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Stiffness of the human foot and evolution of the transverse arch

Nature volume 579, pages 97–100 (2020)Cite this article

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Abstract

The stiff human foot enables an efficient push-off when walking or running, and was critical for the evolution of bipedalism1,2,3,4,5,6. The uniquely arched morphology of the human midfoot is thought to stiffen it5,6,7,8,9, whereas other primates have flat feet that bend severely in the midfoot7,10,11. However, the relationship between midfoot geometry and stiffness remains debated in foot biomechanics12,13, podiatry14,15 and palaeontology4,5,6. These debates centre on the medial longitudinal arch5,6 and have not considered whether stiffness is affected by the second, transverse tarsal arch of the human foot16. Here we show that the transverse tarsal arch, acting through the inter-metatarsal tissues, is responsible for more than 40% of the longitudinal stiffness of the foot. The underlying principle resembles a floppy currency note that stiffens considerably when it curls transversally. We derive a dimensionless curvature parameter that governs the stiffness contribution of the transverse tarsal arch, demonstrate its predictive power using mechanical models of the foot and find its skeletal correlate in hominin feet. In the foot, the material properties of the inter-metatarsal tissues and the mobility of the metatarsals may additionally influence the longitudinal stiffness of the foot and thus the curvature–stiffness relationship of the transverse tarsal arch. By analysing fossils, we track the evolution of the curvature parameter among extinct hominins and show that a human-like transverse arch was a key step in the evolution of human bipedalism that predates the genus Homo by at least 1.5 million years. This renewed understanding of the foot may improve the clinical treatment of flatfoot disorders, the design of robotic feet and the study of foot function in locomotion.

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Fig. 1: Transverse curvature and stiffness.
Fig. 2: Curvature-induced stiffness in mechanical models of hominin feet.
Fig. 3: Three-point bending test on a cadaveric human foot.
Fig. 4: Transverse curvature of extant and extinct feet.

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Data availability

The data supporting the findings of this study are available within the paper and its Supplementary Information.

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Acknowledgements

We thank D. Lieberman for discussions on the manuscript and S. Piazza for constructive contributions. Access to skeletal specimens was provided by G. Aronsen, K. Zyskowski, E. Sargis, Yale Biological Anthropology Laboratories and the Yale Peabody Museum. K. J. Meacham III provided experimental support. S. James helped with figures. Funding support came from the Human Frontier Science Program.

Author information

Author notes

  1. Dhiraj K. Singh

    Present address: Engineering Mechanics Unit, Jawaharlal Nehru Centre for Advanced Scientific Research, Bangalore, India

  2. These authors contributed equally: Carolyn M. Eng, Marcelo Dias

Authors and Affiliations

  1. Department of Mechanical Engineering and Materials Science, Yale University, New Haven, CT, USA

    Madhusudhan Venkadesan, Ali Yawar & Carolyn M. Eng

  2. School of Science, Aalto University, Espoo, Finland

    Marcelo A. Dias

  3. Nordic Institute for Theoretical Physics (NORDITA), Stockholm, Sweden

    Marcelo A. Dias

  4. Department of Engineering, Aarhus University, Aarhus, Denmark

    Marcelo A. Dias

  5. Nonlinear and Non-equilibrium Physics Unit, OIST Graduate University, Onna, Japan

    Dhiraj K. Singh & Mahesh M. Bandi

  6. Department of Orthopaedics and Rehabilitation, Yale University, New Haven, CT, USA

    Steven M. Tommasini & Andrew H. Haims

  7. Department of Radiology and Biomedical Imaging, Yale University, New Haven, CT, USA

    Andrew H. Haims

  8. Mathematics Institute, University of Warwick, Coventry, UK

    Shreyas Mandre

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Contributions

M.V., M.M.B. and S.M. conceived the study; A.Y. and M.V. designed the foot mimics and A.Y. performed the experiments; C.M.E., A.Y. and M.V. designed, collected and analysed the data from the cadaveric experiments in consultation with S.M.T.; A.Y., C.M.E. and M.V. collected and analysed the morphometric data from cadaveric and living human feet in consultation with A.H.H.; M.A.D. and S.M. performed the mathematical modelling in consultation with M.V.; D.K.S. and M.M.B. performed the shell experiments in consultation with M.V.; M.V. planned and wrote the paper; M.V. planned and prepared the figures and tables; M.V., A.Y., S.M., M.A.D. and M.M.B. wrote the Supplementary Information; and all authors contributed to editing the paper.

Corresponding authors

Correspondence to Madhusudhan Venkadesan, Mahesh M. Bandi or Shreyas Mandre.

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The authors declare no competing interests.

