An in silico model of the capturing of magnetic nanoparticles in tumour spheroids in the presence of flow

One of the main challenges in improving the efficacy of conventional chemotherapeutic drugs is that they do not reach the cancer cells at sufficiently high doses while at the same time affecting healthy tissue and causing significant side effects and suffering in cancer patients. To overcome this deficiency, magnetic nanoparticles as transporter systems have emerged as a promising approach to achieve more specific tumour targeting. Drug-loaded magnetic nanoparticles can be directed to the target tissue by applying an external magnetic field. However, the magnetic forces exerted on the nanoparticles fall off rapidly with distance, making the tumour targeting challenging, even more so in the presence of flowing blood or interstitial fluid. We therefore present a computational model of the capturing of magnetic nanoparticles in a test setup: our model includes the flow around the tumour, the magnetic forces that guide the nanoparticles, and the transport within the tumour. We show how a model for the transport of magnetic nanoparticles in an external magnetic field can be integrated with a multiphase tumour model based on the theory of porous media. Our approach based on the underlying physical mechanisms can provide crucial insights into mechanisms that cannot be studied conclusively in experimental research alone. Such a computational model enables an efficient and systematic exploration of the nanoparticle design space, first in a controlled test setup and then in more complex in vivo scenarios. As an effective tool for minimising costly trial-and-error design methods, it expedites translation into clinical practice to improve therapeutic outcomes and limit adverse effects for cancer patients.

Machine learning instructed microfluidic synthesis of curcumin-loaded liposomes

The association of machine learning (ML) tools with the synthesis of nanoparticles has the potential to streamline the development of more efficient and effective nanomedicines. The continuous-flow synthesis of nanoparticles via microfluidics represents an ideal playground for ML tools, where multiple engineering parameters - flow rates and mixing configurations, type and concentrations of the reagents - contribute in a non-trivial fashion to determine the resultant morphological and pharmacological attributes of nanomedicines. Here we present the application of ML models towards the microfluidic-based synthesis of liposomes loaded with a model hydrophobic therapeutic agent, curcumin. After generating over 200 different liposome configurations by systematically modulating flow rates, lipid concentrations, organic:water mixing volume ratios, support-vector machine models and feed-forward artificial neural networks were trained to predict, respectively, the liposome dispersity/stability and size. This work presents an initial step towards the application and cultivation of ML models to instruct the microfluidic formulation of nanoparticles.

Tumour growth: An approach to calibrate parameters of a multiphase porous media model based on in vitro observations of Neuroblastoma spheroid growth in a hydrogel microenvironment

To unravel processes that lead to the growth of solid tumours, it is necessary to link knowledge of cancer biology with the physical properties of the tumour and its interaction with the surrounding microenvironment. Our understanding of the underlying mechanisms is however still imprecise. We therefore developed computational physics-based models, which incorporate the interaction of the tumour with its surroundings based on the theory of porous media. However, the experimental validation of such models represents a challenge to its clinical use as a prognostic tool. This study combines a physics-based model with in vitro experiments based on microfluidic devices used to mimic a three-dimensional tumour microenvironment. By conducting a global sensitivity analysis, we identify the most influential input parameters and infer their posterior distribution based on Bayesian calibration. The resulting probability density is in agreement with the scattering of the experimental data and thus validates the proposed workflow. This study demonstrates the huge challenges associated with determining precise parameters with usually only limited data for such complex processes and models, but also demonstrates in general how to indirectly characterise the mechanical properties of neuroblastoma spheroids that cannot feasibly be measured experimentally.

