John E. A. Bertram

THE ‘LAW OF BONE TRANSFORMATION’: A CASE OF CRYING WOLFF?

Transformation der Knochen, describing in full his understanding of the link between mechanical loading and bone form, developed from many years of investigation (Wolff, 1868, 1869, 1870, 1874, 1884a, b, 1891, 1892). T h e basis of the work was Wolffs general theory of bone transformation : Every change in the.. . function of a bone.. . is followed by certain definite changes in.. . internal structure and external conformation in accordance with mathematical laws. (Treharne, 198 I ) .

Bone modeling during growth: dynamic strain equilibrium in the chick tibiotarsus.

SummaryBone loading was quantified, usingin vivo strain recordings, in the tibiotarsus of growing chicks at 4,8, 12, and 17 weeks of age. The animals were exercised on a treadmil at 35% of their maximum running speed for 15 minutes/day.In vivo bone strains were recorded at six sites on the tibiotarsus. Percentages of the bones length and a percentage of top running speed were used to define functionally equivalent sites on the bone, and a consistent exercise level over the period of growth was studied. The pattern of bone strain defined in terms of strain magnitude, sign, and orientation remained unchanged from 4–17 weeks of age, a period when bone mass and length increased 10-fold and threefold, respectively. Our findings support the hypothesis that bones model (and remodel) during growth to achieve and maintain a similar distribution of dynamic strains at functionally equivalent sites. Because strain magnitude and sign (tensile versus compressive) differed among recording sites, these data also suggest that cellular responses to strain-mediated stimuli differ from site to site within a bone.

Constrained optimization in human walking: cost minimization and gait plasticity.

SUMMARY As walking speed increases, consistent relationships emerge between the three determinant parameters of walking, speed, step frequency and step length. However, when step length or step frequency are predetermined rather than speed, different relationships are spontaneously selected. This result is expected if walking parameters are selected to optimize to an underlying objective function, known as the constrained optimization hypothesis. The most likely candidate for the objective function is metabolic cost per distance traveled, where the hypothesis predicts that the subject will minimize the cost of travel under a given gait constraint even if this requires an unusual step length and frequency combination. In the current study this is tested directly by measuring the walking behavior of subjects constrained systematically to determined speeds, step frequencies or step lengths and comparing behavior to predictions derived directly from minimization of measured metabolic cost. A metabolic cost surface in speed-frequency space is derived from metabolic rate for 10 subjects walking at 49 speed-frequency conditions. Optimization is predicted from the iso-energetic cost contours derived from this surface. Substantial congruence is found between the predicted and observed behavior using the cost of walking per unit distance. Although minimization of cost per distance appears to dominate walking control, certain notable differences from predicted behavior suggest that other factors must also be considered. The results of these studies provide a new perspective on the integration of walking cost with neuromuscular control, and provide a novel approach to the investigation of the control features involved in gait parameter selection.

Size-dependent differential scaling in branches: the mechanical design of trees revisited

SummarySize is a key factor in determining the mechanical and functional properties of any structure. Allometric analysis allows the comparison of dimensional form at different scales. Previous descriptions of branch scaling have attempted to define a single uniform relationship governing the proportions of trees and branches over their entire size range. A new general model of scaling in woody plants is proposed in which those plants and portions of plants below a certain critical size scale allometrically becoming more slender as size increases, while those above this limit become more robust as size increases. The basis for this differential scaling is the relationship between the bending mechanics of branches, the absolute size of the branch and possibly conductance requirements. Evidence for this model is derived from a review of size scaling in trees and shrubs and from a complete analysis of a silver maple (Acersaccharinum) 13 m in height and of 370 kg wet mass.

A collisional perspective on quadrupedal gait dynamics.

The analysis of terrestrial locomotion over the past half century has focused largely on strategies of mechanical energy recovery used during walking and running. In contrast, we describe the underlying mechanics of legged locomotion as a collision-like interaction that redirects the centre of mass (CoM). We introduce the collision angle, determined by the angle between the CoM force and velocity vectors, and show by computing the collision fraction, a ratio of actual to potential collision, that the quadrupedal walk and gallop employ collision-reduction strategies while the trot permits greater collisions. We provide the first experimental evidence that a collision-based approach can differentiate quadrupedal gaits and quantify interspecific differences. Furthermore, we show that this approach explains the physical basis of a commonly used locomotion metric, the mechanical cost of transport. Collision angle and collision fraction provide a unifying analysis of legged locomotion which can be applied broadly across animal size, leg number and gait.

Constrained optimization in human running

SUMMARY Walking humans spontaneously select different speed, frequency and step length combinations, depending on which of these three parameters is specified. This behavior can be explained by constrained optimization of cost of transport (metabolic cost/distance) where cost of transport is seen as the main component of an underlying objective function that is minimized within the limitations of specified constraints. It is then of interest to ask whether or not such results are specific to walking only, or indicate a more general feature of locomotion control. The current study examines running gait selection within the framework of constrained optimization by comparing self-selected running gaits to the gaits predicted by constrained optimization of a cost surface constructed from cost data available in the literature. Normalizing speed and frequency values in the behavioral data by preferred speed and frequency reduced inter-subject variability and made group behavioral trends more visible. Although actual behavior did not coincide exactly with running cost optimization, self-selected gait and predictions from the general human cost surface did agree to within the 95% confidence interval and the region of minimal cost+0.005 ml O2 kg-1 m-1. This was similar to the level of agreement between actual and predicted behavior observed in walking. Thus, there seems to be substantial evidence to suggest that (i) selection of gait parameters in running can largely be predicted using constrained optimization, and (ii) general cost surfaces can be constructed using metabolic data from one group that will largely predict the behavior of other groups.

Correction: Reducing gravity takes the bounce out of running (doi:10.1242/jeb.162024)

There were two errors published in J. Exp. Biol. (2018) 221 , [jeb162024][1] ([[doi:10.1242/jeb.162024][3]][3]). First, a single coefficient A was used to denote what should have been three separate proportionality constants. Three distinct uses of A were: A 1: E freq= A ( g / V ) k , used in Eqn