Comparison between spring network models and continuum constitutive laws: application to the large deformation of a capsule in shear flow

Computational analysis of cancer cell adhesion in curved vessels affected by wall shear stress for prediction of metastatic spreading

Introduction: The dynamics of circulating tumor cells (CTCs) within blood vessels play a pivotal role in predicting metastatic spreading of cancer within the body. However, the limited understanding and method to quantitatively investigate the influence of vascular architecture on CTC dynamics hinders our ability to predict metastatic process effectively. To address this limitation, the present study was conducted to investigate the influence of blood vessel tortuosity on the behaviour of CTCs, focusing specifically on establishing methods and examining the role of shear stress in CTC-vessel wall interactions and its subsequent impact on metastasis. Methods: We computationally simulated CTC behaviour under various shear stress conditions induced by vessel tortuosity. Our computational model, based on the lattice Boltzmann method (LBM) and a coarse-grained spectrin-link membrane model, efficiently simulates blood plasma dynamics and CTC deformability. The model incorporates fluid-structure interactions and receptor-ligand interactions crucial for CTC adhesion using the immersed boundary method (IBM). Results: Our findings reveal that uniform shear stress in straight vessels leads to predictable CTC-vessel interactions, whereas in curved vessels, asymmetrical flow patterns and altered shear stress create distinct adhesion dynamics, potentially influencing CTC extravasation. Quantitative analysis shows a 25% decrease in the wall shear stress in low-shear regions and a 58.5% increase in the high-shear region. We observed high-shear regions in curved vessels to be potential sites for increased CTC adhesion and extravasation, facilitated by elevated endothelial expression of adhesion molecules. This phenomenon correlates with the increased number of adhesion bonds, which rises to approximately 40 in high-shear regions, compared to around 12 for straight vessels and approximately 5-6 in low-shear regions. The findings also indicate an optimal cellular stiffness necessary for successful CTC extravasation in curved vessels. Discussion: By the quantitative assessment of the risk of CTC extravasation as a function of vessel tortuosity, our study offers a novel tool for the prediction of metastasis risk to support the development of personalized therapeutic interventions based on individual vascular characteristics and tumor cell properties.

Lattice Boltzmann simulation of deformable fluid-filled bodies: progress and perspectives

With the rapid development of studies involving droplet microfluidics, drug delivery, cell detection, and microparticle synthesis, among others, many scientists have invested significant efforts to model the flow of these fluid-filled bodies. Motivated by the intricate coupling between hydrodynamics and the interactions of fluid-filled bodies, several methods have been developed. The objective of this review is to present a compact foundation of the methods used in the literature in the context of lattice Boltzmann methods. For hydrodynamics, we focus on the lattice Boltzmann method due to its specific ability to treat time- and spatial-dependent boundary conditions and to incorporate new physical models in a computationally efficient way. We split the existing methods into two groups with regard to the interfacial boundary: fluid-structure and fluid-fluid methods. The fluid-structure methods are characterised by the coupling between fluid dynamics and mechanics of the flowing body, often used in applications involving membranes and similar flexible solid boundaries. We further divide fluid-structure-based methods into two subcategories, those which treat the fluid-structure boundary as a continuum medium and those that treat it as a discrete collection of individual springs and particles. Next, we discuss the fluid-fluid methods, particularly useful for the simulations of fluid-fluid interfaces. We focus on models for immiscible droplets and their interaction in a suspending fluid and describe benchmark tests to validate the models for fluid-filled bodies.

Experimental measurement and numerical modeling of deformation behavior of breast cancer cells passing through constricted microfluidic channels

During the multistep process of metastasis, cancer cells encounter various mechanical forces which make them deform drastically. Developing accurate in-silico models, capable of simulating the interactions between the mechanical forces and highly deformable cancer cells, can pave the way for the development of novel diagnostic and predictive methods for metastatic progression. Spring-network models of cancer cell, empowered by our recently proposed identification approach, promises a versatile numerical tool for developing experimentally validated models that can simulate complex interactions at cellular scale. Using this numerical tool, we presented spring-network models of breast cancer cells that can accurately replicate the experimental data of deformation behavior of the cells flowing in a fluidic domain and passing narrow constrictions comparable to microcapillary. First, using high-speed imaging, we experimentally studied the deformability of breast cancer cell lines with varying metastatic potential (MCF-7 (less invasive), SKBR-3 (medium-high invasive), and MDA-MB-231 (highly invasive)) in terms of their entry time to a constricted microfluidic channel. We observed that MDA-MB-231, that has the highest metastatic potential, is the most deformable cell among the three. Then, by focusing on this cell line, experimental measurements were expanded to two more constricted microchannel dimensions. The experimental deformability data in three constricted microchannel sizes for various cell sizes, enabled accurate identification of the unknown parameters of the spring-network model of the breast cancer cell line (MDA-MB-231). Our results show that the identified parameters depend on the cell size, suggesting the need for a systematic procedure for identifying the size-dependent parameters of spring-network models of cells. As the numerical results show, the presented cell models can simulate the entry process of the cell into constricted channels with very good agreements with the measured experimental data.

