Investigation of the spatiotemporal responses of nanoparticles in tumor tissues with a small-scale mathematical model

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.

Multi-scale modelling of nanoparticle delivery and heat transport in vascularised tumours

We focus on modelling of cancer hyperthermia driven by the application of the magnetic field to iron oxide nanoparticles. We assume that the particles are interacting with the tumour environment by extravasating from the vessels into the interstitial space. We start from Darcy's and Stokes' problems in the interstitial and fluid vessels compartments. Advection-diffusion of nanoparticles takes place in both compartments (as well as uptake in the tumour interstitium), and a heat source proportional to the concentration of nanoparticles drives heat diffusion and convection in the system. The system under consideration is intrinsically multi-scale. The distance between adjacent vessels (the micro-scale) is much smaller than the average tumour size (the macro-scale). We then apply the asymptotic homogenisation technique to retain the influence of the micro-structure on the tissue scale distribution of heat and particles. We derive a new system of homogenised partial differential equations (PDEs) describing blood transport, delivery of nanoparticles and heat transport. The new model comprises a double Darcy's law, coupled with two double advection-diffusion-reaction systems of PDEs describing fluid, particles and heat transport and mass, drug and heat exchange. The role of the micro-structure is encoded in the coefficients of the model, which are to be computed solving appropriate periodic problems. We show that the heat distribution is impaired by increasing vessels' tortuosity and that regularization of the micro-vessels can produce a significant increase (1-2 degrees) in the maximum temperature. We quantify the impact of modifying the properties of the magnetic field depending on the vessels' tortuosity.

Kinetics of Nanomedicine in Tumor Spheroid as an In Vitro Model System for Efficient Tumor-Targeted Drug Delivery With Insights From Mathematical Models

Numerous strategies have been developed to treat cancer conventionally. Most importantly, chemotherapy shows its huge promise as a better treatment modality over others. Nonetheless, the very complex behavior of the tumor microenvironment frequently impedes successful drug delivery to the tumor sites that further demands very urgent and effective distribution mechanisms of anticancer drugs specifically to the tumor sites. Hence, targeted drug delivery to tumor sites has become a major challenge to the scientific community for cancer therapy by assuring drug effects to selective tumor tissue and overcoming undesired toxic side effects to the normal tissues. The application of nanotechnology to the drug delivery system pays heed to the design of nanomedicine for specific cell distribution. Aiming to limit the use of traditional strategies, the adequacy of drug-loaded nanocarriers (i.e., nanomedicine) proves worthwhile. After systemic blood circulation, a typical nanomedicine follows three levels of disposition to tumor cells in order to exhibit efficient pharmacological effects induced by the drug candidates residing within it. As a result, nanomedicine propounds the assurance towards the improved bioavailability of anticancer drug candidates, increased dose responses, and enhanced targeted efficiency towards delivery and distribution of effective therapeutic concentration, limiting toxic concentration. These aspects emanate the proficiency of drug delivery mechanisms. Understanding the potential tumor targeting barriers and limiting conditions for nanomedicine extravasation, tumor penetration, and final accumulation of the anticancer drug to tumor mass, experiments with in vivo animal models for nanomedicine screening are a key step before it reaches clinical translation. Although the study with animals is undoubtedly valuable, it has many associated ethical issues. Moreover, individual experiments are very expensive and take a longer time to conclude. To overcome these issues, nowadays, multicellular tumor spheroids are considered a promising in vitro model system that proposes better replication of in vivo tumor properties for the future development of new therapeutics. In this review, we will discuss how tumor spheroids could be used as an in vitro model system to screen nanomedicine used in targeted drug delivery, aiming for better therapeutic benefits. In addition, the recent proliferation of mathematical modeling approaches gives profound insight into the underlying physical principles and produces quantitative predictions. The hierarchical tumor structure is already well decorous to be treated mathematically. To study targeted drug delivery, mathematical modeling of tumor architecture, its growth, and the concentration gradient of oxygen are the points of prime focus. Not only are the quantitative models circumscribed to the spheroid, but also the role of modeling for the nanoparticle is equally inevitable. Abundant mathematical models have been set in motion for more elaborative and meticulous designing of nanomedicine, addressing the question regarding the objective of nanoparticle delivery to increase the concentration and the augmentative exposure of the therapeutic drug molecule to the core. Thus, to diffuse the dichotomy among the chemistry involved, biological data, and the underlying physics, the mathematical models play an indispensable role in assisting the experimentalist with further evaluation by providing the admissible quantitative approach that can be validated. This review will provide an overview of the targeted drug delivery mechanism for spheroid, using nanomedicine as an advantageous tool.

