for such a solution, we present an achievement procedure. That is, when assuming that the target planning is totally unacceptable and partly acceptable except for one group of dose constraints, the objective is to get a solution located as close as possible to the remaining-one group unacceptable constraint while satisfying the partly acceptable situation. In considering the dynamics in the state space

Schematic diagram of strategy for searching for nearly optimal solution.

This paper is organized as follows. In Section

In the rest of this section, we introduce notations that will be used below:

The radiation oncologist performs IMRT planning to determine the PTV to be irradiated and the OARs to be spared. With IMRT, let

Let

The IMRT planning system

For the given

For the given

We define that the IMRT planning system is acceptable if there exists a common beam set such that the dose distributions in PTVs and OARs are partly dose-volume and mean-dose acceptable for all dose-volume and mean-dose constraints.

The IMRT planning system

If the IMRT planning system is consistent, then it is acceptable. We are interested in the situation where the system is inconsistent and acceptable. In this paper, the problem of dose-volume and mean-dose constraint optimization in IMRT planning is defined to obtain the unknown variable

For describing the system, let us define the functions

This section provides the definition of the dose-volume and mean-dose constraint optimization method and its theoretical results. Consider an initial-value problem for the nonlinear differential equation

We give theoretical results for the behavior of the solution to the dynamical system in Equation (

If we choose the initial value

Since the system can be written as

Next, we prove the stability of equilibrium in the set

If the IMRT planning system is acceptable, then

Any point

To treat any

Then, the calculation of the derivative is reduced to the following:

The last inequality is supported by

A numerical discretization of differential equations describing the system in Equation (

One can design a different combination of vector field and Lyapunov function from that of Equations (

Consider an IMRT planning system

For an IMRT planning system

The first step of the procedure is to solve iterative solutions to the difference equation in Equation (

The next step is to examine the iterative algorithm in Equation (

To evaluate the proposed method, we examine two IMRT treatment planning problems: “acceptable planning” and “remaining-one unacceptable planning.” Figure

Phantom images of (a) C-shape for acceptable planning and (b) extended C-shape simulating the head and neck for remaining-one unacceptable planning. The red region shows PTV, and the blue and green regions indicate OARs.

Table

The prescribed constraints and equivalent parameters for acceptable planning (

Assigned region (colored region in Figure | Organ | Constraint | Equivalent parameter |
---|---|---|---|

PTV (red) | PTV | ( | |

( | |||

OAR (blue) | Core | ( |

The prescribed constraints and equivalent parameters for partly unacceptable planning (

Assigned region (colored region in Figure | Organ | Constraint | Equivalent parameter |
---|---|---|---|

PTV (red) | PTV | ( | |

( | |||

OAR (blue) | Spinal cord | ( | |

OAR (green) | Parotid | ( | |

( |

We used the iterative solutions

We first examined solving the problem of acceptable IMRT planning using the harder condition defined by AAPM, as shown in Table

DVH of (a) PTV with

Time evolution of divergence

The evolutions of the dose-volume proportion rates

Time evolutions of (a)

The proposed Procedure

Figure

DVH of (a) PTV with

DVH of (a) PTV with

DVH of (a) PTV with

For evaluating the effectiveness of the proposed procedure quantitatively, we drew the graph in Figure

Time evolution of distances

We show how selecting the parameter value affects the performance with respect to decreasing the cost function. Figure

Relation between

For handling dose-volume and mean-dose constraints directly in the optimization of IMRT treatment inverse planning, we have proposed a novel iterative algorithm derived from the discretization of a continuous-time dynamical system. The proposed system has an advantage in that the stability of an equilibrium corresponding to the desired optimal solution to the inverse problem is able to be proven using the Lyapunov theorem. Through numerical experiments with the C-shape phantom (AAPM TG-119) for an acceptable planning and an extended C-shape phantom simulating the head and neck for remaining-one unacceptable planning, we confirmed that we can obtain an optimal solution and a nearly optimal solution located as close as possible to the remaining-one unacceptable constraint.

Our approach presented in this paper enables us to develop an iterative method of IMRT optimization by discretizing a continuous-time system in which the global stability of a desired equilibrium is guaranteed based on the dynamical system theory. The advantage of the approach is due to the Lyapunov theorem that establishes the stability of equilibrium if there exists a Lyapunov function even for a hybrid dynamical system. The direct construction of iterative algorithms with a theoretical guarantee of convergence for a given objective function including inequalities with percentages is generally difficult. Although a drawback of the dynamical system approach is that finding a combination of a vector field and Lyapunov function is generally a hard problem, we succeeded in obtaining a proof for the acceptable set and designing an iterative optimization method for an IMRT treatment planning including dose-volume constraints.

All data used to support the findings of this study are included within the paper.

The authors declare no conflicts of interest regarding the publication of this paper.

This research was partially supported by JSPS KAKENHI Grant Number 18K04169.