Modelling the early growth of ductal carcinoma in situ of the breast

The growth of a tumour in a rigid walled cylindrical duct is examined in order to model the initial stages of tumour cell expansion in ductal carcinoma in situ (DCIS) of the breast. A nutrient-limited growth model is formulated, in which cell movement is described by a Stokes flow constitutive relation. The effects on the shape of the tumour boundary of the material properties (i.e. the viscosity) and the extent to which the cells adhere to the duct wall are studied using numerical and asymptotic methods. It is shown how stable, non-planar, interface configurations result and that, during these initial stages, before the duct wall has been breached, few cells die and a nutrient-rich model is usually sufficient to capture the behaviour. Finally, we discuss the relevance of this approach to DCIS and suggest possible avenues for further work. Send offprint requests to:S.J. Franks at Centre of Mathematical Medicine.     

An anisotropic-viscoplastic model of plant cell morphogenesis by tip growth

Plant cell morphogenesis depends critically on two processes: the deposition of new wall material at the cell surface and the mechanical deformation of this material by the stresses resulting from the cell's turgor pressure. We developed a model of plant cell morphogenesis that is a first attempt at integrating these two processes. The model is based on the theories of thin shells and anisotropic viscoplasticity. It includes three sets of equations that give the connection between wall stresses, wall strains and cell geometry. We present an algorithm to solve these equations numerically. Application of this simulation approach to the morphogenesis of tip-growing cells illustrates how the viscoplastic properties of the cell wall affect the shape of the cell at steady state. The same simulation approach was also used to reproduce morphogenetic transients such as the initiation of tip growth and other non-steady changes in cell shape. Finally, we show that the mechanical anisotropy built into the model is required to account for observed patterns of wall expansion in plant cells.     

Modelling Circulating Tumour Cells for Personalised Survival Prediction in Metastatic Breast Cancer

Ductal carcinoma is one of the most common cancers among women, and the main cause of death is the formation of metastases. The development of metastases is caused by cancer cells that migrate from the primary tumour site (the mammary duct) through the blood vessels and extravasating they initiate metastasis. Here, we propose a multi-compartment model which mimics the dynamics of tumoural cells in the mammary duct, in the circulatory system and in the bone. Through a branching process model, we describe the relation between the survival times and the four markers mainly involved in metastatic breast cancer (EPCAM, CD47, CD44 and MET). In particular, the model takes into account the gene expression profile of circulating tumour cells to predict personalised survival probability. We also include the administration of drugs as bisphosphonates, which reduce the formation of circulating tumour cells and their survival in the blood vessels, in order to analyse the dynamic changes induced by the therapy.We analyse the effects of circulating tumour cells on the progression of the disease providing a quantitative measure of the cell driver mutations needed for invading the bone tissue. Our model allows to design intervention scenarios that alter the patient-specific survival probability by modifying the populations of circulating tumour cells and it could be extended to other cancer metastasis dynamics.     

Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation

The pulsatile flow in an anatomically realistic compliant human carotid bifurcation was simulated numerically. Pressure and mass flow waveforms in the carotid arteries were obtained from an individual subject using non-invasive techniques. The geometry of the computational model was reconstructed from magnetic resonance angiograms. Maps of time-average wall shear stress, contours of velocity in the flow field as well as wall movement and tensile stress on the arterial wall are all presented. Inconsistent with previous findings from idealised geometry models, flow in the carotid sinus is dominated by a strong helical flow accompanied by a single secondary vortex motion. This type of flow is induced primarily by the asymmetry and curvature of the in vivo geometry. Flow simulations have been carried out under the rigid wall assumption and for the compliant wall, respectively. Comparison of the results demonstrates the quantitative influence of the vessel wall motion. Generally there is a reduction in the magnitude of wall shear stress, with its degree depending on location and phase of the cardiac cycle. The region of slow or reversed flow was greater, in both spatial and temporal terms in the compliant model, but the global characteristics of the flow and stress patterns remain unchanged. The analysis of mechanical stresses on the vessel surface shows a complicated stress field. Stress concentration occurs at both the anterior and posterior aspects of the proximal internal bulb. These are also regions of low wall shear stress. The comparison of computed and measured wall movement generally shows good agreement.     

