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

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.

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.

Bio-Mechanical Model of Osteosarcoma Tumor Microenvironment: A Porous Media Approach

Osteosarcoma is the most common malignant bone tumor in children and adolescents with a poor prognosis. To describe the progression of osteosarcoma, we expanded a system of data-driven ODE from a previous study into a system of Reaction-Diffusion-Advection (RDA) equations and coupled it with Biot equations of poroelasticity to form a bio-mechanical model. The RDA system includes the spatio-temporal information of the key components of the tumor microenvironment. The Biot equations are comprised of an equation for the solid phase, which governs the movement of the solid tumor, and an equation for the fluid phase, which relates to the motion of cells. The model predicts the total number of cells and cytokines of the tumor microenvironment and simulates the tumor's size growth. We simulated different scenarios using this model to investigate the impact of several biomedical settings on tumors' growth. The results indicate the importance of macrophages in tumors' growth. Particularly, we have observed a high co-localization of macrophages and cancer cells, and the concentration of tumor cells increases as the number of macrophages increases.

A PDE Model of Breast Tumor Progression in MMTV-PyMT Mice

The evolution of breast tumors greatly depends on the interaction network among different cell types, including immune cells and cancer cells in the tumor. This study takes advantage of newly collected rich spatio-temporal mouse data to develop a data-driven mathematical model of breast tumors that considers cells' location and key interactions in the tumor. The results show that cancer cells have a minor presence in the area with the most overall immune cells, and the number of activated immune cells in the tumor is depleted over time when there is no influx of immune cells. Interestingly, in the case of the influx of immune cells, the highest concentrations of both T cells and cancer cells are in the boundary of the tumor, as we use the Robin boundary condition to model the influx of immune cells. In other words, the influx of immune cells causes a dominant outward advection for cancer cells. We also investigate the effect of cells' diffusion and immune cells' influx rates in the dynamics of cells in the tumor micro-environment. Sensitivity analyses indicate that cancer cells and adipocytes' diffusion rates are the most sensitive parameters, followed by influx and diffusion rates of cytotoxic T cells, implying that targeting them is a possible treatment strategy for breast cancer.

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.

Drug delivery: Experiments, mathematical modelling and machine learning

We address the problem of determining from laboratory experiments the data necessary for a proper modeling of drug delivery and efficacy in anticancer therapy. There is an inherent difficulty in extracting the necessary parameters, because the experiments often yield an insufficient quantity of information. To overcome this difficulty, we propose to combine real experiments, numerical simulation, and Machine Learning (ML) based on Artificial Neural Networks (ANN), aiming at a reliable identification of the physical model factors, e.g. the killing action of the drug. To this purpose, we exploit the employed mathematical-numerical model for tumor growth and drug delivery, together with the ANN - ML procedure, to integrate the results of the experimental tests and feed back the model itself, thus obtaining a reliable predictive tool. The procedure represents a hybrid data-driven, physics-informed approach to machine learning. The physical and mathematical model employed for the numerical simulations is without extracellular matrix (ECM) and healthy cells because of the experimental conditions we reproduce.

From tumour perfusion to drug delivery and clinical translation of in silico cancer models

In silico cancer models have demonstrated great potential as a tool to improve drug design, optimise the delivery of drugs to target sites in the host tissue and, hence, improve therapeutic efficacy and patient outcome. However, there are significant barriers to the successful translation of in silico technology from bench to bedside. More precisely, the specification of unknown model parameters, the necessity for models to adequately reflect in vivo conditions, and the limited amount of pertinent validation data to evaluate models' accuracy and assess their reliability, pose major obstacles in the path towards their clinical translation. This review aims to capture the state-of-the-art in in silico cancer modelling of vascularised solid tumour growth, and identify the important advances and barriers to success of these models in clinical oncology. Particular emphasis has been put on continuum-based models of cancer since they - amongst the class of mechanistic spatio-temporal modelling approaches - are well-established in simulating transport phenomena and the biomechanics of tissues, and have demonstrated potential for clinical translation. Three important avenues in in silico modelling are considered in this contribution: first, since systemic therapy is a major cancer treatment approach, we start with an overview of the tumour perfusion and angiogenesis in silico models. Next, we present the state-of-the-art in silico work encompassing the delivery of chemotherapeutic agents to cancer nanomedicines through the bloodstream, and then review continuum-based modelling approaches that demonstrate great promise for successful clinical translation. We conclude with a discussion of what we view to be the key challenges and opportunities for in silico modelling in personalised and precision medicine.

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.

New formulation of the Gompertz equation to describe the kinetics of untreated tumors

BACKGROUND

Different equations have been used to describe and understand the growth kinetics of undisturbed malignant solid tumors. The aim of this paper is to propose a new formulation of the Gompertz equation in terms of different parameters of a malignant tumor: the intrinsic growth rate, the deceleration factor, the apoptosis rate, the number of cells corresponding to the tumor latency time, and the fractal dimensions of the tumor and its contour.

METHODS

Furthermore, different formulations of the Gompertz equation are used to fit experimental data of the Ehrlich and fibrosarcoma Sa-37 tumors that grow in male BALB/c/Cenp mice. The parameters of each equation are obtained from these fittings.

RESULTS

The new formulation of the Gompertz equation reveals that the initial number of cancerous cells in the conventional Gompertz equation is not a constant but a variable that depends nonlinearly on time and the tumor deceleration factor. In turn, this deceleration factor depends on the apoptosis rate of tumor cells and the fractal dimensions of the tumor and its irregular contour.

CONCLUSIONS

It is concluded that this new formulation has two parameters that are directly estimated from the experiment, describes well the growth kinetics of unperturbed Ehrlich and fibrosarcoma Sa-37 tumors, and confirms the fractal origin of the Gompertz formulation and the fractal property of tumors.

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.