Phase-field model of bilipid membrane electroporation.

This paper proposes a new model of membrane electropermeabilisation that combines the water content of the membrane and the transmembrane voltage. Interestingly, thanks to a well defined free-energy of the membrane, we somehow generalise the seminal approach of Chizmadzhev, Weaver and Krassowska, getting rid of the geometrical cylindrical assumption upon which most of the current electroporation models are based. Our approach is physically relevant and we recover a surface diffusion equation of the lipid phase proposed by Leguèbe et al. in a previous phenomenological model. We also perform a fine analysis of the involved nonlocal operators in two simple configurations (a spherical membrane and a flat periodic membrane) that enables us to compare the time constants of the phenomenon in spherical and flat membranes. An accurate splitting scheme combined with Fast Fourier Transforms is developed for efficient computations of the model. Our numerical results enable us to make a link between the molecular dynamics simulations of membrane permeabilisation and the experimental observations on vesicles and cells.

Macro-scale models for fluid flow in tumour tissues: impact of microstructure properties.

Understanding the dynamics underlying fluid transport in tumour tissues is of fundamental importance to assess processes of drug delivery. Here, we analyse the impact of the tumour microscopic properties on the macroscopic dynamics of vascular and interstitial fluid flow. More precisely, we investigate the impact of the capillary wall permeability and the hydraulic conductivity of the interstitium on the macroscopic model arising from formal asymptotic 2-scale techniques. The homogenization technique allows us to derive two macroscale tissue models of fluid flow that take into account the microscopic structure of the vessels and the interstitial tissue. Different regimes were derived according to the magnitude of the vessel wall permeability and the interstitial hydraulic conductivity. Importantly, we provide an analysis of the properties of the models and show the link between them. Numerical simulations were eventually performed to test the models and to investigate the impact of the microstructure on the fluid transport. Future applications of our models include their calibration with real imaging data to investigate the impact of the tumour microenvironment on drug delivery.

Physiological changes may dominate the electrical properties of liver during reversible electroporation: Measurements and modelling.

This study presents electrical measurements (both conductivity during the pulses and impedance spectroscopy before and after) performed in liver tissue of mice during electroporation with classical electrochemotherapy conditions (8 pulses of 100 µs duration). A four-needle electrode arrangement inserted in the tissue was used for the measurements. The undesirable effects of the four-electrode geometry, notably concerning its sensitivity, were quantified and discussed showing how the electrode geometry chosen for the measurements can impact the results. Numerical modelling was applied to the information collected during the pulse, and to the impedance spectra acquired before and after the pulses sequence. Our results show that the numerical results were not consistent, suggesting that other collateral phenomena not considered in the model are at work during electroporation in vivo. We show how the modification in the volume of the intra and extra cellular media, likely caused by the vascular lock effect, could at least partially explain the recorded impedance evolution. In the present study we demonstrate the significant impact that physiological effects have on impedance changes following electroporation at the tissue scale and the potential need of introducing them into the numerical models. The code for the numerical model is publicly available at https://gitlab.inria.fr/poignard/4-electrode-system.

Population modeling of tumor growth curves and the reduced Gompertz model improve prediction of the age of experimental tumors.

Tumor growth curves are classically modeled by means of ordinary differential equations. In analyzing the Gompertz model several studies have reported a striking correlation between the two parameters of the model, which could be used to reduce the dimensionality and improve predictive power. We analyzed tumor growth kinetics within the statistical framework of nonlinear mixed-effects (population approach). This allowed the simultaneous modeling of tumor dynamics and inter-animal variability. Experimental data comprised three animal models of breast and lung cancers, with 833 measurements in 94 animals. Candidate models of tumor growth included the exponential, logistic and Gompertz models. The exponential and-more notably-logistic models failed to describe the experimental data whereas the Gompertz model generated very good fits. The previously reported population-level correlation between the Gompertz parameters was further confirmed in our analysis (R2 > 0.92 in all groups). Combining this structural correlation with rigorous population parameter estimation, we propose a reduced Gompertz function consisting of a single individual parameter (and one population parameter). Leveraging the population approach using Bayesian inference, we estimated times of tumor initiation using three late measurement timepoints. The reduced Gompertz model was found to exhibit the best results, with drastic improvements when using Bayesian inference as compared to likelihood maximization alone, for both accuracy and precision. Specifically, mean accuracy (prediction error) was 12.2% versus 78% and mean precision (width of the 95% prediction interval) was 15.6 days versus 210 days, for the breast cancer cell line. These results demonstrate the superior predictive power of the reduced Gompertz model, especially when combined with Bayesian estimation. They offer possible clinical perspectives for personalized prediction of the age of a tumor from limited data at diagnosis. The code and data used in our analysis are publicly available at https://github.com/cristinavaghi/plumky.

