Quantitative analysis of tumour spheroid structure

Identification of MCM4 and PRKDC as new regulators of osteosarcoma cell dormancy based on 3D cell cultures

Dormancy is a potential way for tumors to develop drug resistance and escape treatment. However, the mechanisms involved in cancer dormancy remain poorly understood. This is mainly because there is no in vitro culture model making it possible to spontaneously induce dormancy. In this context, the present work proposes the use of three-dimensional (3D) spheroids developed from osteosarcoma cell lines as a relevant model for studying cancer dormancy. MNNG-HOS, SaOS-2, 143B, MG-63, U2OS and SJSA-1 cell lines were cultured in 3D using the Liquid Overlay Technique (LOT). Dormancy was studied by staining cancer cells with a lipophilic dye (DiD), and long-term DiD+ cells were considered as dormant cancer cells. The role of the extracellular matrix in inducing dormancy was investigated by embedding cells into methylcellulose or Geltrex™. Gene expression of DiD+ cells was assessed with a Nanostring™ approach and the role of the genes detected in dormancy was validated by a transient down-expression model using siRNA treatment. Proliferation was measured using fluorescence microscopy and the xCELLigence technology. We observed that MNNG-HOS, 143B and MG-G3 cell lines had a reduced proliferation rate in 3D compared to 2D. U2OS cells had an increased proliferation rate when they were cultured in Geltrex™ compared to other 3D culture methods. Using 3D cultures, a transcriptomic signature of dormancy was obtained and showed a decreased expression of 18 genes including ETV4, HELLS, ITGA6, MCM4, PRKDC, RAD21 and UBE2T. The treatment with siRNA targeting these genes showed that cancer cell proliferation was reduced when the expression of ETV4 and MCM4 were decreased, whereas proliferation was increased when the expression of RAD21 was decreased. 3D culture facilitates the maintenance of dormant cancer cells characterized by a reduced proliferation and less differential gene expression as compared to proliferative cells. Further studies of the genes involved has enabled us to envisage their role in regulating cell proliferation.

Principles of Hanging Drop Method (Spheroid Formation) in Cell Culture

A type of three-dimensional (3D) cell culture models which is simple and easy is hanging drop method. The hanging drop method emerges as a pivotal technique with diverse applications in cancer research and cell biology. This method facilitates the formation of multicellular spheroids, providing a unique environment for studying cell behavior dynamics. The hanging drop method's theoretical underpinning relies on gravity-enforced self-assembly, allowing for cost-effective, reproducible 3D cell cultures with controlled spheroid sizes. The advantages of this approach include its efficiency in producing cellular heterogeneity, particularly in non-adherent 3D cultures, and its ability to create hypoxic spheroids, making it a suitable model for studying cancer. Moreover, the hanging drop method has proven valuable in investigating various aspects such as tissue structure, signaling pathways, immune activation of cancer cells, and notably, cell proliferation. Researchers have utilized the hanging drop method to explore the dynamics of cell proliferation, studying the effects of mesenchymal stem cells (MSC) secretome on cancer cells. The method's application involves co-culturing different cell lines, assessing spheroid formations, and quantifying their sizes over time. These studies have unveiled intricate cell behavior dynamics, demonstrating how the MSC secretome influences cancer cell growth and viability within a three-dimensional co-culture paradigm.

Patient-derived melanoma models

Melanoma is a very aggressive, rapidly metastasizing tumor that has been studied intensively in the past regarding the underlying genetic and molecular mechanisms. More recently developed treatment modalities have improved response rates and overall survival of patients. However, the majority of patients suffer from secondary treatment resistance, which requires in depth analyses of the underlying mechanisms. Here, melanoma models based on patients-derived material may play an important role. Consequently, a plethora of different experimental techniques have been developed in the past years. Among these are 3D and 4D culture techniques, organotypic skin reconstructs, melanoma-on-chip models and patient-derived xenografts, Every technique has its own strengths but also weaknesses regarding throughput, reproducibility, and reflection of the human situation. Here, we provide a comprehensive overview of currently used techniques and discuss their use in different experimental settings.

