Balkan endemic nephropathy (BEN) is a chronic tubulointerstitial disease associated with urothelial cancer, which affects people living in the alluvial plains along the tributaries of the Danube River. Challenges of studying BEN using the epidemiological method are multiple. The natural history from exposure to occurrence of the disease may take many years. The early stages of BEN are not easily detectable clinically, as the disease is asymptomatic until a significant decline in function occurs, and even then symptoms are usually non-specific. The natural history of BEN is complex, possibly with multiple risk factors operating both at the stage of initiation of renal damage and in its progression. In BEN, genetic susceptibility is due to multiple genes of small effects, gene-gene interactions, and gene-environment interactions of complex nature that are difficult to assess with current study designs. BEN is now kidney disease of the old people, and many risk factors for disease such as smoking, alcohol consumption, obesity, and diabetes could contribute to the kidney damage. Evidence is presented that environmental rather than genetic factors play a decisive role in the etiopathogenesis of BEN. Aristolochic acid, described as a culprit of BEN in 1959, is confirmed in 2007 by the molecular biology methods. Mycotoxins and polycyclic aromatic hydrocarbons, leached from lignites and found in the vicinity of endemic settlements, deserve further investigation. Despite advances in understanding the epidemiology of BEN, more research is needed on the patterns of BEN over time and between places, and on identifying the contributions of modifiable risk factors in initiating and hastening progression of BEN in order to improve the scope for preventing BEN. Primary prevention is still at the beginning. Knowledge accumulated in the fifty years of BEN research and new data about prevention and treatment of chronic kidney disease reveal several effective methods in secondary and tertiary prevention of BEN. Genetic epidemiology could establish the relative size of the genetic effect in relation to other sources of variation in disease risk (i.e., environmental effects such as intrauterine environment, physical and chemical effects, as well as behavioral and social aspects). Public health authorities in the several countries having aristolochic acid nephropathy should take immediate measures for reducing dietary exposure of residents to Aristolochia.

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We propose an adaptation of the partitioning method for determination of the MoorePenrose inverse of a matrix augmented by a block-column matrix. A simplified implementation of the partitioning method on specific Toeplitz matrices is obtained. The idea for observing this type of Toeplitz matrices lies in the fact that they appear in the linear motion blur models in which blurring matrices (representing the convolution kernels) are known in advance. The advantage of the introduced method is a significant reduction in the computational time required to calculate the Moore-Penrose inverse of specific Toeplitz matrices of an arbitrary size. The method is implemented in M A T L A B , and illustrative examples are presented.

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