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.

Abstract. The physics of solid tumor growth can be considered at three distinct size scales: the tumor scale, the cell-extracellular matrix (ECM) scale and the sub-cellular scale. In this paper we consider the tumor scale in the interest of eventually developing a system-level understanding of the progression of cancer. At this scale, cell populations and chemical species are best treated as concentration fields that vary with time and space. The cells have chemo-mechanical interactions with each other and with the ECM, consume glucose and oxygen that are transported through the tumor, and create chemical byproducts. We present a continuum mathematical model for the biochemical dynamics and mechanics that govern tumor growth. The biochemical dynamics and mechanics also engender free energy changes that serve as universal measures for comparison of these processes. Within our mathematical framework we therefore consider the free energy inequality, which arises from the first and second laws of thermodynamics. With the model we compute preliminary estimates of the free energy rates of a growing tumor in its pre-vascular stage by using currently available data from single cells and multicellular tumor spheroids. BackgroundThe progression of a tumor involves (a) cell proliferation, (b) cell motility, (c) metabolism by which the cells consume glucose and oxygen and create byproducts, (d) mechanical interactions between cancer cells, the ECM and surrounding tissues, and (e) mass transport of chemical species to and through the tumor. Each of these processes has a physically-distinct contribution to the free energy rate in the developing tumor. Complex biophysical interactions between these processes are more broadly observable at the tumor scale than in single-cell studies. Additionally, as we demonstrate in this communication, tumor scale studies have the potential of identifying the relevant questions regarding energy rates that must be considered at the lower, cell-ECM and sub-cellular scales. Using the tumor scale studies, it is of interest to track the free energy rates and thereby gain a system-level understanding of the processes listed above in developing tumors. More broadly, we argue in the Discussion of this paper that there is an interest in combining free energy studies at the tumor, cell-ECM and sub-cellular scales. This will reveal how the energetics change with time and state of the tumor between these different scales and between the processes underlying tumor growth to affect the progression of the cancer.

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In this paper, we address some modelling issues related to biological growth. Our treatment is based on a formulation for growth that was proposed within the context of mixture theory (J Mech Phys Solids 52:1595-1625, 2004). We aim to make this treatment more appropriate for the physics of porous soft tissues, paying particular attention to the nature of fluid transport, and mechanics of fluid and solid phases. The interactions between transport and mechanics have significant implications for growth and swelling. We also reformulate the governing differential equations for reaction-transport of solutes to represent the incompressibility constraint on the fluid phase of the tissue. This revision enables a straightforward implementation of numerical stabilisation for the advection-dominated limit of these equations. A finite element implementation with operator splitting is used to solve the coupled, non-linear partial differential equations that arise from the theory. We carry out a numerical and analytic study of the convergence of the operator splitting scheme subject to strain- and stress-homogenisation of the mechanics of fluid-solid interactions. A few computations are presented to demonstrate aspects of the physical mechanisms, and the numerical performance of the formulation.