Heterogeneity Profoundly Alters Emergent Stress Fields in Constrained Multicellular Systems

Stress fields emerging from the transfer of forces between cells within multicellular systems are increasingly being recognized as major determinants of cell fate. Current analytical and numerical models used for the calculation of stresses within cell monolayers assume homogeneous contractile and mechanical cellular properties; however, cell behavior varies by region within constrained tissues. Here, we show the impact of heterogeneous cell properties on resulting stress fields that guide cell phenotype and apoptosis. Using circular micropatterns, we measured biophysical metrics associated with cell mechanical stresses. We then computed cell-layer stress distributions using finite element contraction models and monolayer stress microscopy. In agreement with previous studies, cell spread area, alignment, and traction forces increase, whereas apoptotic activity decreases, from the center of cell layers to the edge. The distribution of these metrics clearly indicates low cell stress in central regions and high cell stress at the periphery of the patterns. However, the opposite trend is predicted by computational models when homogeneous contractile and mechanical properties are assumed. In our model, utilizing heterogeneous cell-layer contractility and elastic moduli values based on experimentally measured biophysical parameters, we calculate low cell stress in central areas and high anisotropic stresses in peripheral regions, consistent with the biometrics. These results clearly demonstrate that common assumptions of uniformity in cell contractility and stiffness break down in postconfluence confined multicellular systems. This work highlights the importance of incorporating regional variations in cell mechanical properties when estimating emergent stress fields from collective cell behavior.     

Residual stress generation and necrosis formation in multi-cell tumour spheroids

We consider how cell proliferation and death generate residual stresses within a multi-cell tumour spheroid (MCTS). Previous work by Jones and co-workers [8] has shown that isotropic growth in a purely elastic MCTS produces growth induced stresses which eventually become unbounded, and hence are physically unrealistic. Since viscoelastic materials show stress relaxation under a fixed deformation we consider the effect of the addition of a small amount of viscosity to the elastic system by examining formation of equilibrium stress profiles within a Maxwell type viscoelastic MCTS. A model of necrosis formation based upon that proposed by Please and co-workers (see [16] [17] [18]) is then presented in which necrosis forms under conditions of adverse mechanical stress rather than in regions of extreme chemical stress as is usually assumed. The influence of rheology on necrosis formation is then investigated, and it is shown that the excessive stress generated in the purely elastic tumour can be relieved either by the addition of some viscosity to the system or by accounting for an inner necrotic interface with an appropriate stress boundary condition.     

Loss of stability-a new model for stress relaxation in plant cell walls

This study addresses the mechanism of wall stress relaxation in growing plant cells. The current viscoelastic model of cell wall relaxation, which dates from the work of Preston, Cleland, Lockhart, and others in the 1960s, has serious shortcomings. It has been shown however that the theory of loss of stability (LOS) can be applied to materials in tension, leading to the conclusion that the relaxation of stresses in the walls of any pressure vessel is rigorously modeled using LOS. We propose that LOS also provides a more appropriate and versatile model of stress relaxation in growing plant cells. We argue that when treated as a manifestation of LOS, the regulation of cell turgor has a rigorous and demonstrable basis in the geometrical and physical properties of the cell wall and the cell's ability to import water. Thus plant cell growth can be regarded as an inherently self-limiting process, tunable by biochemical or structural means. Lastly, despite the current limitations of our model, we apply direct measurement of elastic modulus, wall thickness and cell radius obtained from cylindrical Chara corallina cells to generate an initial calculation of critical pressures in a hypothetical spherical cell with the same material properties.     

Viscoelasticity as a Biomarker for High-Throughput Flow Cytometry

The mechanical properties of living cells are a label-free biophysical marker of cell viability and health; however, their use has been greatly limited by low measurement throughput. Although examining individual cells at high rates is now commonplace with fluorescence activated cell sorters, development of comparable techniques that nondestructively probe cell mechanics remains challenging. A fundamental hurdle is the signal response time. Where light scattering and fluorescence signatures are virtually instantaneous, the cell stress relaxation, typically occurring on the order of seconds, limits the potential speed of elastic property measurement. To overcome this intrinsic barrier to rapid analysis, we show here that cell viscoelastic properties measured at frequencies far higher than those associated with cell relaxation can be used as a means of identifying significant differences in cell phenotype. In these studies, we explore changes in erythrocyte mechanical properties caused by infection with Plasmodium falciparum and find that the elastic response alone fails to detect malaria at high frequencies. At timescales associated with rapid assays, however, we observe that the inelastic response shows significant changes and can be used as a reliable indicator of infection, establishing the dynamic viscoelasticity as a basis for nondestructive mechanical analogs of current high-throughput cell classification methods.     