Numerical optimization of plasmid DNA delivery combined with hyaluronidase injection for electroporation protocol.

BACKGROUND AND OBJECTIVE: The paper focuses on the numerical strategies to optimize a plasmid DNA delivery protocol, which combines hyaluronidase and electroporation. METHODS: A well-defined continuum mechanics model of muscle porosity and advanced numerical optimization strategies have been used, to propose a substantial improvement of a pre-existing experimental protocol of DNA transfer in mice. Our work suggests that a computational model might help in the definition of innovative therapeutic procedures, thanks to the fine tuning of all the involved experimental steps. This approach is particularly interesting in optimizing complex and costly protocols, to make in vivo DNA therapeutic protocols more effective. RESULTS: Our preliminary work suggests that computational model might help in the definition of innovative therapeutic protocol, thanks to the fine tuning of all the involved operations. CONCLUSIONS: This approach is particularly interesting in optimizing complex and costly protocols for which the number of degrees of freedom prevents a experimental test of the possible configuration.

Numerical workflow of irreversible electroporation for deep-seated tumor.

The paper provides a numerical workflow, based on the 'real-life' clinical workflow of irreversible electroporation (IRE) performed for the treatment of deep-seated liver tumors. Thanks to a combination of numerical modeling, image registration algorithm and clinical data, our numerical workflow enables to provide the distribution of the electric field as effectively delivered by the clinical IRE procedure. As a proof of concept, we show on a specific clinical case of IRE ablation of liver tumor that clinical data could be advantageously combined to numerical simulations in a near future, in order to give to the interventional radiologists information on the effective IRE ablation. We also corroborate the simulated treated region with the post-treatment MRI performed 3 d after the treatment.

A Continuum Mechanics Model of Enzyme-Based Tissue Degradation in Cancer Therapies.

We propose a mathematical model to describe enzyme-based tissue degradation in cancer therapies. The proposed model combines the poroelastic theory of mixtures with the transport of enzymes or drugs in the extracellular space. The effect of the matrix-degrading enzymes on the tissue composition and its mechanical response are accounted for. Numerical simulations in 1D, 2D and axisymmetric (3D) configurations show how an injection of matrix-degrading enzymes alters the porosity of a biological tissue. We eventually exhibit numerically the main consequences of a matrix-degrading enzyme pretreatment in the framework of chemotherapy: the removal of the diffusive hindrance to the penetration of therapeutic molecules in tumors and the reduction of interstitial fluid pressure which improves transcapillary transport. Both effects are consistent with previous biological observations.

Experimental estimation of stored stress within spherical microtissues : What can and cannot be inferred from cutting experiments.

Biological tissues accumulate mechanical stress during their growth. The mere measurement of the stored stress is not an easy task. We address here the spherical case and our experiments consist in performing an incision of a spherical microtissue (tumor spheroid) grown in vitro. On the theoretical part we derive a compatibility condition on the stored stress in spherical symmetry, which imposes a relation between the circumferential and radial stored stress. The numerical implementation uses the hyperelastic model of Ciarlet and Geymonat. A parametric study is performed to assess the influence of each parameter on the shape of the domain after the incision. As a conclusion, the total radial stored stress can be confidently estimated from the measurement of the opening after incision. We validate the approach with experimental data.

Viscoelastic modeling of the fusion of multicellular tumor spheroids in growth phase.

BACKGROUND: Since several decades, the experiments have highlighted the analogy of fusing cell aggregates with liquid droplets. The physical macroscopic models have been derived under incompressible assumptions. The aim of this paper is to provide a 3D model of growing spheroids, which is more relevant regarding embryo cell aggregates or tumor cell spheroids. METHODS: We extend the past approach to a compressible 3D framework in order to account for the tumor spheroid growth. We exhibit the crucial importance of the effective surface tension, and of the inner pressure of the spheroid to describe precisely the fusion. The experimental data were obtained on spheroids of colon carcinoma human cells (HCT116 cell line). After 3 or 6 days of culture, two identical spheroids were transferred in one well and their fusion was monitored by live videomicroscopy acquisition each 2 h during 72 h. From these images the neck radius and the diameter of the assembly of the fusing spheroids are extracted. RESULTS: The numerical model is fitted with the experiments. It is worth noting that the time evolution of both neck radius and spheroid diameter are quantitatively obtained. The interesting feature lies in the fact that such measurements characterise the macroscopic rheological properties of the tumor spheroids. CONCLUSIONS: The experimental determination of the kinetics of neck radius and overall diameter during spheroids fusion characterises the rheological properties of the spheroids. The consistency of the model is shown by fitting the model with two different experiments, enhancing the importance of both surface tension and cell proliferation. GENERAL SIGNIFICANCE: The paper sheds new light on the macroscopic rheological properties of tumor spheroids. It emphasizes the role of the surface tension and the inner pressure in the fusion of growing spheroid. Under geometrical assumptions, the model reduces to a 2-parameter differential equation fit with experimental measurements. The 3-D partial differential system makes it possible to study the fusion of spheroids in non-symmetrical or more general frameworks.

