Quantification of brain morphology has become an important cornerstone in understanding brain structure. Measures of cortical morphology such as thickness and surface area are frequently used to compare groups of subjects or characterise longitudinal changes. However, such measures are often treated as independent from each other. A recently described scaling law, derived from a statistical physics model of cortical folding, demonstrates that there is a tight covariance between three commonly used cortical morphology measures: cortical thickness, total surface area, and exposed surface area. We show that assuming the independence of cortical morphology measures can hide features and potentially lead to misinterpretations. Using the scaling law, we account for the covariance between cortical morphology measures and derive novel independent measures of cortical morphology. By applying these new measures, we show that new information can be gained; in our example we show that distinct morphological alterations underlie healthy ageing compared to temporal lobe epilepsy, even on the coarse level of a whole hemisphere. We thus provide a conceptual framework for characterising cortical morphology in a statistically valid and interpretable manner, based on theoretical reasoning about the shape of the cortex.

The primate cerebral cortex can take on a bewildering diversity of shapes and sizes within and across species, whilst maintaining archetypal qualities that make it instantly recognisable as a "brain". Here we present a new way of expressing the shape of a cortex explicitly as the hierarchical composition of structures across spatial scales. In computational simulations, as one successively removes sulci and gyri smaller than a specified scale, the cortices of 11 primate species are gradually coarse-grained into less folded brains until lyssencephaly (no folding). We show that this process, in all cases, occurs along a common scale-free morphometric trajectory overlapping with other mammalian species, indicating that these cortices are not only approximately fractal in shape, but also, strikingly, are approximations of the same archetypal fractal shape. These results imply the existence of a single universal gyrification mechanism that operates in a scalefree manner on cortical folds of all sizes, and that there are surprisingly few effective degrees of freedom through which cortical shapes can be selected for by evolution. Finally, we demonstrate that this new understanding can be of practical use: biological processes can now be interrogated in a highly scale-dependent way for increased sensitivity and precision. To our knowledge, this is the most parsimonious universal description of the brain's shape that is at the same time mechanistically insightful, practically useful, and in full agreement with empirical data across species and individuals.

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The universal scaling law of cortical morphology describes cortical folding as the covariance of average grey matter thickness, pial surface area, and exposed surface area. It applies for mammalian species, humans, and across lobes, however it remains to be shown that local cortical folding obeys the same rules. Here, we develop a method to obtain morphological measures for small regions across the cortex and correct surface areas by curvature to account for differences in patch size, resulting in a map of local morphology. It enables a near-pointwise analysis of morphological variables and their regional changes due to processes such as healthy ageing. We confirm empirically that the theorised covariance of morphological measures still holds at this level of local partition sizes as predicted, justifying the use of independent variables derived from the scaling law to identify regional differences in folding, subject-specific abnormalities, and local effects of ageing.