A Lipid-Structured Model of Atherosclerosis with Macrophage Proliferation.

Atherosclerotic plaques are fatty deposits that form in the walls of major arteries and are one of the major causes of heart attacks and strokes. Macrophages are the main immune cells in plaques and macrophage dynamics influence whether plaques grow or regress. Macrophage proliferation is a key process in atherosclerosis, particularly in the development of mid-stage plaques, but very few mathematical models include proliferation. In this paper we reframe the lipid-structured model of Ford et al. (J Theor Biol 479:48-63, 2019. https://doi.org/10.1016/j.jtbi.2019.07.003 ) to account for macrophage proliferation. Proliferation is modelled as a non-local decrease in the lipid structural variable. Steady state analysis indicates that proliferation assists in reducing eventual necrotic core lipid content and spreads the lipid load of the macrophage population amongst the cells. The contribution of plaque macrophages from proliferation relative to recruitment from the bloodstream is also examined. The model suggests that a more proliferative plaque differs from an equivalent (defined as having the same lipid content and cell numbers) recruitment-dominant plaque in the way lipid is distributed amongst the macrophages. The macrophage lipid distribution of an equivalent proliferation-dominant plaque is less skewed and exhibits a local maximum near the endogenous lipid content.

HDL and plaque regression in a multiphase model of early atherosclerosis.

Atherosclerosis is a chronic disease of the arteries characterised by the accumulation of lipids and lipid-engorged cells in the artery wall. Early plaque growth is aggravated by the deposition of low density lipoproteins (LDL) in the wall and the subsequent immune response. High density lipoproteins (HDL) counterbalance the effects of LDL by accepting cholesterol from macrophages and removing it from the plaque. In this paper, we develop a free boundary multiphase model to investigate the effects of LDL and HDL on early plaque development. We examine how the rates of LDL and HDL deposition affect cholesterol accumulation in macrophages, and how this impacts cell death rates and emigration. We identify a region of LDL-HDL parameter space where plaque growth stabilises for low LDL and high HDL influxes, due to macrophage emigration and HDL clearance that counterbalances the influx of new cells and cholesterol. We explore how the efferocytic uptake of dead cells and the recruitment of new macrophages affect plaque development for a range of LDL and HDL influxes. Finally, we consider how changes in the LDL-HDL profile can change the course of plaque development. We show that changes towards lower LDL and higher HDL can slow plaque growth and even induce regression. We find that these changes have less effect on larger, more established plaques, and that temporary changes will only slow plaque growth in the short term.

A Lipid-Structured Model of Atherosclerotic Plaque Macrophages with Lipid-Dependent Kinetics.

Atherosclerotic plaques are fatty growths in artery walls that cause heart attacks and strokes. Plaque formation is driven by macrophages that are recruited to the artery wall. These cells consume and remove blood-derived lipids, such as modified low-density lipoprotein. Ineffective lipid removal, due to macrophage death and other factors, leads to the accumulation of lipid-loaded macrophages and formation of a necrotic lipid core. Experimental observations suggest that macrophage functionality varies with the extent of lipid loading. However, little is known about the influence of macrophage lipid loads on plaque fate. Extending work by Ford et al. (J Theor Biol 479:48-63, 2019) and Chambers et al. (A lipid-structured model of atherosclerosis with macrophage proliferation, 2022), we develop a plaque model where macrophages are structured by their ingested lipid load and behave in a lipid-dependent manner. The model considers several macrophage behaviours, including recruitment to and emigration from the artery wall; proliferation and apotosis; ingestion of plaque lipids; and secondary necrosis of apoptotic cells. We consider apoptosis, emigration and proliferation to be lipid-dependent and we model these effects using experimentally informed functions of the internalised lipid load. Our results demonstrate that lipid-dependent macrophage behaviour can substantially alter plaque fate by changing both the total quantity of lipid in the plaque and the distribution of lipid between the live cells, dead cells and necrotic core. The consequences of macrophage lipid-dependence are often unpredictable because lipid-dependent effects introduce subtle, nonlinear interactions between the modelled cell behaviours. These observations highlight the importance of mathematical modelling in unravelling the complexities of macrophage lipid accumulation during atherosclerotic plaque formation.

