Mechanics of Cell Growth

A morpho-viscoelasticity theory for growth in proliferating aggregates

Despite significant research efforts in the continuum modeling of biological growth, certain aspects have been overlooked. For instance, numerous investigations have examined the influence of morphogenetic cell behaviors, like division and intercalation, on the mechanical response of passive (non-growing) tissues. Yet, their impact on active growth dynamics remains inadequately explored. A key reason for this inadequacy stems from challenges in the continuum treatment of cell-level processes. While some coarse-grained models have been proposed to address these shortcomings, a focus on cell division and cell expansion has been missing, rendering them unusable when it comes to modeling growth. Moreover, existing studies are limited to two-dimensional tissues and are yet to be formally extended to three-dimensional multicellular systems. To address these limitations, we here present a generalized multiscale model for three-dimensional aggregates that accounts for complex morphogenetic movements that include division, expansion, and intercalation. The proposed continuum theory thus allows for a comprehensive exploration into the growth and dissipation mechanics of proliferating aggregates, such as spheroids and organoids.

Continuum Growth Mechanics: Reconciling Two Common Frameworks

The objective of this study was to investigate whether the two most common growth mechanics modeling frameworks, the constrained-mixture growth model and the kinematic growth model, could be reconciled mathematically. The purpose of this effort was to provide practical guidelines for potential users of these modeling frameworks. Results showed that the kinematic growth model is mathematically consistent with a special form of the constrained-mixture growth model, where only one generation of a growing solid exists at any given time, overturning its entire solid mass at each instant of growth in order to adopt the reference configuration dictated by the growth deformation. The thermodynamics of the kinematic growth model, along with the specialized constrained-mixture growth model, requires a cellular supply of chemical energy to allow deposition of solid mass under a stressed state. A back-of-the-envelope calculation shows that the amount of chemical energy required to sustain biological growth under these models is negligibly small, when compared to the amount of energy normally consumed daily by the human body. In conclusion, this study successfully reconciled the two most popular growth theories for biological growth and explained the special circumstances under which the constrained-mixture growth model reduces to the kinematic growth model.

Computational study of biomechanical drivers of renal cystogenesis

Renal cystogenesis is the pathological hallmark of autosomal dominant polycystic kidney disease, caused by PKD1 and PKD2 mutations. The formation of renal cysts is a common manifestation in ciliopathies, a group of syndromic disorders caused by mutation of proteins involved in the assembly and function of the primary cilium. Cystogenesis is caused by the derailment of the renal tubular architecture and tissue deformation that eventually leads to the impairment of kidney function. However, the biomechanical imbalance of cytoskeletal forces that are altered in cells with Pkd1 mutations has never been investigated, and its nature and extent remain unknown. In this computational study, we explored the feasibility of various biomechanical drivers of renal cystogenesis by examining several hypothetical mechanisms that may promote morphogenetic markers of cystogenesis. Our objective was to provide physics-based guidance for our formulation of hypotheses and our design of experimental studies investigating the role of biomechanical disequilibrium in cystogenesis. We employed the finite element method to explore the role of (1) wild-type versus mutant cell elastic modulus; (2) contractile stress magnitude in mutant cells; (3) localization and orientation of contractile stress in mutant cells; and (4) sequence of cell contraction and cell proliferation. Our objective was to identify the factors that produce the characteristic tubular cystic growth. Results showed that cystogenesis occurred only when mutant cells contracted along the apical-basal axis, followed or accompanied by cell proliferation, as long as mutant cells had comparable or lower elastic modulus than wild-type cells, with their contractile stresses being significantly greater than their modulus. Results of these simulations allow us to focus future in vitro and in vivo experimental studies on these factors, helping us formulate physics-based hypotheses for renal tubule cystogenesis.

