Convolutional neural networks (CNN) are increasingly used for image classification tasks. In general, the architectures of these networks are set ad hoc with little justification for selecting various components, such as the number of layers, layer depth, and convolution settings. In this work, we develop a structured approach to explore and select architectures that provide optimal classification performance. This was developed with an IRB-approved data set with 9,216 2-D maximum intensity projection (MIP) MRI breast images, containing breast cancer malignant or benign classes. This data set was divided into 7,372 training, 922 validation, and 922 test images. The architecture search method employs a genetic algorithm to find optimal ResNet-based classification architectures through crossover and mutation. Each generation, new model architectures are created from the genetic algorithm and trained with supervised machine learning. This search method identifies the model with the highest area under the ROC curve (AUC). The genetic algorithm produced an optimal model architecture which classifies malignancy in images with 76% accuracy and achieves an AUC score of .83. This approach offers a rational framework for automatic architecture exploration, potentially leading to a set of more efficient and generalizable CNN-based classifiers.