Delivery of Anticancer Drugs Using Microbubble-Assisted Ultrasound in a 3D Spheroid Model

Tumor spheroids are promising three-dimensional (3D) in vitro tumor models for the evaluation of drug delivery methods. The design of noninvasive and targeted drug methods is required to improve the intratumoral bioavailability of chemotherapeutic drugs and reduce their adverse off-target effects. Among such methods, microbubble-assisted ultrasound (MB-assisted US) is an innovative modality for noninvasive targeted drug delivery. The aim of the present study is to evaluate the efficacy of this US modality for the delivery of bleomycin, doxorubicin, and irinotecan in colorectal cancer (CRC) spheroids. MB-assisted US permeabilized the CRC spheroids to propidium iodide, which was used as a drug model without affecting their growth and viability. Histological analysis and electron microscopy revealed that MB-assisted US affected only the peripheral layer of the CRC spheroids. The acoustically mediated bleomycin delivery induced a significant decrease in CRC spheroid growth in comparison to spheroids treated with bleomycin alone. However, this US modality did not improve the therapeutic efficacy of doxorubicin and irinotecan on CRC spheroids. In conclusion, this study demonstrates that tumor spheroids are a relevant approach to evaluate the efficacy of MB-assisted US for the delivery of chemotherapeutics.

Predicting Radiotherapy Patient Outcomes with Real-Time Clinical Data Using Mathematical Modelling

Longitudinal tumour volume data from head-and-neck cancer patients show that tumours of comparable pre-treatment size and stage may respond very differently to the same radiotherapy fractionation protocol. Mathematical models are often proposed to predict treatment outcome in this context, and have the potential to guide clinical decision-making and inform personalised fractionation protocols. Hindering effective use of models in this context is the sparsity of clinical measurements juxtaposed with the model complexity required to produce the full range of possible patient responses. In this work, we present a compartment model of tumour volume and tumour composition, which, despite relative simplicity, is capable of producing a wide range of patient responses. We then develop novel statistical methodology and leverage a cohort of existing clinical data to produce a predictive model of both tumour volume progression and the associated level of uncertainty that evolves throughout a patient's course of treatment. To capture inter-patient variability, all model parameters are patient specific, with a bootstrap particle filter-like Bayesian approach developed to model a set of training data as prior knowledge. We validate our approach against a subset of unseen data, and demonstrate both the predictive ability of our trained model and its limitations.

Real-Time Cell Cycle Imaging in a 3D Cell Culture Model of Melanoma, Quantitative Analysis, Optical Clearing, and Mathematical Modeling

Aberrant cell cycle progression is a hallmark of solid tumors. Therefore, cell cycle analysis is an invaluable technique to study cancer cell biology. However, cell cycle progression has been most commonly assessed by methods that are limited to temporal snapshots or that lack spatial information. In this chapter, we describe a technique that allows spatiotemporal real-time tracking of cell cycle progression of individual cells in a multicellular context. The power of this system lies in the use of 3D melanoma spheroids generated from melanoma cells engineered with the fluorescent ubiquitination-based cell cycle indicator (FUCCI). This technique, combined with mathematical modeling, allows us to gain further and more detailed insight into several relevant aspects of solid cancer cell biology, such as tumor growth, proliferation, invasion, and drug sensitivity.

Formation and Growth of Co-Culture Tumour Spheroids: New Compartment-Based Mathematical Models and Experiments

Co-culture tumour spheroid experiments are routinely performed to investigate cancer progression and test anti-cancer therapies. Therefore, methods to quantitatively characterise and interpret co-culture spheroid growth are of great interest. However, co-culture spheroid growth is complex. Multiple biological processes occur on overlapping timescales and different cell types within the spheroid may have different characteristics, such as differing proliferation rates or responses to nutrient availability. At present there is no standard, widely-accepted mathematical model of such complex spatio-temporal growth processes. Typical approaches to analyse these experiments focus on the late-time temporal evolution of spheroid size and overlook early-time spheroid formation, spheroid structure and geometry. Here, using a range of ordinary differential equation-based mathematical models and parameter estimation, we interpret new co-culture experimental data. We provide new biological insights about spheroid formation, growth, and structure. As part of this analysis we connect Greenspan's seminal mathematical model to co-culture data for the first time. Furthermore, we generalise a class of compartment-based spheroid mathematical models that have previously been restricted to one population so they can be applied to multiple populations. As special cases of the general model, we explore multiple natural two population extensions to Greenspan's seminal model and reveal biological mechanisms that can describe the internal dynamics of growing co-culture spheroids and those that cannot. This mathematical and statistical modelling-based framework is well-suited to analyse spheroids grown with multiple different cell types and the new class of mathematical models provide opportunities for further mathematical and biological insights.