Modeling Inelastic Responses Using Constrained Reactive Mixtures

This study reviews the progression of our research, from modeling growth theories for cartilage tissue engineering, to the formulation of constrained reactive mixture theories to model inelastic responses in any solid material, such as theories for damage mechanics, viscoelasticity, plasticity, and elasto-plastic damage. In this framework, multiple solid generations α can co-exist at any given time in the mixture. The oldest generation is denoted by α = s and is called the master generation, whose reference configuration X s is observable. The solid generations α are all constrained to share the same velocity v s , but may have distinct reference configurations X α . An important element of this formulation is that the time-invariant mapping F α s = ∂ X α / ∂ X s between these reference configurations is a function of state, whose mathematical formulation is postulated by constitutive assumption. Thus, reference configurations X α are not observable ( α ≠ s ) . This formulation employs only observable state variables, such as the deformation gradient F s of the master generation and the referential mass concentrations ρ r α of each generation, in contrast to classical formulations of inelastic responses which rely on internal state variable theory, requiring evolution equations for those hidden variables. In constrained reactive mixtures, the evolution of the mass concentrations is governed by the axiom of mass balance, using constitutive models for the mass supply densities ρ ˆ r α . Classical and constrained reactive mixture approaches share considerable mathematical analogies, as they both introduce a multiplicative decomposition of the deformation gradient, also requiring evolution equations to track some of the state variables. However, they also differ at a fundamental level, since one adopts only observable state variables while the other introduces hidden state variables. In summary, this review presents an alternative foundational approach to the modeling of inelastic responses in solids, grounded in the classical framework of mixture theory.

Continuum Thermodynamics of Constrained Reactive Mixtures

Mixture theory models continua consisting of multiple constituents with independent motions. In constrained mixtures, all constituents share the same velocity but they may have different reference configurations. The theory of constrained reactive mixtures was formulated to analyze growth and remodeling in living biological tissues. It can also reproduce and extend classical frameworks of damage mechanics and viscoelasticity under isothermal conditions, when modeling bonds that can break and reform. This study focuses on establishing the thermodynamic foundations of constrained reactive mixtures under more general conditions, for arbitrary reactive processes where temperature varies in time and space. By incorporating general expressions for reaction kinetics, it is shown that the residual dissipation statement of the Clausius-Duhem inequality must include a reactive power density, while the axiom of energy balance must include a reactive heat supply density. Both of these functions are proportional to the molar production rate of a reaction, and they depend on the chemical potentials of the mixture constituents. We present novel formulas for the classical thermodynamic concepts of energy of formation and heat of reaction, making it possible to evaluate the heat supply generated by reactive processes from the knowledge of the specific free energy of mixture constituents as well as the reaction rate. We illustrate these novel concepts with mixtures of ideal gases, and isothermal reactive damage mechanics and viscoelasticity, as well as reactive thermoelasticity. This framework facilitates the analysis of reactive tissue biomechanics and physiological and biomedical engineering processes where temperature variations cannot be neglected.

A Direct Comparison of Node and Element-Based Finite Element Modeling Approaches to Study Tissue Growth

Finite element analysis is a useful tool to model growth of biological tissues and predict how growth can be impacted by stimuli. Previous work has simulated growth using node-based or element-based approaches, and this implementation choice may influence predicted growth, irrespective of the applied growth model. This study directly compared node-based and element-based approaches to understand the isolated impact of implementation method on growth predictions by simulating growth of a bone rudiment geometry, and determined what conditions produce similar results between the approaches. We used a previously reported node-based approach implemented via thermal expansion and an element-based approach implemented via osmotic swelling, and we derived a mathematical relationship to relate the growth resulting from these approaches. We found that material properties (modulus) affected growth in the element-based approach, with growth completely restricted for high modulus values relative to the growth stimulus, and no restriction for low modulus values. The node-based approach was unaffected by modulus. Node- and element-based approaches matched marginally better when the conversion coefficient to relate the approaches was optimized based on the results of initial simulations, rather than using the theoretically predicted conversion coefficient (median difference in node position 0.042 cm versus 0.052 cm, respectively). In summary, we illustrate here the importance of the choice of implementation approach for modeling growth, provide a framework for converting models between implementation approaches, and highlight important considerations for comparing results in prior work and developing new models of tissue growth.