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Peer review information Nature thanks Stephen Piazza and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data figures and tables

Extended Data Fig. 1 Illustrated anatomy of the foot.

a, Identification of the bones of the foot that are referred to in the main text. The cuneiforms, cuboid and the navicular are collectively referred to as the tarsal bones. b, The plantar fascia, a tough elastic band, extends from the calcaneus to the distal end of the phalanges. The fascia split and rejoin at multiple locations. c, The long plantar, short plantar and calcaneonavicular ligaments are located in the midfoot and are primarily longitudinally oriented. The deep and superficial transverse metatarsal ligaments are examples of stiff, transversally oriented elastic tissues between the metatarsals. Anatomical images are from Primal Pictures.

Extended Data Fig. 2 Mathematical and computational analysis of continuum elastic shells.

a, The shell is clamped at one end and loaded with a knife edge at the other. It is of length L, width w, thickness t and has radius of curvature R (curvature c = 1/R). b, The free end displaces by a height δz on loading and reaction forces at the clamped end resist deformation. c, A cross-sectional view of the shell shows the location of the neutral plane, if the shell were to act as an elastic beam. d, Out-of-plane (z-axis) displacement profile for one numerical simulation of a shell (L = 0.1 m, w = 0.05 m, t = 0.003 m, R = 0.03 m). Most of the displacement happens close to the loaded edge, unlike an elastic beam. e, The stress component σxx is shown as a colour map of the undeformed shell. In an elastic beam, the intersection of the neutral plane with the shell (c) would exactly match the locations of zero stress. Because of curvature-induced in-plane stretching, the zero-stress curve differs from the neutral plane predictions in the vicinity of the loaded edge and—to a lesser extent—near the clamped boundary.

Extended Data Fig. 3 Experimental characterization of arched shells.

a, The experimental set-up used in stiffness measurements. b, A magnification of the shell from underneath shows how a curvature-matched edge-loading attachment was used to mimic a theoretical knife edge. A curvature-matched clamp was fixed and glued to the other end of the shell. c, Representative data that show the linearity of the force–displacement data. The best-fit quadratic is indistinguishable from the linear fit to within sensor resolution. d, e, The Young’s modulus (d) and Poisson’s ratio (e) of the PDMS material used to fabricate the shells were estimated from simultaneous stress and strain measurements during an extension test of a rectangular PDMS block.

Extended Data Fig. 4 Design and characterization of discrete mechanical foot mimics.

a, Experimental arrangement for load–displacement measurements. The distal loading platforms for the three metatarsals are staggered in height so that all three metatarsals are loaded vertically despite the transverse curvature. In hominin feet, this is accomplished by the metatarsal torsion. b, Side view of a single metatarsal showing length L and thickness t of the foot mimics. The effect of thickness is to provide a moment arm for the longitudinal spring and thus affect the rotational stiffness of the hinge. c, Mimics with three different thicknesses were fabricated and the thickness was estimated using load–displacement measurements on curvature-free flat mimics. The accuracy of the estimated thickness values are evaluated by plotting the predicted stiffness based on the thickness estimates against the measured stiffness. Details of the thickness estimation technique and statistics of the stiffness–stiffness correlation are provided in Supplementary Information 4.4.

Extended Data Fig. 5 Effect of cutting the transverse springs in mechanical foot mimics.

Stiffness of transversally curved foot mimics lacking the transverse inter-metatarsal springs (T−) is strongly correlated with the stiffness of flat mimics with intact transverse inter-metatarsal springs.

Extended Data Fig. 6 Transverse curvature of hominin feet.

a, Definitions of length L and width w. b, Definition of the thickness t. The fourth metatarsal is highlighted in green. The distal heads of the metatarsals rest flat on the ground and the proximal heads are raised away from the ground to different degrees because of the TTA. c, Schematic showing the accrual of torsion on the lateral metatarsals about their long axis. The curvature of the TTA was estimated using the torsion of the fourth metatarsal θMT4. In addition, the average curvature was also estimated using the angle of the normal to the dorsal surface of the fourth metatarsal θdorsal, as measured in the midfoot (Supplementary equation (5.3)). d, Linear regression of the two methods to estimate TTA curvature. Details of the curvature estimation procedure and statistical results of the regression are provided in Supplementary Information 5.1.

Extended Data Table 1 The estimated work during foot deformation
Full size table
Extended Data Table 2 Foot morphometrics for extant species
Full size table
Extended Data Table 3 Fossil morphometric data
Full size table

Supplementary information

Supplementary Information

Supplementary notes regarding (S1) Foot stiffness, (S2) Mathematical modelling of continuum shells, (S3) Experiments on continuum shells, (S4) Mechanical mimics of the foot, and (S5) Human and other hominin feet.

Reporting Summary

Source data

for Fig. 4 are comprised of 3D surface reconstructions of the fourth metatarsal from volumetric CT scans of 6 feet from human volunteers, and 6 cadaveric feet. Each filename follows the format <specimen ID>-MT4.stl.

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Venkadesan, M., Yawar, A., Eng, C.M. et al. Stiffness of the human foot and evolution of the transverse arch. Nature 579, 97–100 (2020). https://doi.org/10.1038/s41586-020-2053-y

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