Eric P. Salathe

A biomechanical model of the foot

The foot is modeled as a statically indeterminate structure supporting its load at the heads of the five metatarsals and the tuberosity of the calcaneous. The distribution of support is determined through an analysis of the deformations caused in the structure as a result of the forces at these locations. The analysis includes the effect of the plantar aponeurosis and takes into account the deformation of the metatarsals and bending of the joints. A parametric study is presented to illustrate the behavior of the solution under a broad range of conditions.

Effect of Varying Arch Height with and Without the Plantar Fascia on the Mechanical Properties of the Foot

A biomechanical model was used to calculate the mechanical properties of the foot at a load of 683 newtons, while changing arch height with and without the plantar fascia. An increase in arch height from 20 mm to 60 mm decreased predicted vertical displacement of the foot from 11.8 mm to 5.5 mm with the plantar fascia intact and from 13.5 mm to 7.5 mm without the plantar fascia. The amount of horizontal elongation decreased from 8.6 mm to 8.4 mm with the plantar fascia and increased from 9.8 mm to 11.7 mm without. A 60-mm arch height yielded a 40% increase in horizontal elongation and a 36% increase in vertical displacement when the plantar fascia was cut, whereas a 20-mm arch height yielded a 13% increase in horizontal elongation and a 14% increase in vertical displacement. A change in arch height from 20 mm to 60 mm increased stiffness of the foot with and without the plantar fascia.

A mathematical analysis of fluid movement across capillary walls

Abstract A mathematical analysis of the exchange of fluid between a blood capillary and the surrounding tissue is presented. The diameter and permeability of the capillary are assumed to vary along its length in accordance with observation, and fluid movement across the capillary wall is assumed to be governed by a generalization of Starlings law. The motion of the interstitial fluid obeys a nonlinear form of Darcys law in which the porosity and hydrodynamic conductivity of the tissue vary with interstitial fluid pressure. An asymptotic method of solution is developed for a simplified problem in order to establish techniques applicable to the general case. The results are used to discuss some specific examples of physiological interest.

A biomechanical analysis of posterior tibial tendon dysfunction, medial displacement calcaneal osteotomy and flexor digitorum longus transfer in adult acquired flat foot

BACKGROUND Biomechanical models have been used to study stress in the metatarsals, subtalar motion, lateral column lengthening and subtalar arthroereisis. Posterior tibial tendon dysfunction has been associated with increased loads in the arch of the acquired flat foot. We examine whether a 10 millimeter (mm) medial displacement calcaneal osteotomy and flexor digitorum longus transfer to the navicular reduces these increased loads in the flat foot. METHODS The response of a normal foot, a foot with posterior tibial tendon dysfunction, and a flat foot to an applied load of 683Newton was analyzed using a multi-segment biomechanical model. The distribution of load on the metatarsals, the moment about each joint, the force on each of the plantar ligaments and the muscle forces were computed. FINDINGS Posterior tibial tendon dysfunction results in increased load on the medial arch, which may cause the foot to flatten. A 10mm medial displacement calcaneal osteotomy substantially decreases the load on the first metatarsal and the moment at the talo-navicular joint and increases the load on the fifth metatarsal and the calcaneal-cuboid joint. Adding the flexor digitorum longus transfer to the medial displacement calcaneal osteotomy has only a small effect on the flattened foot. INTERPRETATION Our biomechanical analysis illustrates that when the foot becomes flat, the force on the talo-navicular joint increases substantially from its value for the normal foot, and that medial displacement calcaneal osteotomy can reduce this increased force back toward the value occurring in the normal foot. This study provides a biomechanical rationale for medial displacement calcaneal osteotomy treatments for posterior tibial tendon dysfunction.

