Logistic modeling is a common tool for multi-metric modeling. In our previous work (Deasy and El Naqa 2007; El Naqa et al. 2006a), a logit transformation was used:
where n is the number of cases (patients), and x _i is a vector of the input variable values (i.e., image features) used to predict f(x _i ) for outcome y _i (i.e., TCP or NTCP) of the i _th:

where d is the number of model variables, and the β’s are the set of model coefficients determined by maximizing the probability that the data gave rise to the observations.

Machine learning represents a wide class of artificial intelligence techniques (e.g., neural networks, decision trees, support vector machines), which are able to emulate human intelligence by learning the surrounding environment from the given input data. These methods are increasingly being utilized in radiation oncology because of their ability to detect nonlinear patterns in the data (El Naqa et al. 2015a). This is due to their ability to detect complex patterns in heterogeneous datasets with superior results when compared to state of the art in each of these disciplines. In particular, neural networks were extensively investigated to model post-radiation treatment outcomes for cases of lung injury (Munley et al. 1999; Su et al. 2005) and biochemical failure and rectal bleeding in prostate cancer (Gulliford et al. 2004; Tomatis et al. 2012). A rather more robust approach of machine learning methods is kernel-based methods and its favorite technique of support vector machines (SVMs), which are universal constructive learning procedures based on the statistical learning theory (Vapnik 1998). Learning is defined in this context as estimating dependencies from data (Hastie et al. 2001).

There are two common types of learning: supervised and unsupervised. Supervised learning is used to estimate an unknown (input, output) mapping from known (input, output) samples (e.g., classification or regression). In unsupervised learning, only input samples are given to the learning system (e.g., clustering or dimensionality reduction). In image-based outcome modeling, we focus mainly on supervised learning, wherein the endpoints of the treatments such as TCP or NTCP are provided by experienced oncologists in our case.

For discrimination between patients who are at low risk versus patients who are at high risk of radiation therapy, the main idea of SVM would be to separate these two classes with “hyperplanes” that maximize the margin between them in the nonlinear feature space defined by implicit kernel mapping as shown in Fig. 14.3. The objective here is to minimize the bounds on the generalization error of a model on unseen data before rather than minimizing the mean-square error over the training dataset itself (data fitting). Mathematically, the optimization problem could be formulated as minimizing the following cost function:
subject to the constraint:
where wis a weighting vector and Φ(⋅)is a nonlinear mapping function. The ζ _i represents the tolerance error allowed for each sample to be on the wrong side of the margin (called hinge loss). Note that minimization of the first term in Eq. (14.3) increases the separation (margin) between the two classes, whereas minimization of the second term improves fitting accuracy. The trade-off between complexity (or margin separation) and fitting error is controlled by the regularization parameter C. However, such a nonlinear formulation would suffer from the curse of dimensionality (i.e., the dimensions of the problem become too large to solve) (Haykin 1999; Hastie et al. 2001). Therefore, the dual optimization problem is solved instead of Eq. (14.3), which is convex complexity becomes dependent only on the number of samples and not on the dimensionality of the feature space. The prediction function in this case is characterized by only a subset of the training data, each of which are then known as “support vectors” s _i :
where n _s is the number of support vectors (i.e., samples at the boundary as in fig. 14.3), α _i are the dual coefficients determined by quadratic programming, and K(⋅, ⋅) is the kernel function. Typical kernels (mapping functionals) include:
where c is a constant, q is the order of the polynomial, and σ is the width of the radial basis functions. Note that the kernel in these cases acts as a similarity function between sample points in the feature space. Moreover, kernels enjoy closure properties, i.e., one can create admissible composite kernels by weighted addition and multiplication of elementary kernels. This flexibility allows for the construction of a neural network by using a combination of sigmoidal kernels. Alternatively, one could choose a logistic regression equivalent kernel by proper choice of the objective function in Eq. (14.3). In Fig. 14.4, we show an example for the application of machine learning to predict local failure in lung cancer (El Naqa et al. 2010).

In a retrospective study of 30 non-small cell lung cancer (NSCLC) patients, 30 features were extracted from both PET and CT images with and without motion correction as shown in Fig. 14.5.

The features included tumor volume; SUV/HU measurements, such as mean, minimum, maximum, and the standard deviation; IVH metrics; and texture-based features such as energy, contrast, local homogeneity, and entropy. The data corrected for motion artifacts based on a population-averaged probability spread function (PSF) using deconvolution methods derived from four 4D CT datasets (El Naqa et al. 2006b). An example of such features in this case is shown in Fig. 14.6.

Using modeling approaches described in Sect. 14.2.2 and implemented in the DREES software, Fig. 14.7 shows the results for predicting local failure, which consisted of a model of two parameters from features from both PET and CT based on intensity volume histograms provided the best.