Modelling daily weight variation in honey bee hives.

A quantitative understanding of the dynamics of bee colonies is important to support global efforts to improve bee health and enhance pollination services. Traditional approaches focus either on theoretical models or data-centred statistical analyses. Here we argue that the combination of these two approaches is essential to obtain interpretable information on the state of bee colonies and show how this can be achieved in the case of time series of intra-day weight variation. We model how the foraging and food processing activities of bees affect global hive weight through a set of ordinary differential equations and show how to estimate the parameters of this model from measurements on a single day. Our analysis of 10 hives at different times shows that the estimation of crucial indicators of the health of honey bee colonies are statistically reliable and fall in ranges compatible with previously reported results. The crucial indicators, which include the amount of food collected (foraging success) and the number of active foragers, may be used to develop early warning indicators of colony failure.

Macrophage Anti-inflammatory Behaviour in a Multiphase Model of Atherosclerotic Plaque Development.

Atherosclerosis is an inflammatory disease characterised by the formation of plaques, which are deposits of lipids and cholesterol-laden macrophages that form in the artery wall. The inflammation is often non-resolving, due in large part to changes in normal macrophage anti-inflammatory behaviour that are induced by the toxic plaque microenvironment. These changes include higher death rates, defective efferocytic uptake of dead cells, and reduced rates of emigration. We develop a free boundary multiphase model for early atherosclerotic plaques, and we use it to investigate the effects of impaired macrophage anti-inflammatory behaviour on plaque structure and growth. We find that high rates of cell death relative to efferocytic uptake results in a plaque populated mostly by dead cells. We also find that emigration can potentially slow or halt plaque growth by allowing material to exit the plaque, but this is contingent on the availability of live macrophage foam cells in the deep plaque. Finally, we introduce an additional bead species to model macrophage tagging via microspheres, and we use the extended model to explore how high rates of cell death and low rates of efferocytosis and emigration prevent the clearance of macrophages from the plaque.

A new lipid-structured model to investigate the opposing effects of LDL and HDL on atherosclerotic plaque macrophages.

Atherosclerotic plaques form in artery walls due to a chronic inflammatory response driven by lipid accumulation. A key component of the inflammatory response is the interaction between monocyte-derived macrophages and extracellular lipid. Although concentrations of low-density lipoprotein (LDL) and high-density lipoprotein (HDL) particles in the blood are known to affect plaque progression, their impact on the lipid load of plaque macrophages remains unexplored. In this paper, we develop a lipid-structured mathematical model to investigate the impact of blood LDL/HDL levels on plaque composition, and lipid distribution in plaque macrophages. A reduced subsystem, derived by summing the equations of the full model, describes the dynamics of biophysical quantities relating to plaque composition (e.g. total number of macrophages, total amount of intracellular lipid). We also derive a continuum approximation of the model to facilitate analysis of the macrophage lipid distribution. The results, which include time-dependent numerical solutions and asymptotic analysis of the unique steady state solution, indicate that plaque lipid content is sensitive to the influx of LDL relative to HDL capacity. The macrophage lipid distribution evolves in a wave-like manner towards an equilibrium profile which may be monotone decreasing, quasi-uniform or unimodal, attaining its maximum value at a non-zero lipid level. Our model also reveals that macrophage uptake may be severely impaired by lipid accumulation. We conclude that lipid accumulation in plaque macrophages may serve as a partial explanation for the defective uptake of apoptotic cells (efferocytosis) often reported in atherosclerotic plaques.

Modelling Preferential Phagocytosis in Atherosclerosis: Delineating Timescales in Plaque Development.