Dynamical modeling of tissue electroporation.

In this paper, we propose a new dynamical model of tissue electroporation. The model is based on equivalent circuit approach at the tissue. Considering two current densities from cells and extracellular matrix, we identify the macroscopic homogenised contribution of the cell membranes. Our approach makes it possible to define a macroscopic homogenised electric field and a macroscopic homogenised transmembrane potential. This provides a direct link between the cell scale electroporation models and the tissue models. Finite element method adapted to the new non-linear model of tissue electroporation is used to compare experiments with simulations. Adapting the phenomenological electroporation model of Leguèbe et al. to the tissue scale, we calibrate the tissue model with experimental data. This makes two steps appear in the tissue electroporation process, as for cells. The new insight of the model lies in the well-established equivalent circuit approach to provide a homogenised version of cell scale models. Our approach is tightly linked to numerical homogenisation strategies adapted to bioelectrical tissue modeling.

Tumor growth model of ductal carcinoma: from in situ phase to stroma invasion.

This paper aims at modeling breast cancer transition from the in situ stage -when the tumor is confined to the duct- to the invasive phase. Such a transition occurs thanks to the degradation of the duct membrane under the action of specific enzymes so-called matrix metalloproteinases (MMPs). The model consists of advection-reaction equations that hold in the duct and in the surrounding tissue, in order to describe the proliferation and the necrosis of the cancer cells in each subdomain. The divergence of the velocity is given by the increase of the cell densities. Darcy law is imposed in order to close the system. The key-point of the modeling lies in the description of the transmission conditions across the duct. Nonlinear Kedem-Katchalsky transmission conditions across the membrane describe the discontinuity of the pressure as a linear function of the flux. These transmission conditions make it possible to describe the transition from the in situ stage to the invasive phase at the macroscopic level. More precisely, the membrane permeability increases with respect to the local concentration of MMPs. The cancer cells are no more confined to the duct and the tumor invades the surrounding tissue. The model is enriched by the description of nutrients concentration, tumor necrosis factors, and MMPs production. The mathematical model is implemented in a 3D C++-code, which is based on well-adapted finite difference schemes on Cartesian grid. The membrane interface is described by a level-set, and the transmission conditions are precisely approached at the second order thanks to well-suited sharp stencils. Our continuous approach provides new significant insights in the macroscopic modeling of the breast cancer phase transition, due to the membrane degradation by MMP enzymes.

Spatial modelling of tumour drug resistance: the case of GIST liver metastases.

This work is devoted to modelling gastrointestinal stromal tumour metastases to the liver, their growth and resistance to therapies. More precisely, resistance to two standard treatments based on tyrosine kinase inhibitors (imatinib and sunitinib) is observed clinically. Using observations from medical images (CT scans), we build a spatial model consisting in a set of non-linear partial differential equations. After calibration of its parameters with clinical data, this model reproduces qualitatively and quantitatively the spatial tumour evolution of one specific patient. Important features of the growth such as the appearance of spatial heterogeneities and the therapeutical failures may be explained by our model. We then investigate numerically the possibility of optimizing the treatment in terms of progression-free survival time and minimum tumour size reachable by varying the dose of the first treatment. We find that according to our model, the progression-free survival time reaches a plateau with respect to this dose. We also demonstrate numerically that the spatial structure of the tumour may provide much more insights on the cancer cell activities than the standard RECIST criteria, which only consists in the measurement of the tumour diameter. Finally, we discuss on the non-predictivity of the model using only CT scans, in the sense that the early behaviour of the lesion is not sufficient to predict the response to the treatment.

Free boundary problem for cell protrusion formations: theoretical and numerical aspects.

In this paper, a free boundary problem for cell protrusion formation is studied theoretically and numerically. The cell membrane is precisely described thanks to a level set function, whose motion is due to specific signalling pathways. The aim is to model the chemical interactions between the cell and its environment, in the process of invadopodia or pseudopodia formation. The model consists of Laplace equation with Dirichlet condition inside the cell coupled to Laplace equation with Neumann condition in the outer domain. The actin polymerization is accounted for as the gradient of the inner signal, which drives the motion of the interface. We prove the well-posedness of our free boundary problem under a sign condition on the datum. This criterion ensures the consistency of the model, and provides conditions to focus on for any enrichment of the model. We then propose a new first order Cartesian finite-difference method to solve the problem. We eventually exhibit the main biological features that can be accounted for by the model: the formation of thin and elongated protrusions as for invadopodia, or larger protrusion as for pseudopodia, depending on the source term in the equation. The model provides the theoretical and numerical grounds for single cell migration modeling, whose formulation is valid in 2D and 3D. In particular, specific chemical reactions that occurred at the cell membrane could be precisely described in forthcoming works.