An analytical model describing the mechanics of erythrocyte membrane wrapping during active invasion of a plasmodium falciparum merozoite

The invasion of a merozoite into an erythrocyte by membrane wrapping is a hallmark of malaria pathogenesis. The invasion involves biomechanical interactions whereby the merozoite exerts actomyosin-based forces to push itself into and through the erythrocyte membrane while concurrently inducing biochemical damage to the erythrocyte membrane. Whereas the biochemical damage process has been investigated, the detailed mechanistic understanding of the invasion mechanics remains limited. Thus, the current study aimed to develop a mathematical model describing the mechanical factors involved in the merozoite invasion into an erythrocyte and explore the invasion mechanics. A shell theory model was developed comprising constitutive, equilibrium and governing equations of the deformable erythrocyte membrane to predict membrane mechanics during the wrapping of an entire non-deformable ellipsoidal merozoite. Predicted parameters include principal erythrocyte membrane deformations and stresses, wrapping and indentation forces, and indentation work. The numerical investigations considered two limits for the erythrocyte membrane deformation during wrapping (4% and 51% areal strain) and erythrocyte membrane phosphorylation (decrease of membrane elastic modulus from 1 to 0.5 kPa). For an intact erythrocyte, the maximum indentation force was 1 and 8.5 pN, and the indentation work was 1.92 × 10^-18 and 1.40 × 10^-17 J for 4% and 51% areal membrane strain. Phosphorylation damage in the erythrocyte membrane reduced the required indentation work by 50% to 0.97 × 10^-18 and 0.70 × 10^-17 J for 4% and 51% areal strain. The current study demonstrated the developed model's feasibility to provide new knowledge on the physical mechanisms of the merozoite invasion process that contribute to the invasion efficiency towards the discovery of new invasion-blocking anti-malaria drugs.

Large Growth Deformations of Thin Tissue Using Solid-Shells

Simulating large scale expansion of thin structures, such as in growing leaves, is challenging. Solid-shells have a number of potential advantages over conventional thin-shell methods, but have thus far only been investigated for small plastic deformation cases. In response, we present a new general-purpose FEM growth framework for handling a wide range of challenging growth scenarios using the solid-shell element. Solid-shells are a middle-ground between traditional volume and thin-shell elements where volumetric characteristics are retained while being treatable as a 2D manifold much like thin-shells. These elements are adaptable to accommodate the many techniques that are required for simulating large and intricate plastic deformations, including morphogen diffusion, plastic embedding, strain-aware adaptive remeshing, and collision handling. We demonstrate the capabilities of growing solid-shells in reproducing buckling, rippling, curling, and collision deformations, relevant towards animating growing leaves, flowers, and other thin structures. Solid-shells are compared side-by-side with thin-shells to examine their bending behavior and runtime performance. The experiments demonstrate that solid-shells are a viable alternative to thin-shells for simulating large and intricate growth deformations.

A systematic approach for developing mechanistic models for realistic simulation of cancer cell motion and deformation

Understanding and predicting metastatic progression and developing novel diagnostic methods can highly benefit from accurate models of the deformability of cancer cells. Spring-based network models of cells can provide a versatile way of integrating deforming cancer cells with other physical and biochemical phenomena, but these models have parameters that need to be accurately identified. In this study we established a systematic method for identifying parameters of spring-network models of cancer cells. We developed a genetic algorithm and coupled it to the fluid-solid interaction model of the cell, immersed in blood plasma or other fluids, to minimize the difference between numerical and experimental data of cell motion and deformation. We used the method to create a validated model for the human lung cancer cell line (H1975), employing existing experimental data of its deformation in a narrow microchannel constriction considering cell-wall friction. Furthermore, using this validated model with accurately identified parameters, we studied the details of motion and deformation of the cancer cell in the microchannel constriction and the effects of flow rates on them. We found that ignoring the viscosity of the cell membrane and the friction between the cell and wall can introduce remarkable errors.