Global sensitivity analysis based on Gaussian-process metamodelling for complex biomechanical problems

Biomechanical models often need to describe very complex systems, organs or diseases, and hence also include a large number of parameters. One of the attractive features of physics-based models is that in those models (most) parameters have a clear physical meaning. Nevertheless, the determination of these parameters is often very elaborate and costly and shows a large scatter within the population. Hence, it is essential to identify the most important parameters (worth the effort) for a particular problem at hand. In order to distinguish parameters which have a significant influence on a specific model output from non-influential parameters, we use sensitivity analysis, in particular the Sobol method as a global variance-based method. However, the Sobol method requires a large number of model evaluations, which is prohibitive for computationally expensive models. We therefore employ Gaussian processes as a metamodel for the underlying full model. Metamodelling introduces further uncertainty, which we also quantify. We demonstrate the approach by applying it to two different problems: nanoparticle-mediated drug delivery in a complex, multiphase tumour-growth model, and arterial growth and remodelling. Even relatively small numbers of evaluations of the full model suffice to identify the influential parameters in both cases and to separate them from non-influential parameters. The approach also allows the quantification of higher-order interaction effects. We thus show that a variance-based global sensitivity analysis is feasible for complex, computationally expensive biomechanical models. Different aspects of sensitivity analysis are covered including a transparent declaration of the uncertainties involved in the estimation process. Such a global sensitivity analysis not only helps to massively reduce costs for experimental determination of parameters but is also highly beneficial for inverse analysis of such complex models.

A continuum mechanical framework for modeling tumor growth and treatment in two- and three-phase systems

The growth and treatment of tumors is an important problem to society that involves the manifestation of cellular phenomena at length scales on the order of centimeters. Continuum mechanical approaches are being increasingly used to model tumors at the largest length scales of concern. The issue of how to best connect such descriptions to smaller-scale descriptions remains open. We formulate a framework to derive macroscale models of tumor behavior using the thermodynamically constrained averaging theory (TCAT), which provides a firm connection with the microscale and constraints on permissible forms of closure relations. We build on developments in the porous medium mechanics literature to formulate fundamental entropy inequality expressions for a general class of three-phase, compositional models at the macroscale. We use the general framework derived to formulate two classes of models, a two-phase model and a three-phase model. The general TCAT framework derived forms the basis for a wide range of potential models of varying sophistication, which can be derived, approximated, and applied to understand not only tumor growth but also the effectiveness of various treatment modalities.

Validation and parameter optimization of a hybrid embedded/homogenized solid tumor perfusion model

The goal of this paper is to investigate the validity of a hybrid embedded/homogenized in-silico approach for modeling perfusion through solid tumors. The rationale behind this novel idea is that only the larger blood vessels have to be explicitly resolved while the smaller scales of the vasculature are homogenized. As opposed to typical discrete or fully resolved 1D-3D models, the required data can be obtained with in-vivo imaging techniques since the morphology of the smaller vessels is not necessary. By contrast, the larger vessels, whose topology and structure is attainable noninvasively, are resolved and embedded as one-dimensional inclusions into the three-dimensional tissue domain which is modeled as a porous medium. A sound mortar-type formulation is employed to couple the two representations of the vasculature. We validate the hybrid model and optimize its parameters by comparing its results to a corresponding fully resolved model based on several well-defined metrics. These tests are performed on a complex data set of three different tumor types with heterogeneous vascular architectures. The correspondence of the hybrid model in terms of mean representative elementary volume blood and interstitial fluid pressures is excellent with relative errors of less than 4%. Larger, but less important and explicable errors are present in terms of blood flow in the smaller, homogenized vessels. We finally discuss and demonstrate how the hybrid model can be further improved to apply it for studies on tumor perfusion and the efficacy of drug delivery.

Extension of a multiphase tumour growth model to study nanoparticle delivery to solid tumours

One of the main challenges in increasing the efficacy of conventional chemotherapeutics is the fact that they do not reach cancerous cells at a sufficiently high dosage. In order to remedy this deficiency, nanoparticle-based drugs have evolved as a promising novel approach to more specific tumour targeting. Nevertheless, several biophysical phenomena prevent the sufficient penetration of nanoparticles in order to target the entire tumour. We therefore extend our vascular multiphase tumour growth model, enabling it to investigate the influence of different biophysical factors on the distribution of nanoparticles in the tumour microenvironment. The novel model permits the examination of the interplay between the size of vessel-wall pores, the permeability of the blood-vessel endothelium and the lymphatic drainage on the delivery of particles of different sizes. Solid tumours develop a non-perfused core and increased interstitial pressure. Our model confirms that those two typical features of solid tumours limit nanoparticle delivery. Only in case of small nanoparticles is the transport dominated by diffusion, and particles can reach the entire tumour. The size of the vessel-wall pores and the permeability of the blood-vessel endothelium have a major impact on the amount of delivered nanoparticles. This extended in-silico tumour growth model permits the examination of the characteristics and of the limitations of nanoparticle delivery to solid tumours, which currently complicate the translation of nanoparticle therapy to a clinical stage.

