The Modified Mathematical Model of the Pathogenesis of Urolithiasis: Add Calculi Dissolution Effect

Abstract

The first mathematical model of the process leading to the onset of urolithiasis so as to clarify how a variety of factors affecting urolithiasis influence the pathogenesis quantitatively was derived. Then conditions for not causing the onset of urolithiasis based on the mathematical model were quantitatively discussed. The background from which this mathematical model was derived was as follows. So far various studies for the cause of the onset of urolithiasis has been made and the factors influencing the pathogenesis has been almost clear. On the other hand, though the understanding of individual factor influencing the pathogenesis has progressed biologically and clinically, theoretical study of the integrated dynamics leading to the calculus of urolithiasis through the crystal growth and aggregation from the crystal nucleation using a mathematical model has not been made yet. In the mathematical model, the process leading to the onset of urolithiasis is divided into the following three processes. (1) formation of crystal nuclei. (2) formation of calculi by growth of crystal nuclei. (3) bonding of calculi to urinary tract cells and growth of calculi. In the first mathematical model, the process of dissolving calculi was not taken into account in the process (3) above. However, in clinical, treatment for dissolving calculi using a stone-dissolving drug is also performed. Therefore, in the mathematical model of the pathogenesis of urolithiasis the calculi dissolution effect must be also taken into account. In this study, the modified mathematical model of the pathogenesis of urolithiasis taking the calculi dissolution effect into account is derived and the nature is examined. Through the analysis of the modified mathematical model and the results of numerical simulation, the conditions for suppressing the calculus growth was modified analytically and numerically. And the dependence of the growth of the calculus on the reaction rate constant concerning dissolution of the calculus, the volume of the urinary tract or the flow rate of urine was also clarified analytically and numerically. In particular, it was shown that if the calculi adhered to the urinary tract, increasing the flow rate or reducing the urinary tract volume would not contribute to the suppression of the calculi growth very much.