Pushing Droplet Through a Porous Medium

I use a mechanical model of a soft body to study the dynamics of an individual fluid droplet in a random, non-wettable porous medium. The model of droplet relies on the spring–mass system with pressure. I run hundreds of independent simulations. I average droplets trajectories and calculate the averaged tortuosity of the porous domain. Results show that porous media tortuosity increases with decreasing porosity, similar to single-phase fluid study, but the form of this relationship is different.

Transition to stress focusing for locally curved sheets

A rectangular thin elastic sheet is deformed by forcing a contact between two points at the middle of its length. A transition to buckling with stress focusing is reported for the sheets sufficiently narrow with a critical width proportional to the sheet length with an exponent 2/3 in the small thickness limit. Additionally, a spring network model is solved to explore the thick sheet limit and to validate the scaling behavior of the transition in the thin sheet limit. The numerical results reveal that buckling does not exist for the thickest sheets, and a stability criterion is established for the buckling of a curved sheet.

Constitutive Model of Erythrocyte Membranes with Distributions of Spectrin Orientations and Lengths

We present an analytical hyperelastic constitutive model of the red blood cell (erythrocyte) membrane based on recently improved characterizations of density and microscopic structure of its spectrin network from proteomics and cryo-electron tomography. The model includes distributions of both orientations and natural lengths of spectrin and updated copy numbers of proteins. By applying finite deformation to the spectrin network, we obtain the total free energy and stresses in terms of invariants of shear and area deformation. We generalize an expression of the initial shear modulus, which is independent of the number of molecular orientations within the network and also derive a simplified version of the model. We apply the model and its simplified version to analyze micropipette aspiration computationally and analytically and explore the effect of local cytoskeletal density change. We also explore the discrepancies among shear modulus values measured using different experimental techniques reported in the literature. We find that the model exhibits hardening behavior and can explain many of these discrepancies. Moreover, we find that the distribution of natural lengths plays a crucial role in the hardening behavior when the correct copy numbers of proteins are used. The initial shear modulus values we obtain using our current model (5.9-15.6 pN/μm) are close to the early estimates (6-9 pN/μm). This new, to our knowledge, constitutive model establishes a direct connection between the molecular structure of spectrin networks and constitutive laws and also defines a new picture of a much denser spectrin network than assumed in prior studies.

Simulation of circulating tumor cell transport and adhesion in cell suspensions in microfluidic devices

Understanding cell transport and adhesion dynamics under flow is important for many biotransport problems. We investigated the influence of cell size, ligand coating density, micropost size, and intercellular collisions on circulating tumor cell adhesion and transport in microfluidic devices. The cells were modeled as coarse-grained cell membranes and the adhesion was modeled as pairwise interacting potentials, while the fluid was solved using the lattice Boltzmann method. The coupling between the cell and the fluid was achieved through the immersed boundary method. The cell showed transient rolling adhesion in high shear regions and firm adhesion in low shear regions. The adhesive force for rolling cells on a micropost was increasing before the cell reached the crest of the post and then decreasing afterward. The adhesive strength for cells increases with ligand coating density. Cell trajectories in a microfluidic device with a shifted post design were studied as well. At low concentrations, the majority of the cells follow streamlines closely. However, the intercellular collision and collision from red blood cells impacted the cell trajectories. An L 2 norm of | e | was defined to characterize the difference between the cell trajectories and the associated streamlines. It was shown that | e | L 2 increases with micropost sizes and cell concentrations.

Biomimetic surgical mesh to replace fascia with tunable force-displacement

Here we mimic the mechanical properties of native fascia to design surgical mesh for fascia replacement. Despite the widespread acceptance of synthetic materials as tissue scaffolds for pelvic floor disorders, mechanical property mismatch between mesh and adjacent native tissue drives fibrosis and erosion, leading the FDA to remove several surgical meshes from the market. However, autologous tissue does not induce either fibrosis or adjacent tissue erosion, suggesting the potential for biomimetic surgical mesh. In this study, we determined the design rules for mesh that mimics native fascia by mathematically modeling multi-component polymer networks, composed of elastin-like and collagen-like fibers, using a spring-network model. To validate the model, we measured the stress-strain curves of native bovine and nonhuman primate (Macaca mulatta) abdominal fascia in both toe and linear regions. We find that the stiffer collagen-like fibers must remain limp until the elastin-like fibers extend to the initial length of spanning collagen-like fibers under uniaxial tension. Comparing model results to experiment determines the product of fiber volume fraction and elastic modulus, a critical design parameter. Dual fiber mesh with mechanical properties that mimic fascia are feasible. These results have broad application to a wide range of soft tissue replacements including ~200,000 surgeries/year for pelvic floor disorders, because standard-of-care mesh contain only stiffer polymers that behave more like collagen than native tissue.