Viscoelasticity as a Biomarker for High-Throughput Flow Cytometry

The mechanical properties of living cells are a label-free biophysical marker of cell viability and health; however, their use has been greatly limited by low measurement throughput. Although examining individual cells at high rates is now commonplace with fluorescence activated cell sorters, development of comparable techniques that nondestructively probe cell mechanics remains challenging. A fundamental hurdle is the signal response time. Where light scattering and fluorescence signatures are virtually instantaneous, the cell stress relaxation, typically occurring on the order of seconds, limits the potential speed of elastic property measurement. To overcome this intrinsic barrier to rapid analysis, we show here that cell viscoelastic properties measured at frequencies far higher than those associated with cell relaxation can be used as a means of identifying significant differences in cell phenotype. In these studies, we explore changes in erythrocyte mechanical properties caused by infection with Plasmodium falciparum and find that the elastic response alone fails to detect malaria at high frequencies. At timescales associated with rapid assays, however, we observe that the inelastic response shows significant changes and can be used as a reliable indicator of infection, establishing the dynamic viscoelasticity as a basis for nondestructive mechanical analogs of current high-throughput cell classification methods.     

Macromolecular biophysics of the plant cell wall: Concepts and methodology

Plant cell walls provide form and mechanical strength to the living plant, but the relationship between their complex architecture and their remarkable ability to withstand external stress is not well understood. Primary cell walls are adapted to withstand tensile stresses while secondary cell walls also need to withstand compressive stresses. Therefore, while primary cell walls can with advantage be flexible and elastic, secondary cell walls must be rigid to avoid buckling under compressive loads. In addition, primary cell walls must be capable of growth and are subjected to cell separation forces at the cell corners. To understand how these stresses are resisted by cell walls, it will be necessary to find out how the walls deform internally under load, and how rigid are specific constituents of each type of cell wall. The most promising spectroscopic techniques for this purpose are solid-state nuclear magnetic resonance (NMR), and Fourier-transform infrared (FTIR) and Raman microscopy. By NMR relaxation experiments, it is possible to probe thermal motion in each cell-wall component. Novel adaptations of FTIR and Raman spectroscopy promise to allow mechanical stress and strain upon specific polymers to be examined in situ within the cell wall.     

Dynamic patterns of mechanical stimulation co-localise with growth and cell proliferation during morphogenesis in the avian embryonic knee joint

Muscle contractions begin in early embryonic life, generating forces that regulate the correct formation of the skeleton. In this paper we test the hypothesis that the biophysical stimulation generated by muscle forces may be a causative factor for the changes in shape of the knee joint as it grows. We do this by predicting the spatial and temporal patterns of biophysical stimuli, where cell proliferation and rudiment shape changes occur within the emerging tissues of the joint over time. We used optical projection tomography (OPT) to create anatomically accurate finite element models of the embryonic knee at three time points (stages) of development. OPT was also used to locate muscle attachment sites and AFM was used to determine material properties. An association was found between the emergence of joint shape, cell proliferation and the pattern of biophysical stimuli generated by embryonic muscle contractions. Elevated rates of growth and cell proliferation in the medial condyle were found to co-localise with elevated patterns of biophysical stimuli including maximum principal stresses and fluid flow, throughout the time period studied, indicating that cartilage growth and chondrocyte proliferation in the epiphysis is potentially related to local patterns of biophysical stimuli. The development of the patella and articular cartilages, which is known to be affected by in ovo immobilisation, could be contributed to by specific patterns of fluid flow, pore pressure and stress in the joint interzone. This suggests that both cartilage growth and tissue differentiation in the embryonic joint is regulated by specific patterns of biophysical stimuli and that these stimuli are needed for the correct development of the joint.     

