Focus on the Physics of Cancer

Despite the spectacular achievements of molecular biology in the second half of the twentieth century and the crucial advances it permitted in cancer research, the fight against cancer has brought some disillusions. It is nowadays more and more apparent that getting a global picture of the very diverse and interlinked aspects of cancer development necessitates, in synergy with these achievements, other perspectives and investigating tools. In this undertaking, multidisciplinary approaches that include quantitative sciences in general and physics in particular play a crucial role. This 'focus on' collection contains 19 articles representative of the diversity and state-of-the-art of the contributions that physics can bring to the field of cancer research.

Many cell movements occur via polymerization of the actin cytoskeleton beneath the plasma membrane at the front of the cell, forming a protrusion called a lamellipodium, while myosin contraction squeezes forward the back of the cell. In what is known as the 'molecular clutch' description of cell motility, forward movement results from the engagement of the acto-myosin motor with cell-matrix adhesions, thus transmitting force to the substrate and producing movement. However during cell translocation, clutch engagement is not perfect, and as a result, the cytoskeleton slips with respect to the substrate, undergoing backward (retrograde) flow in the direction of the cell body. Retrograde flow is therefore inversely proportional to cell speed and depends on adhesion and acto-myosin dynamics. Here we asked whether the molecular clutch was a general mechanism by measuring motility and retrograde flow for the Caenorhabditis elegans sperm cell in different adhesive conditions. These cells move by adhering to the substrate and emitting a dynamic lamellipodium, but the sperm cell does not contain an acto-myosin cytoskeleton. Instead the lamellipodium is formed by the assembly of major sperm protein, which has no biochemical or structural similarity to actin. We find that these cells display the same molecular clutch characteristics as acto-myosin containing cells. We further show that retrograde flow is produced both by cytoskeletal assembly and contractility in these cells. Overall this study shows that the molecular clutch hypothesis of how polymerization is transduced into motility via adhesions is a general description of cell movement regardless of the composition of the cytoskeleton.

Using an Atomic Force Microscope (AFM) with a 5.3 μm diameter spherical probe, we determined mechanical properties of individual human mammary epithelial cells. The cells were derived from a pair of cell lines that mimic cell progression through four phases of neoplastic transformation: normal (non-transformed), immortal, tumorigenic, and metastatic. Measurements on cells in all four phases were taken over both the cytoplasmic and nuclear regions. Moreover, the measurements were made for cells in different microenvironments as related to cell–cell contacts: isolated cells; cells residing on the periphery of a contiguous cell monolayer; and cells on the inside of a contiguous cell monolayer. By fitting the AFM force versus indentation curves to a Hertz model, we determined the pseudo-elastic Young's modulus, E. Combining all data for the cellular subregions (over nucleus and cytoplasm) and the different cell microenvironments, we obtained stiffness values for normal, immortal, tumorigenic, and metastatic cells of 870 Pa, 870 Pa, 490 Pa, and 580 Pa, respectively. That is, cells become softer as they advance to the tumorigenic phase and then stiffen somewhat in the final step to metastatic cells. We also found a distinct contrast in the influence of a cell's microenvironment on cell stiffness. Normal mammary epithelial cells inside a monolayer are stiffer than peripheral cells, which are stiffer than isolated cells. However, the microenvironment had a slight, opposite effect on tumorigenic and little effect on immortal and metastatic cell stiffness. Thus, the stiffness of cancer cells is less sensitive to the microenvironment than normal cells. Our results show that the mechanical properties of a cell can depend on cancer progression and microenvironment (cell–cell interactions).

Many cell types, including neurons, astrocytes and other cells of the central nervous system, respond to changes in the extracellular matrix or substrate viscoelasticity, and increased tissue stiffness is a hallmark of several disease states, including fibrosis and some types of cancers. Whether the malignant tissue in brain, an organ that lacks the protein-based filamentous extracellular matrix of other organs, exhibits the same macroscopic stiffening characteristic of breast, colon, pancreatic and other tumors is not known. In this study we show that glioma cells, like normal astrocytes, respond strongly in vitro to substrate stiffness in the range of 100 to 2000 Pa, but that macroscopic (mm to cm) tissue samples isolated from human glioma tumors have elastic moduli in the order of 200 Pa that are indistinguishable from those of normal brain. However, both normal brain and glioma tissues increase their shear elastic moduli under modest uniaxial compression, and glioma tissue stiffens more strongly under compression than normal brain. These findings suggest that local tissue stiffness has the potential to alter glial cell function, and that stiffness changes in brain tumors might arise not from increased deposition or crosslinking of the collagen-rich extracellular matrix, but from pressure gradients that form within the tumors in vivo.