An approach for vascular tumor growth based on a hybrid embedded/homogenized treatment of the vasculature within a multiphase porous medium model

The aim of this work is to develop a novel computational approach to facilitate the modeling of angiogenesis during tumor growth. The preexisting vasculature is modeled as a 1D inclusion and embedded into the 3D tissue through a suitable coupling method, which allows for nonmatching meshes in 1D and 3D domain. The neovasculature, which is formed during angiogenesis, is represented in a homogenized way as a phase in our multiphase porous medium system. This splitting of models is motivated by the highly complex morphology, physiology, and flow patterns in the neovasculature, which are challenging and computationally expensive to resolve with a discrete, 1D angiogenesis and blood flow model. Moreover, it is questionable if a discrete representation generates any useful additional insight. By contrast, our model may be classified as a hybrid vascular multiphase tumor growth model in the sense that a discrete, 1D representation of the preexisting vasculature is coupled with a continuum model describing angiogenesis. It is based on an originally avascular model which has been derived via the thermodynamically constrained averaging theory. The new model enables us to study mass transport from the preexisting vasculature into the neovasculature and tumor tissue. We show by means of several illustrative examples that it is indeed capable of reproducing important aspects of vascular tumor growth phenomenologically.

Coupling tumor growth and bio distribution models

We couple a tumor growth model embedded in a microenvironment, with a bio distribution model able to simulate a whole organ. The growth model yields the evolution of tumor cell population, of the differential pressure between cell populations, of porosity of ECM, of consumption of nutrients due to tumor growth, of angiogenesis, and related growth factors as function of the locally available nutrient. The bio distribution model on the other hand operates on a frozen geometry but yields a much refined distribution of nutrient and other molecules. The combination of both models will enable simulating the growth of a tumor in a whole organ, including a realistic distribution of therapeutic agents and allow hence to evaluate the efficacy of these agents.

A monolithic multiphase porous medium framework for (a-)vascular tumor growth

We present a dynamic vascular tumor model combining a multiphase porous medium framework for avascular tumor growth in a consistent Arbitrary Lagrangian Eulerian formulation and a novel approach to incorporate angiogenesis. The multiphase model is based on Thermodynamically Constrained Averaging Theory and comprises the extracellular matrix as a porous solid phase and three fluid phases: (living and necrotic) tumor cells, host cells and the interstitial fluid. Angiogenesis is modeled by treating the neovasculature as a proper additional phase with volume fraction or blood vessel density. This allows us to define consistent inter-phase exchange terms between the neovasculature and the interstitial fluid. As a consequence, transcapillary leakage and lymphatic drainage can be modeled. By including these important processes we are able to reproduce the increased interstitial pressure in tumors which is a crucial factor in drug delivery and, thus, therapeutic outcome. Different coupling schemes to solve the resulting five-phase problem are realized and compared with respect to robustness and computational efficiency. We find that a fully monolithic approach is superior to both the standard partitioned and a hybrid monolithic-partitioned scheme for a wide range of parameters. The flexible implementation of the novel model makes further extensions (e.g., inclusion of additional phases and species) straightforward.

Predicting the role of microstructural and biomechanical cues in tumor growth and spreading

A continuum porous media model is developed for elucidating the role of the mechanical cues in regulating tumor growth and spreading. It is shown that stiffer matrices and higher cell-matrix adhesion limit tumor growth and spreading toward the surrounding tissue. Higher matrix porosities, conversely, favor the growth and the dissemination of tumor cells. This model could be used for predicting the response of malignant masses to novel therapeutic agents affecting directly the tumor microenvironment and its micromechanical cues.

