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 uxes of mass, and terms for momentum and energy transfer between species are introduced to enhance the classical balance laws. The transported species include: (i) a uid phase, and (ii) 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 uxes from the concentration gradient, chemical potential gradient, stress gradient, body force and inertia have not emerged in a uniÿed fashion from previous formulations of the problem. The present work demonstrates these e ects 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. ?

Purpose To determine the association between Prostate Imaging Reporting and Data System (PI-RADS) version 2 scores and prostate cancer (PCa) in a cohort of patients undergoing biopsy of transition zone (TZ) lesions. Materials and Methods A total of 634 TZ lesions in 457 patients were identified from a prospectively maintained database of consecutive patients undergoing prostate magnetic resonance imaging. Prostate lesions were retrospectively categorized with the PI-RADS version 2 system by two readers in consensus who were blinded to histopathologic findings. The proportion of cancer detection for all PCa and for clinically important PCa (Gleason score ≥3+4) for each PI-RADS version 2 category was determined. The performance of PI-RADS version 2 in cancer detection was evaluated. Results For PI-RADS category 2 lesions, the overall proportion of cancers was 4% (one of 25), without any clinically important cancer. For PI-RADS category 3, 4, and 5 lesions, the overall proportion of cancers was 22.2% (78 of 352), 39.1% (43 of 110), and 87.8% (129 of 147), respectively, and the proportion of clinically important cancers was 11.1% (39 of 352), 29.1% (32 of 110), and 77.6% (114 of 147), respectively. Higher PI-RADS version 2 scores were associated with increasing likelihood of the presence of clinically important PCa (P < .001). Differences were found in the percentage of cancers in the PI-RADS category between PI-RADS 3 and those upgraded to PI-RADS 4 based on diffusion-weighted imaging for clinically important cancers (proportion for clinically important cancers for PI-RADS 3 and PI-RADS 3+1 were 11.1% [39 of 352] and 30.8% [28 of 91], respectively; P < .001). Conclusion Higher PI-RADS version 2 scores are associated with a higher proportion of clinically important cancers in the TZ. PI-RADS category 2 lesions rarely yield PCa, and their presence does not justify targeted biopsy.

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.

We discuss the roles of continuum linear elasticity and atomistic calculations in determining the formation volume and the strain energy of formation of a point defect in a crystal. Our considerations bear special relevance to defect formation under stress. The elasticity treatment is based on the Green's function solution for a center of contraction or expansion in an anisotropic solid. It makes possible the precise definition of a formation volume tensor and leads to an extension of Eshelby's result for the work done by an external stress during the transformation of a continuum inclusion (Proc. Roy. Soc. Lond. Ser. A, 241(1226), 376, 1957 ). Parameters necessary for a complete continuum calculation of elastic fields around a point defect are obtained by comparing with an atomistic solution in the far field. However, an elasticity result makes it possible to test the validity of the formation volume that is obtained via atomistic calculations under various boundary conditions. It also yields the correction term for formation volume calculated under these boundary conditions. Using two types of boundary conditions commonly employed in atomistic calculations, a comparison is also made of the strain energies of formation predicted by continuum elasticity and atomistic calculations. The limitations of the continuum linear elastic treatment are revealed by comparing with atomistic calculations of the formation volume and strain energies of small crystals enclosing point defects.

Remodelling of biological tissue, due to changes in microstructure, is treated in the continuum mechanical setting. Microstructural change is expressed as an evolution of the reference configuration. This evolution is expressed as a point-to-point map from the reference configuration to a remodelled configuration. A "preferred" change in configuration is considered in the form of a globally incompatible tangent map. This field could be experimentally determined, or specified from other insight. Issues of global compatibility and evolution equations for the resulting configurations are addressed. It is hypothesized that the tissue reaches local equilibrium with respect to changes in microstructure. A governing differential equation and boundary conditions are obtained for the microstructural changes by posing the problem in a variational setting. The Eshelby stress tensor, a separate configurational stress, and thermodynamic driving (material) forces arise in this formulation, which is recognized as describing a process of self-assembly. An example is presented to illustrate the theoretical framework.