Biomechanical study of stress in the fifth metatarsal

OBJECTIVE: The stress throughout the fifth metatarsal was determined under various loading conditions, in order to better understand the causes of fractures to this bone. DESIGN: A mathematical approach was taken, in which the stresses were analysed using the methods of beam theory. BACKGROUND: Finite element analysis has frequently been used to determine the stress in bones. Beam theory provides an easier method for determining the force and moment resultant in any cross-section. The distribution of stress throughout the cross-section can then be found by solving certain partial differential equations defined on the cross-sections. METHODS: Cross-sections of the bone were obtained by slicing a mould, into which the bone was placed, at numerous intervals along its length. Analytic expressions describing each cross-section were obtained by fitting a Fourier series to a sequence of points along the boundary. RESULTS: The maximum stress found in the fifth metatarsal resulted from an oblique load, and had a magnitude less than would occur in a subject during normal walking. CONCLUSIONS: Since the magnitude of the stress is submaximal, this study lends theoretical support to the clinical observation that the diaphyseal fracture is indeed a stress fracture. RELEVANCE: Our analysis adds a biomechanical rationale to the pathomechanics of diaphyseal stress fractures of the fifth metatarsal. It suggests that inversion during repetitive activities may predispose the foot to fractures at a predictable location.

Mathematical analysis of oxygen transport to tissue

Abstract The mathematical analysis of steady state oxygen distribution in a Krogh cylinder is presented in this paper. Perturbation techniques are used to determine the effect of axial diffusion when the diameter of the Krogh cylinder is small compared to its length. Separate asymptotic expansions are developed for the arterial end, the central region, and the venous end of the Krogh cylinder. Solutions are obtained for each of these regions, and they are combined, using the methods of matched asymptotic expansions, to obtain a solution that is uniformly valid throughout the entire Krogh cylinder. The accuracy of the perturbation analysis is examined by comparing the approximate solution with the exact solution for the case of a linear oxyhemoglobin dissociation relationship. A simple analytic criterion is derived for determining when axial diffusion is important and when it can be neglected.

Subtalar Pronation — Relationship to the Medial Longitudinal Arch Loading in the Normal Foot

A three-dimensional biomechanical model was used to calculate the mechanical response of the foot to a load of 683 Newtons with the subtalar joint in the neutral position, at five degrees of pronation, and at five degrees of supination. Pronation causes the forefoot to evert, increasing the load borne by the first metatarsal. This results in a 47% increase in the moment about the talonavicular joint and a 58% increase in the moment about the navicular-medial cuneiform joint. Subtalar joint supination causes the forefoot to invert and results in a 55% increase in the moment about the calcaneal-cuboid joint.

Mathematical modeling of oxygen transport in skeletal muscle

Abstract Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.

Medial displacement calcaneal osteotomy reduces the excess forces in the medial longitudinal arch of the flat foot.

OBJECTIVE The hypothesis tested was that the increased load on the medial arch of the flat foot can be reduced through a medial displacement calcaneal osteotomy. DESIGN A three-dimensional, biomechanical, multisegment model was used in conjunction with experimental data from the literature. BACKGROUND Biomechanical models have been used to study the plantar fascia, medial arch height, subtalar motion and distribution of forces in the foot. METHODS Responses of a normal foot, a flat foot and a flat foot with a medial displacement calcaneal osteotomy to an applied load of 683 Newtons were analyzed, and the distribution of support among the metatarsal heads and moment about various joints were computed. RESULTS Compared to the normal foot, our flat foot model shifts the distribution of support from the lateral to the medial side, decreasing support provided by the fifth metatarsal from 11% to 1% of the total load, increasing support provided by the first metatarsal from 12% to 22% and increasing the moment about the talo-navicular joint from 20 to 28 Newton-meters. A ten millimeter medial displacement calcaneal osteotomy shifts support back toward the lateral side, with 11% provided by the fifth metatarsal and 13% by the first metatarsal. The moment at the talo-navicular joint decreases to eighteen Newton-meters. CONCLUSION Our analysis indicates that a ten millimeter medial displacement calcaneal osteotomy in a flat foot model decreases the load on the medial arch.

The role of myoglobin in retarding oxygen depletion in skeletal muscle.

Myoglobin retards the development of anoxia in a poorly perfused region of skeletal muscle by facilitating diffusion into this region from adjacent normally perfused regions and by releasing bound oxygen directly into the tissue. We examine these phenomena by analyzing a mathematical model of time-dependent myoglobin-facilitated oxygen transport. The governing equations are solved using similarity transformations and multiple-scale techniques. We find that when perfusion of a region is suddenly decreased, oxygen depletion is significantly retarded by direct release of myoglobin-bound oxygen into the tissue and that myoglobin-facilitated diffusion of oxygen from adjacent regions becomes significant at very low oxygen concentration.