Evaluating the influence of mechanical stress on anticancer treatments through a multiphase porous media model

Drug resistance is one of the leading causes of poor therapy outcomes in cancer. As several chemotherapeutics are designed to target rapidly dividing cells, the presence of a low-proliferating cell population contributes significantly to treatment resistance. Interestingly, recent studies have shown that compressive stresses acting on tumor spheroids are able to hinder cell proliferation, through a mechanism of growth inhibition. However, studies analyzing the influence of mechanical compression on therapeutic treatment efficacy have still to be performed. In this work, we start from an existing mathematical model for avascular tumors, including the description of mechanical compression. We introduce governing equations for transport and uptake of a chemotherapeutic agent, acting on cell proliferation. Then, model equations are adapted for tumor spheroids and the combined effect of compressive stresses and drug action is investigated. Interestingly, we find that the variation in tumor spheroid volume, due to the presence of a drug targeting cell proliferation, considerably depends on the compressive stress level of the cell aggregate. Our results suggest that mechanical compression of tumors may compromise the efficacy of chemotherapeutic agents. In particular, a drug dose that is effective in reducing tumor volume for stress-free conditions may not perform equally well in a mechanically compressed environment.

A tumor growth model with deformable ECM

Existing tumor growth models based on fluid analogy for the cells do not generally include the extracellular matrix (ECM), or if present, take it as rigid. The three-fluid model originally proposed by the authors and comprising tumor cells (TC), host cells (HC), interstitial fluid (IF) and an ECM, considered up to now only a rigid ECM in the applications. This limitation is here relaxed and the deformability of the ECM is investigated in detail. The ECM is modeled as a porous solid matrix with Green-elastic and elasto-visco-plastic material behavior within a large strain approach. Jauman and Truesdell objective stress measures are adopted together with the deformation rate tensor. Numerical results are first compared with those of a reference experiment of a multicellular tumor spheroid (MTS) growing in vitro, then three different tumor cases are studied: growth of an MTS in a decellularized ECM, growth of a spheroid in the presence of host cells and growth of a melanoma. The influence of the stiffness of the ECM is evidenced and comparison with the case of a rigid ECM is made. The processes in a deformable ECM are more rapid than in a rigid ECM and the obtained growth pattern differs. The reasons for this are due to the changes in porosity induced by the tumor growth. These changes are inhibited in a rigid ECM. This enhanced computational model emphasizes the importance of properly characterizing the biomechanical behavior of the malignant mass in all its components to correctly predict its temporal and spatial pattern evolution.

A two-phase model of plantar tissue: a step toward prediction of diabetic foot ulceration

A new computational model, based on the thermodynamically constrained averaging theory, has been recently proposed to predict tumor initiation and proliferation. A similar mathematical approach is proposed here as an aid in diabetic ulcer prevention. The common aspects at the continuum level are the macroscopic balance equations governing the flow of the fluid phase, diffusion of chemical species, tissue mechanics, and some of the constitutive equations. The soft plantar tissue is modeled as a two-phase system: a solid phase consisting of the tissue cells and their extracellular matrix, and a fluid one (interstitial fluid and dissolved chemical species). The solid phase may become necrotic depending on the stress level and on the oxygen availability in the tissue. Actually, in diabetic patients, peripheral vascular disease impacts tissue necrosis; this is considered in the model via the introduction of an effective diffusion coefficient that governs transport of nutrients within the microvasculature. The governing equations of the mathematical model are discretized in space by the finite element method and in time domain using the θ-Wilson Method. While the full mathematical model is developed in this paper, the example is limited to the simulation of several gait cycles of a healthy foot.

Averaging Theory for Description of Environmental Problems: What Have We Learned?