Mesoscale simulation of soft particles with tunable contact angle in multicomponent fluids

Soft particles at fluid interfaces play an important role in many aspects of our daily life, such as the food industry, paints and coatings, and medical applications. Analytical methods are not capable of describing the emergent effects of the complex dynamics of suspensions of many soft particles, whereas experiments typically either only capture bulk properties or require invasive methods. Computational methods are therefore a great tool to complement experimental work. However, an efficient and versatile numerical method is needed to model dense suspensions of many soft particles. In this article we propose a method to simulate soft particles in a multicomponent fluid, both at and near fluid-fluid interfaces, based on the lattice Boltzmann method, and characterize the error stemming from the fluid-structure coupling for the particle equilibrium shape when adsorbed onto a fluid-fluid interface. Furthermore, we characterize the influence of the preferential contact angle of the particle surface and the particle softness on the vertical displacement of the center of mass relative to the fluid interface. Finally, we demonstrate the capability of our model by simulating a soft capsule adsorbing onto a fluid-fluid interface with a shear flow parallel to the interface, and the covering of a droplet suspended in another fluid by soft particles with different wettability.

Structural modelling of the cardiovascular system

Computational modelling of the cardiovascular system offers much promise, but represents a truly interdisciplinary challenge, requiring knowledge of physiology, mechanics of materials, fluid dynamics and biochemistry. This paper aims to provide a summary of the recent advances in cardiovascular structural modelling, including the numerical methods, main constitutive models and modelling procedures developed to represent cardiovascular structures and pathologies across a broad range of length and timescales; serving as an accessible point of reference to newcomers to the field. The class of so-called hyperelastic materials provides the theoretical foundation for the modelling of how these materials deform under load, and so an overview of these models is provided; comparing classical to application-specific phenomenological models. The physiology is split into components and pathologies of the cardiovascular system and linked back to constitutive modelling developments, identifying current state of the art in modelling procedures from both clinical and engineering sources. Models which have originally been derived for one application and scale are shown to be used for an increasing range and for similar applications. The trend for such approaches is discussed in the context of increasing availability of high performance computing resources, where in some cases computer hardware can impact the choice of modelling approach used.

Numerical simulation of cell squeezing through a micropore by the immersed boundary method

The deformability of cells has been used as a biomarker to detect circulating tumor cells (CTCs) from patient blood sample using microfluidic devices with microscale pores. Successful separations of CTCs from a blood sample requires careful design of the micropore size and applied pressure. This paper presented a parametric study of cell squeezing through micropores with different size and pressure. Different membrane compressibility modulus was used to characterize the deformability of varying cancer cells. Nucleus effect was also considered. It shows that the cell translocation time though the micropore increases with cell membrane compressibility modulus and nucleus stiffness. Particularly, it increases exponentially as the micropore diameter or pressure decreases. The simulation results such as the cell squeezing shape and translocation time agree well with experimental observations. The simulation results suggest that special care should be taken in applying Laplace-Young equation (LYE) to microfluidic design due to the nonuniform stress distribution and membrane bending resistance.

Theory and algorithms to compute Helfrich bending forces: a review

Cell membranes are vital to shield a cell's interior from the environment. At the same time they determine to a large extent the cell's mechanical resistance to external forces. In recent years there has been considerable interest in the accurate computational modeling of such membranes, driven mainly by the amazing variety of shapes that red blood cells and model systems such as vesicles can assume in external flows. Given that the typical height of a membrane is only a few nanometers while the surface of the cell extends over many micrometers, physical modeling approaches mostly consider the interface as a two-dimensional elastic continuum. Here we review recent modeling efforts focusing on one of the computationally most intricate components, namely the membrane's bending resistance. We start with a short background on the most widely used bending model due to Helfrich. While the Helfrich bending energy by itself is an extremely simple model equation, the computation of the resulting forces is far from trivial. At the heart of these difficulties lies the fact that the forces involve second order derivatives of the local surface curvature which by itself is the second derivative of the membrane geometry. We systematically derive and compare the different routes to obtain bending forces from the Helfrich energy, namely the variational approach and the thin-shell theory. While both routes lead to mathematically identical expressions, so-called linear bending models are shown to reproduce only the leading order term while higher orders differ. The main part of the review contains a description of various computational strategies which we classify into three categories: the force, the strong and the weak formulation. We finally give some examples for the application of these strategies in actual simulations.

