We explore the possibility to use an alternative pointwise regularity exponent for multifractal analysis, the weak scaling exponent introduced by Yves Meyer. Indeed, it is defined in the general setting of tempered distributions so that no a priori assumption needs to be verified in order to use it (in contradistinction with the classical Hölder exponent or the p-exponent). This study requires the investigation of new multiresolution quantities, the \((\theta , \omega )\) -leaders, in replacement for the classical wavelet leaders or p-leaders. On the mathematical side, this paper is partly review, and partly research: we collect the relevant results concerning the weak scaling exponent already obtained, and we complement them. We illustrate this investigation on physiological data collected on marathon runners. The weak scaling exponent is relevant in this context because the cadence of runners cannot be modelled by locally bounded or even locally \(L^p\) functions so that previously used exponents cannot be used. We will explore the additional information that a multifractal analysis based on the weak scaling exponent yields on our understanding of the physiological mechanisms involved during a marathon race.