Bad seeds produce bad crops: A single stage-process of breast carcinogenesis

It is a commonly held belief that human breast carcinogenesis is a multi-stage-process, and that progression from pre-invasion to invasion is triggered by overproduction of proteolytic enzymes that cause degradation of the basement membrane. These assumptions are hard to reconcile with two critical facts: (1) a subset of normal appearing tissues share a similar immunohistochemical or genetic profile with malignant counterparts and (2) a vast majority of in situ tumors express high levels of proteolytic enzymes, while only 10–30% of untreated in situ tumors progress to invasion. These facts argue that alternative pathways may play more direct roles in tumor progression and invasion in some cases.Loss of the myoepithelial (ME) cell layer is the most distinct sign associated with invasion. Our recent studies revealed that a subset of normal appearing duct clusters harbored a high frequency of focal ME cell layer disruptions (FMCLD). The residual ME cells of these duct clusters had significantly reduced expression of tumor suppressors, elevated rates of apoptosis and infiltration of immunoreactive cells, and the epithelial cell clusters overlying these disruptions had a significantly elevated frequency of tumor-associated phenotypes.Based on these and other findings, we have proposed that these morphologically normal appearing duct clusters are derived from genetically damaged stem cells, and could progress directly to invasion or metastasis through two pathways: (1) the entire ME basal cell layer is gradually degenerated or disappeared, allowing direct physical contact of epithelial cells with stromal and immunoreactive cells, which induce invasive properties without morphological alterations and (2) ER negative cell clusters overlying FMCLD retain the potential for multi-lineage differentiation that continuously proliferate and provide new cells and their own vascular structures for invasion and metastasis.     

Mechanical Control of Bacterial Cell Shape

In bacteria, cytoskeletal filament bundles such as MreB control the cell morphology and determine whether the cell takes on a spherical or a rod-like shape. Here we use a theoretical model to describe the interplay of cell wall growth, mechanics, and cytoskeletal filaments in shaping the bacterial cell. We predict that growing cells without MreB exhibit an instability that favors rounded cells. MreB can mechanically reinforce the cell wall and prevent the onset of instability. We propose that the overall bacterial shape is determined by a dynamic turnover of cell wall material that is controlled by mechanical stresses in the wall. The model affirms that morphological transformations with and without MreB are reversible, and quantitatively describes the growth of irregular shapes and cells undergoing division. The theory also suggests a unique coupling between mechanics and chemistry that can control organismal shapes in general.     

Flow behaviour in an asymmetric compliant experimental model for abdominal aortic aneurysm

An experimental study was carried out on asymmetrical abdominal aortic aneurysm (AAA) to analyse the physiological flows involved. Velocity measurements were performed using particle image velocimetry. Resting and exercise flow rates were investigated in models with rigid and compliant walls to assess the parameters affecting the flow behaviour. The secondary flow patterns, and especially the evolution of the vortices within the AAA, were found to be highly dependent on both the flow waveforms and the wall behaviour. Vortices impacts on the distal walls of the AAA occur in the compliant model and can increase the local pressure on the AAA walls and thus increase the wall stresses; AAA wall stresses are one of the most important factors contributing to ruptured aneurysm.     

Inspiratory flow in the nose: a model coupling flow and vasoerectile tissue distensibility.

We have developed a discrete multisegmental model describing the coupling between inspiratory flow and nasal wall distensibility. This model is composed of 14 individualized compliant elements, each with its own relationship between cross-sectional area and transmural pressure. Conceptually, this model is based on flow limitation induced by the narrowing of duct due to collapsing pressure. For a given inspiratory pressure and for a given compliance distribution, this model predicts the area profile and inspiratory flow. Acoustic rhinometry and posterior rhinomanometry were used to determine the initial geometric area and mechanical characteristics of each element. The proposed model, used under steady-state conditions, is able to simulate the pressure-flow relationship observed in vivo under normal conditions (4 subjects) and under pathological conditions (4 vasomotor rhinitis and 3 valve syndrome subjects). Our results suggest that nasal wall compliance is an essential parameter to understand the nasal inspiratory flow limitation phenomenon and the associated increase of resistance that is well known to physiologists. By predicting the functional pressure-flow relationship, this model could be a useful tool for the clinician to evaluate the potential effects of treatments.     

Mathematical modelling of radiotherapy strategies for early breast cancer

Targeted intraoperative radiotherapy (Targit) is a new concept of partial breast irradiation where single fraction radiotherapy is delivered directly to the tumour bed. Apart from logistic advantages, this strategy minimizes the risk of missing the tumour bed and avoids delay between surgery and radiotherapy. It is presently being compared with the standard fractionated external beam radiotherapy (EBRT) in randomized trials. In this paper we present a mathematical model for the growth and invasion of a solid tumour into a domain of tissue (in this case breast tissue), and then a model for surgery and radiation treatment of this tumour. We use the established linear-quadratic (LQ) model to compute the survival probabilities for both tumour cells and irradiated breast tissue and then simulate the effects of conventional EBRT and Targit. True local recurrence of the tumour could arise either from stray tumour cells, or the tumour bed that harbours morphologically normal cells having a predisposition to genetic changes, such as a loss of heterozygosity (LOH) in genes that are crucial for tumourigenesis, e.g. tumour suppressor genes (TSGs). Our mathematical model predicts that the single high dose of radiotherapy delivered by Targit would result in eliminating all these sources of recurrence, whereas the fractionated EBRT would eliminate stray tumour cells, but allow (by virtue of its very schedule) the cells with LOH in TSGs or cell-cycle checkpoint genes to pass on low-dose radiation-induced DNA damage and consequently mutations that may favour the development of a new tumour. The mathematical model presented here is an initial attempt to model a biologically complex phenomenon that has until now received little attention in the literature and provides a 'proof of principle' that it is possible to produce clinically testable hypotheses on the effects of different approaches of radiotherapy for breast cancer.     

