A Mathematical Modelling Study of Chemotactic Dynamics in Cell Cultures: The Impact of Spatio-temporal Heterogeneity.

As motivated by studies of cellular motility driven by spatiotemporal chemotactic gradients in microdevices, we develop a framework for constructing approximate analytical solutions for the location, speed and cellular densities for cell chemotaxis waves in heterogeneous fields of chemoattractant from the underlying partial differential equation models. In particular, such chemotactic waves are not in general translationally invariant travelling waves, but possess a spatial variation that evolves in time, and may even oscillate back and forth in time, according to the details of the chemotactic gradients. The analytical framework exploits the observation that unbiased cellular diffusive flux is typically small compared to chemotactic fluxes and is first developed and validated for a range of exemplar scenarios. The framework is subsequently applied to more complex models considering the chemoattractant dynamics under more general settings, potentially including those of relevance for representing pathophysiology scenarios in microdevice studies. In particular, even though solutions cannot be constructed in all cases, a wide variety of scenarios can be considered analytically, firstly providing global insight into the important mechanisms and features of cell motility in complex spatiotemporal fields of chemoattractant. Such analytical solutions also provide a means of rapid evaluation of model predictions, with the prospect of application in computationally demanding investigations relating theoretical models and experimental observation, such as Bayesian parameter estimation.

De-risking clinical trial failure through mechanistic simulation.

Drug development typically comprises a combination of pre-clinical experimentation, clinical trials, and statistical data-driven analyses. Therapeutic failure in late-stage clinical development costs the pharmaceutical industry billions of USD per year. Clinical trial simulation represents a key derisking strategy and combining them with mechanistic models allows one to test hypotheses for mechanisms of failure and to improve trial designs. This is illustrated with a T-cell activation model, used to simulate the clinical trials of IMA901, a short-peptide cancer vaccine. Simulation results were consistent with observed outcomes and predicted that responses are limited by peptide off-rates, peptide competition for dendritic cell (DC) binding, and DC migration times. These insights were used to hypothesise alternate trial designs predicted to improve efficacy outcomes. This framework illustrates how mechanistic models can complement clinical, experimental, and data-driven studies to understand, test, and improve trial designs, and how results may differ between humans and mice.

Predicted limited redistribution of T cells to secondary lymphoid tissue correlates with increased risk of haematological malignancies in asplenic patients.

The spleen, a secondary lymphoid tissue (SLT), has an important role in generation of adaptive immune responses. Although splenectomy remains a common procedure, recent studies reported poor prognosis and increased risk of haematological malignancies in asplenic patients. The high baseline trafficking of T lymphocytes to splenic tissue suggests splenectomy may lead to loss of blood-borne malignant immunosurveillance that is not compensated for by the remaining SLT. To date, no quantitative analysis of the impact of splenectomy on the human T cell trafficking dynamics and tissue localisation has been reported. We developed a quantitative computational model that describes organ distribution and trafficking of human lymphocytes to explore the likely impact of splenectomy on immune cell distributions. In silico splenectomy resulted in an average reduction of T cell numbers in SLT by 35% (95%CI 0.12-0.97) and a comparatively lower, 9% (95%CI 0.17-1.43), mean decrease of T cell concentration in SLT. These results suggest that the surveillance capacity of the remaining SLT insufficiently compensates for the absence of the spleen. This may, in part, explain haematological malignancy risk in asplenic patients and raises the question of whether splenectomy has a clinically meaningful impact on patient responses to immunotherapy.

Quantifying the limits of CAR T-cell delivery in mice and men.