Atherosclerotic plaques develop over a long time and can cause heart attacks and strokes. There are no simple mathematical models that capture the different timescales of rapid macrophage and lipid dynamics and slow plaque growth. We propose a simple ODE model for lipid dynamics that includes macrophage preference for ingesting apoptotic material and modified low-density lipoproteins (modLDL) over ingesting necrotic material. We use multiple timescale analysis to show that if the necrosis rate is small then the necrotic core in the model plaque may continue to develop slowly even when the lipid levels in plaque macrophages, apoptotic material and modLDL appear to have reached equilibrium. We use the model to explore the effect of macrophage emigration, apoptotic cell necrosis, total rate of macrophage phagocytosis and modLDL influx into the plaque on plaque lipid accumulation.

A multiphase model of growth factor-regulated atherosclerotic cap formation.

Atherosclerosis is characterised by the growth of fatty plaques in the inner artery wall. In mature plaques, vascular smooth muscle cells (SMCs) are recruited from adjacent tissue to deposit a collagenous cap over the fatty plaque core. This cap isolates the thrombogenic plaque content from the bloodstream and prevents the clotting cascade that leads to myocardial infarction or stroke. Despite the protective role of the cap, the mechanisms that regulate cap formation and maintenance are not well understood. It remains unclear why some caps become stable, while others become vulnerable to rupture. We develop a multiphase PDE model with non-standard boundary conditions to investigate collagen cap formation by SMCs in response to diffusible growth factor signals from the endothelium. Platelet-derived growth factor stimulates SMC migration, proliferation and collagen degradation, while transforming growth factor (TGF)-[Formula: see text] stimulates SMC collagen synthesis and inhibits collagen degradation. The model SMCs respond haptotactically to gradients in the collagen phase and have reduced rates of migration and proliferation in dense collagenous tissue. The model, which is parameterised using in vivo and in vitro experimental data, reproduces several observations from plaque growth in mice. Numerical and analytical results demonstrate that a stable cap can be formed by a relatively small SMC population and emphasise the critical role of TGF-[Formula: see text] in effective cap formation. These findings provide unique insight into the mechanisms that may lead to plaque destabilisation and rupture. This work represents an important step towards the development of a comprehensive in silico plaque model.

Poor hive thermoregulation produces an Allee effect and leads to colony collapse.

In recent years the honey bee industry has been experiencing increased loss of hives. The accumulation of multiple stressors on a hive potentially drives hive loss in various ways, including winter loss and colony collapse disorder. One of these stressors is the breakdown of thermoregulation inside the hive. For pupae to develop correctly into healthy adult bees, the temperature within the hive must be regulated by the hive bees to within a narrow range that ensures optimal development. Suboptimal development in adults affects their brain and flight muscles so bees becomes inefficient foragers with shorter life spans. We model the effect of thermoregulation on hive health using a system of delay differential equations that show that thermoregulatory stress has the capacity to drive colony loss in the model via a saddle-node bifurcation with an associated Allee effect.

A Spatially Resolved and Quantitative Model of Early Atherosclerosis.

Atherosclerosis is a major burden for all societies, and there is a great need for a deeper understanding of involved key inflammatory, immunological and biomechanical processes. A decisive step for the prevention and medical treatment of atherosclerosis is to predict what conditions determine whether early atherosclerotic plaques continue to grow, stagnate or become regressive. The driving biological and mechanobiological mechanisms that determine the stability of plaques are yet not fully understood. We develop a spatially resolved and quantitative mathematical model of key contributors of early atherosclerosis. The stability of atherosclerotic model plaques is assessed to identify and classify progression-prone and progression-resistant atherosclerotic regions based on measurable or computable in vivo inputs, such as blood cholesterol concentrations and wall shear stresses. The model combines Darcy's law for the transmural flow through vessels walls, the Kedem-Katchalsky equations for endothelial fluxes of lipoproteins, a quantitative model of early plaque formation from a recent publication and a novel submodel for macrophage recruitment. The parameterization and analysis of the model suggest that the advective flux of lipoproteins through the endothelium is decisive, while the influence of the advective transport within the artery wall is negligible. Further, regions in arteries with an approximate wall shear stress exposure below 20% of the average exposure and their surroundings are potential regions where progression-prone atherosclerotic plaques develop.

