Maximum Density Path Research Articles

Algorithm for the Length-Constrained Maximum-Density Path Problem in a Tree with Uniform Edge Lengths

SUMMARY Given an edge-weighted tree with n vertices and a positive integer L, the length-constrained maximum-density path problem is to find a path of length at least L with maximum density in the tree. The density of a path is the sum of the weights of the edges in the path divided by the number of edges in the path. We present an O(n) time algorithm for the problem. The previously known algorithms run in O(nL )o rO(nlogn) time.

Diffusion weighted imaging-based maximum density path analysis and classification of Alzheimer's disease

Characterizing brain changes in Alzheimer's disease (AD) is important for patient prognosis and for assessing brain deterioration in clinical trials. In this diffusion weighted imaging study, we used a new fiber-tract modeling method to investigate white matter integrity in 50 elderly controls (CTL), 113 people with mild cognitive impairment, and 37 AD patients. After clustering tractography using a region-of-interest atlas, we used a shortest path graph search through each bundle's fiber density map to derive maximum density paths (MDPs), which we registered across subjects. We calculated the fractional anisotropy (FA) and mean diffusivity (MD) along all MDPs and found significant MD and FA differences between AD patients and CTL subjects, as well as MD differences between CTL and late mild cognitive impairment subjects. MD and FA were also associated with widely used clinical scores. As an MDP is a compact low-dimensional representation of white matter organization, we tested the utility of diffusion tensor imaging measures along these MDPs as features for support vector machine based classification of AD.

Automatic clustering and population analysis of white matter tracts using maximum density paths

We introduce a framework for population analysis of white matter tracts based on diffusion-weighted images of the brain. The framework enables extraction of fibers from high angular resolution diffusion images (HARDI); clustering of the fibers based partly on prior knowledge from an atlas; representation of the fiber bundles compactly using a path following points of highest density (maximum density path; MDP); and registration of these paths together using geodesic curve matching to find local correspondences across a population. We demonstrate our method on 4-Tesla HARDI scans from 565 young adults to compute localized statistics across 50 white matter tracts based on fractional anisotropy (FA). Experimental results show increased sensitivity in the determination of genetic influences on principal fiber tracts compared to the tract-based spatial statistics (TBSS) method. Our results show that the MDP representation reveals important parts of the white matter structure and considerably reduces the dimensionality over comparable fiber matching approaches.

Structural abnormality of the corticospinal tract in major depressive disorder.

BackgroundScientists are beginning to document abnormalities in white matter connectivity in major depressive disorder (MDD). Recent developments in diffusion-weighted image analyses, including tractography clustering methods, may yield improved characterization of these white matter abnormalities in MDD. In this study, we acquired diffusion-weighted imaging data from MDD participants and matched healthy controls. We analyzed these data using two tractography clustering methods: automated fiber quantification (AFQ) and the maximum density path (MDP) procedure. We used AFQ to compare fractional anisotropy (FA; an index of water diffusion) in these two groups across major white matter tracts. Subsequently, we used the MDP procedure to compare FA differences in fiber paths related to the abnormalities in major fiber tracts that were identified using AFQ.ResultsFA was higher in the bilateral corticospinal tracts (CSTs) in MDD (p’s < 0.002). Secondary analyses using the MDP procedure detected primarily increases in FA in the CST-related fiber paths of the bilateral posterior limbs of the internal capsule, right superior corona radiata, and the left external capsule.ConclusionsThis is the first study to implicate the CST and several related fiber pathways in MDD. These findings suggest important new hypotheses regarding the role of CST abnormalities in MDD, including in relation to explicating CST-related abnormalities to depressive symptoms and RDoC domains and constructs.

ATLAS-BASED FIBER CLUSTERING FOR MULTI-SUBJECT ANALYSIS OF HIGH ANGULAR RESOLUTION DIFFUSION IMAGING TRACTOGRAPHY.

High angular resolution diffusion imaging (HARDI) allows in vivo analysis of the white matter structure and connectivity. Based on orientation distribution functions (ODFs) that represent the directionality of water diffusion at each point in the brain, tractography methods can recover major axonal pathways. This enables tract-based analysis of fiber integrity and connectivity. For multi-subject comparisons, fibers may be clustered into bundles that are consistently found across subjects. To do this, we scanned 20 young adults with HARDI at 4 T. From the reconstructed ODFs, we performed whole-brain tractography with a novel Hough transform method. We then used measures of agreement between the extracted 3D curves and a co-registered probabilistic DTI atlas to select key pathways. Using median filtering and a shortest path graph search, we derived the maximum density path to compactly represent each tract in the population. With this tract-based method, we performed tract-based analysis of fractional anisotropy, and assessed how the chosen tractography algorithm influenced the results. The resulting method may expedite population-based statistical analysis of HARDI and DTI.

An optimal algorithm for the maximum-density path in a tree

We studied the problem of finding the maximum-density path in a tree. By spine decomposition and a linear-time algorithm for the maximum density segment problem, we developed an O ( n log n ) time algorithm, which improves the previously best result of O ( n log 2 n ) by using centroid decomposition. We also show the time complexity is optimal in the algebraic computation tree model.

Finding a maximum-density path in a tree under the weight and length constraints

Let T = ( V , E ) be a tree with n nodes such that each node v is associated with a value-weight pair ( val v , w v ) , where the value val v is a real number and the weight w v is a positive integer. The density of a path P = 〈 v 1 , v 2 , … , v k 〉 is defined as ∑ i = 1 k val i / ∑ i = 1 k w i . The weight of P, denoted by w ( P ) , is ∑ i = 1 k w i . Given a tree of n nodes, two integers w min and w max , and a length lower bound L, we present a pseudo-polynomial O ( w max n L ) -time algorithm to find a maximum-density path P in T such that w min ⩽ w ( P ) ⩽ w max and the length of P is at least L.

Finding a length-constrained maximum-sum or maximum-density subtree and its application to logistics

We study the problem of finding a length-constrained maximum-density path in a tree with weight and length on each edge. This problem was proposed in [R.R. Lin, W.H. Kuo, K.M. Chao, Finding a length-constrained maximum-density path in a tree, Journal of Combinatorial Optimization 9 (2005) 147–156] and solved in O ( n U ) time when the edge lengths are positive integers, where n is the number of nodes in the tree and U is the length upper bound of the path. We present an algorithm that runs in O ( n log 2 n ) time for the generalized case when the edge lengths are positive real numbers, which indicates an improvement when U = Ω ( log 2 n ) . The complexity is reduced to O ( n log n ) when edge lengths are uniform. In addition, we study the generalized problems of finding a length-constrained maximum-sum or maximum-density subtree in a given tree or graph, providing algorithmic and complexity results.

Finding a Length-Constrained Maximum-Density Path in a Tree

Let T = (V,E,w) be an undirected and weighted tree with node set V and edge set E, where w(e) is an edge weight function for e ∈ E. The density of a path, say e1, e2,..., e k , is defined as ∑ k i = 1 w(e i )/k. The length of a path is the number of its edges. Given a tree with n edges and a lower bound L where 1≤ L ≤ n, this paper presents two efficient algorithms for finding a maximum-density path of length at least L in O(nL) time. One of them is further modified to solve some special cases such as full m-ary trees in O(n) time.