CAR (Chimeric Antigen Receptor) T cells have demonstrated clinical success for the treatment of multiple lymphomas and leukaemias, but not for various solid tumours, despite promising data from murine models. Lower effective CAR T-cell delivery rates to human solid tumours compared to haematological malignancies in humans and solid tumours in mice might partially explain these divergent outcomes. We used anatomical and physiological data for human and rodent circulatory systems to calculate the typical perfusion of healthy and tumour tissues, and estimated the upper limits of immune cell delivery rates across different organs, tumour types and species. Estimated maximum delivery rates were up to 10 000-fold greater in mice than humans yet reported CAR T-cell doses are typically only 10-100-fold lower in mice, suggesting that the effective delivery rates of CAR T cells into tumours in clinical trials are far lower than in corresponding mouse models. Estimated delivery rates were found to be consistent with published positron emission tomography data. Results suggest that higher effective human doses may be needed to drive efficacy comparable to mouse solid tumour models, and that lower doses should be tested in mice. We posit that quantitation of species and organ-specific delivery and homing of engineered T cells will be key to unlocking their potential for solid tumours.

Identifying and characterising the impact of excitability in a mathematical model of tumour-immune interactions.

We study a five-compartment mathematical model originally proposed by Kuznetsov et al. (1994) to investigate the effect of nonlinear interactions between tumour and immune cells in the tumour microenvironment, whereby immune cells may induce tumour cell death, and tumour cells may inactivate immune cells. Exploiting a separation of timescales in the model, we use the method of matched asymptotics to derive a new two-dimensional, long-timescale, approximation of the full model, which differs from the quasi-steady-state approximation introduced by Kuznetsov et al. (1994), but is validated against numerical solutions of the full model. Through a phase-plane analysis, we show that our reduced model is excitable, a feature not traditionally associated with tumour-immune dynamics. Through a systematic parameter sensitivity analysis, we demonstrate that excitability generates complex bifurcating dynamics in the model. These are consistent with a variety of clinically observed phenomena, and suggest that excitability may underpin tumour-immune interactions. The model exhibits the three stages of immunoediting - elimination, equilibrium, and escape, via stable steady states with different tumour cell concentrations. Such heterogeneity in tumour cell numbers can stem from variability in initial conditions and/or model parameters that control the properties of the immune system and its response to the tumour. We identify different biophysical parameter targets that could be manipulated with immunotherapy in order to control tumour size, and we find that preferred strategies may differ between patients depending on the strength of their immune systems, as determined by patient-specific values of associated model parameters.

Combining Mathematical Models With Experimentation to Drive Novel Mechanistic Insights Into Macrophage Function.

This perspective outlines an approach to improve mechanistic understanding of macrophages in inflammation and tissue homeostasis, with a focus on human inflammatory bowel disease (IBD). The approach integrates wet-lab and in-silico experimentation, driven by mechanistic mathematical models of relevant biological processes. Although wet-lab experimentation with genetically modified mouse models and primary human cells and tissues have provided important insights, the role of macrophages in human IBD remains poorly understood. Key open questions include: (1) To what degree hyperinflammatory processes (e.g., gain of cytokine production) and immunodeficiency (e.g., loss of bacterial killing) intersect to drive IBD pathophysiology? and (2) What are the roles of macrophage heterogeneity in IBD onset and progression? Mathematical modeling offers a synergistic approach that can be used to address such questions. Mechanistic models are useful for informing wet-lab experimental designs and provide a knowledge constrained framework for quantitative analysis and interpretation of resulting experimental data. The majority of published mathematical models of macrophage function are based either on animal models, or immortalized human cell lines. These experimental models do not recapitulate important features of human gastrointestinal pathophysiology, and, therefore are limited in the extent to which they can fully inform understanding of human IBD. Thus, we envision a future where mechanistic mathematical models are based on features relevant to human disease and parametrized by richer human datasets, including biopsy tissues taken from IBD patients, human organ-on-a-chip systems and other high-throughput clinical data derived from experimental medicine studies and/or clinical trials on IBD patients.

Feather arrays are patterned by interacting signalling and cell density waves.

