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Computers --- Mathématiques --- Ordinateurs --- Wiskunde --- 512 --- Algebra --- -LISP (Computer program language) --- 681.3*I12 --- List processing computer language --- List processing (Electronic computers) --- Mathematics --- Mathematical analysis --- Data processing --- Algorithms: algebraic algorithms; nonalgebraic algorithms; analysis of algorithms (Algebraic manipulation; computing methodologies) --- LISP (Computer program language) --- Data processing. --- LISP (Computer program language). --- 681.3*I12 Algorithms: algebraic algorithms; nonalgebraic algorithms; analysis of algorithms (Algebraic manipulation; computing methodologies) --- 512 Algebra --- REDUCE.

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Every day people pass tiny crystals without even noticing it, as a result of their kidneys filtering waste and toxins from their bloodstream and excreting them to the urine. This process is essential to the healthy functioning of the human body, though physiological deviations can cause these crystals to grow to the size of stones large enough to obstruct the urinary tract and cause severe pain. This condition is called kidney stone disease or nephrolithiasis, and has even been found in mummies dating back to 4000 B.C. More recently, nephrolithiasis has been gaining incidence worldwide with up to 13% of the population in the West suffering from it. Once a person has passed a significantly sized stone, with or without medical intervention, their chances of forming another critical stone rise significantly. These 'stoneformers' often suffer from conditions which cause their urine to get supersaturated with stone forming minerals or lack certain inhibitors of stone formation. Aside from genetic factors, dietary measures like a high calcium or low fluid intake can also increase the risk of stone formation. Medical efforts are geared towards preventing the occurrence of these symptomatic renal stones, since symptoms of renal obstruction often come very sudden and accompanied by a lot of pain. Depending on the size of the stone, the treatments can range from shockwave therapy to surgical removal. Globally, these acute treatments are amounting to an ever growing sum of costs for patients, as well as time invested by medical staff. In order to facilitate the prediction of a patient's risk at renal stone formation, the process can be modelled using a Population Balance Equation. This equation mathematically represents the effects of nucleation, growth and agglomeration on the population density of the renal crystals. Once the urine is supersaturated with stone forming dietary minerals, crystallization of these components occurs and they precipitate to form nucleates. This is followed by growth and agglomeration which occur at different rates depending on the change in concentration and inhibitors present. This creates a population of crystals of different sizes, which changes in time. A 24-hour urine collection of a subject provides them with a biochemical profile which reflects the concentrations of the stoneforming components and inhibitors. Using these as input values, the kidneys can be modelled after a semi-batch reactor in which crystallization occurs to various degrees. In combination with relevant experimental data from literature, the population balance equation can be solved using a numerical technique based on the moments of the crystal size distribution. The solution of this model provides a size distribution of the crystal population which can indicate the comparative risk of forming a critical kidney stone.