Flexural rigidity of pressurized model notochords in regular packing patterns

The biomechanics of embryonic notochords are studied using an elastic membrane model. An initial study varying internal pressure and stiffness ratio determines tension and geometric ratios as a function of internal pressure, membrane stiffness ratio, and cell packing pattern. A subsequent three-point bending study determines flexural rigidity as a function of internal pressure, configuration, and orientation. Flexural rigidity is found to be independent of membrane stiffness ratio. Controlling for number and volume of cells and their internal pressure, the eccentric staircase pattern of cell packing has more than double the flexural rigidity of the radially symmetric bamboo pattern. Moreover, the eccentric staircase pattern is found to be more than twice as stiff in lateral bending than in dorsoventral bending. This suggests a mechanical advantage to the eccentric WT staircase pattern of the embryonic notochord, over patterns with round cross-section.

Morphoelastic models discriminate between different mechanisms of left-right asymmetric stomach morphogenesis

The mechanisms by which the vertebrate stomach undergoes its evolutionarily conserved leftward bending remain incompletely understood. Although the left and right sides of the organ are known to possess different gene expression patterns and undergo distinct morphogenetic events, the physical mechanisms by which these differences generate morphological asymmetry remain unclear. Here, we develop a continuum model of asymmetric stomach morphogenesis. Using a morphoelastic framework, we investigate the morphogenetic implications of a variety of hypothetical, tissue-level growth differences between the left and right sides of a simplified tubular organ. Simulations reveal that, of the various differential growth mechanisms tested, only one category is consistent with the leftward stomach curvature observed in wild-type embryos: equal left and right volumetric growth rates, coupled with transversely isotropic tissue thinning on the left side. Simulating this mechanism in a defined region of the model over a longer period of growth leads to mature stomach-like curvatures.

Physical models of notochord cell packing reveal how tension ratios determine morphometry

The physical and geometric aspects of notochords are investigated using a model of finite-length notochords, with interior vacuolated cells arranged in two common packing configurations, and sheath modeled as homogeneous and thin. The key ratios governing packing patterns and eccentricity are number of cells per unit length λ and cell tension ratio Γ. By analyzing simulations that vary Γ and total number of cells N, we find that eccentricity, λ, and internal pressure approach consistent asymptotic values away from the tapering ends, as N increases. The length of the tapering ends is quantified as a function of Γ and pattern. Formulas are derived for geometric ratios, pressure, and energy as functions of Γ and pattern. These observations on the relationship between mechanics, geometry, and pattern provide a framework for further work which may provide insight into the roles of mechanosensing and pressure-volume regulation in the notochord.

Bubble packing, eccentricity, and notochord development

This paper develops a theoretical basis for the observed relationship between cell arrangements in notochords and analog physical models, and the eccentricity of their cross sections. Three models are developed and analyzed, of the mechanics of cell packing in sheaths. The key ratios governing the packing patterns and eccentricity are cells per unit length λ, tension ratio Γ, and eccentricity e. For flexible and semi-flexible sheaths, the optimal packing pattern shifts from "bamboo", with a symmetric cross section, to "staircase", with an eccentric cross section, at a critical value λ = 1.13. In rigid tubes, this threshold is lowered as imposed eccentricity is increased. Patterns can be observed which are not optimal; pattern transitions may occur below or above the critical λ values. The eccentricity of staircase patterns in flexible and semi-flexible tubes is found to be dependent on the tension ratio Γ, increasing as sheath tension decreases relative to interior cell tension. A novel "serpentine" packing pattern appears for low Γ near the critical λ. The developmental utility of enforcing notochord eccentricity is discussed, as well as potential mechanisms for such control.

Does viral load alter behavior of the bee parasite Varroa destructor?

The invasive mite Varroa destructor has negatively impacted global apiculture, by being a vector for many viruses of the honey bee (Apis mellifera). Until now, most studies have been limited to varroa-honey bee or virus-honey bee interactions. The aim of this study is to bridge the important research gap of varroa-virus interactions by correlating varroa behavior with viral load. Ten-minute video recordings of 200 varroa mites were analyzed, and average speeds of the mites were compared to individual qPCR viral loads for deformed wing virus (DWV) and sacbrood virus (SBV). Statistically significant models reveal that colony, DWV, and SBV all might play a role in mite behavior, suggesting that the varroa-virus interaction needs to be an integral part of future studies on honey bee pathogens.

Tissue self-organization underlies morphogenesis of the notochord

The notochord is a conserved axial structure that in vertebrates serves as a hydrostatic scaffold for embryonic axis elongation and, later on, for proper spine assembly. It consists of a core of large fluid-filled vacuolated cells surrounded by an epithelial sheath that is encased in extracellular matrix. During morphogenesis, the vacuolated cells inflate their vacuole and arrange in a stereotypical staircase pattern. We investigated the origin of this pattern and found that it can be achieved purely by simple physical principles. We are able to model the arrangement of vacuolated cells within the zebrafish notochord using a physical model composed of silicone tubes and water-absorbing polymer beads. The biological structure and the physical model can be accurately described by the theory developed for the packing of spheres and foams in cylinders. Our experiments with physical models and numerical simulations generated several predictions on key features of notochord organization that we documented and tested experimentally in zebrafish. Altogether, our data reveal that the organization of the vertebrate notochord is governed by the density of the osmotically swelling vacuolated cells and the aspect ratio of the notochord rod. We therefore conclude that self-organization underlies morphogenesis of the vertebrate notochord.This article is part of the Theo Murphy meeting issue on 'Mechanics of development'.