Feathers are arranged in a precise pattern in avian skin. They first arise during development in a row along the dorsal midline, with rows of new feather buds added sequentially in a spreading wave. We show that the patterning of feathers relies on coupled fibroblast growth factor (FGF) and bone morphogenetic protein (BMP) signalling together with mesenchymal cell movement, acting in a coordinated reaction-diffusion-taxis system. This periodic patterning system is partly mechanochemical, with mechanical-chemical integration occurring through a positive feedback loop centred on FGF20, which induces cell aggregation, mechanically compressing the epidermis to rapidly intensify FGF20 expression. The travelling wave of feather formation is imposed by expanding expression of Ectodysplasin A (EDA), which initiates the expression of FGF20. The EDA wave spreads across a mesenchymal cell density gradient, triggering pattern formation by lowering the threshold of mesenchymal cells required to begin to form a feather bud. These waves, and the precise arrangement of feather primordia, are lost in the flightless emu and ostrich, though via different developmental routes. The ostrich retains the tract arrangement characteristic of birds in general but lays down feather primordia without a wave, akin to the process of hair follicle formation in mammalian embryos. The embryonic emu skin lacks sufficient cells to enact feather formation, causing failure of tract formation, and instead the entire skin gains feather primordia through a later process. This work shows that a reaction-diffusion-taxis system, integrated with mechanical processes, generates the feather array. In flighted birds, the key role of the EDA/Ectodysplasin A receptor (EDAR) pathway in vertebrate skin patterning has been recast to activate this process in a quasi-1-dimensional manner, imposing highly ordered pattern formation.

Integrated method for quantitative morphometry and oxygen transport modeling in striated muscle.

Identifying structural limitations in O2 transport is primarily restricted by current methods employed to characterize the nature of physiological remodeling. Inadequate resolution or breadth of available data has impaired development of routine diagnostic protocols and effective therapeutic strategies. Understanding O2 transport within striated muscle faces major challenges, most notably in quantifying how well individual fibers are supplied by the microcirculation, which has necessitated exploring tissue O2 supply using theoretical modeling of diffusive exchange. With capillary domains identified as a suitable model for the description of local O2 supply and requiring less computation than numerically calculating the trapping regions that are supplied by each capillary via biophysical transport models, we sought to design a high-throughput method for histological analysis. We present an integrated package that identifies optimal protocols for identification of important input elements, processing of digitized images with semiautomated routines, and incorporation of these data into a mathematical modeling framework with computed output visualized as the tissue partial pressure of O2 (Po2) distribution across a biopsy sample. Worked examples are provided using muscle samples from experiments involving rats and humans. NEW & NOTEWORTHY Progress in quantitative morphometry and analytical modeling has tended to develop independently. Real diagnostic power lies in harnessing both disciplines within one user-friendly package. We present a semiautomated, high-throughput tool for determining muscle phenotype from biopsy material, which also provides anatomically relevant input to quantify tissue oxygenation, in a coherent package not previously available to nonspecialist investigators.

Theoretical Insights into the Retinal Dynamics of Vascular Endothelial Growth Factor in Patients Treated with Ranibizumab, Based on an Ocular Pharmacokinetic/Pharmacodynamic Model.