Free and Interfacial Boundaries in Individual-Based Models of Multicellular Biological systems

Coordination of cell behaviour is key to a myriad of biological processes including tissue morphogenesis, wound healing, and tumour growth. As such, individual-based computational models, which explicitly describe inter-cellular interactions, are commonly used to model collective cell dynamics. However, when using individual-based models, it is unclear how descriptions of cell boundaries affect overall population dynamics. In order to investigate this we define three cell boundary descriptions of varying complexities for each of three widely used off-lattice individual-based models: overlapping spheres, Voronoi tessellation, and vertex models. We apply our models to multiple biological scenarios to investigate how cell boundary description can influence tissue-scale behaviour. We find that the Voronoi tessellation model is most sensitive to changes in the cell boundary description with basic models being inappropriate in many cases. The timescale of tissue evolution when using an overlapping spheres model is coupled to the boundary description. The vertex model is demonstrated to be the most stable to changes in boundary description, though still exhibits timescale sensitivity. When using individual-based computational models one should carefully consider how cell boundaries are defined. To inform future work, we provide an exploration of common individual-based models and cell boundary descriptions in frequently studied biological scenarios and discuss their benefits and disadvantages.

Mera: A scalable high throughput automated micro-physiological system

There is an urgent need for scalable Microphysiological Systems (MPS's)1 that can better predict drug efficacy and toxicity at the preclinical screening stage. Here we present Mera, an automated, modular and scalable system for culturing and assaying microtissues with interconnected fluidics, inbuilt environmental control and automated image capture. The system presented has multiple possible fluidics modes. Of these the primary mode is designed so that cells may be matured into a desired microtissue type and in the secondary mode the fluid flow can be re-orientated to create a recirculating circuit composed of inter-connected channels to allow drugging or staining. We present data demonstrating the prototype system Mera using an Acetaminophen/HepG2 liver microtissue toxicity assay with Calcein AM and Ethidium Homodimer (EtHD1) viability assays. We demonstrate the functionality of the automated image capture system. The prototype microtissue culture plate wells are laid out in a 3 × 3 or 4 × 10 grid format with viability and toxicity assays demonstrated in both formats. In this paper we set the groundwork for the Mera system as a viable option for scalable microtissue culture and assay development.

Rationally Designed Enzyme-Resistant Peptidic Assemblies for Plasma Membrane Targeting in Cancer Treatment

Peptides are being increasingly important for subcellular targeted cancer treatment to improve specificity and reverse multidrug resistance. However, there has been yet any report on targeting plasma membrane (PM) through self-assembling peptides. A simple synthetic peptidic molecule (tF4) is developed. It is revealed that tF4 is carboxyl esterase-resistant and self-assembles into vesical nanostructures. Importantly, tF4 assemblies interact with PM through orthogonal hydrogen bonding and hydrophobic interaction to regulate cancer cellular functions. Mechanistically, tF4 assemblies induce stress fiber formation, cytoskeleton reconstruction, and death receptor 4/5 (DR4/5) expression in cancer cells. DR4/5 triggers extrinsic caspase-8 signaling cascade, resulting in cell death. The results provide a new strategy for developing enzyme-resistant and PM-targeting peptidic molecules against cancer.