Impact of external medium conductivity on cell membrane electropermeabilization by microsecond and nanosecond electric pulses.

The impact of external medium conductivity on the efficiency of the reversible permeabilisation caused by pulsed electric fields was investigated. Pulses of 12 ns, 102 ns or 100 µs were investigated. Whenever permeabilisation could be detected after the delivery of one single pulse, media of lower conductivity induced more efficient reversible permeabilisation and thus independently of the medium composition. Effect of medium conductivity can however be hidden by some saturation effects, for example when pulses are cumulated (use of trains of 8 pulses) or when the detection method is not sensitive enough. This explains the contradicting results that can be found in the literature. The new data are complementary to those of one of our previous study in which an opposite effect of the conductivity was highlighted. It stresses that the conductivity of the medium influences the reversible permeabilization by several ways. Moreover, these results clearly indicate that electropermeabilisation does not linearly depend on the energy delivered to the cells.

Cell membrane permeabilization by 12-ns electric pulses: Not a purely dielectric, but a charge-dependent phenomenon.

Electric pulses of a few nanoseconds in duration can induce reversible permeabilization of cell membrane and cell death. Whether these effects are caused by ionic or purely dielectric phenomena is still discussed. We address this question by studying the impact of conductivity of the pulsing buffer on the effect of pulses of 12 ns and 3.2 MV/m on the DC-3F mammalian cell line. When pulses were applied in a high-conductivity medium (1.5 S/m), cells experienced both reversible electropermeabilization and cell death. On the contrary, no effect was observed in the low-conductivity medium (0.1 S/m). Possible artifacts due to differences in viscosity, temperature increase or electrochemical reactions were excluded. The influence of conductivity reported here suggests that charges still play a role, even for 12-ns pulses. All theoretical models agree with this experimental observation, since all suggest that only high-conductivity medium can induce a transmembrane voltage high enough to induce pore creation, in turn. However, most models fail to describe why pulse accumulation is experimentally required to observe biological effects. They mostly show no increase of permeabilization with accumulation of pulses. Currently, only one model properly describes pulse accumulation by modeling diffusion of the altered membrane regions.

"Classical" electropermeabilization modeling at the cell scale.

The aim of this paper is to provide new models of cell electropermeabilization involving only a few parameters. A static and a dynamical model, which are based on the description of the electric potential in a biological cell, are derived. Existence and uniqueness results are provided for each differential system, and an accurate numerical method to compute the solution is described. We then present numerical simulations that corroborate the experimental observations, providing the consistency of the modeling. We emphasize that our new models involve very few parameters, compared with the most achieved models of Neu and Krassowska (Phys Rev E 53(3):3471-3482, 1999) and DeBruin and Krassowska (Biophys J 77:1225-1233, 1999), but they provide the same qualitative results. Thus, these models will facilitate drastically the forthcoming inverse problem solving, which will consist in fitting them with the experiments.

Modeling of the migration of endothelial cells on bioactive micropatterned polymers.

In this paper, a macroscopic model describing endothelial cell migration on bioactive micropatterned polymers is presented. It is based on a system of partial differential equations of Patlak-Keller-Segel type that describes the evolution of the cell densities. The model is studied mathematically and numerically. We prove existence and uniqueness results of the solution to the differential system. We also show that fundamental physical properties such as mass conservation, positivity and boundedness of the solution are satisfied. The numerical study allows us to show that the modeling results are in good agreement with the experiments.

Ion fluxes, transmembrane potential, and osmotic stabilization: a new dynamic electrophysiological model for eukaryotic cells.

Survival of mammalian cells is achieved by tight control of cell volume, while transmembrane potential has been known to control many cellular functions since the seminal work of Hodgkin and Huxley. Regulation of cell volume and transmembrane potential have a wide range of implications in physiology, from neurological and cardiac disorders to cancer and muscle fatigue. Therefore, understanding the relationship between transmembrane potential, ion fluxes, and cell volume regulation has become of great interest. In this paper we derive a system of differential equations that links transmembrane potential, ionic concentrations, and cell volume. In particular, we describe the dynamics of the cell within a few seconds after an osmotic stress, which cannot be done by the previous models in which either cell volume was constant or osmotic regulation instantaneous. This new model demonstrates that both membrane potential and cell volume stabilization occur within tens of seconds of changes in extracellular osmotic pressure. When the extracellular osmotic pressure is constant, the cell volume varies as a function of transmembrane potential and ion fluxes, thus providing an implicit link between transmembrane potential and cell volume. Experimental data provide results that corroborate the numerical simulations of the model in terms of time-related changes in cell volume and dynamics of the phenomena. This paper can be seen as a generalization of previous electrophysiological results, since under restrictive conditions they can be derived from our model.