Neovascular age-related macular degeneration (wet AMD) results from the pathological angiogenesis of choroidal capillaries, which leak fluid within or below the macular region of the retina. The current standard of care for treating wet AMD utilizes intravitreal injections of anti-VEGF antibodies or antibody fragments to suppress ocular vascular endothelial growth factor (VEGF) levels. While VEGF suppression has been demonstrated in wet AMD patients by serial measurements of free-VEGF concentrations in aqueous humor samples, it is presumed that anti-VEGF molecules also permeate across the inner limiting membrane (ILM) of the retina as well as the retinal pigmented epithelium (RPE) and suppress VEGF levels in the retina and/or choroidal regions. The latter effects are inferred from serial optical coherence tomography (OCT) measurements of fluid in the retinal and sub-retinal spaces. In order to gain theoretical insights to the dynamics of retinal levels of free-VEGF following intravitreal injection of anti-VEGF molecules, we have extended our previous two-compartment pharmacokinetic/pharmacodynamic (PK/PD) model of ranibizumab-VEGF suppression in vitreous and aqueous humors to a three-compartment model that includes the retinal compartment. In the new model, reference values for the macromolecular permeability coefficients between retina and vitreous ( pILM) and between retina and choroid ( pRPE) were estimated from PK data obtained in rabbit. With these values, the three-compartment model was used to re-analyze the aqueous humor levels of free-VEGF obtained in wet AMD patients treated with ranibizumab and to compare them to the simulated retinal levels of free-VEGF, including the observed variability in PK and PD. We have also used the model to explore the impact of varying pILM and pRPE to assess the case in which an anti-VEGF molecule is impermeable to the ILM and to assess the potential effects of AMD pathology on the RPE barrier. Our simulations show that, for the reference values of pILM and pRPE, the simulated duration of VEGF suppression in the retina is approximately 50% shorter than the observed duration of VEGF suppression in the aqueous humor, a finding that may explain the short duration of suppressed disease activity in the "high anti-VEGF demand" patients reported by Fauser and Muether ( Br. J. Ophthalmol. 2016, 100, 1494-1498 ). At 10-fold lower values of pRPE, the durations of VEGF suppression in the retina and aqueous humor are comparable. Lastly we have used the model to explore the impact of dose and binding parameters on the duration and depth of VEGF suppression in the aqueous and retinal compartments. Our simulations with the three-compartment PK/PD model provide new insights into inter-patient variability in response to anti-VEGF therapy and offer a mechanistic framework for developing treatment regimens and molecules that may prolong the duration of retinal VEGF suppression.

Changes in the retreatment radiation tolerance of the spinal cord with time after the initial treatment.

PURPOSE: To estimate, from experimental data, the retreatment radiation 'tolerances' of the spinal cord at different times after initial treatment. MATERIALS AND METHODS: A model was developed to show the relationship between the biological effective doses (BEDs) for two separate courses of treatment with the BED of each course being expressed as a percentage of the designated 'retreatment tolerance' BED value, denoted [Formula: see text] and [Formula: see text]. The primate data of Ang et al. ( 2001 ) were used to determine the fitted parameters. However, based on rodent data, recovery was assumed to commence 70 days after the first course was complete, and with a non-linear relationship to the magnitude of the initial BED (BEDinit). RESULTS: The model, taking into account the above processes, provides estimates of the retreatment tolerance dose after different times. Extrapolations from the experimental data can provide conservative estimates for the clinic, with a lower acceptable myelopathy incidence. Care must be taken to convert the predicted [Formula: see text] value into a formal BED value and then a practical dose fractionation schedule. CONCLUSIONS: Used with caution, the proposed model allows estimations of retreatment doses with elapsed times ranging from 70 days up to three years after the initial course of treatment.

An in silico model of cytotoxic T-lymphocyte activation in the lymph node following short peptide vaccination.

Tumour immunotherapy is dependent upon activation and expansion of tumour-targetting immune cells, known as cytotoxic T-lymphocytes (CTLs). Cancer vaccines developed in the past have had limited success and the mechanisms resulting in failure are not well characterized. To elucidate these mechanisms, we developed a human-parametrized, in silico, agent-based model of vaccination-driven CTL activation within a clinical short-peptide vaccination context. The simulations predict a sharp transition in the probability of CTL activation, which occurs with variation in the separation rate (or off-rate) of tumour-specific immune response-inducing peptides (cognate antigen) from the major histocompatibility class I (MHC-I) receptors of dendritic cells (DCs) originally at the vaccination site. For peptides with MHC-I off-rates beyond this transition, it is predicted that no vaccination strategy will lead to successful expansion of CTLs. For slower off-rates, below the transition, the probability of CTL activation becomes sensitive to the numbers of DCs and T cells that interact subsequent to DC migration to the draining lymph node of the vaccination site. Thus, the off-rate is a key determinant of vaccine design.

A Mechanistic Model of the Intravitreal Pharmacokinetics of Large Molecules and the Pharmacodynamic Suppression of Ocular Vascular Endothelial Growth Factor Levels by Ranibizumab in Patients with Neovascular Age-Related Macular Degeneration.