The common bile duct ligation in rat: a relevant in vivo model to study the role of mechanical stress on cell and matrix behaviour

Common bile duct ligation leads to bile accumulation and liver fibrosis. In this model, little attention has been dedicated to the modification of the common bile duct. We have studied by histochemistry and immunohistochemistry, 3 and 5 days after ligation, the connective tissue modifications of the common bile duct wall. After bile duct ligation, compared with normal bile duct, a strong increase of the bile duct diameter, due to bile stasis, and a thickness of the bile duct wall were observed; numerous myofibroblasts expressing α-smooth muscle actin appeared in parallel with the detection of many proliferating connective tissue cells. These myofibroblasts secreted very early high amount of elastic fibre components, elastin and fibrillin-1. Elastic fibre increase was also observed close to the epithelial cell layer. Procollagen type III deposition was also induced 3 days after ligation but decreased thereafter, underlining that myofibroblasts modify their synthesis of extracellular matrix components to comply with the request. We show here that common bile duct ligation represents an invaluable model to study myofibroblastic differentiation and extracellular matrix adaptation produced by an acute mechanical stress.     

Intercellular adhesion and cancer invasion: a discrete simulation using the extended Potts model

We develop a discrete model of malignant invasion using a thermodynamic argument. An extension of the Potts model is used to simulate a population of malignant cells experiencing interactions due to both homotypic and heterotypic adhesion while also secreting proteolytic enzymes and experiencing a haptotactic gradient. In this way we investigate the influence of changes in cell-cell adhesion on the invasion process. We demonstrate that the morphology of the invading front is influenced by changes in the adhesiveness parameters, and detail how the invasiveness of the tumour is related to adhesion. We show that cell-cell adhesion has less of an influence on invasion compared with cell-medium adhesion, and that increases in both proteolytic enzyme secretion rate and the coefficient of haptotaxis act in synergy to promote invasion. We extend the simulation by including proliferation, and, following experimental evidence, develop an algorithm for cell division in which the mitotic rate is explicitly related to changes in the relative magnitudes of homotypic and heterotypic adhesiveness. We show that although an increased proliferation rate usually results in an increased depth of invasion into the extracellular matrix, it does not invariably do so, and may, indeed, cause invasiveness to be reduced.     

Computational comparison of aortic root stresses in presence of stentless and stented aortic valve bio-prostheses

We provide a computational comparison of the performance of stentless and stented aortic prostheses, in terms of aortic root displacements and internal stresses. To this aim, we consider three real patients; for each of them, we draw the two prostheses configurations, which are characterized by different mechanical properties and we also consider the native configuration. For each of these scenarios, we solve the fluid–structure interaction problem arising between blood and aortic root, through Finite Elements. In particular, the Arbitrary Lagrangian–Eulerian formulation is used for the numerical solution of the fluid-dynamic equations and a hyperelastic material model is adopted to predict the mechanical response of the aortic wall and the two prostheses. The computational results are analyzed in terms of aortic flow, internal wall stresses and aortic wall/prosthesis displacements; a quantitative comparison of the mechanical behavior of the three scenarios is reported. The numerical results highlight a good agreement between stentless and native displacements and internal wall stresses, whereas higher/non-physiological stresses are found for the stented case.     

Secretion of phosphoglycerate kinase from tumour cells is controlled by oxygen-sensing hydroxylases

Solid tumour cells employ glycolytic enzymes including phosphoglycerate kinase (PGK) to make ATP when their supply of oxygen is limiting. PGK is also secreted by tumour cells and facilitates cleavage of disulfide bonds in plasmin, which triggers proteolytic release of the angiogenesis inhibitor, angiostatin. Although PGK production by tumour cells was enhanced by hypoxia, its secretion was inhibited. Inhibition of secretion correlated with decrease in angiostatin formation by the tumour cells. In contrast, hypoxia did not inhibit the secretion of the angiogenesis activator, vascular endothelial cell growth factor (VEGF). PGK secretion was reversed by normoxia and was under control of the oxygen-sensing protein hydroxylases, as inhibitors of this class of enzymes mimicked the effect of hypoxia on PGK secretion. Direct hydroxylation of PGK was not the mechanism by which the protein hydroxylases controlled its secretion. These findings show that production and secretion of PGK are regulated separately and indicate that oxygen and the protein hydroxylases can control not only gene expression but also protein secretion.     