Presurgical, non-invasive methods of differentiating brain tumors have remained unsatisfactory even for specialized academic hospitals. Despite major advances in clinical and neuroradiological diagnostic techniques, the majority of neurooncology patients still need to undergo a brain biopsy for diagnosis. Recent single cell experiments suggested that biomechanical cell properties might be very sensitive in detecting cellular malignancy. Accordingly, we investigated magnetic resonance elastography (MRE) as an investigative tool for the clinical routine diagnostic work-up of intracranial neoplasm. In order to obtain sufficient spatial resolution for the biomechanical characterization of intracranial tumors, we modified a recently introduced least-squares solution of the stationary wave equation, facilitating stable solutions of the magnitude |G*| and the phase angle φ of the complex shear modulus G*. MRE was added to a routine diagnostic or presurgical neuroradiological magnetic resonance imaging work-up in 16 prospective patients and it was well tolerated in all cases. Our preliminary tumor MRE data revealed alterations in viscoelastic constants, e.g. a loss of stiffness in malignancies compared to healthy reference tissue, or benign variants. Based on larger studies on selected tumor entities to establish threshold and reference values for future diagnostic purposes, MRE may thus provide a predictive marker for tumor malignancy and thereby contribute to an early non-invasive clinical assessment of suspicious cerebral lesions.

Interfaces between stratified epithelia and their supporting stromas commonly exhibit irregular shapes. Undulations are particularly pronounced in dysplastic tissues and typically evolve into long, finger-like protrusions in carcinomas. In previous work (Basan et al 2011 Phys. Rev. Lett. 106 158101), we demonstrated that an instability arising from viscous shear stresses caused by the constant flow due to cell turnover in the epithelium could drive this phenomenon. While interfacial tension between the two tissues as well as mechanical resistance of the stroma tend to maintain a flat interface, an instability occurs for sufficiently large viscosity, cell-division rate and thickness of the dividing region in the epithelium. Here, extensions of this work are presented, where cell division in the epithelium is coupled to the local concentration of nutrients or growth factors diffusing from the stroma. This enhances the instability by a mechanism similar to that of the Mullins–Sekerka instability in single-diffusion processes of crystal growth. We furthermore present the instability for the generalized case of a viscoelastic stroma.

Heterogeneities in the perfusion of solid tumors prevent optimal delivery of nanotherapeutics. Clinical imaging protocols for obtaining patient-specific data have proven difficult to implement. It is challenging to determine which perfusion features hold greater prognostic value and to relate measurements to vessel structure and function. With the advent of systemically administered nanotherapeutics whose delivery is dependent on overcoming diffusive and convective barriers to transport, such knowledge is increasingly important. We describe a framework for the automated evaluation of vascular perfusion curves measured at the single vessel level. Primary tumor fragments, collected from triple-negative breast cancer patients and grown as xenografts in mice, were injected with fluorescence contrast and monitored using intravital microscopy. The time to arterial peak and venous delay, two features whose probability distributions were measured directly from time-series curves, were analyzed using a fuzzy c-mean supervised classifier in order to rank individual tumors according to their perfusion characteristics. The resulting rankings correlated inversely with experimental nanoparticle accumulation measurements, enabling the modeling of nanotherapeutics delivery without requiring any underlying assumptions about tissue structure or function, or heterogeneities contained therein. With additional calibration, these methodologies may enable the investigation of nanotherapeutics delivery strategies in a variety of tumor models.

Glioblastomas are highly diffuse, malignant tumors that have so far evaded clinical treatment. The strongly invasive behavior of cells in these tumors makes them very resistant to treatment, and for this reason both experimental and theoretical efforts have been directed toward understanding the spatiotemporal pattern of tumor spreading. Although usual models assume a standard diffusion behavior, recent experiments with cell cultures indicate that cells tend to move in directions close to that of glioblastoma invasion, thus indicating that a biased random walk model may be much more appropriate. Here we show analytically that, for realistic parameter values, the speeds predicted by biased dispersal are consistent with experimentally measured data. We also find that models beyond reaction–diffusion–advection equations are necessary to capture this substantial effect of biased dispersal on glioblastoma spread.

Intermediate filaments play a key role in cell mechanics, providing cells with compliance to small deformations and reinforcing them when large forces are applied. Here, we present a study of networks of keratin intermediate filaments in living cells under the influence of external forces. We expose the cells to controlled shear forces applied by microflow and investigate the response of the keratin network in situ. Our results show that bundle dynamics are reduced upon the application of shear flow. It is likely that cytoskeletal cross-talk is involved in this shear stress response via actin–keratin coupling.