The Comparative Invasiveness of Endometriotic Cell Lines to Breast and Endometrial Cancer Cell Lines

Endometriosis is an invasive condition that affects 10% of women (and people assigned as female at birth) worldwide. The purpose of this study was to characterize the relative invasiveness of three available endometriotic cell lines (EEC12Z, iEc-ESCs, tHESCs) to cancer cell lines (MDA-MB-231, SW1353 and EM-E6/E7/TERT) and assess whether the relative invasiveness was consistent across different invasion assays. All cell lines were subjected to transwell, spheroid drop, and spheroid-gel invasion assays, and stained for vimentin, cytokeratin, E-Cadherin and N-Cadherin to assess changes in expression. In all assays, endometriotic cell lines showed comparable invasiveness to the cancer cell lines used in this study, with no significant differences in invasiveness identified. EEC12Z cells that had invaded within the assay periods showed declines in E-Cadherin expression compared to cells that had not invaded within the assay period, without significant changes in N-Cadherin expression, which may support the hypothesis that an epithelial-to-mesenchymal transition is an influence on the invasiveness shown by this peritoneal endometriosis cell line.

Applications and Advances of Multicellular Tumor Spheroids: Challenges in Their Development and Analysis

Biomedical research requires both in vitro and in vivo studies in order to explore disease processes or drug interactions. Foundational investigations have been performed at the cellular level using two-dimensional cultures as the gold-standard method since the early 20th century. However, three-dimensional (3D) cultures have emerged as a new tool for tissue modeling over the last few years, bridging the gap between in vitro and animal model studies. Cancer has been a worldwide challenge for the biomedical community due to its high morbidity and mortality rates. Various methods have been developed to produce multicellular tumor spheroids (MCTSs), including scaffold-free and scaffold-based structures, which usually depend on the demands of the cells used and the related biological question. MCTSs are increasingly utilized in studies involving cancer cell metabolism and cell cycle defects. These studies produce massive amounts of data, which demand elaborate and complex tools for thorough analysis. In this review, we discuss the advantages and disadvantages of several up-to-date methods used to construct MCTSs. In addition, we also present advanced methods for analyzing MCTS features. As MCTSs more closely mimic the in vivo tumor environment, compared to 2D monolayers, they can evolve to be an appealing model for in vitro tumor biology studies.

Replacement, Reduction, and Refinement of Animal Experiments in Anticancer Drug Development: The Contribution of 3D In Vitro Cancer Models in the Drug Efficacy Assessment

In the last decades three-dimensional (3D) in vitro cancer models have been proposed as a bridge between bidimensional (2D) cell cultures and in vivo animal models, the gold standards in the preclinical assessment of anticancer drug efficacy. 3D in vitro cancer models can be generated through a multitude of techniques, from both immortalized cancer cell lines and primary patient-derived tumor tissue. Among them, spheroids and organoids represent the most versatile and promising models, as they faithfully recapitulate the complexity and heterogeneity of human cancers. Although their recent applications include drug screening programs and personalized medicine, 3D in vitro cancer models have not yet been established as preclinical tools for studying anticancer drug efficacy and supporting preclinical-to-clinical translation, which remains mainly based on animal experimentation. In this review, we describe the state-of-the-art of 3D in vitro cancer models for the efficacy evaluation of anticancer agents, focusing on their potential contribution to replace, reduce and refine animal experimentations, highlighting their strength and weakness, and discussing possible perspectives to overcome current challenges.

Development of a scoring function for comparing simulated and experimental tumor spheroids

Progress continues in the field of cancer biology, yet much remains to be unveiled regarding the mechanisms of cancer invasion. In particular, complex biophysical mechanisms enable a tumor to remodel the surrounding extracellular matrix (ECM), allowing cells to invade alone or collectively. Tumor spheroids cultured in collagen represent a simplified, reproducible 3D model system, which is sufficiently complex to recapitulate the evolving organization of cells and interaction with the ECM that occur during invasion. Recent experimental approaches enable high resolution imaging and quantification of the internal structure of invading tumor spheroids. Concurrently, computational modeling enables simulations of complex multicellular aggregates based on first principles. The comparison between real and simulated spheroids represents a way to fully exploit both data sources, but remains a challenge. We hypothesize that comparing any two spheroids requires first the extraction of basic features from the raw data, and second the definition of key metrics to match such features. Here, we present a novel method to compare spatial features of spheroids in 3D. To do so, we define and extract features from spheroid point cloud data, which we simulated using Cells in Silico (CiS), a high-performance framework for large-scale tissue modeling previously developed by us. We then define metrics to compare features between individual spheroids, and combine all metrics into an overall deviation score. Finally, we use our features to compare experimental data on invading spheroids in increasing collagen densities. We propose that our approach represents the basis for defining improved metrics to compare large 3D data sets. Moving forward, this approach will enable the detailed analysis of spheroids of any origin, one application of which is informing in silico spheroids based on their in vitro counterparts. This will enable both basic and applied researchers to close the loop between modeling and experiments in cancer research.