Intravitreal injection of anti-VEGF (vascular endothelial growth factor) antibodies or antibody fragments has been shown to be a highly effective treatment for neovascular age-related macular degeneration (wet AMD). The ocular half-life (t1/2) of these large molecules, determined in ocular fluids or derived from serum levels, varies with molecular size and is larger in humans than in preclinical animal species. The high affinity binding of VEGF to these molecules lowers the free concentration of VEGF and reduces its occupancy on VEGF receptors in ocular tissues. To understand the biophysical determinants of t1/2 for anti-VEGF antibodies and the time-course of VEGF in ocular fluids, we developed a mechanistic model of intravitreal pharmacokinetics (IVT PK) for anti-VEGF antibodies and combined it with a mechanistic model of the pharmacodynamics (RVR PD) of VEGF suppression by ranibizumab, an anti-VEGF recombinant, humanized monoclonal antibody fragment (Fab). Our IVT PK model predicts that the ocular t1/2 of a large molecule will be approximately four-times the calculated value of its vitreous diffusion time (Tdiff), defined as rvit(2)/6D, where rvit is the radius of the vitreous chamber in that species (modeled as a sphere), and D is the diffusion coefficient of the molecule in physiological saline at 37 °C obtained from the Stokes-Einstein relation. This prediction is verified from a compilation of data and calculations on various large molecules in the human, monkey, rabbit, and rat and is consistent with the reported t1/2 values of ranibizumab in humans (mean value 7.9 days) and the calculated Tdiff of 1.59 days. Our RVR PD model is based on the publication of Saunders et al. (Br. J. Ophthalmol. 2015, 99, 1554-1559) who reported data on the time-course of VEGF levels in aqueous humor samples obtained from 31 patients receiving ranibizumab treatment for wet AMD and developed a compartmental mathematical model to describe the VEGF suppression profiles. We modified Saunders' model with the known 2:1 stoichiometry of ranibizumab-VEGF binding and included the association and dissociation kinetics of the binding reactions. Using the RVR PD model, we reanalyzed Saunders' data to estimate the in vivo dissociation constant (KD) between ranibizumab and VEGF. Our analysis demonstrates the delicate interrelationship between the in vivo KD value and the intravitreal half-life and yields an in vivo KD estimate that is appreciably larger than the in vitro KD estimates reported in the literature. Potential explanations for the difference between the in vivo and in vitro KD values, which appear to reflect the different methodologies and experimental conditions, are discussed. We conclude that the combined mechanistic model of IVT PK and RVR PD provides a useful framework for simulating the effects of dose, KD, and the molecular weight of VEGF-binding molecules on the duration of VEGF suppression.

Glucose-lactate metabolic cooperation in cancer: insights from a spatial mathematical model and implications for targeted therapy.

A recent study has hypothesised a glucose-lactate metabolic symbiosis between adjacent hypoxic and oxygenated regions of a developing tumour, and proposed a treatment strategy to target this symbiosis. However, in vivo experimental support remains inconclusive. Here we develop a minimal spatial mathematical model of glucose-lactate metabolism to examine, in principle, whether metabolic symbiosis is plausible in human tumours, and to assess the potential impact of inhibiting it. We find that symbiosis is a robust feature of our model system-although on the length scale at which oxygen supply is diffusion-limited, its occurrence requires very high cellular metabolic activity-and that necrosis in the tumour core is reduced in the presence of symbiosis. Upon simulating therapeutic inhibition of lactate uptake, we predict that targeted treatment increases the extent of tissue oxygenation without increasing core necrosis. The oxygenation effect is correlated strongly with the extent of wild-type hypoxia and only weakly with wild-type symbiotic behaviour, and therefore may be promising for radiosensitisation of hypoxic, lactate-consuming tumours even if they do not exhibit a spatially well-defined symbiosis. Finally, we conduct in vitro experiments on the U87 glioblastoma cell line to facilitate preliminary speculation as to where highly malignant tumours might fall in our parameter space, and find that these experiments suggest a weakly symbiotic regime for U87 cells, thus raising the new question of what relationship might exist between symbiosis and tumour malignancy.

