Modelling count data with partial differential equation models in biology.

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

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.

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.

Measuring and Modelling the Epithelial- Mesenchymal Hybrid State in Cancer: Clinical Implications.

The epithelial-mesenchymal (E/M) hybrid state has emerged as an important mediator of elements of cancer progression, facilitated by epithelial mesenchymal plasticity (EMP). We review here evidence for the presence, prognostic significance, and therapeutic potential of the E/M hybrid state in carcinoma. We further assess modelling predictions and validation studies to demonstrate stabilised E/M hybrid states along the spectrum of EMP, as well as computational approaches for characterising and quantifying EMP phenotypes, with particular attention to the emerging realm of single-cell approaches through RNA sequencing and protein-based techniques.

Mechanical Pressure Driving Proteoglycan Expression in Mammographic Density: a Self-perpetuating Cycle?

Regions of high mammographic density (MD) in the breast are characterised by a proteoglycan (PG)-rich fibrous stroma, where PGs mediate aligned collagen fibrils to control tissue stiffness and hence the response to mechanical forces. Literature is accumulating to support the notion that mechanical stiffness may drive PG synthesis in the breast contributing to MD. We review emerging patterns in MD and other biological settings, of a positive feedback cycle of force promoting PG synthesis, such as in articular cartilage, due to increased pressure on weight bearing joints. Furthermore, we present evidence to suggest a pro-tumorigenic effect of increased mechanical force on epithelial cells in contexts where PG-mediated, aligned collagen fibrous tissue abounds, with implications for breast cancer development attributable to high MD. Finally, we summarise means through which this positive feedback mechanism of PG synthesis may be intercepted to reduce mechanical force within tissues and thus reduce disease burden.

The role of mechanical interactions in EMT.

The detachment of cells from the boundary of an epithelial tissue and the subsequent invasion of these cells into surrounding tissues is important for cancer development and wound healing, and is strongly associated with the epithelial-mesenchymal transition (EMT). Chemical signals, such as TGF-ß, produced by surrounding tissue can be uptaken by cells and induce EMT. In this work, we present a novel cell-based discrete mathematical model of mechanical cellular relaxation, cell proliferation, and cell detachment driven by chemically-dependent EMT in an epithelial tissue. A continuum description of the model is then derived in the form of a novel nonlinear free boundary problem. Using the discrete and continuum models we explore how the coupling of chemical transport and mechanical interactions influences EMT, and postulate how this could be used to help control EMT in pathological situations.

Mechanical Cell Competition in Heterogeneous Epithelial Tissues.

Mechanical cell competition is important during tissue development, cancer invasion, and tissue ageing. Heterogeneity plays a key role in practical applications since cancer cells can have different cell stiffness and different proliferation rates than normal cells. To study this phenomenon, we propose a one-dimensional mechanical model of heterogeneous epithelial tissue dynamics that includes cell-length-dependent proliferation and death mechanisms. Proliferation and death are incorporated into the discrete model stochastically and arise as source/sink terms in the corresponding continuum model that we derive. Using the new discrete model and continuum description, we explore several applications including the evolution of homogeneous tissues experiencing proliferation and death, and competition in a heterogeneous setting with a cancerous tissue competing for space with an adjacent normal tissue. This framework allows us to postulate new mechanisms that explain the ability of cancer cells to outcompete healthy cells through mechanical differences rather than an intrinsic proliferative advantage. We advise when the continuum model is beneficial and demonstrate why naively adding source/sink terms to a continuum model without considering the underlying discrete model may lead to incorrect results.

Mechanical Model of Dexterous Continuum Manipulators with Compliant Joints and Tendon/External Force Interactions.

Dexterous continuum manipulators (DCMs) have been widely adopted for minimally- and less-invasive surgery. During the operation, these DCMs interact with surrounding anatomy actively or passively. The interaction force will inevitably affect the tip position and shape of DCMs, leading to potentially inaccurate control near critical anatomy. In this paper, we demonstrated a 2D mechanical model for a tendon actuated, notched DCM with compliant joints. The model predicted deformation of the DCM accurately in the presence of tendon force, friction force, and external force. A partition approach was proposed to describe the DCM as a series of interconnected rigid and flexible links. Beam mechanics, taking into consideration tendon interaction and external force on the tip and the body, was applied to obtain the deformation of each flexible link of the DCM. The model results were compared with experiments for free bending as well as bending in the presence of external forces acting at either the tip or body of the DCM. The overall mean error of tip position between model predictions and all of the experimental results was 0.62±0.41mm. The results suggest that the proposed model can effectively predict the shape of the DCM.