Population dynamics in tumor growth modeling

A new approach based on local interaction between cancer and tissue cells was applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and the further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) were inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotone function of their local concentration and thus of the local curvature of the tumor surface.     

Stress-mediated progression of solid tumors: effect of mechanical stress on tissue oxygenation,cancer cell proliferation,and drug delivery

Oxygen supply plays a central role in cancer cell proliferation. While vascular density increases at the early stages of carcinogenesis, mechanical solid stresses developed during growth compress tumor blood vessels and, thus, drastically reduce not only the supply of oxygen, but also the delivery of drugs at inner tumor regions. Among other effects, hypoxia and reduced drug delivery compromise the efficacy of radiation and chemo/nanotherapy, respectively. In the present study, we developed a mathematical model of tumor growth to investigate the interconnections among tumor oxygenation that supports cancer cell proliferation, the heterogeneous accumulation of mechanical stresses owing to tumor growth, the non-uniform compression of intratumoral blood vessels due to the mechanical stresses, and the insufficient delivery of oxygen and therapeutic agents because of vessel compression. We found that the high vascular density and increased cancer cell proliferation often observed in the periphery compared to the interior of a tumor can be attributed to heterogeneous solid stress accumulation. Highly vascularized peripheral regions are also associated with greater oxygenation compared with the compressed, less vascularized inner regions. We also modeled the delivery of drugs of two distinct sizes, namely chemotherapy and nanomedicine. Model predictions suggest that drug delivery is affected negatively by vessel compression independently of the size of the therapeutic agent. Finally, we demonstrated the applicability of our model to actual geometries, employing a breast tumor model derived from MR images.     

Role of Constitutive Behavior and Tumor-Host Mechanical Interactions in the State of Stress and Growth of Solid Tumors

Mechanical forces play a crucial role in tumor patho-physiology. Compression of cancer cells inhibits their proliferation rate, induces apoptosis and enhances their invasive and metastatic potential. Additionally, compression of intratumor blood vessels reduces the supply of oxygen, nutrients and drugs, affecting tumor progression and treatment. Despite the great importance of the mechanical microenvironment to the pathology of cancer, there are limited studies for the constitutive modeling and the mechanical properties of tumors and on how these parameters affect tumor growth. Also, the contribution of the host tissue to the growth and state of stress of the tumor remains unclear. To this end, we performed unconfined compression experiments in two tumor types and found that the experimental stress-strain response is better fitted to an exponential constitutive equation compared to the widely used neo-Hookean and Blatz-Ko models. Subsequently, we incorporated the constitutive equations along with the corresponding values of the mechanical properties - calculated by the fit - to a biomechanical model of tumor growth. Interestingly, we found that the evolution of stress and the growth rate of the tumor are independent from the selection of the constitutive equation, but depend strongly on the mechanical interactions with the surrounding host tissue. Particularly, model predictions - in agreement with experimental studies - suggest that the stiffness of solid tumors should exceed a critical value compared with that of the surrounding tissue in order to be able to displace the tissue and grow in size. With the use of the model, we estimated this critical value to be on the order of 1.5. Our results suggest that the direct effect of solid stress on tumor growth involves not only the inhibitory effect of stress on cancer cell proliferation and the induction of apoptosis, but also the resistance of the surrounding tissue to tumor expansion.     

Population dynamics in modeling tumor growth

A new approach based on local interaction between cancer and tissue cells is applied to the problem of the onset and growth of solid tumors in homogeneous tissues and effects associated with dramatic changes in tumor growth after crossing the boundary between different tissues. The characteristic sizes and growth rates of spherical tumors, the points of the beginning and the end of spherical growth, and further development of complex structures from the spherical ones (rough interface between the tumor and the host tissue, elongate outgrowths, dendritic structures, and metastases) are inferred assuming that the reproduction rate of a population of cancer cells is a nonmonotonous function of their local concentration and thus of the local curvature of the tumor surface. The growth behavior changes dramatically when the tumor crosses a boundary between two tissues.     