Comparing methods for modelling spreading cell fronts.

Spreading cell fronts play an essential role in many physiological processes. Classically, models of this process are based on the Fisher-Kolmogorov equation; however, such continuum representations are not always suitable as they do not explicitly represent behaviour at the level of individual cells. Additionally, many models examine only the large time asymptotic behaviour, where a travelling wave front with a constant speed has been established. Many experiments, such as a scratch assay, never display this asymptotic behaviour, and in these cases the transient behaviour must be taken into account. We examine the transient and the asymptotic behaviour of moving cell fronts using techniques that go beyond the continuum approximation via a volume-excluding birth-migration process on a regular one-dimensional lattice. We approximate the averaged discrete results using three methods: (i) mean-field, (ii) pair-wise, and (iii) one-hole approximations. We discuss the performance of these methods, in comparison to the averaged discrete results, for a range of parameter space, examining both the transient and asymptotic behaviours. The one-hole approximation, based on techniques from statistical physics, is not capable of predicting transient behaviour but provides excellent agreement with the asymptotic behaviour of the averaged discrete results, provided that cells are proliferating fast enough relative to their rate of migration. The mean-field and pair-wise approximations give indistinguishable asymptotic results, which agree with the averaged discrete results when cells are migrating much more rapidly than they are proliferating. The pair-wise approximation performs better in the transient region than does the mean-field, despite having the same asymptotic behaviour. Our results show that each approximation only works in specific situations, thus we must be careful to use a suitable approximation for a given system, otherwise inaccurate predictions could be made.

A general reaction-diffusion model of acidity in cancer invasion.

We model the metabolism and behaviour of a developing cancer tumour in the context of its microenvironment, with the aim of elucidating the consequences of altered energy metabolism. Of particular interest is the Warburg Effect, a widespread preference in tumours for cytosolic glycolysis rather than oxidative phosphorylation for glucose breakdown, as yet incompletely understood. We examine a candidate explanation for the prevalence of the Warburg Effect in tumours, the acid-mediated invasion hypothesis, by generalising a canonical non-linear reaction-diffusion model of acid-mediated tumour invasion to consider additional biological features of potential importance. We apply both numerical methods and a non-standard asymptotic analysis in a travelling wave framework to obtain an explicit understanding of the range of tumour behaviours produced by the model and how fundamental parameters govern the speed and shape of invading tumour waves. Comparison with conclusions drawn under the original system--a special case of our generalised system--allows us to comment on the structural stability and predictive power of the modelling framework.

Incorporating spatial correlations into multispecies mean-field models.

In biology, we frequently observe different species existing within the same environment. For example, there are many cell types in a tumour, or different animal species may occupy a given habitat. In modeling interactions between such species, we often make use of the mean-field approximation, whereby spatial correlations between the locations of individuals are neglected. Whilst this approximation holds in certain situations, this is not always the case, and care must be taken to ensure the mean-field approximation is only used in appropriate settings. In circumstances where the mean-field approximation is unsuitable, we need to include information on the spatial distributions of individuals, which is not a simple task. In this paper, we provide a method that overcomes many of the failures of the mean-field approximation for an on-lattice volume-excluding birth-death-movement process with multiple species. We explicitly take into account spatial information on the distribution of individuals by including partial differential equation descriptions of lattice site occupancy correlations. We demonstrate how to derive these equations for the multispecies case and show results specific to a two-species problem. We compare averaged discrete results to both the mean-field approximation and our improved method, which incorporates spatial correlations. We note that the mean-field approximation fails dramatically in some cases, predicting very different behavior from that seen upon averaging multiple realizations of the discrete system. In contrast, our improved method provides excellent agreement with the averaged discrete behavior in all cases, thus providing a more reliable modeling framework. Furthermore, our method is tractable as the resulting partial differential equations can be solved efficiently using standard numerical techniques.

