A new mathematical approach for handling DVH criteria in IMRT planning

Abstract

The appropriate handling of planning criteria on the cumulative dose-volume histogram (DVH) is a highly problematic issue in intensity-modulated radiation therapy (IMRT) plan optimization. The nonconvexity of DVH criteria and globality of the resulting optimization problems complicate the design of suitable optimization methods, which feature numerical efficiency, reliable convergence and optimality of the results. This work examines the mathematical structure of DVH criteria and proves the valuable properties of isotonicity/antitonicity, connectedness, invexity and sufficiency of the Karush–Kuhn–Tucker condition. These properties facilitate the use of efficient and goal-oriented optimization methods. An exemplary algorithmic realization with feasible direction methods gives rise to a functional framework for interactive IMRT planning on DVH criteria. Numerical examples on real world planning cases prove its practical capability.

Appendix: Mathematical terminology

Definition 1

(Isotone/antitone functions [35, Section 2.4]) A function

$$\begin{aligned} g: \mathbb {R}^I&\longrightarrow \mathbb {R}, \nonumber \\ \mathbf {x}&\longmapsto g(\mathbf {x}) \nonumber \end{aligned}$$

is called isotone, if for all $\mathbf {x}, \mathbf {x}' \in \mathbb {R}^I$ with $\mathbf {x} \le \mathbf {x}'$, which means $x_k \le x'_k$ in each component, the corresponding function values fulfill $g(\mathbf {x}) \le g(\mathbf {x}')$. It is called antitone, if the same implication holds with $g(\mathbf {x}) \ge g(\mathbf {x}')$.

Definition 2

(Path-connected sets [35, Section 4.2]) A set $\mathcal {X} \subseteq \mathbb {R}^I$ is path-connected, if for all $\mathbf {x}, \mathbf {x}' \in \mathcal {X}$ there is a continuous function

$$\begin{aligned} \Lambda _{\mathbf {x}, \mathbf {x}'}: [0,1]&\longrightarrow \mathbb {R}^I, \nonumber \\ \lambda&\longmapsto \Lambda (\lambda ) \nonumber \end{aligned}$$

with $\Lambda _{\mathbf {x}, \mathbf {x}'}(0) = \mathbf {x}$, $\Lambda _{\mathbf {x}, \mathbf {x}'}(1) = \mathbf {x}'$ and $\Lambda _{\mathbf {x}, \mathbf {x}'}(\lambda ) \in \mathcal {X}$ for each $\lambda$.

Definition 3

(Connected functions, [35, Section 4.2]) A function

$$\begin{aligned} g: \mathbb {R}^I&\longrightarrow \mathbb {R}, \nonumber \\ \mathbf {x}&\longmapsto g(\mathbf {x}) \nonumber \end{aligned}$$

is called connected, if its sublevelsets

$$\begin{aligned} \mathcal {X}_{\le u}(g) = \left\{ \mathbf {x} \in \mathbb {R}^I: \ g(\mathbf {x}) \le u \right\} \end{aligned}$$

are path-connected in the sense of Definition 2 for all value bounds $u \in \mathbb {R}$.

Definition 4

(Invex functions, [36, 37]) A function

$$\begin{aligned} g: \mathbb {R}^I&\longrightarrow \mathbb {R}, \nonumber \\ \mathbf {x}&\longmapsto g(\mathbf {x}) \nonumber \end{aligned}$$

is called invex, if it is continuously differentiable and

• $\nabla g(\mathbf {x}) = 0$ if and only if $\mathbf {x}$ is a global minimum of $g$, or equivalently

• there is a kernel function $\Theta _g: \mathbb {R}^I \times \mathbb {R}^I \rightarrow \mathbb {R}^I$, such that for all $\mathbf {x}, \mathbf {x}' \in \mathbb {R}^I$

  $$\begin{aligned} g(\mathbf {x}') - g(\mathbf {x}) \ge \Theta _g(\mathbf {x},\mathbf {x}')^T \cdot \nabla g(\mathbf {x}) \end{aligned}$$