A lipid-structured model for macrophage populations in atherosclerotic plaques.

Atherosclerosis is a chronic inflammatory disease driven by the accumulation of pro-inflammatory, lipid-loaded macrophages at sites inside artery walls. These accumulations lead to the development of atherosclerotic plaques. The rupture of plaques that contain lipid-rich necrotic cores can trigger heart attacks and strokes via occlusion of blood vessels. We construct and analyse a system of partial integro-differential equations that model lipid accumulation by macrophages, the generation of apoptotic cells and the formation of the necrotic core. The model accounts for the following cell behaviours: monocyte recruitment into the plaque and differentiation into macrophages; macrophage ingestion of low density lipoproteins (LDL) and of apoptotic cells and necrotic material; lipid offloading to high density lipoproteins (HDL); macrophage emigration; and apoptosis of macrophages and necrosis of apoptotic cells. With this model, we study how changes in parameters predict the characteristic features of plaque pathology. In particular, we find the qualitative form of lipid distribution across the macrophage population and show that high lipid loads can occur in the absence of LDL ingestion. We also demonstrate the importance of macrophage emigration in mitigating and resolving inflammation and plaque lipid accumulation.

Efferocytosis perpetuates substance accumulation inside macrophage populations.

In both cells and animals, cannibalism can transfer harmful substances from the consumed to the consumer. Macrophages are immune cells that consume their own dead via a process called cannibalistic efferocytosis. Macrophages that contain harmful substances are found at sites of chronic inflammation, yet the role of cannibalism in this context remains unexplored. Here we take mathematical and experimental approaches to study the relationship between cannibalistic efferocytosis and substance accumulation in macrophages. Through mathematical modelling, we deduce that substances which transfer between individuals through cannibalism will concentrate inside the population via a coalescence process. This prediction was confirmed for macrophage populations inside a closed system. We used image analysis of whole slide photomicrographs to measure both latex microbead and neutral lipid accumulation inside murine bone marrow-derived macrophages (104-[Formula: see text]) following their stimulation into an inflammatory state ex vivo. While the total number of phagocytosed beads remained constant, cell death reduced cell numbers and efferocytosis concentrated the beads among the surviving macrophages. As lipids are also conserved during efferocytosis, these cells accumulated lipid derived from the membranes of dead and consumed macrophages (becoming macrophage foam cells). Consequently, enhanced macrophage cell death increased the rate and extent of foam cell formation. Our results demonstrate that cannibalistic efferocytosis perpetuates exogenous (e.g. beads) and endogenous (e.g. lipids) substance accumulation inside macrophage populations. As such, cannibalism has similar detrimental consequences in both cells and animals.

A two-phase model of early fibrous cap formation in atherosclerosis.

Atherosclerotic plaque growth is characterised by chronic, non-resolving inflammation that promotes the accumulation of cellular debris and extracellular fat in the inner artery wall. This material is highly thrombogenic, and plaque rupture can lead to the formation of blood clots that occlude major arteries and cause myocardial infarction or stroke. In advanced plaques, vascular smooth muscle cells (SMCs) are recruited from deeper in the artery wall to synthesise a cap of fibrous tissue that stabilises the plaque and sequesters the thrombogenic plaque content from the bloodstream. The fibrous cap provides crucial protection against the clinical consequences of atherosclerosis, but the mechanisms of cap formation are poorly understood. In particular, it is unclear why certain plaques become stable and robust while others become fragile and dangerously vulnerable to rupture. We develop a multiphase model with non-standard boundary conditions to investigate early fibrous cap formation in the atherosclerotic plaque. The model is parameterised using data from a range of in vitro and in vivo studies, and includes highly nonlinear mechanisms of SMC proliferation and migration in response to an endothelium-derived chemical signal. We demonstrate that the model SMC population naturally evolves towards a steady-state, and predict a rate of cap formation and a final plaque SMC content consistent with experimental observations in mice. Parameter sensitivity simulations show that SMC proliferation makes a limited contribution to cap formation, and demonstrate that stable cap formation relies primarily on a critical balance between the rates of SMC recruitment to the plaque, chemotactic SMC migration within the plaque and SMC loss by apoptosis or phenotype change. This model represents the first detailed in silico study of fibrous cap formation in atherosclerosis, and establishes a multiphase modelling framework that can be readily extended to investigate many other aspects of plaque development.