A Craniomaxillofacial Surgical Assistance Workstation for Enhanced Single-Stage Reconstruction Using Patient-Specific Implants.

BACKGROUND: Craniomaxillofacial reconstruction with patient-specific, customized craniofacial implants (CCIs) is ideal for skeletal defects involving areas of aesthetic concern-the non-weight-bearing facial skeleton, temporal skull, and/or frontal-forehead region. Results to date are superior to a variety of "off-the-shelf" materials, but require a protocol computed tomography scan and preexisting defect for computer-assisted design/computer-assisted manufacturing of the CCI. The authors developed a craniomaxillofacial surgical assistance workstation to address these challenges and intraoperatively guide CCI modification for an unknown defect size/shape. METHODS: First, the surgeon designed an oversized CCI based on his/her surgical plan. Intraoperatively, the surgeon resected the bone and digitized the resection using a navigation pointer. Next, a projector displayed the limits of the craniofacial bone defect onto the prefabricated, oversized CCI for the size modification process; the surgeon followed the projected trace to modify the implant. A cadaveric study compared the standard technique (n = 1) to the experimental technique (n = 5) using surgical time and implant fit. RESULTS: The technology reduced the time and effort needed to resize the oversized CCI by an order of magnitude as compared with the standard manual resizing process. Implant fit was consistently better for the computer-assisted case compared with the control by at least 30%, requiring only 5.17 minutes in the computer-assisted cases compared with 35 minutes for the control. CONCLUSION: This approach demonstrated improvement in surgical time and accuracy of CCI-based craniomaxillofacial reconstruction compared with previously reported methods. The craniomaxillofacial surgical assistance workstation will provide craniofacial surgeons a computer-assisted technology for effective and efficient single-stage reconstruction when exact craniofacial bone defect sizes are unknown.

Quantifying "Softness" of Organic Coatings on Gold Nanoparticles Using Correlated Small-Angle X-ray and Neutron Scattering.

Small-angle X-ray and neutron scattering provide powerful tools to selectively characterize the inorganic and organic components of hybrid nanomaterials. Using hydrophobic gold nanoparticles coated with several commercial and dendritic thiols, the size of the organic layer on the gold particles is shown to increase from 1.2 to 4.1 nm. A comparison between solid-state diffraction from self-assembled lattices of nanoparticles and the solution data from neutron scattering suggests that engineering softness/deformability in nanoparticle coatings is less straightforward than simply increasing the organic size. The "dendritic effect" in which higher generations yield increasingly compact molecules explains changes in the deformability of organic ligand shells.

Shape Tracking of a Dexterous Continuum Manipulator Utilizing Two Large Deflection Shape Sensors.

Dexterous continuum manipulators (DCMs) can largely increase the reachable region and steerability for minimally and less invasive surgery. Many such procedures require the DCM to be capable of producing large deflections. The real-time control of the DCM shape requires sensors that accurately detect and report large deflections. We propose a novel, large deflection, shape sensor to track the shape of a 35 mm DCM designed for a less invasive treatment of osteolysis. Two shape sensors, each with three fiber Bragg grating sensing nodes is embedded within the DCM, and the sensors' distal ends fixed to the DCM. The DCM centerline is computed using the centerlines of each sensor curve. An experimental platform was built and different groups of experiments were carried out, including free bending and three cases of bending with obstacles. For each experiment, the DCM drive cable was pulled with a precise linear slide stage, the DCM centerline was calculated, and a 2D camera image was captured for verification. The reconstructed shape created with the shape sensors is compared with the ground truth generated by executing a 2D-3D registration between the camera image and 3D DCM model. Results show that the distal tip tracking accuracy is 0.40 ± 0.30 mm for the free bending and 0.61 ± 0.15 mm, 0.93 ± 0.05 mm and 0.23 ± 0.10 mm for three cases of bending with obstacles. The data suggest FBG arrays can accurately characterize the shape of large-deflection DCMs.

Modeling Cable and Guide Channel Interaction in a High-Strength Cable-Driven Continuum Manipulator.