Development of a 3D Tumor Spheroid Model from the Patient's Glioblastoma Cells and Its Study by Metabolic Fluorescence Lifetime Imaging

Patient-specific in vitro tumor models are a promising platform for studying the mechanisms of oncogenesis and personalized selection of drugs. In case of glial brain tumors, development and use of such models is particularly relevant as the effectiveness of such tumor treatment remains extremely unsatisfactory. The aim of the study was to develop a model of a 3D tumor glioblastoma spheroid based on a patient's surgical material and to study its metabolic characteristics by means of fluorescence lifetime imaging microscopy of metabolic coenzymes.

Materials and Methods

The study was conducted with tumor samples from patients diagnosed with glioblastoma (Grade IV). To create spheroids, primary cultures were isolated from tumor tissue samples; the said cultures were characterized morphologically and immunocytochemically, and then planted into round-bottom ultra low-adhesion plates. The number of cells for planting was chosen empirically. The characteristics of the growth of cell cultures were compared with spheroids from glioblastomas of patients with U373 MG stable line of human glioblastoma. Visualization of autofluorescence of metabolic coenzymes of nicotinamide adenine dinucleotide (phosphate) NAD(P)H and flavin adenine dinucleotide (FAD) in spheroids was performed by means of an LSM 880 laser scanning microscope (Carl Zeiss, Germany) with a FLIM module (Becker & Hickl GmbH, Germany). The autofluorescence decay parameters were studied under normoxic and hypoxic conditions (3.5% О₂).

Results

An original protocol for 3D glioblastoma spheroids cultivation was developed. Primary glial cultures from surgical material of patients were obtained and characterized. The isolated glioblastoma cells had a spindle-shaped morphology with numerous processes and a pronounced granularity of cytoplasm. All cultures expressed glial fibrillary acidic protein (GFAP). The optimal seeding dose of 2000 cells per well was specified; its application results in formation of spheroids with a dense structure and stable growth during 7 days. The FLIM method helped to establish that spheroid cells from the patient material had a generally similar metabolism to spheroids from the stable line, however, they demonstrated more pronounced metabolic heterogeneity. Cultivation of spheroids under hypoxic conditions revealed a transition to a more glycolytic type of metabolism, which is expressed in an increase in the contribution of the free form of NAD(P)H to fluorescence decay.

Conclusion

The developed model of tumor spheroids from patients' glioblastomas in combination with the FLIM can serve as a tool to study characteristics of tumor metabolism and develop predictive tests to evaluate the effectiveness of antitumor therapy.

Advancements in melanoma cancer metastasis models

Metastatic melanoma is a complex and deadly disease. Due to its complexity, the development of novel therapeutic strategies to inhibit metastatic melanoma remains an outstanding challenge. Our ability to study metastasis is advanced with the development of in vitro and in vivo models that better mimic the different steps of the metastatic cascade beginning from primary tumor initiation to final metastatic seeding. In this review, we provide a comprehensive summary of in vitro models, in vivo models, and in silico platforms to study the individual steps of melanoma metastasis. Furthermore, we highlight the advantages and limitations of each model and discuss the challenges of how to improve current models to enhance translation for melanoma cancer patients and future therapies.