Collagen bundle morphometry in skin and scar tissue: a novel distance mapping method provides superior measurements compared to Fourier analysis.

Histopathological evaluations of fibrotic processes require the characterization of collagen morphology in terms of geometrical features such as bundle orientation thickness and spacing. However, there are currently no reliable and valid techniques of measuring bundle thickness and spacing. Hence, two objective methods quantifying the collagen bundle thickness and spacing were tested for their reliability and validity: Fourier first-order maximum analysis and Distance Mapping, with the latter constituting a newly developed morphometric technique. Histological slides were constructed and imaged from 50 scar and 50 healthy human skin biopsies and subsequently analyzed by two observers to determine the interobserver reliability via the intraclass correlation coefficient. An intraclass correlation coefficient larger than 0.7 is considered as representing good reliability. The interobserver reliability for the Fourier first-order maximum and for the Distance Mapping algorithms, respectively, showed an intraclass correlation coefficient above 0.72 and 0.89. Additionally, we performed an assessment of validity in the form of responsiveness, in particular, demonstrating medium to excellent results via a calculation of the effect size, highlighting that both methods are sensitive enough to measure a treatment effect in clinical practice. In summary, two reliable and valid measurement methods were demonstrated for collagen bundle morphometry for the first time. Due to its superior reliability and more useful measures (bundle thickness and bundle spacing), Distance Mapping emerges as the preferred and more practical method. Nevertheless, in the future, both methods can be used for reliable and valid collagen morphometry of skin and scars, whereas further applications evaluating the quantitative microscopy of other fibrotic processes are anticipated.

The influence of toxicity constraints in models of chemotherapeutic protocol escalation.

The prospect of exploiting mathematical and computational models to gain insight into the influence of scheduling on cancer chemotherapeutic effectiveness is increasingly being considered. However, the question of whether such models are robust to the inclusion of additional tumour biology is relatively unexplored. In this paper, we consider a common strategy for improving protocol scheduling that has foundations in mathematical modelling, namely the concept of dose densification, whereby rest phases between drug administrations are reduced. To maintain a manageable scope in our studies, we focus on a single cell cycle phase-specific agent with uncomplicated pharmacokinetics, as motivated by 5-Fluorouracil-based adjuvant treatments of liver micrometastases. In particular, we explore predictions of the effectiveness of dose densification and other escalations of the protocol scheduling when the influence of toxicity constraints, cell cycle phase specificity and the evolution of drug resistance are all represented within the modelling. For our specific focus, we observe that the cell cycle and toxicity should not simply be neglected in modelling studies. Our explorations also reveal the prediction that dose densification is often, but not universally, effective. Furthermore, adjustments in the duration of drug administrations are predicted to be important, especially when dose densification in isolation does not yield improvements in protocol outcomes.

A mathematical model of tumour and blood pHe regulation: The HCO3-/CO2 buffering system.

Malignant tumours are characterised by a low, acidic extracellular pH (pHe) which facilitates invasion and metastasis. Previous research has proposed the potential benefits of manipulating systemic pHe, and recent experiments have highlighted the potential for buffer therapy to raise tumour pHe, prevent metastases, and prolong survival in laboratory mice. To examine the physiological regulation of tumour buffering and investigate how perturbations of the buffering system (via metabolic/respiratory disorders or changes in parameters) can alter tumour and blood pHe, we develop a simple compartmentalised ordinary differential equation model of pHe regulation by the HCO3-/CO2 buffering system. An approximate analytical solution is constructed and used to carry out a sensitivity analysis, where we identify key parameters that regulate tumour pHe in both humans and mice. From this analysis, we suggest promising alternative and combination therapies, and identify specific patient groups which may show an enhanced response to buffer therapy. In addition, numerical simulations are performed, validating the model against well-known metabolic/respiratory disorders and predicting how these disorders could change tumour pHe.