Advances in Water Resources has been a prime archival source for implementation of averaging theories in changing the scale at which processes of importance in environmental modeling are described. Thus in celebration of the 35th year of this journal, it seems appropriate to assess what has been learned about these theories and about their utility in describing systems of interest. We review advances in understanding and use of averaging theories to describe porous medium flow and transport at the macroscale, an averaged scale that models spatial variability, and at the megascale, an integral scale that only considers time variation of system properties. We detail physical insights gained from the development and application of averaging theory for flow through porous medium systems and for the behavior of solids at the macroscale. We show the relationship between standard models that are typically applied and more rigorous models that are derived using modern averaging theory. We discuss how the results derived from averaging theory that are available can be built upon and applied broadly within the community. We highlight opportunities and needs that exist for collaborations among theorists, numerical analysts, and experimentalists to advance the new classes of models that have been derived. Lastly, we comment on averaging developments for rivers, estuaries, and watersheds.

Averaging Theory for Description of Environmental Problems: What Have We Learned?

Advances in Water Resources has been a prime archival source for implementation of averaging theories in changing the scale at which processes of importance in environmental modeling are described. Thus in celebration of the 35th year of this journal, it seems appropriate to assess what has been learned about these theories and about their utility in describing systems of interest. We review advances in understanding and use of averaging theories to describe porous medium flow and transport at the macroscale, an averaged scale that models spatial variability, and at the megascale, an integral scale that only considers time variation of system properties. We detail physical insights gained from the development and application of averaging theory for flow through porous medium systems and for the behavior of solids at the macroscale. We show the relationship between standard models that are typically applied and more rigorous models that are derived using modern averaging theory. We discuss how the results derived from averaging theory that are available can be built upon and applied broadly within the community. We highlight opportunities and needs that exist for collaborations among theorists, numerical analysts, and experimentalists to advance the new classes of models that have been derived. Lastly, we comment on averaging developments for rivers, estuaries, and watersheds.

A multiphase model for three-dimensional tumor growth

Several mathematical formulations have analyzed the time-dependent behaviour 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 (TCAT). A tumor mass is treated as a multiphase medium composed of an extracellular matrix (ECM); tumor cells (TC), which may become necrotic depending on the nutrient concentration and tumor phase pressure; healthy cells (HC); and an interstitial fluid (IF) 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 (MTS) 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 behaviour: initially, the rapidly growing tumor cells 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 tumor cells 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 tumor and healthy cells to the ECM, and the less favourable transport of nutrients. In particular, for tumor cells adhering less avidly to the ECM, the healthy tissue is progressively displaced as the malignant mass grows, whereas tumor cell 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 3D geometry. It is shown that tumor cells tend to migrate among adjacent vessels seeking new oxygen and nutrient. 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 tumor cell proliferation.

Tumor growth modeling from the perspective of multiphase porous media mechanics

Multiphase porous media mechanics is used for modeling tumor growth, using governing equations obtained via the thermodynamically constrained averaging theory (TCAT). This approach incorporates the interaction of more phases than legacy tumor growth models. The tumor is treated as a multiphase system composed of an extracellular matrix, tumor cells which may become necrotic depending on nutrient level and pressure, healthy cells and an interstitial fluid which transports nutrients. The governing equations are numerically solved within a Finite Element framework for predicting the growth rate of the tumor mass, and of its individual components, as a function of the initial tumor-to-healthy cell ratio, nutrient concentration, and mechanical strain. Preliminary results are shown.

Numerical Modelling of Thermo–Hydro–Mechanical (THM) in Deforming Porous Media for Subsurface Systems

