Fracture Detection Using Max-Flow Min-Cut

Abstract

In this chapter, we propose an alternative technique for detection and localization of mandibular fractures using the concepts underlying network flow. As mentioned previously, the fractures mandibular could be either (a) hairline or minor, denoting situations where the broken bone fragments are not visibly out of alignment or have incurred very little relative displacement, or (b) major, denoting situations where the broken fragments are clearly displaced relative to each other. In the previous chapter, we modeled a minor or hairline fracture as a stochastic degradation of a hypothetical intact mandible. Here, we model a fracture as a discontinuity or cut in the flow of intensities between two designated points, termed as the source and sink in a directed graph or flow network. A fracture is detected by determining a minimum cut in the flow network using the well-known Maximum-Flow Minimum-Cut (Max-Flow Min-Cut) algorithm by Ford and Fulkerson. This approach for identification and localization of fractures is shown to yield more promising results in the case of minor fractures while requiring very little preprocessing of the input image data. We first model a sequence of 2D CT image slices as a collection of independent 2D directed graphs and execute the max-flow min-cut algorithm on each such directed graph. Later, we model the sequence of 2D CT image slices containing a fractured mandible as one complete 3D directed graph and run the same max-flow min-cut algorithm on it. The max-flow min-cut algorithm is shown to be successful for both 2D flow network and 3D flow network representations. The flow network is constructed based on the knowledge of the geometry of the human mandible and the fracture pattern. Although, simple capacity functions are designed as edge weights in the flow network representation, the network flow-based scheme is shown to be effective in the detection of minor fractures.