A continuum treatment of growth in biological tissue: The coupling of mass transport and mechanics

Growth (and resorption) of biological tissue is formulated in the continuum setting. The treatment is macroscopic, rather than cellular or sub-cellular. Certain assumptions that are central to classical continuum mechanics are revisited, the theory is reformulated, and consequences for balance laws and constitutive relations are deduced. The treatment incorporates multiple species. Sources and fluxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (\romannumeral 1) a fluid phase, and (\romannumeral 2) the precursors and byproducts of the reactions that create and break down tissue. A notable feature is that the full extent of coupling between mass transport and mechanics emerges from the thermodynamics. Contributions to fluxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a unified fashion from previous formulations of the problem. The present work demonstrates these effects via a physically-consistent treatment. The presence of multiple, interacting species requires that the formulation be consistent with mixture theory. This requirement has far-reaching consequences. A preliminary numerical example is included to demonstrate some aspects of the coupled formulation.Comment: 29 pages, 11 figures, accepted for publication in Journal of the Mechanics and Physics of Solids. See journal for final versio

Stellar Mixing and the Primordial Lithium Abundance

We compare the properties of recent samples of the lithium abundances in halo stars to one another and to the predictions of theoretical models including rotational mixing, and we examine the data for trends with metal abundance. We find from a KS test that in the absence of any correction for chemical evolution, the Ryan, Norris, & Beers (1999} sample is fully consistent with mild rotational mixing induced depletion and, therefore, with an initial lithium abundance higher than the observed value. Tests for outliers depend sensitively on the threshold for defining their presence, but we find a 10$--$45% probability that the RNB sample is drawn from the rotationally mixed models with a 0.2 dex median depletion (with lower probabilities corresponding to higher depletion factors). When chemical evolution trends (Li/H versus Fe/H) are treated in the linear plane we find that the dispersion in the RNB sample is not explained by chemical evolution; the inferred bounds on lithium depletion from rotational mixing are similar to those derived from models without chemical evolution. We find that differences in the equivalent width measurements are primarily responsible for different observational conclusions concerning the lithium dispersion in halo stars. The standard Big Bang Nucleosynthesis predicted lithium abundance which corresponds to the deuterium abundance inferred from observations of high-redshift, low-metallicity QSO absorbers requires halo star lithium depletion in an amount consistent with that from our models of rotational mixing, but inconsistent with no depletion.Comment: 39 pages, 9 figures; submitted Ap

Biological remodelling: Stationary energy, configurational change, internal variables and dissipation

Remodelling is defined as an evolution of microstructure or variations in the configuration of the underlying manifold. The manner in which a biological tissue and its subsystems remodel their structure is treated in a continuum mechanical setting. While some examples of remodelling are conveniently modelled as evolution of the reference configuration (Case I), others are more suited to an internal variable description (Case II). In this paper we explore the applicability of stationary energy states to remodelled systems. A variational treatment is introduced by assuming that stationary energy states are attained by changes in microstructure via one of the two mechanisms--Cases I and II. An example is presented to illustrate each case. The example illustrating Case II is further studied in the context of the thermodynamic dissipation inequality.Comment: 24 pages, 4 figures. Replaced version has corrections to typos in equations, and the corresponding correct plot of the solution--all in Section

Domain-wall fermions with U(1)U(1) dynamical gauge fields

We have carried out a numerical simulation of a domain-wall model in $(2+1)$-dimensions, in the presence of a dynamical gauge field only in an extra dimension, corresponding to the weak coupling limit of a ( 2-dimensional ) physical gauge coupling. Using a quenched approximation we have investigated this model at $\beta_{s} ( = 1 / g^{2}_{s} ) =$ 0.5 ( ``symmetric'' phase), 1.0, and 5.0 (``broken'' phase), where $g_s$ is the gauge coupling constant of the extra dimension. We have found that there exists a critical value of a domain-wall mass $m_{0}^{c}$ which separates a region with a fermionic zero mode on the domain-wall from the one without it, in both symmetric and broken phases. This result suggests that the domain-wall method may work for the construction of lattice chiral gauge theories.Comment: 27 pages (11 figures), latex (epsf style-file needed