Individual-based and continuum models of growing cell populations: a comparison

In this paper we compare two alternative theoretical approaches for simulating the growth of cell aggregates in vitro: individual cell (agent)-based models and continuum models. We show by a quantitative analysis of both a biophysical agent-based and a continuum mechanical model that for densely packed aggregates the expansion of the cell population is dominated by cell proliferation controlled by mechanical stress. The biophysical agent-based model introduced earlier (Drasdo and Hoehme in Phys Biol 2:133-147, 2005) approximates each cell as an isotropic, homogeneous, elastic, spherical object parameterised by measurable biophysical and cell-biological quantities and has been shown by comparison to experimental findings to explain the growth patterns of dense monolayers and multicellular spheroids. Both models exhibit the same growth kinetics, with initial exponential growth of the population size and aggregate diameter followed by linear growth of the diameter and power-law growth of the cell population size. Very sparse monolayers can be explained by a very small or absent cell-cell adhesion and large random cell migration. In this case the expansion speed is not controlled by mechanical stress but by random cell migration and can be modelled by the Fisher-Kolmogorov-Petrovskii-Piskounov (FKPP) reaction-diffusion equation. The growth kinetics differs from that of densely packed aggregates in that the initial spread, as quantified by the radius of gyration, is diffusive. Since simulations of the lattice-free agent-based model in the case of very large random migration are too long to be practical, lattice-based cellular automaton (CA) models have to be used for a quantitative analysis of sparse monolayers. Analysis of these dense monolayers leads to the identification of a critical parameter of the CA model so that eventually a hierarchy of three model types (a detailed biophysical lattice-free model, a rule-based cellular automaton and a continuum approach) emerge which yield the same growth pattern for dense and sparse cell aggregates.     

Integrity of actin fibers and microtubules influences metastatic tumor cell adhesion

Tumor cell adhesion within host organ microvasculature, its stabilization and invasion into host organ parenchyma appear to be important steps during formation of distant metastasis. These interactions of circulating tumor cells with the host organs occur in the presence of fluid shear forces and soluble and cellular environmental conditions of the blood that can modulate their cellular responses and possibly their metastatic efficiency. Cytoskeletal components, such as actin filaments and microtubules, can regulate biophysical characteristics and cellular signaling of the circulating cells. Therefore, we investigated the role of these cytoskeletal structures for early steps during metastasis formation in vivo and in vitro. Using an intravital observation technique, tumor cell adhesion of colon carcinoma cells within the hepatic microcirculation of rats and their invasion into liver parenchyma was observed. Disruption of actin filaments increased cell adhesion, whereas tubulin disruption inhibited adhesive interactions in vivo. The impairment of the cytoskeleton modulated adhesion-mediated cell signaling via focal adhesion kinase (FAK) and paxillin under flow conditions in vitro. In the presence of fluid flow, focal adhesions were enlarged and hyperphosphorylated, whereas stress fibers were reduced compared to static cell adhesion. Disruption of microtubules, however, partially inhibited these effects. Combining the in vivo and in vitro results, our study suggested that changes in cell rigidity and avidity of cell adhesion molecules after disruption of cytoskeletal components appear to be more important for initial adhesive interactions in vivo than their interference with adhesion-mediated cellular signal transduction.     

Explicit Separation of Growth and Motility in a New Tumor Cord Model

We investigate a new model of tumor growth in which cell motility is considered an explicitly separate process from growth. Bulk tumor expansion is modeled by individual cell motility in a density-dependent diffusion process. This model is implemented in the context of an in vivo system, the tumor cord. We investigate numerically microscale density distributions of different cell classes and macroscale whole tumor growth rates as functions of the strength of transitions between classes. Our results indicate that the total tumor growth follows a classical von Bertalanffy growth profile, as many in vivo tumors are observed to do. This provides a quick validation for the model hypotheses. The microscale and macroscale properties are both sensitive to fluctuations in the transition parameters, and grossly adopt one of two phenotypic profiles based on their parameter regime. We analyze these profiles and use the observations to classify parameter regimes by their phenotypes. This classification yields a novel hypothesis for the early evolutionary selection of the metastatic phenotype by selecting against less motile cells which grow to higher densities and may therefore induce local collapse of the vascular network.     