Kajian aliran berbilang–fasa dan aliran haba dalam bahantara berliang separa tepu menjadi semakin penting dalam kejuruteraan geomekanik persekitaran kerana ia berkait rapat dengan pengukuhan bahantara berliang itu di dalam zon tidak tepu. Satu model berangka telah dikembangkan untuk menghuraikan masalah terganding haba–hidro–mekanikal (THM) dalam bahan berliang boleh ubah bentuk dengan aliran dua–fasa. Hubungan yang disarankan oleh Brooks dan Corey telah digunakan kepada tekanan rerambut, ketepuan air dan kebolehtelapan air serta gas. Satu kajian lanjutan dibuat ke atas model berangka tersebut berasaskan kod COMES–GEO untuk menyelesaikan masalah yang berlaku dalam zon tak tepu di medan telaga Kg. Puteh, Kota Bharu, Kelantan iaitu sebuah akuifer cetek yang berpotensi untuk mengeluarkan bekalan air bumi terbanyak di daerah Kota Bharu. Untuk menunjukkan model dan prosedur penyelesaiannya, rumusan pelaksanaan berangka dan contoh–contoh masalah dibincangkan di dalam kajian ini. Kata kunci: Aliran berbilang-fasa; aliran haba; bahan berliang boleh ubah bentuk; model berangka;zon tidak tepu; haba-hidro-mekanikal. The study of multiphase flow and heat flow in partially saturated porous media is important in environmental geomechanics engineering because of its relevance to consolidation of porous media in unsaturated zone. A numerical model which describes the thermo–hydro–mechanical (THM) coupled problems in deformable porous material with two–phase flow has been developed. The relationships between capillary pressure, saturation of water and relative permeabilities of water and gas, proposed by Brooks and Corey was used. An extended study of the numerical model, based on the COMES–GEO code was conducted recently to solve unsaturated problems in local condition of Kg. Puteh wellfield, Kota Bahru. This site is a potential shallow aquifer which contribute to the largest groundwater supply in Kota Bahru, Kelantan. Some numerical investigation on the proposed formulation is discussed with illustrative example problems to demonstrate solution procedures and validating of the model. Key words: Multiphase flow; heat flow; deforming porous media; numerical model; unsaturated zone; thermo-hydro-mechanical.

Biomechanical Multiphase Approaches in Soft Tissues

Presence of fluids drives mechanical response in biological, soft hydrated tissues. A multiphase approach has been developed to allow for a fully-comprehensive, physically based view of their history-dependent response in static and dynamics. The related formulation considers both geometric and material non-linearities. In the present work, the non-linear aspects related to fluid flows are investigated. Low speed flows of liquid phase depend on the permeability of the deforming solid matrix; permeability is naturally related to void ratio, but further dependences on other variables can be explicitly added. Two different biomechanical problems are considered and the results are presented here. In the first case, a human lumbar intervertebral segment is investigated in healthy and degenerate condition: higher permeability and lower mechanical properties of disc affect the capability to withstand an imposed constant compressive load in time. The permeability is related only to void ratio and is not affected by the overpressure that dissipates with time. The second case refers to the trabecular meshwork in the eye; this tissue has a rule in the regulation of outflow of the aqueous humour from the eye: a variation in permeability determines obviously a variation of internal ocular pressure to guarantee the assumed constant outflow, and on the other side, pressure changes affect permeability distribution and thus final pressure changes. This effect is particularly significant with glaucoma, and thus the increment of intraocular pressure can be related to a progressive reduction in tissue permeability. The modelling of this effect can be improved by introducing a proper permeability-pressure relationship in addition to the permeability-void ratio dependency.

Optimizing particle size for targeting diseased microvasculature: from experiments to artificial neural networks

Background

Nanoparticles with different sizes, shapes, and surface properties are being developed for the early diagnosis, imaging, and treatment of a range of diseases. Identifying the optimal configuration that maximizes nanoparticle accumulation at the diseased site is of vital importance. In this work, using a parallel plate flow chamber apparatus, it is demonstrated that an optimal particle diameter (d_opt) exists for which the number (n_s) of nanoparticles adhering to the vessel walls is maximized. Such a diameter depends on the wall shear rate (S). Artificial neural networks are proposed as a tool to predict n_s as a function of S and particle diameter (d), from which to eventually derive d_opt. Artificial neural networks are trained using data from flow chamber experiments. Two networks are used, ie, ANN231 and ANN2321, exhibiting an accurate prediction for n_s and its complex functional dependence on d and S. This demonstrates that artificial neural networks can be used effectively to minimize the number of experiments needed without compromising the accuracy of the study. A similar procedure could potentially be used equally effectively for in vivo analysis.