Cell spheroids as a versatile research platform: formation mechanisms, high throughput production, characterization and applications

Three-dimensional cell culture has tremendous advantages to closely mimic the in vivo architecture and microenvironment of healthy tissue and organs, as well as of solid tumors. Spheroids are currently the most attractive 3D model to produce uniform reproducible cell structures as well as a potential basis for engineering large tissues and complex organs. In this review we discuss, from an engineering perspective, processes to obtain uniform 3D cell spheroids, comparing dynamic and static cultures and considering aspects such as mass transfer and shear stress. In addition, computational and mathematical modelling of complex cell spheroid systems are discussed. The non-cell-adhesive hydrogel-based method and dynamic cell culture in bioreactors are focused in detail and the myriad of developed spheroid characterization techniques is presented. The main bottlenecks and weaknesses are discussed, especially regarding the analysis of morphological parameters, cell quantification and viability, gene expression profiles, metabolic behavior and high-content analysis. Finally, a vast set of applications of spheroids as tools for in vitro study model systems is examined, including drug screening, tissue formation, pathologies development, tissue engineering and biofabrication, 3D bioprinting and microfluidics, together with their use in high-throughput platforms.

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.

Thermosensitive Liposome-Mediated Drug Delivery in Chemotherapy: Mathematical Modelling for Spatio-temporal Drug Distribution and Model-Based Optimisation

Thermosensitive liposome-mediated drug delivery has shown promising results in terms of improved therapeutic efficacy and reduced side effects compared to conventional chemotherapeutics. In order to facilitate our understanding of the transport mechanisms and their complex interplays in the drug delivery process, computational models have been developed to simulate the multiple steps involved in liposomal drug delivery to solid tumours. In this study we employ a multicompartmental model for drug-loaded thermosensitive liposomes, with an aim to identify the key transport parameters in determining therapeutic dosing and outcomes. The computational model allows us to not only examine the temporal and spatial variations of drug concentrations in the different compartments by utilising the tumour cord concept, but also assess the therapeutic efficacy and toxicity. In addition, the influences of key factors on systemic plasma concentration and intracellular concentration of the active drug are investigated; these include different chemotherapy drugs, release rate constants and heating duration. Our results show complex relationships between these factors and the predicted therapeutic outcome, making it difficult to identify the "best" parameter set. To overcome this challenge, a model-based optimisation method is proposed in an attempt to find a set of release rate constants and heating duration that can maximise intracellular drug concentration while minimising systemic drug concentration. Optimisation results reveal that under the operating conditions and ranges examined, the best outcome would be achieved with a low drug release rate at physiological temperature, combined with a moderate to high release rate at mild hyperthermia and 1 h heating after injection.

Numerical modeling of nanodrug distribution in tumors with heterogeneous vasculature

The distribution and accumulation of nanoparticle dosage in a tumor are important in evaluating the effectiveness of cancer treatment. The cell survival rate can quantify the therapeutic effect, and the survival rates after multiple treatments are helpful to evaluate the efficacy of a chemotherapy plan. We developed a mathematical tumor model based on the governing equations describing the fluid flow and particle transport to investigate the drug transportation in a tumor and computed the resulting cumulative concentrations. The cell survival rate was calculated based on the cumulative concentration. The model was applied to a subcutaneous tumor with heterogeneous vascular distributions. Various sized dextrans and doxorubicin were respectively chosen as the nanodrug carrier and the traditional chemotherapeutic agent for comparison. The results showed that: 1) the largest nanoparticle drug in the current simulations yielded the highest cumulative concentration in the well vascular region, but second lowest in the surrounding normal tissues, which implies it has the best therapeutic effect to tumor and at the same time little harmful to normal tissue; 2) on the contrary, molecular chemotherapeutic agent produced the second lowest cumulative concentration in the well vascular tumor region, but highest in the surrounding normal tissue; 3) all drugs have very small cumulative concentrations in the tumor necrotic region, where drug transport is solely through diffusion. This might mean that it is hard to kill tumor stem cells hiding in it. The current model indicated that the effectiveness of the anti-tumor drug delivery was determined by the interplay of the vascular density and nanoparticle size, which governs the drug transport properties. The use of nanoparticles as anti-tumor drug carriers is generally a better choice than molecular chemotherapeutic agent because of its high treatment efficiency on tumor cells and less damage to normal tissues.

