The Burden Matrix

The Burden matrix (Burden, 1989 Todeschini and Consonni, 2000, 2009 Consonni and Todeschini, 2012) is a special vertex- and edge-weighted adjacency matrix, denoted by BM. It is defined as 

Several generalizations of the Burden matrix have been proposed by, e.g., Ivanciuc (2001) and Sheridan (2002). Ivanciuc generalized the definition of the Burden matrix by replacing the atomic numbers Z with a vertex weighting scheme Wf. 

The Burden matrix can be used to produce the unique numbering for atoms in a molecule and for substructure searches. The Sheridan modification of the Burden matrix appears to be useful in the design of drug-like compounds (Todeschini and Consonni, 2000, 2009). 

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The - augmented adjacency matrix is an example of a weighted adjacency matrix where only diagonal elements are replaced by empirical parameters representing heteroatoms. The - Burden matrix is another interesting weighted adjacency matrix. 

Table SIS Atom electronegativity values for the Burden matrix.

Table S16 Some Burden eigenvalues derived from the Burden matrix weighted by relative atomic masses for the data set of phenethylamines (Appendix C - Set 2).

The Burden matrix is another interesting weighted adjacency matrix from which Burden eigenvalues are computed and used in QSAR/QS PR modeling. This is defined as [Burden, 1989]... 

A generalization of the Burden matrix was proposed in DRAGON descriptors where instead of the atomic numbers Z, atomic masses (m), van der Waals volumes (v), Sanderson electronegativities (e), and polarizabilities (p) are used as the weighting schemes for graph vertices. Thus, a general definition of the Burden matrix in terms of a vertex weighting scheme W is... 

This kind of generalization was also proposed by Ivanciuc [ Ivanciuc, 200 If], who suggested to use in the main diagonal local vertex invariants instead of the atomic numbers Z. Moreover, an extension of the Burden matrix was proposed to derive BCUT descriptors. 

An alternative to fingerprint based similarities are those based on BCUTs (Burden, CAS, University of Texas). This method uses a modified connectivity matrix (the Burden matrix) onto which are mapped atomic descriptors (such as atomic mass and polarizability) and connectivity information. The eigenvectors of this matrix represent a compressed summary of the information in the matrix and are used to describe a molecule. Typically 5-6 BCUT descriptors suffice to describe the chemical space of a set of molecules, and the space is usually partitioned into distinct bins , with each molecule assigned to the appropriate partition. In this format, similarity calculations become very simple molecules which are mapped into the same partition are similar. As an alternative, one could use larger numbers of molecular properties and a correlation vector approach. 

From the beginning of 14C studies, bone was burdened with a marginal status as a sample type. It was missing from the list of sample materials which Libby initially recommended [10]. He and other researchers discouraged its use for the reason that the carbon content and specifically the organic carbon content, was low even in relatively recent bone and because it was a very porous structure potentially subject to chemical alteration and presumably to contamination. It was concluded that bone would systematically violate the third assumption of the 14C method as listed in Table 1. (It should be noted that "burned bone" was highly recommended. However, the sample material was the carbonized hair, skin, and other tissue rather than the bone matrix itself.)... 

Other substances that can accumulate within the body include poorly soluble particulates that are deposited in the alveolar region of the lungs, substances that bind irreversibly to endogenous proteins, and certain metals and ions that interact with the crystal matrix of bone. The properties of these substances are such that the body cannot readily remove them hence they gradually build up with successive exposures and the body burden can be maintained for long periods of time. 

The Burden modified eigenvalues are the largest absolute eigenvalues of the above defined matrix ... 

Chemical heterogeneity in the solid matrix of aquifers is usually not well characterized or appreciated. However, chemical heterogeneity can play an important role in the assessment of model uncertainty (e.g., Zhu and Burden, 2001). 

Highlights The picture that emerges from the urinalysis studies described earlier for unexposed and exposed populations quite clearly represents the state of the art. Methods that require extensive sample pretreatment, involving destruction of all the organic matter in the urine matrix that is usually followed by ion-exchange to separate and preconcentrated the uranium, are labor intensive, slow, expensive, and usually do not yield full information on the isotopic composition of uranium. These are now almost universally replaced by a variety of simple sample preparation procedures that are suitable for ICPMS analysis. Some methods require simple dilution and the use of an internal standard and others even allow direct introduction of raw urine into the ICPMS (with flow injection to minimize salt burden). In addition to superb sensitivity, the ICPMS-based method readily supply detailed isotopic composition, sometimes also of the minor isotopes ( U and U) not only of the U/ U ratio. It should also be noted that the result of the urinalysis is usually used in the bio-kinetic models to assess the body burden of uranium and that these models contain... 

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