Physiology is found to be replete with fractal time series, for example, heart beats, respiration rate, walking and cerebral blood flow are all fractal processes. The evolution of such fractal stochastic phenomena is found to be well described by fractional partial differential equations for the evolution of the probability density and fractional Langevin equations for the evolution of particle trajectories. These ‘modern’ descriptions of complexity require the application of the fractional calculus, to capture the long-term memory (inverse power law) observed in complex time series, which we interpret in the context of a number of biomedical phenomena.

We review how the traditional mathematics of random walks, Langevin equations and Fokker-Planck equations are replaced by fractional random walks, fractional Langevin equations and fractional phase space equations. These fractional operators are necessary for a proper description of the calculus of medicine, since they are required to determine the time evolution of fractal phenomena. We apply the methods of the fractional calculus to heart beat, stride interval, blood flow to the brain and breath interval time series.