We present here the first evidence of mechanical penetration by a metastatic cancer cell. During metastasis, the invasive cancer-cell penetrates tissue and extracellular matrix, changes shape and applies force. These applied forces, in turn, depend on substrate stiffness and degradability. The initial stage of metastatic penetration comprises substrate indentation, which, however, has not yet been studied. Hence, we evaluate the evolution of indentation, focusing on differences relating to the metastatic potential (MP) of the cells and substrate stiffness. We found that metastatic cells attain a mushroom-like morphology and then, over several hours, repeatedly indent the substrate in a manner suggestive of a special role for the nucleus. Cells with higher MP have previously been shown to be softer internally and externally than those with lower MP yet, paradoxically, applied stronger forces. Cells of higher MP develop stronger forces on gels stiff enough to provide grip handles yet soft enough to indent, whereas benign cells did not indent substrates at all. These findings provide insight into the central role of physical forces in the initial stages of metastatic penetration and reveal new targets for treatment.

Although understanding the collective migration of cells, such as that seen in epithelial sheets, is essential for understanding diseases such as metastatic cancer, this motion is not yet as well characterized as individual cell migration. Here we adapt quantitative metrics used to characterize the flow and deformation of soft matter to contrast different types of motion within a migrating sheet of cells. Using a finite-time Lyapunov exponent (FTLE) analysis, we find that—in spite of large fluctuations—the flow field of an epithelial cell sheet is not chaotic. Stretching of a sheet of cells (i.e. positive FTLE) is localized at the leading edge of migration and increases when the cells are more highly stimulated. By decomposing the motion of the cells into affine and non-affine components using the metric $D_{\min }^2$, we quantify local plastic rearrangements and describe the motion of a group of cells in a novel way. We find an increase in plastic rearrangements with increasing cell densities, whereas inanimate systems tend to exhibit less non-affine rearrangements with increasing density.

We present a mathematical model for the protrusion of lamellipodia in motile cells. The model lamellipodium consists of a viscoelastic actin gel in the bulk and a dynamic boundary layer of newly polymerized filaments at the leading edge called the semiflexible region (SR). The density of filaments in the SR can increase due to nucleation of new filaments and decrease due to capping and severing of existing filaments. Following on from previous publications, we present important approximations that make the model feasible and accessible to fast computational analysis. It reveals that there are three qualitatively different parameter regimes: a stable, stationarily protruding lamellipodium; a stable lamellipodium showing oscillatory motion of the leading edge; and zero filament density and no stable lamellipodium. Hence, the model defines criteria for the existence of lamellipodia and the ability of cells to move effectively, and we discuss which parameter changes can induce transitions between the different states. Furthermore, stable lamellipodia have to be able to exert and withstand substantial forces. We can fit the experimentally measured dynamic force–velocity relation that describes how cells can adapt to increasing external forces when encountering an obstacle in their environment during motion. Moreover, we predict a different stationary force–velocity relation that should apply if cells experience a constant force, e.g. exerted by the surrounding tissue.

Cancer metastasis, the process of cancer cell migration from a primary to distal location, typically leads to a poor patient prognosis. Hematogenous metastasis is initiated by intravasation of circulating tumor cells (CTCs) into the bloodstream, which are then believed to adhere to the luminal surface of the endothelium and extravasate into distal locations. Apoptotic agents such as tumor necrosis factor apoptosis-inducing ligand (TRAIL), whether in soluble ligand form or expressed on the surface of natural killer cells, have shown promise in treating CTCs to reduce the probability of metastasis. The role of hemodynamic shear forces in altering the cancer cell response to apoptotic agents has not been previously investigated. Here, we report that human colon cancer COLO 205 and prostate cancer PC-3 cells exposed to a uniform fluid shear stress in a cone-and-plate viscometer become sensitized to TRAIL-induced apoptosis. Shear-induced sensitization directly correlates with the application of fluid shear stress, and TRAIL-induced apoptosis increases in a fluid shear stress force- and time-dependent manner. In contrast, TRAIL-induced necrosis is not affected by the application fluid shear stress. Interestingly, fluid shear stress does not sensitize cancer cells to apoptosis when treated with doxorubicin, which also induces apoptosis in cancer cells. Caspase inhibition experiments reveal that shear stress-induced sensitization to TRAIL occurs via caspase-dependent apoptosis. These results suggest that physiological fluid shear forces can modulate receptor-mediated apoptosis of cancer cells in the presence of apoptotic agents.