This paper presents several mechanical models of a high-strength cable-driven dexterous manipulator designed for surgical procedures. A stiffness model is presented that distinguishes between contributions from the cables and the backbone. A physics-based model incorporating cable friction is developed and its predictions are compared with experimental data. The data show that under high tension and high curvature, the shape of the manipulator deviates significantly from a circular arc. However, simple parametric models can fit the shape with good accuracy. The motivating application for this study is to develop a model so that shape can be predicted using easily measured quantities such as tension, so that real-time navigation may be performed, especially in minimally-invasive surgical procedures, while reducing the need for hazardous imaging methods such as fluoroscopy.

Ancillary procedures necessary for translational research in experimental craniomaxillofacial surgery.

INTRODUCTION: Swine are often regarded as having analogous facial skeletons to humans and therefore serve as an ideal animal model for translational investigation. However, there is a dearth of literature describing the pertinent ancillary procedures required for craniomaxillofacial research. With this in mind, our objective was to evaluate all necessary procedures required for perioperative management and animal safety related to experimental craniomaxillofacial surgical procedures such as orthotopic, maxillofacial transplantation. METHODS: Miniature swine (n = 9) were used to investigate perioperative airway management, methods for providing nutrition, and long-dwelling intravenous access. Flap perfusion using near-infrared laser angiography and facial nerve assessment with electromyoneurography were explored. RESULTS: Bivona tracheostomy was deemed appropriate versus Shiley because soft, wire-reinforced tubing reduced the incidence of tracheal necrosis. Percutaneous endoscopic gastrostomy tube, as opposed to esophagostomy, provided a reliable route for postoperative feeding. Femoral venous access with dorsal tunneling proved to be an ideal option being far from pertinent neck vessels. Laser angiography was beneficial for real-time evaluation of graft perfusion. Facial electromyoneurography techniques for tracing capture were found most optimal using percutaneous leads near the oral commissure.Experience shows that ancillary procedures are critical, and malpositioning of devices may lead to irreversible sequelae with premature animal death. CONCLUSIONS: Face-jaw-teeth transplantation in swine is a complicated procedure that demands special attention to airway, feeding, and intravascular access. It is critical that each ancillary procedure be performed by a dedicated team familiar with relevant anatomy and protocol. Emphasis should be placed on secure skin-level fixation for all tube/lines to minimize risk for dislodgement. A reliable veterinarian team is invaluable and critical for long-term success.

Preliminary development of a workstation for craniomaxillofacial surgical procedures: introducing a computer-assisted planning and execution system.

INTRODUCTION: Facial transplantation represents one of the most complicated scenarios in craniofacial surgery because of skeletal, aesthetic, and dental discrepancies between donor and recipient. However, standard off-the-shelf vendor computer-assisted surgery systems may not provide custom features to mitigate the increased complexity of this particular procedure. We propose to develop a computer-assisted surgery solution customized for preoperative planning, intraoperative navigation including cutting guides, and dynamic, instantaneous feedback of cephalometric measurements/angles as needed for facial transplantation and other related craniomaxillofacial procedures. METHODS: We developed the Computer-Assisted Planning and Execution (CAPE) workstation to assist with planning and execution of facial transplantation. Preoperative maxillofacial computed tomography (CT) scans were obtained on 4 size-mismatched miniature swine encompassing 2 live face-jaw-teeth transplants. The system was tested in a laboratory setting using plastic models of mismatched swine, after which the system was used in 2 live swine transplants. Postoperative CT imaging was obtained and compared with the preoperative plan and intraoperative measures from the CAPE workstation for both transplants. RESULTS: Plastic model tests familiarized the team with the CAPE workstation and identified several defects in the workflow. Live swine surgeries demonstrated utility of the CAPE system in the operating room, showing submillimeter registration error of 0.6 ± 0.24 mm and promising qualitative comparisons between intraoperative data and postoperative CT imaging. CONCLUSIONS: The initial development of the CAPE workstation demonstrated that integration of computer planning and intraoperative navigation for facial transplantation are possible with submillimeter accuracy. This approach can potentially improve preoperative planning, allowing ideal donor-recipient matching despite significant size mismatch, and accurate surgical execution for numerous types of craniofacial and orthognathic surgical procedures.

Implementing measurement error models with mechanistic mathematical models in a likelihood-based framework for estimation, identifiability analysis and prediction in the life sciences.

Throughout the life sciences, we routinely seek to interpret measurements and observations using parametrized mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis and prediction, called profile-wise analysis. This frequentist approach to uncertainty quantification for mechanistic models leverages the profile likelihood for targeting parameters and understanding their influence on predictions. Case studies, motivated by simple caricature models routinely used in systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.

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.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics.

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

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.

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.

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.