Flow structures and red blood cell dynamics in arteriole of dilated or constricted cross section

Vessel with 'circular' or 'star-shaped' cross sections are studied, representing respectively dilated or constricted cases where endothelial cells smoothly line or bulge into the lumen. Computational haemodynamics simulations are carried out on idealised periodic arteriole-sized vessels, with red blood cell 'tube' hematocrit value=24%. A further simulation of a single red blood cell serves for comparison purposes. The bulk motion of the red blood cells reproduces well-known effects, including the presence of a cell-free layer and the apparent shear-thinning non-Newtonian rheology. The velocity flow field is analysed in a Lagrangian reference frame, relative to any given red blood cell, hence removing the bulk coaxial motion and highlighting instead the complex secondary flow patterns. An aggregate formation becomes apparent, continuously rearranging and dynamic, brought about by the inter-cellular fluid mechanics interactions and the deformability properties of the cells. The secondary flow field induces a vacillating radial migration of the red blood cells. At different radial locations, the red blood cells express different residence times, orientation and shape. The shear stresses exerted by the flow on the vessel wall are influenced by the motion of red blood cells, despite the presence of the cell-free layer. Spatial (and temporal) variations of wall shear stress patters are observed, especially for the 'circular' vessel. The 'star-shaped' vessel bears considerable stress at the protruding endothelial cell crests, where the stress vectors are coaxially aligned. The bulging endothelial cells hence regularise the transmission of stresses on the vessel wall.

Hemodynamics in the microcirculation and in microfluidics

Hemodynamics in microcirculation is important for hemorheology and several types of circulatory disease. Although hemodynamics research has a long history, the field continues to expand due to recent advancements in numerical and experimental techniques at the micro-and nano-scales. In this paper, we review recent computational and experimental studies of blood flow in microcirculation and microfluidics. We first focus on the computational studies of red blood cell (RBC) dynamics, from the single cellular level to mesoscopic multiple cellular flows, followed by a review of recent computational adhesion models for white blood cells, platelets, and malaria-infected RBCs, in which the cell adhesion to the vascular wall is essential for cellular function. Recent developments in optical microscopy have enabled the observation of flowing blood cells in microfluidics. Experimental particle image velocimetry and particle tracking velocimetry techniques are described in this article. Advancements in micro total analysis system technologies have facilitated flowing cell separation with microfluidic devices, which can be used for biomedical applications, such as a diagnostic tool for breast cancer or large intestinal tumors. In this paper, cell-separation techniques are reviewed for microfluidic devices, emphasizing recent advances and the potential of this fast-evolving research field in the near future.

Simulation of malaria-infected red blood cells in microfluidic channels: Passage and blockage

Malaria-infected red blood cells (iRBCs) become less deformable with the progression of infection and tend to occlude microcapillaries. This process has been investigated in vitro using microfluidic channels. The objective of this paper is to provide a quantitative basis for interpreting the experimental observations of iRBC occlusion of microfluidic channels. Using a particle-based model for the iRBC, we simulate the traverse of iRBCs through a converging microfluidic channel and explore the progressive loss of cell deformability due to three factors: the stiffening of the membrane, the reduction of the cell's surface-volume ratio, and the growing solid parasites inside the cell. When examined individually, each factor tends to hinder the passage of the iRBC and lengthen the transit time. Moreover, at sufficient magnitude, each may lead to obstruction of narrow microfluidic channels. We then integrate the three factors into a series of simulations that mimic the development of malaria infection through the ring, trophozoite, and schizont stages. These simulations successfully reproduce the experimental observation that with progression of infection, the iRBC transitions from passage to blockage in larger and larger channels. The numerical results suggest a scheme for quantifying iRBC rigidification through microfluidic measurements of the critical pressure required for passage.

Reorientation of a nonspherical capsule in creeping shear flow

The dynamics of a capsule and a biological cell is of great interest in chemical engineering and bioengineering. Although the dynamics of a rigid spheroid is well understood by Jeffery's theory, that of a spheroidal capsule remains unclear. In this Letter, the motion of a spheroidal capsule or a red blood cell in creeping shear flow is investigated. The results show that the orientation of a nonspherical capsule is variant under time reversal, though that of a rigid spheroid is invariant. Surprisingly, the alignment of a nonspherical capsule over a long time duration shows a transition depending on the shear rate, which can be utilized for a particle-alignment technique. These findings form a fundamental basis of the suspension mechanics of capsules and biological cells.