Several mathematical formulations have analyzed the time-dependent behavior of a tumor mass. However, most of these propose simplifications that compromise the physical soundness of the model. Here, multiphase porous media mechanics is extended to model tumor evolution, using governing equations obtained via the thermodynamically constrained averaging theory. A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TCs), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HCs); and an interstitial fluid for the transport of nutrients. The equations are solved by a finite element method to predict the growth rate of the tumor mass as a function of the initial tumor-to-healthy cell density ratio, nutrient concentration, mechanical strain, cell adhesion and geometry. Results are shown for three cases of practical biological interest such as multicellular tumor spheroids (MTSs) and tumor cords. First, the model is validated by experimental data for time-dependent growth of an MTS in a culture medium. The tumor growth pattern follows a biphasic behavior: initially, the rapidly growing TCs tend to saturate the volume available without any significant increase in overall tumor size; then, a classical Gompertzian pattern is observed for the MTS radius variation with time. A core with necrotic cells appears for tumor sizes larger than 150 μm, surrounded by a shell of viable TCs whose thickness stays almost constant with time. A formula to estimate the size of the necrotic core is proposed. In the second case, the MTS is confined within a healthy tissue. The growth rate is reduced, as compared to the first case—mostly due to the relative adhesion of the TCs and HCs to the ECM, and the less favorable transport of nutrients. In particular, for HCs adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas TC infiltration is predicted for the opposite condition. Interestingly, the infiltration potential of the tumor mass is mostly driven by the relative cell adhesion to the ECM. In the third case, a tumor cord model is analyzed where the malignant cells grow around microvessels in a three-dimensional geometry. It is shown that TCs tend to migrate among adjacent vessels seeking new oxygen and nutrients. This model can predict and optimize the efficacy of anticancer therapeutic strategies. It can be further developed to answer questions on tumor biophysics, related to the effects of ECM stiffness and cell adhesion on TC proliferation.

The process of cancer cell invasion through the extracellular matrix (ECM) of connective tissue plays a prominent role in tumor progression and is based fundamentally on biomechanics. Cancer cell invasion usually requires cell adhesion to the ECM through the cell-matrix adhesion receptors integrins. The expression of the αvβ3 integrin is increased in several tumor types and is consistently associated with increased metastasis formation in patients. The hypothesis was that the αvβ3 integrin expression increases the invasiveness of cancer cells through increased cellular stiffness, and increased cytoskeletal remodeling dynamics. Here, the invasion of cancer cells with different αvβ3 integrin expression levels into dense three-dimensional (3D) ECMs has been studied. Using a cell sorter, two subcell lines expressing either high or low amounts of αvβ3 integrins (αvβ3^high or αvβ3^low cells, respectively) have been isolated from parental MDA-MB-231 breast cancer cells. αvβ3^high cells showed a threefold increased cell invasion compared to αvβ3^low cells. Similar results were obtained for A375 melanoma, 786-O kidney and T24 bladder carcinoma cells, and cells in which the β3 integrin subunit was knocked down using specific siRNA. To investigate whether contractile forces are essential for αvβ3 integrin-mediated increased cellular stiffness and subsequently enhanced cancer cell invasion, invasion assays were performed in the presence of myosin light chain kinase inhibitor ML-7 and Rho kinase inhibitor Y27632. Indeed, cancer cell invasiveness was reduced after addition of ML-7 and Y27632 in αvβ3^high cells but not in αvβ3^low cells. Moreover, after addition of the contractility enhancer calyculin A, an increase in pre-stress in αvβ3^low cells was observed, which enhanced cellular invasiveness. In addition, inhibition of the Src kinase, STAT3 or Rac1 strongly reduced the invasiveness of αvβ3^high cells, whereas the invasiveness of β3 specific knock-down cells and αvβ3^low cells was not altered. In summary, these results suggest that the αvβ3 integrin enhances cancer cell invasion through increased cellular stiffness and enhanced cytoskeletal remodeling dynamics, which enables the cells to generate and transmit contractile forces to overcome the steric hindrance of 3D ECMs.

