. Author manuscript; available in PMC: 2013 Feb 7.

Published in final edited form as: Phys Med Biol. 2012 Jan 6;57(3):591–608. doi: 10.1088/0031-9155/57/3/591

Table 1.

Objective/Constraint function definitions for a structure that has |V| voxels in volume V.

Objective f(d)	Nominal	Robust	Expected
Mean dose	$\frac{1}{∣ V ∣} Σ_{i \in V} d_{i}^{1}$	$\min ∕ \max_{I = 1}^{K} \frac{1}{∣ V ∣} Σ_{i \in V} d_{i}^{I}$	$\frac{1}{K} Σ_{I = 1}^{K} (w^{I} \frac{1}{∣ V ∣} Σ_{i \in V} d_{i}^{I})$
Min dose	$\min_{i \in V} d_{i}^{1}$	$\min_{I = 1}^{K} \min_{i \in V} d_{i}^{I}$
Max dose	$\max_{i \in V} d_{i}^{1}$	$\max_{I = 1}^{K} \min_{i \in V} d_{i}^{I}$
Overdose ramp	$\frac{1}{∣ V ∣} Σ_{i \in V} \max (0, d_{i}^{1} - d^{p r e s})$	$\max_{I = 1}^{K} \frac{1}{∣ V ∣} Σ_{i \in V} \max (0, d_{i}^{I} - d^{p r e s})$
Underdose ramp	$\frac{1}{∣ V ∣} Σ_{i \in V} \max (0, d^{p r e s} - d_{i}^{1})$	$\max_{I = 1}^{K} \frac{1}{∣ V ∣} Σ_{i \in V} \max (0, d^{p r e s} - d_{i}^{I})$