The kidney plays an indispensable role in the regulation of whole-organism water balance, electrolyte balance, and acid-base balance, and in the excretion of metabolic wastes and toxins. In this paper, we review representative mathematical models that have been developed to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, and regulation of renal oxygen transport. We discuss how such modeling efforts have significantly expanded our understanding of renal function in both health and disease. 1. Introduction The kidneys are organs that serve a number of essential regulatory roles. Most of us know that our kidneys function as filters, removing metabolic wastes and toxins from the blood and excreting them through the urine. But the kidneys also serve other essential functions. Through various regulatory mechanisms, the kidneys help maintain the body’s water balance, electrolyte balance, and acid-base balance. Additionally, the kidneys produce or activate hormones that are involved in erythrogenesis, calcium metabolism, and the regulation of blood pressure and blood flow. Most mammalian kidneys have three major sections: the cortex, the outer medulla, and the inner medulla. The outer and inner medulla are collectively referred to as the medulla. The outer medulla may be divided into the outer stripe and the inner stripe. Despite decades of experimental efforts, some aspects of the fundamental kidney functions remain yet to be fully unexplained. For example, the processes by which a concentrated urine is produced by the mammalian kidney (or, more specifically, the production of a substantial concentrating effect in the inner medulla) when the animal is deprived of water remains one of the longest-standing mysteries in traditional physiology. In conjunction with experimental work, mathematical models have helped to test, confirm, refute, or suggest a number of hypotheses related to the urine concentrating mechanism [2]. This paper will describe modeling efforts that have sought to better understand kidney physiology and pathophysiology, including the regulation of glomerular filtration, the regulation of renal blood flow by means of the tubuloglomerular feedback mechanisms and of the myogenic mechanism, the urine concentrating mechanism, and regulation of renal oxygen transport. 2. How Is Glomerular Filtration Regulated? The functional unit of the
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