A Quantitative Model of Early Atherosclerotic Plaques Parameterized Using In Vitro Experiments.

There are a growing number of studies that model immunological processes in the artery wall that lead to the development of atherosclerotic plaques. However, few of these models use parameters that are obtained from experimental data even though data-driven models are vital if mathematical models are to become clinically relevant. We present the development and analysis of a quantitative mathematical model for the coupled inflammatory, lipid and macrophage dynamics in early atherosclerotic plaques. Our modeling approach is similar to the biologists' experimental approach where the bigger picture of atherosclerosis is put together from many smaller observations and findings from in vitro experiments. We first develop a series of three simpler submodels which are least-squares fitted to various in vitro experimental results from the literature. Subsequently, we use these three submodels to construct a quantitative model of the development of early atherosclerotic plaques. We perform a local sensitivity analysis of the model with respect to its parameters that identifies critical parameters and processes. Further, we present a systematic analysis of the long-term outcome of the model which produces a characterization of the stability of model plaques based on the rates of recruitment of low-density lipoproteins, high-density lipoproteins and macrophages. The analysis of the model suggests that further experimental work quantifying the different fates of macrophages as a function of cholesterol load and the balance between free cholesterol and cholesterol ester inside macrophages may give valuable insight into long-term atherosclerotic plaque outcomes. This model is an important step toward models applicable in a clinical setting.

Nonlinear dynamics of early atherosclerotic plaque formation may determine the efficacy of high density lipoproteins (HDL) in plaque regression.

We use a computational model to explore the effect of foam cell accumulation on plaque regression following an increase in high density lipoprotein (HDL) influx into the plaque. Atherosclerotic plaque formation is the outcome of cellular and cytokine responses to low density lipoproteins (LDL) that penetrate the artery wall following an injury to the endothelium and become modified. We modelled the cells and cytokines that are most important in plaque formation using partial differential equations. The model includes monocytes and macrophages, foam cells, macrophage chemoattractants, endothelium-stimulating cytokines, modified low density lipoproteins (mod LDL) and HDL. We included interactions both at the endothelium surface and inside the artery wall. The model predicts that when HDL influx into a well-established plaque with large numbers of foam cells is increased, the plaque may not regress but may continue to grow at a slower rate. If HDL influx is increased when a model plaque is recently established and has fewer foam cells, then the plaque does regress. If modLDL influx into the plaque is lowered at the same time that HDL influx increased or the capacity of the HDL to remove cholesterol from foam cells is increased, then the plaque is more likely to regress. The predictions of the model are in qualitative agreement with experimental studies in mice and rabbits. The results suggest that the intrinsic dynamics of reverse cholesterol transport by HDL are important in determining the success of HDL raising in promoting plaque regression.

Keratin-14-Positive Precursor Cells Spawn a Population of Migratory Corneal Epithelia that Maintain Tissue Mass throughout Life.

The dynamics of epithelial stem cells (SCs) that contribute to the formation and maintenance of the cornea are poorly understood. Here, we used K14CreERT2-Confetti (Confetti) mice, sophisticated imaging, and computational modeling to trace the origins and fate of these cells during embryogenesis and adult life. We show that keratin-14 (K14+)-expressing progenitors are defined and widely distributed across the E16.5 cornea, after which they undergo cycles of proliferation and dispersal prior to eyelid opening. K14+ clonal patches disappear from the central cornea and are replaced by limbal-derived K14+ streaks, a finding that aligned with bromodeoxyuridine label-retaining studies. We also elucidated the mechanism by which SC clones are lost during life and propose this is due to population asymmetry and neutral drift. Finally, we established that the occurrence of an equatorial migratory mid-line is a consequence of apoptosis in a narrow nasal-temporal region, the site where eyelids meet during blinking.

