Biomedical Engineering Reference
In-Depth Information
with its own weight, and there are likely to be 5 or more beams. That
means that IMRT requires many thousands of additional variables,
over and above those needed for uniform-beam radiation therapy. All
together, these form a vast hyper-space of treatment variables and the
both the score and the constraints are functions of all of those
variables
If only one variable were to be optimized, one could plot the score on
the ordinate versus the value of that variable on the abscissa of a two-
dimensional graph such as that portrayed in Figure 9.6 below, and
search for the lowest point. If two variables were to be optimized,
the score could be represented as a surface in a three-dimensional
perspective plot such as that shown in Figure 9.7 below. The score
would be represented as a sort of “landscape” with hills and valleys in
it, within which one wants to find the lowest point in the deepest
valley. But, we have no ability to portray a function of thousands or
more variables graphically. Nevertheless, we can speak conceptually
of the search landscape as a hyper-dimensional world.
How can one hope to have any possibility of success, given the
vastness of the hyperspace which must be searched? That there is
hope is due to several reasons. First, one virtually always selects only
a subset of the variables for optimization, while fixing others such as
the modality, number, direction, and shape of the beams beforehand,
thereby reducing the dimensionality of the hyper-space that must be
searched. Second, especially when biophysical quantities such as
TCP and NTCP are used in the score function, or when particular
choices are made about what parameters to optimize, the score
function varies quite smoothly throughout the search space. For the
most part it doesn't jump wildly around, so one may not need to look
at closely spaced points. The third reason for the possibility of
success is that, though it is very unlikely that one will in fact find a
global extremum in a finite time, one may well find a good solution.
In a sense, the possibility of success comes from the acceptability of
failure.
The search itself
There is a vast literature on the subject of maximization or
minimization. There are many very different, and all fascinating,
methods that have been developed. And, as you might expect, there
are often variants of a given method. Here I will only address two
types of search techniques, without in any way giving a full