Proliferation Cycle Transcriptomic Signatures are Strongly associated With Gastric Cancer Patient Survival

Gastric cancer is one of the most heterogeneous tumors with multi-level molecular disturbances. Sustaining proliferative signaling and evading growth suppressors are two important hallmarks that enable the cancer cells to become tumorigenic and ultimately malignant, which enable tumor growth. Discovering and understanding the difference in tumor proliferation cycle phenotypes can be used to better classify tumors, and provide classification schemes for disease diagnosis and treatment options, which are more in line with the requirements of today's precision medicine. We collected 691 eligible samples from The Cancer Genome Atlas (TCGA) and Gene Expression Omnibus (GEO) database, combined with transcriptome data, to explore different heterogeneous proliferation cycle phenotypes, and further study the potential genomic changes that may lead to these different phenotypes in this study. Interestingly, two subtypes with different clinical and biological characteristics were identified through cluster analysis of gastric cancer transcriptome data. The repeatability of the classification was confirmed in an independent Gene Expression Omnibus validation cohort, and consistent phenotypes were observed. These two phenotypes showed different clinical outcomes, and tumor mutation burden. This classification helped us to better classify gastric cancer patients and provide targeted treatment based on specific transcriptome data.

A multi-layered model of human skin elucidates mechanisms of wrinkling in the forehead

Skin wrinkling, especially in the facial area, is a prominent sign of aging and is a growing area of research aimed at developing cosmetics and dermatological treatments. To better understand and treat undesirable skin wrinkles, it is vitally important to elucidate the underlying mechanisms of skin wrinkling, a largely mechanical process. Human skin, a multi-layer composite, has six mechanically distinct layers: from the outermost inward they are the stratum corneum, viable epidermis, dermal-epidermal-junction, papillary dermis, reticular dermis, and hypodermis. To better address the through-thickness hierarchy, and the development of wrinkling within this complicated hierarchy, we established a six-layered model of human skin realized with finite element modeling, by leveraging available morphological and biomechanical data on human skin of the forehead. Exercising our new model we aimed to quantify the effects of three potential mechanisms of wrinkle formation: (1) skin compression due to muscle contraction (dynamic wrinkles); (2) age-related volumetric tissue loss (static wrinkles); and (3) the combined effects of both mechanisms. Since hydration of the stratum corneum significantly affects its stiffness we also aimed to quantify the influence its hydration with these three potential mechanisms of wrinkle formation. Our six-layered skin model, combined with the proposed wrinkling mechanisms, successfully predicts the formation of dynamic and static wrinkles in the forehead consistent with the experimental literature. We observed three wrinkling modes in the forehead where the deepest wrinkles could reach to the reticular dermis. With further refinement our new six-layered model of human skin can be applied to study other region-specific wrinkle types such as the "crow's feet" and the nasolabial folds.

A multi-layered computational model for wrinkling of human skin predicts aging effects

The development and progression of wrinkles from young to aged human skin relates to both structural and mechanical changes induced by aging. Here we aim to better understand the interaction of skin's layered morphology with dynamic wrinkles predicted in young and aged skin. First, we compare the predictions of wrinkling from 3-D finite element models of human skin including two to six distinct and anatomically motivated layers. Second, we perform parametric analyses using our six-layered model to determine how age-related changes in the architecture of human skin affect dynamic surface wrinkling. Specifically, we consider the following aging-related changes in the morphology of skin: flattening of the dermal-epidermal junction (DEJ) interface; thinning of both the viable epidermis (VE) and the reticular dermis (RD); and thickening of the papillary dermis (PD). We use skin compression to model dynamic, expressional wrinkles due to muscle contraction, and volumetric tissue loss to model effects of aging in wrinkling simulations. Our results highlight the role of skin's multi-layered structure in the modeling of wrinkling formation. Our six-layered model, consisting of all of the mechanical layers, predicts deep wrinkles with better fidelity than models including fewer layers. From our parametric study, applying our six-layered model, we conclude that: (1) the relative thicknesses of the layers in the epidermis or dermis significantly influences surface wrinkling in skin; and, (2) flattening of the DEJ with aging enhances surface wrinkling. Thinning of VE increases the relative stiffness of the epidermis and thus enhances dynamic wrinkling, while thickening of PD or thinning of RD has the same effect by reducing the equivalent stiffness of the substrate. Consequently, strategies to minimize wrinkling could maintain the undulating morphology of the DEJ, thereby delaying dynamic wrinkling and delaying the propagation of buckling into the deeper dermis or hypodermis. Additional strategies to minimize wrinkling could target preventing the VE and RD from thinning or preventing the PD from thickening.