A Model for the Spread of an Invasive Weed, Tradescantia fluminensis.

Tradescantia fluminensis is an invasive weed and a serious threat to native forests in eastern Australia and New Zealand. Current methods of eradication are often ineffective, so understanding the growth mechanisms of Tradescantia is important in formulating better control strategies. We present a partial differential equation (PDE) model for Tradescantia growth and spatial proliferation that accounts for Tradescantia's particular creeping and branching morphology, and the impact of self-shading on plant growth. This is the first PDE model to represent a weed that spreads via a creeping growth habit rather than by seed dispersal. We use a travelling wave analysis to investigate how Tradescantia extends to colonise new territory. Numerical simulations and analysis show that the model provides a good qualitative representation of the behaviour of this plant. This model provides a foundation for assessing different control and eradication strategies for Tradescantia.

Self-organized centripetal movement of corneal epithelium in the absence of external cues.

Maintaining the structure of the cornea is essential for high-quality vision. In adult mammals, corneal epithelial cells emanate from stem cells in the limbus, driven by an unknown mechanism towards the centre of the cornea as cohesive clonal groups. Here we use complementary mathematical and biological models to show that corneal epithelial cells can self-organize into a cohesive, centripetal growth pattern in the absence of external physiological cues. Three conditions are required: a circumferential location of stem cells, a limited number of cell divisions and mobility in response to population pressure. We have used these complementary models to provide explanations for the increased rate of centripetal migration caused by wounding and the potential for stem cell leakage to account for stable transplants derived from central corneal tissue, despite the predominantly limbal location of stem cells.

Bifurcation and dynamics in a mathematical model of early atherosclerosis: How acute inflammation drives lesion development.

We present here a mathematical model describing the primary mechanisms that drive the early stages of atherosclerosis. This involves the interactions between modified low density lipoprotein (LDL), monocytes/macrophages, cytokines and foam cells. This model suggests that there is an initial inflammatory phase associated with atherosclerotic lesion development and a longer, quasi-static process of plaque development inside the arterial wall that follows the initial transient. We will show results that show how different LDL concentrations in the blood stream and different immune responses can affect the development of a plaque. Through numerical bifurcation analysis, we show the existence of a fold bifurcation when the flux of LDL from the blood is sufficiently high. By analysing the model presented in this paper, we gain a greater insight into this inflammatory response qualitatively and quantitatively.

Rapid behavioral maturation accelerates failure of stressed honey bee colonies.

Many complex factors have been linked to the recent marked increase in honey bee colony failure, including pests and pathogens, agrochemicals, and nutritional stressors. It remains unclear, however, why colonies frequently react to stressors by losing almost their entire adult bee population in a short time, resulting in a colony population collapse. Here we examine the social dynamics underlying such dramatic colony failure. Bees respond to many stressors by foraging earlier in life. We manipulated the demography of experimental colonies to induce precocious foraging in bees and used radio tag tracking to examine the consequences of precocious foraging for their performance. Precocious foragers completed far fewer foraging trips in their life, and had a higher risk of death in their first flights. We constructed a demographic model to explore how this individual reaction of bees to stress might impact colony performance. In the model, when forager death rates were chronically elevated, an increasingly younger forager force caused a positive feedback that dramatically accelerated terminal population decline in the colony. This resulted in a breakdown in division of labor and loss of the adult population, leaving only brood, food, and few adults in the hive. This study explains the social processes that drive rapid depopulation of a colony, and we explore possible strategies to prevent colony failure. Understanding the process of colony failure helps identify the most effective strategies to improve colony resilience.