Collective cell migration is an important feature of wound healing, as well as embryonic and tumor development. The origin of collective cell migration is mainly intercellular interactions through effects such as a line tension preventing cells from detaching from the boundary. In contrast, in this study, we show for the first time that the formation of a constant cell front of a monolayer can also be maintained by the dynamics of the underlying migrating single cells. Ballistic motion enables the maintenance of the integrity of the sheet, while a slowed down dynamics and glass-like behavior cause jamming of cells at the front when two monolayers—even of the same cell type—meet. By employing a velocity autocorrelation function to investigate the cell dynamics in detail, we found a compressed exponential decay as described by the Kohlrausch–William–Watts function of the form $C(\delta x)_{t} \sim \exp {(-(x/x_{0}(t))^{\beta (t)})}$, with 1.5 ⩽ β(t) ⩽ 1.8. This clearly shows that although migrating cells are an active, non-equilibrium system, the cell monolayer behaves in a glass-like way, which requires jamming as a part of intercellular interactions. Since it is the dynamics which determine the integrity of the cell sheet and its front for weakly interacting cells, it becomes evident why changes of the migratory behavior during epithelial to mesenchymal transition can result in the escape of single cells and metastasis.

In this paper, we explore how potential biomechanical influences on cell cycle entrance and cell migration affect the growth dynamics of cell populations. We consider cell populations growing in free, granular and tissue-like environments using a mathematical single-cell-based model. In a free environment we study the effect of pushing movements triggered by proliferation versus active pulling movements of cells stretching cell–cell contacts on the multi-cellular kinetics and the cell population morphotype. By growing cell clones embedded in agarose gel or cells of another type, one can mimic aspects of embedding tissues. We perform simulation studies of cell clones expanding in an environment of granular objects and of chemically inert cells. In certain parameter ranges, we find the formation of invasive fingers reminiscent of viscous fingering. Since the simulation studies are highly computation-time consuming, we mainly study one-cell-thick monolayers and show that for selected parameter settings the results also hold for multi-cellular spheroids. Finally, we compare our model to the experimentally observed growth dynamics of multi-cellular spheroids in agarose gel.

In most instances, tumors have to push their surroundings in order to grow. Thus, during their development, tumors must be able to both exert and sustain mechanical stresses. Using a novel experimental procedure, we study quantitatively the effect of an applied mechanical stress on the long-term growth of a spherical cell aggregate. Our results indicate the possibility to modulate tumor growth depending on the applied pressure. Moreover, we demonstrate quantitatively that the cells located in the core of the spheroid display a different response to stress than those in the periphery. We compare the results to a simple numerical model developed for describing the role of mechanics in cancer progression.

Cancer results from a sequence of genetic and epigenetic changes that lead to a variety of abnormal phenotypes including increased proliferation and survival of somatic cells and thus to a selective advantage of pre-cancerous cells. The notion of cancer progression as an evolutionary process has been attracting increasing interest in recent years. A great deal of effort has been made to better understand and predict the progression to cancer using mathematical models; these mostly consider the evolution of a well-mixed cell population, even though pre-cancerous cells often evolve in highly structured epithelial tissues. In this study, we propose a novel model of cancer progression that considers a spatially structured cell population where clones expand via adaptive waves. This model is used to assess two different paradigms of asexual evolution that have been suggested to delineate the process of cancer progression. The standard scenario of periodic selection assumes that driver mutations are accumulated strictly sequentially over time. However, when the mutation supply is sufficiently high, clones may arise simultaneously on distinct genetic backgrounds, and clonal adaptation waves interfere with each other. We find that in the presence of clonal interference, spatial structure increases the waiting time for cancer, leads to a patchwork structure of non-uniformly sized clones and decreases the survival probability of virtually neutral (passenger) mutations, and that genetic distance begins to increase over a characteristic length scale L_c. These characteristic features of clonal interference may help us to predict the onset of cancers with pronounced spatial structure and to interpret spatially sampled genetic data obtained from biopsies. Our estimates suggest that clonal interference likely occurs in the progression of colon cancer and possibly other cancers where spatial structure matters.

Clinical diagnosis of skin cancers is based on several morphological criteria, among which is the presence of microstructures (e.g. dots and nests) sparsely distributed within the tumour lesion. In this study, we demonstrate that these patterns might originate from a phase separation process. In the absence of cellular proliferation, in fact, a binary mixture model, which is used to represent the mechanical behaviour of skin cancers, contains a cell–cell adhesion parameter that leads to a governing equation of the Cahn–Hilliard type. Taking into account a reaction–diffusion coupling between nutrient consumption and cellular proliferation, we show, with both analytical and numerical investigations, that two-phase models may undergo a spinodal decomposition even when considering mass exchanges between the phases. The cell–nutrient interaction defines a typical diffusive length in the problem, which is found to control the saturation of a growing separated domain, thus stabilizing the microstructural pattern. The distribution and evolution of such emerging cluster morphologies, as predicted by our model, are successfully compared to the clinical observation of microstructural patterns in tumour lesions.