Growth and adaptation mechanisms of tumour spheroids with time-dependent oxygen availability

Tumours are subject to external environmental variability. However, in vitro tumour spheroid experiments, used to understand cancer progression and develop cancer therapies, have been routinely performed for the past fifty years in constant external environments. Furthermore, spheroids are typically grown in ambient atmospheric oxygen (normoxia), whereas most in vivo tumours exist in hypoxic environments. Therefore, there are clear discrepancies between in vitro and in vivo conditions. We explore these discrepancies by combining tools from experimental biology, mathematical modelling, and statistical uncertainty quantification. Focusing on oxygen variability to develop our framework, we reveal key biological mechanisms governing tumour spheroid growth. Growing spheroids in time-dependent conditions, we identify and quantify novel biological adaptation mechanisms, including unexpected necrotic core removal, and transient reversal of the tumour spheroid growth phases.

Geometric analysis enables biological insight from complex non-identifiable models using simple surrogates

An enduring challenge in computational biology is to balance data quality and quantity with model complexity. Tools such as identifiability analysis and information criterion have been developed to harmonise this juxtaposition, yet cannot always resolve the mismatch between available data and the granularity required in mathematical models to answer important biological questions. Often, it is only simple phenomenological models, such as the logistic and Gompertz growth models, that are identifiable from standard experimental measurements. To draw insights from complex, non-identifiable models that incorporate key biological mechanisms of interest, we study the geometry of a map in parameter space from the complex model to a simple, identifiable, surrogate model. By studying how non-identifiable parameters in the complex model quantitatively relate to identifiable parameters in surrogate, we introduce and exploit a layer of interpretation between the set of non-identifiable parameters and the goodness-of-fit metric or likelihood studied in typical identifiability analysis. We demonstrate our approach by analysing a hierarchy of mathematical models for multicellular tumour spheroid growth experiments. Typical data from tumour spheroid experiments are limited and noisy, and corresponding mathematical models are very often made arbitrarily complex. Our geometric approach is able to predict non-identifiabilities, classify non-identifiable parameter spaces into identifiable parameter combinations that relate to features in the data characterised by parameters in a surrogate model, and overall provide additional biological insight from complex non-identifiable models.

Utility of the Cerebral Organoid Glioma 'GLICO' Model for Screening Applications

Glioblastoma, a grade IV astrocytoma, is regarded as the most aggressive primary brain tumour with an overall median survival of 16.0 months following the standard treatment regimen of surgical resection, followed by radiotherapy and chemotherapy with temozolomide. Despite such intensive treatment, the tumour almost invariably recurs. This poor prognosis has most commonly been attributed to the initiation, propagation, and differentiation of cancer stem cells. Despite the unprecedented advances in biomedical research over the last decade, the current in vitro models are limited at preserving the inter- and intra-tumoural heterogeneity of primary tumours. The ability to understand and manipulate complex cancers such as glioblastoma requires disease models to be clinically and translationally relevant and encompass the cellular heterogeneity of such cancers. Therefore, brain cancer research models need to aim to recapitulate glioblastoma stem cell function, whilst remaining amenable for analysis. Fortunately, the recent development of 3D cultures has overcome some of these challenges, and cerebral organoids are emerging as cutting-edge tools in glioblastoma research. The opportunity to generate cerebral organoids via induced pluripotent stem cells, and to perform co-cultures with patient-derived cancer stem cells (GLICO model), has enabled the analysis of cancer development in a context that better mimics brain tissue architecture. In this article, we review the recent literature on the use of patient-derived glioblastoma organoid models and their applicability for drug screening, as well as provide a potential workflow for screening using the GLICO model. The proposed workflow is practical for use in most laboratories with accessible materials and equipment, a good first pass, and no animal work required. This workflow is also amenable for analysis, with separate measures of invasion, growth, and viability.

In Vitro Setup for Determination of Nanoparticle-Mediated Magnetic Cell and Drug Accumulation in Tumor Spheroids under Flow Conditions