FEBio: History and Advances

The principal goal of the FEBio project is to provide an advanced finite element tool for the biomechanics and biophysics communities that allows researchers to model mechanics, transport, and electrokinetic phenomena for biological systems accurately and efficiently. In addition, because FEBio is geared toward the research community, the code is designed such that new features can be added easily, thus making it an ideal tool for testing novel computational methods. Finally, because the success of a code is determined by its user base, integral goals of the FEBio project have been to offer support and outreach to our community; to provide mechanisms for dissemination of results, models, and data; and to encourage interaction between users. This review presents the history of the FEBio project, from its initial developments through its current funding period. We also present a glimpse into the future of FEBio.

Reactive Constrained Mixtures for Modeling the Solid Matrix of Biological Tissues

This article illustrates our approach for modeling the solid matrix of biological tissues using reactive constrained mixtures. Several examples are presented to highlight the potential benefits of this approach, showing that seemingly disparate fields of mechanics and chemical kinetics are actually closely interrelated and may be elegantly expressed in a unified framework. Thus, constrained mixture models recover classical theories for fibrous materials with bundles oriented in different directions or having different reference configurations, that produce characteristic fiber recruitment patterns under loading. Reactions that exchange mass among various constituents of a mixture may be used to describe tissue growth and remodeling, which may also alter the material's anisotropy. Similarly, reactions that describe the breaking and reforming of bonds may be used to model free energy dissipation in a viscoelastic material. Therefore, this framework is particularly well suited for modeling biological tissues.

Interstitial growth and remodeling of biological tissues: tissue composition as state variables

Growth and remodeling of biological tissues involves mass exchanges between soluble building blocks in the tissue's interstitial fluid and the various constituents of cells and the extracellular matrix. As the content of these various constituents evolves with growth, associated material properties, such as the elastic modulus of the extracellular matrix, may similarly evolve. Therefore, growth theories may be formulated by accounting for the evolution of tissue composition over time in response to various biological and mechanical triggers. This approach has been the foundation of classical bone remodeling theories that successfully describe Wolff's law by establishing a dependence between Young's modulus and bone apparent density and by formulating a constitutive relation between bone mass supply and the state of strain. The goal of this study is to demonstrate that adding tissue composition as state variables in the constitutive relations governing the stress-strain response and the mass supply represents a very general and straightforward method to model interstitial growth and remodeling in a wide variety of biological tissues. The foundation for this approach is rooted in the framework of mixture theory, which models the tissue as a mixture of multiple solid and fluid constituents. A further generalization is to allow each solid constituent in a constrained solid mixture to have its own reference (stress-free) configuration. Several illustrations are provided, ranging from bone remodeling to cartilage tissue engineering and cervical remodeling during pregnancy.

Continuum mixture models of biological growth and remodeling: past successes and future opportunities

Biological growth processes involve mass exchanges that increase, decrease, or replace material that constitutes cells, tissues, and organs. In most cases, such exchanges alter the structural makeup of the material and consequently affect associated mechanobiological responses to applied loads. Given that the type and extent of changes in structural integrity depend on the different constituents involved (e.g., particular cytoskeletal or extracellular matrix proteins), the continuum theory of mixtures is ideally suited to model the mechanics of growth and remodeling. The goal of this review is twofold: first, to highlight a few illustrative examples that show diverse applications of mixture theory to describe biological growth and/or remodeling; second, to identify some open problems in the fields of modeling soft-tissue growth and remodeling.

Frontiers in growth and remodeling

Unlike common engineering materials, living matter can autonomously respond to environmental changes. Living structures can grow stronger, weaker, larger, or smaller within months, weeks, or days as a result of a continuous microstructural turnover and renewal. Hard tissues can adapt by increasing their density and grow strong. Soft tissues can adapt by increasing their volume and grow large. For more than three decades, the mechanics community has actively contributed to understand the phenomena of growth and remodeling from a mechanistic point of view. However, to date, there is no single, unified characterization of growth, which is equally accepted by all scientists in the field. Here we shed light on the continuum modeling of growth and remodeling of living matter, and give a comprehensive overview of historical developments and trends. We provide a state-of-the-art review of current research highlights, and discuss challenges and potential future directions. Using the example of volumetric growth, we illustrate how we can establish and utilize growth theories to characterize the functional adaptation of soft living matter. We anticipate this review to be the starting point for critical discussions and future research in growth and remodeling, with a potential impact on life science and medicine.