A generalized Fisher equation (GFE) relates the time derivative of the average of the intrinsic rate of growth to its variance. The GFE is an exact mathematical result that has been widely used in population dynamics and genetics, where it originated. Here we demonstrate that the GFE can also be useful in other fields, specifically in chemistry, with models of two chemical reaction systems for which the mechanisms and rate coefficients correspond reasonably well to experiments. A bad fit of the GFE can be a sign of high levels of measurement noise; for low or moderate levels of noise, fulfillment of the GFE is not degraded. Hence, the GFE presents a noise threshold that may be used to test the validity of experimental measurements without requiring any additional information. In a different approach information about the system (model) is included in the calculations. In that case, the discrepancy with the GFE can be used as an optimization criterion for the determination of rate coefficients in a given reaction mechanism. The generalized form of Fisher equation (GFE) holds for temporal functions, which are different from zero (for chemical reaction systems this means strictly positive) with continuous second-order derivatives. These Fisher equations are exact results, which are independent of the detailed kinetics of the process: They are valid whether the evolution equations are linear or nonlinear, or local or nonlocal in space and/or time (2). Here we show that the GFE can be useful in chemical kinetics. This is tested with two chemical reaction systems, for which both the reaction mechanism and the rate coefficients are reasonably well known and correspond to experiments. The use of the GFE is new for chemical kinetics, a subject used in many fields other than chemistry, such as biology, biotechnology, chemical engineering, materials science, etc.In Generalized Form of the Fisher Equation we define the notation to write the GFE as in ref. 5. Then, in Use of the GFE for Testing Experimental Measurements we discuss the effect of noise.If the agreement with the GFE is calculated using experimentally measured concentrations-or concentrations generated with a nominal model-and rates of growth analytically calculated with a set of rate coefficients, the fit to the GFE depends on the values of these rates coefficients. For a given reaction system, optimal rate coefficients yield minimum deviation from the GFE. Hence the GFE can be used as a general criterion in optimization

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Background Chronic total occlusion (CTO) is common in patients with diabetes mellitus. Data on the long‐term outcomes after treatment of CTOs in this high‐risk population are scarce. Aim To compare the long‐term clinical outcomes of CTO revascularization either by coronary artery bypass graft (CABG) or successful percutaneous coronary intervention (PCI) versus optimal medical treatment (MT) alone in patients with diabetes. Methods and Results A total of 538 consecutive patients with diabetes and at least one CTO were identified from 2010 to 2014 in our center. In the present analysis, patients were stratified according to the CTO treatment strategy that was selected. MT was selected in 61% of patients whereas revascularization in the remaining 39%. Patients undergoing revascularization were younger, had higher left ventricular ejection fraction (LVEF), lower ACEF score, and more positive myocardial ischemia detection results compared to the MT group (p < .001).Patients referred for CABG had higher rates of left main disease compared to the PCI and MT groups (32% vs. 3% and 11%, respectively; p < .001). Complete revascularization was more often achieved in the CABG group, compared to the PCI group (62% vs. 32% p < .001). Multivariable analysis showed that revascularization with CABG was associated with lower rates of all‐cause and cardiac mortality rates compared to MT, [hazard ratio (HR) 0.41, 95% confidence interval (CI) 0.25–0.70, p < .001 and HR 0.40, 95% CI 0.20–81, p = .011, respectively]. Successful CTO‐PCI showed a trend towards benefit in all‐cause mortality (HR 0.58, 95% CI 0.33–1.04, p = .06). Conclusion In our registry, CTO revascularization in diabetic patients, especially with CABG, was associated with lower long‐term mortality rates as compared to MT alone.

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