Superparamagnetic iron oxide nanoparticles (SPIONs) are used in nanomedicine as transporter systems for therapeutic cargos, or to magnetize cells to make them magnetically guidable. In cancer treatment, the site-directed delivery of chemotherapeutics or immune effector cells to the tumor can increase the therapeutic efficacy in the target region, and simultaneously reduce toxic side-effects in the rest of the body. To enable the transfer of new methods, such as the nanoparticle-mediated transport from bench to bedside, suitable experimental setups must be developed. In vivo, the SPIONs or SPION-loaded cells must be applied into the blood stream, to finally reach the tumor: consequently, targeting and treatment efficacy should be analyzed under conditions which are as close to in vivo as possible. Here, we established an in vitro method, including tumor spheroids placed in a chamber system under the influence of a magnetic field, and adapted to a peristaltic pump, to mimic the blood flow. This enabled us to analyze the magnetic capture and antitumor effects of magnetically targeted mitoxantrone and immune cells under dynamic conditions. We showed that the magnetic nanoparticle-mediated accumulation increased the anti-tumor effects, and reduced the unspecific distribution of both mitoxantrone and cells. Especially for nanomedical research, investigation of the site-specific targeting of particles, cells or drugs under circulation is important. We conclude that our in vitro setup improves the screening process of nanomedical candidates for cancer treatment.

Computationally efficient framework for diagnosing, understanding and predicting biphasic population growth

Throughout the life sciences, biological populations undergo multiple phases of growth, often referred to as biphasic growth for the commonly encountered situation involving two phases. Biphasic population growth occurs over a massive range of spatial and temporal scales, ranging from microscopic growth of tumours over several days, to decades-long regrowth of corals in coral reefs that can extend for hundreds of kilometres. Different mathematical models and statistical methods are used to diagnose, understand and predict biphasic growth. Common approaches can lead to inaccurate predictions of future growth that may result in inappropriate management and intervention strategies being implemented. Here, we develop a very general computationally efficient framework, based on profile likelihood analysis, for diagnosing, understanding and predicting biphasic population growth. The two key components of the framework are as follows: (i) an efficient method to form approximate confidence intervals for the change point of the growth dynamics and model parameters and (ii) parameter-wise profile predictions that systematically reveal the influence of individual model parameters on predictions. To illustrate our framework we explore real-world case studies across the life sciences.

Efficient inference and identifiability analysis for differential equation models with random parameters

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.

Parameter estimation and uncertainty quantification using information geometry

In this work, we: (i) review likelihood-based inference for parameter estimation and the construction of confidence regions; and (ii) explore the use of techniques from information geometry, including geodesic curves and Riemann scalar curvature, to supplement typical techniques for uncertainty quantification, such as Bayesian methods, profile likelihood, asymptotic analysis and bootstrapping. These techniques from information geometry provide data-independent insights into uncertainty and identifiability, and can be used to inform data collection decisions. All code used in this work to implement the inference and information geometry techniques is available on GitHub.

A stochastic mathematical model of 4D tumour spheroids with real-time fluorescent cell cycle labelling

In vitro tumour spheroids have been used to study avascular tumour growth and drug design for over 50 years. Tumour spheroids exhibit heterogeneity within the growing population that is thought to be related to spatial and temporal differences in nutrient availability. The recent development of real-time fluorescent cell cycle imaging allows us to identify the position and cell cycle status of individual cells within the growing spheroid, giving rise to the notion of a four-dimensional (4D) tumour spheroid. We develop the first stochastic individual-based model (IBM) of a 4D tumour spheroid and show that IBM simulation data compares well with experimental data using a primary human melanoma cell line. The IBM provides quantitative information about nutrient availability within the spheroid, which is important because it is difficult to measure these data experimentally.

Designing and interpreting 4D tumour spheroid experiments

Tumour spheroid experiments are routinely used to study cancer progression and treatment. Various and inconsistent experimental designs are used, leading to challenges in interpretation and reproducibility. Using multiple experimental designs, live-dead cell staining, and real-time cell cycle imaging, we measure necrotic and proliferation-inhibited regions in over 1000 4D tumour spheroids (3D space plus cell cycle status). By intentionally varying the initial spheroid size and temporal sampling frequencies across multiple cell lines, we collect an abundance of measurements of internal spheroid structure. These data are difficult to compare and interpret. However, using an objective mathematical modelling framework and statistical identifiability analysis we quantitatively compare experimental designs and identify design choices that produce reliable biological insight. Measurements of internal spheroid structure provide the most insight, whereas varying initial spheroid size and temporal measurement frequency is less important. Our general framework applies to spheroids grown in different conditions and with different cell types.