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Graph theory

In mathematics graph theory is the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of vertices, nodes, or points which are connected by edges, arcs, or lines. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another; see Graph (discrete mathematics) for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in discrete mathematics.

Graph theory is a promising mathematical tool to study the connectome. However, little research has been undertaken to correlate graph metrics to functional properties of the brain. In a study, Jones et al. reported a unique association between the strength of cortical regions and their function.

Eight structural graphs were constructed within DSI Studio using publicly available imaging data derived from the Human Connectome Project. Whole-brain fiber tractography was performed to quantify the strength of each cortical region comprising our atlas.

Rank-order analysis revealed 27 distinct areas with high average strength, several of which are associated with eloquent cortical functions. Area 4 localizes to the primary motor cortex and is important for fine motor control. Areas 2, 3a and 3b localize to the primary sensory cortex and are involved in primary sensory processing. Areas V1-V4 in the occipital pole are involved in primary visual processing. Several language areas, including area 44, were also found to have high average strength.

Regions of average high strength tend to localize to eloquent areas of the brain, such as the primary sensorimotor cortex, primary visual cortex, and Broca's area. Future studies will examine the dynamic effects of neurologic disease on this metric 1).

Resting state functional magnetic resonance imaging data were acquired using multi echo imaging, echo-planar imaging pre-operatively from five participants each with a right temporal-parietal-occipital glioblastoma. Complex networks analysis was initiated by parcellating the brain into anatomically regions amongst which connections were identified by retaining the most significant correlations between the respective wavelet decomposed time-series.

Key characteristics of complex networks described in healthy controls were preserved in these patients, including ubiquitous small world organization. An exponentially truncated power law fit to the degree distribution predicted findings of general network robustness to injury but with a core of hubs exhibiting disproportionate vulnerability. Tumours produced a consistent reduction in local and long-range connectivity with distinct patterns of connection loss depending on lesion location.

Connectome analysis is a feasible and novel approach to brain mapping in individual patients with brain tumours. Applications to pre-surgical planning include identifying regions critical to network function that should be preserved and visualising connections at risk from tumour resection. In the future one could use such data to model functional plasticity and recovery of cognitive deficits 2).

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graph_theory.txt · Last modified: 2019/11/13 13:48 by administrador