A novel individual-cell-based mathematical model based on multicellular tumour spheroids for evaluating doxorubicin-related delivery in avascular regions

BACKGROUND AND PURPOSE

Effective drug delivery in the avascular regions of tumours, which is crucial for the promising antitumour activity of doxorubicin-related therapy, is governed by two inseparable processes: intercellular diffusion and intracellular retention. To accurately evaluate doxorubicin-related delivery in the avascular regions, these two processes should be assessed together. Here we describe a new approach to such an assessment.

EXPERIMENTAL APPROACH

An individual-cell-based mathematical model based on multicellular tumour spheroids was developed that describes the different intercellular diffusion and intracellular retention kinetics of doxorubicin in each cell layer. The different effects of a P-glycoprotein inhibitor (LY335979) and a hypoxia inhibitor (YC-1) were quantitatively evaluated and compared, in vitro (tumour spheroids) and in vivo (HepG2 tumours in mice). This approach was further tested by evaluating in these models, an experimental doxorubicin derivative, INNO 206, which is in Phase II clinical trials.

KEY RESULTS

Inhomogeneous, hypoxia-induced, P-glycoprotein expression compromised active transport of doxorubicin in the central area, that is, far from the vasculature. LY335979 inhibited efflux due to P-glycoprotein but limited levels of doxorubicin outside the inner cells, whereas YC-1 co-administration specifically increased doxorubicin accumulation in the inner cells without affecting the extracellular levels. INNO 206 exhibited a more effective distribution profile than doxorubicin.

CONCLUSIONS AND IMPLICATIONS

The individual-cell-based mathematical model accurately evaluated and predicted doxorubicin-related delivery and regulation in the avascular regions of tumours. The described framework provides a mechanistic basis for the proper development of doxorubicin-related drug co-administration profiles and nanoparticle development and could avoid unnecessary clinical trials.

3D tumor spheroids: an overview on the tools and techniques used for their analysis

In comparison with 2D cell culture models, 3D spheroids are able to accurately mimic some features of solid tumors, such as their spatial architecture, physiological responses, secretion of soluble mediators, gene expression patterns and drug resistance mechanisms. These unique characteristics highlight the potential of 3D cellular aggregates to be used as in vitro models for screening new anticancer therapeutics, both at a small and large scale. Nevertheless, few reports have focused on describing the tools and techniques currently available to extract significant biological data from these models. Such information will be fundamental to drug and therapeutic discovery process using 3D cell culture models. The present review provides an overview of the techniques that can be employed to characterize and evaluate the efficacy of anticancer therapeutics in 3D tumor spheroids.

Pharmacokinetics of an injectable modified-release 2-hydroxyflutamide formulation in the human prostate gland using a semiphysiologically based biopharmaceutical model

The local distribution of 2-hydroxyflutamide (2-HOF) in prostate tissue after a single intraprostatic injection of a novel parenteral modified-release (MR) formulation in patients with localized prostate cancer was estimated using a semiphysiologically based biopharmaceutical model. Plasma concentration-time profiles for 2-HOF were acquired from a phase II study in 24 patients and the dissolution of the MR formulation was investigated in vitro. Human physiological values and the specific physicochemical properties of 2-HOF were obtained from the literature or calculated via established algorithms. A compartmental modeling approach was adopted for tissue and blood in the prostate gland, where the compartments were modeled as a series of concentric spherical shells contouring the centrally positioned depot formulation. Discrete fluid connections between the blood compartments were described by the representative flow of blood, whereas the mass transport of drug from tissue to tissue and tissue to blood was described by a one-dimensional diffusion approximation. An empirical dissolution approach was adopted for the release of 2-HOF from the formulation. The model adequately described the plasma concentration-time profiles of 2-HOF. Predictive simulations indicated that the local tissue concentration of 2-HOF within a distance of 5 mm from the depot formulation was approximately 40 times higher than that of unbound 2-HOF in plasma. The simulations also indicated that spreading the formulation throughout the prostate gland would expose more of the gland and increase the overall release rate of 2-HOF from the given dose. The increased release rate would initially increase the tissue and plasma concentrations but would also reduce the terminal half-life of 2-HOF in plasma. Finally, an in vitro-in vivo correlation of the release of 2-HOF from the parenteral MR formulation was established. This study shows that intraprostatic 2-HOF concentrations are significantly higher than systemic plasma concentrations and that increased distribution of 2-HOF throughout the gland, using strategic imaging-guided administration, is possible. This novel parenteral MR formulation, thus, facilitates good pharmacological effect while minimizing the risk of side effects.