A single-cell-based model of tumor growth in vitro: monolayers and spheroids

To what extent the growth dynamics of tumors is controlled by nutrients, biomechanical forces and other factors at different stages and in different environments is still largely unknown. Here we present a biophysical model to study the spatio-temporal growth dynamics of two-dimensional tumor monolayers and three-dimensional tumor spheroids as a complementary tool to in vitro experiments. Within our model each cell is represented as an individual object and parametrized by cell-biophysical and cell-kinetic parameters that can all be experimentally determined. Hence our modeling strategy allows us to study which mechanisms on the microscopic level of individual cells may affect the macroscopic properties of a growing tumor. We find the qualitative growth kinetics and patterns at early growth stages to be remarkably robust. Quantitative comparisons between computer simulations using our model and published experimental observations on monolayer cultures suggest a biomechanically-mediated form of growth inhibition during the experimentally observed transition from exponential to sub-exponential growth at sufficiently large tumor sizes. Our simulations show that the same transition during the growth of avascular tumor spheroids can be explained largely by the same mechanism. Glucose (or oxygen) depletion seems to determine mainly the size of the necrotic core but not the size of the tumor. We explore the consequences of the suggested biomechanical form of contact inhibition, in order to permit an experimental test of our model. Based on our findings we propose a phenomenological growth law in early expansion phases in which specific biological small-scale processes are subsumed in a small number of effective parameters.     

A constituent-based model for the nonlinear viscoelastic behavior of ligaments

This paper presents a constitutive model for predicting the nonlinear viscoelastic behavior of soft biological tissues and in particular of ligaments. The constitutive law is a generalization of the well-known quasi-linear viscoelastic theory (QLV) in which the elastic response of the tissue and the time-dependent properties are independently modeled and combined into a convolution time integral. The elastic behavior, based on the definition of anisotropic strain energy function, is extended to the time-dependent regime by means of a suitably developed time discretization scheme. The time-dependent constitutive law is based on the postulate that a constituent-based relaxation behavior may be defined through two different stress relaxation functions: one for the isotropic matrix and one for the reinforcing (collagen) fibers. The constitutive parameters of the viscoelastic model have been estimated by curve fitting the stress relaxation experiments conducted on medial collateral ligaments (MCLs) taken from the literature, whereas the predictive capability of the model was assessed by simulating experimental tests different from those used for the parameter estimation. In particular, creep tests at different maximum stresses have been successfully simulated. The proposed nonlinear viscoelastic model is able to predict the time-dependent response of ligaments described in experimental works (Bonifasi-Lista et al., 2005, J. Orthopaed. Res., 23, pp. 67-76; Hingorani et al., 2004, Ann. Biomed. Eng., 32, pp. 306-312; Provenzano et al., 2001, Ann. Biomed. Eng., 29, pp. 908-214; Weiss et al., 2002, J. Biomech., 35, pp. 943-950). In particular, the nonlinear viscoelastic response which implies different relaxation rates for different applied strains, as well as different creep rates for different applied stresses and direction-dependent relaxation behavior, can be described.     

Emergent behaviors from a cellular automaton model for invasive tumor growth in heterogeneous microenvironments

Understanding tumor invasion and metastasis is of crucial importance for both fundamental cancer research and clinical practice. In vitro experiments have established that the invasive growth of malignant tumors is characterized by the dendritic invasive branches composed of chains of tumor cells emanating from the primary tumor mass. The preponderance of previous tumor simulations focused on non-invasive (or proliferative) growth. The formation of the invasive cell chains and their interactions with the primary tumor mass and host microenvironment are not well understood. Here, we present a novel cellular automaton (CA) model that enables one to efficiently simulate invasive tumor growth in a heterogeneous host microenvironment. By taking into account a variety of microscopic-scale tumor-host interactions, including the short-range mechanical interactions between tumor cells and tumor stroma, degradation of the extracellular matrix by the invasive cells and oxygen/nutrient gradient driven cell motions, our CA model predicts a rich spectrum of growth dynamics and emergent behaviors of invasive tumors. Besides robustly reproducing the salient features of dendritic invasive growth, such as least-resistance paths of cells and intrabranch homotype attraction, we also predict nontrivial coupling between the growth dynamics of the primary tumor mass and the invasive cells. In addition, we show that the properties of the host microenvironment can significantly affect tumor morphology and growth dynamics, emphasizing the importance of understanding the tumor-host interaction. The capability of our CA model suggests that sophisticated in silico tools could eventually be utilized in clinical situations to predict neoplastic progression and propose individualized optimal treatment strategies.