AFM-based dual nano-mechanical phenotypes for cancer metastasis

An enhanced mechanical compliance is considered to be a mechanical indicator for metastatic cancer cells. Our study using atomic force microscopy (AFM) revealed that breast cancer cells agreed well with this hypothesis. However, prostate cancer cells displayed a reverse correlation; less metastatic prostate cancer cells were more mechanically compliant. Two-dimensional AFM force spectroscopy was performed to characterize dual mechanical properties—the cell–substrate adhesion as well as the mechanical compliance. Interestingly, prostate cancer cells displayed a strong positive correlation between the cell–substrate adhesion and metastatic potential. However, there was no clearly observable correlation between the cell–substrate adhesion and the metastatic potential despite variations in mechanical compliance of breast cancer cells. These results suggest that the correlation between the dual mechanical signatures and metastatic potential be uniquely identified for cancer cells originating from different organs. We postulate that this correlation could reveal which step of cancer progression is favorable in terms of physical interaction between cancer cells and micro-environments. We expect that based on the “seed and soil hypothesis”, the identification of the dual mechanical phenotypes, could provide a new insight for understanding how a dominant metastatic site is determined for cancer cells originating from specific organs.     

Improving the Efficiency of Abdominal Aortic Aneurysm Wall Stress Computations

An abdominal aortic aneurysm is a pathological dilation of the abdominal aorta, which carries a high mortality rate if ruptured. The most commonly used surrogate marker of rupture risk is the maximal transverse diameter of the aneurysm. More recent studies suggest that wall stress from models of patient-specific aneurysm geometries extracted, for instance, from computed tomography images may be a more accurate predictor of rupture risk and an important factor in AAA size progression. However, quantification of wall stress is typically computationally intensive and time-consuming, mainly due to the nonlinear mechanical behavior of the abdominal aortic aneurysm walls. These difficulties have limited the potential of computational models in clinical practice. To facilitate computation of wall stresses, we propose to use a linear approach that ensures equilibrium of wall stresses in the aneurysms. This proposed linear model approach is easy to implement and eliminates the burden of nonlinear computations. To assess the accuracy of our proposed approach to compute wall stresses, results from idealized and patient-specific model simulations were compared to those obtained using conventional approaches and to those of a hypothetical, reference abdominal aortic aneurysm model. For the reference model, wall mechanical properties and the initial unloaded and unstressed configuration were assumed to be known, and the resulting wall stresses were used as reference for comparison. Our proposed linear approach accurately approximates wall stresses for varying model geometries and wall material properties. Our findings suggest that the proposed linear approach could be used as an effective, efficient, easy-to-use clinical tool to estimate patient-specific wall stresses.     

Residual strains in conduit arteries

Residual strains and stresses are those that exist in a body when all external loads are removed. Residual strains in arteries can be characterized by the opening angle of the sector-like cross-section which arises when an unloaded ring segment is radially cut. A review of experimental methods for measuring residual strains and the main results about the variation of the opening angle with arterial localization, age, smooth muscle activity, mechanical environment and certain vascular pathologies are presented and discussed. It is shown that, in addition to their well-established ability to homogenize the stress field in the arterial wall, residual strains make arteries more compliant and thereby improve their performance as elastic reservoirs and ensure more effective local control of the arterial lumen by smooth muscle cells. Finally, evidence that, in some cases, residual strains remain in arteries even after they have been cut radially is discussed.     

Secretory and endo/exocytic trafficking in invadopodia formation: The MT1-MMP paradigm

Invadopodia are actin-rich, adhesive protrusions that extend into and remodel the extracellular matrix. They are associated with high levels of pericellular proteolysis and correlate with the invasive capacity of a variety of tumour cells. Invadopodia have, thus, been proposed to recapitulate key events of the metastatic process. Although our understanding of the patho-physiology of invadopodia is still in its infancy, the molecular components and signalling pathways leading to their formation have received increasing attention. Recent studies have revealed that diverse membrane polarized secretory and endo/exocytic trafficking pathways converge at these structures for the delivery, in a temporally controlled and spatially confined manner, of key proteolytic enzymes. Here, we will focus our attention on MT1-MMP, a paradigmatic metalloprotease that is primarily responsible for the proteolytic activity of invadopodia. We propose that the biosynthetic/secretory pathway might be critical for the polarized delivery of MT1-MMP to invadopodia that form as “default response” whenever cells have to deal with extracellular matrix (ECM) of variable composition and stiffness. Conversely, “inducible” endo/exocytic trafficking routes might primarily control the delivery of MT1-MMP to invadopodia when cells need to respond in a fast and transient manner to soluble motogenic factors, rather than the insoluble ECM.