Tissue geometry may govern lung branching mode selection

Lung branching morphogenesis proceeds in three stereotyped modes (domain, planar, and orthogonal branching). Much is known about the molecular players, including growth factors such as fibroblast growth factor 10 but it is unknown how these signals could actuate the different branching patterns. With the aim of identifying mechanisms that may determine the different branching modes, we developed a computational model of the epithelial lung bud and its surrounding mesenchyme. We studied transport of morphogens and localization of morphogen flux at lobe surfaces and lobe edges. We find that a single simple mechanism is theoretically capable of directing an epithelial tubule to elongate, bend, flatten, or bifurcate, depending solely on geometric ratios of the tissues in the vicinity of a growing tubule tip. Furthermore, the same simple mechanism is capable of generating orthogonal or planar branching, depending only on the same geometric ratios.

Quantifying cellular and subcellular stretches in embryonic lung epithelia under peristalsis: where to look for mechanosensing

Peristalsis begins in the lung as soon as the smooth muscle (SM) forms, and persists until birth. As the prenatal lung is filled with liquid, SM action can, through lumen pressure, deform tissues far from the immediately adjacent tissues. Stretching of embryonic tissues has been shown to have potent morphogenetic effects. We hypothesize that these effects are at work in lung morphogenesis. In order to refine that broad hypothesis in a quantitative framework, we geometrically analyse cell shapes in an epithelial tissue, and individual cell deformations resulting from peristaltic waves that completely occlude the airway. Typical distortions can be very large, with opposite orientations in the stalk and tip regions. Apical distortions are always greater than basal distortions. We give a quantitative estimate of the relationship between length of occluded airway and the resulting tissue stretch in the distal tip. We refine our analysis of cell stresses and strains from peristalsis with a simple mechanical model of deformation of cells within an epithelium, which accounts for basic subcellular geometry and material properties. The model identifies likely stress concentrations near the nucleus and at the apical cell-cell junction. The surprisingly large strains of airway peristalsis may serve to rearrange cells and stimulate other mechanosensitive processes by repeatedly aligning cytoskeletal components and/or breaking and reforming lateral cell-cell adhesions. Stress concentrations between nuclei of adjacent cells may serve as a mechanical control mechanism guiding the alignment of nuclei as an epithelium matures.

Quantifying stretch and secretion in the embryonic lung: Implications for morphogenesis

Branching in the embryonic lung is controlled by a variety of morphogens. Mechanics is also believed to play a significant role in lung branching. The relative roles and interactions of these two broad factors are challenging to determine. We considered three hypotheses for explaining why tracheal occlusion triples branching with no overall increase in size. Both hypotheses are based on tracheal occlusion blocking the exit of secretions. (H1) Increased lumen pressure stretches tissues; stretch receptors at shoulders of growing tips increase local rate of branching. (H2) Blocking exit of secretions blocks advective transport of morphogens, leading to (H2a) increased overall concentration of morphogens or (H2b) increased flux of morphogens at specific locations. We constructed and analyzed computational models of tissue stretch and solute transport in a 3D lung geometry. Observed tissue stresses and stretches were predominantly in locations unrelated to subsequent branch locations, suggesting that tissue stretch (H1) is not the mechanism of enhancement of branching. Morphogen concentration in the mesenchyme (H2a) increased with tracheal occlusion, consistent with previously reported results. Morphogen flux at the epithelial surface (H2b) completely changed its distribution pattern when the trachea was occluded, tripling the number of locations at which it was elevated. Our results are consistent with the hypothesis that tracheal occlusion blocks outflow of secretions, leading to a higher number of high-flux locations at branching tips, in turn leading to a large increase in number of branching locations.

Morphogenetic implications of peristaltic fluid-tissue dynamics in the embryonic lung

Peristalsis begins in the lung as soon as the smooth muscle forms, and persists until birth. Since the prenatal lung is liquid-filled, smooth muscle action can deform tissues and transport fluid far from the immediately adjacent tissues. Stretching of embryonic tissues and sensation of internal fluid flows have been shown to have potent morphogenetic effects. We hypothesize that these effects are at work in lung morphogenesis. To place that hypothesis in a quantitative framework, we analyze a model of the fluid-structure interactions between embryonic tissues and lumen fluid resulting from peristaltic waves that partially occlude the airway. We find that if the airway is closed, deformations are synchronized; by contrast, if the trachea is open, maximal occlusion precedes maximal pressure. We perform a parametric analysis of how occlusion, stretch, and flow depend on tissue stiffnesses, smooth muscle force, tissue shape and size, and fluid viscosity. We find that most of these relationships are governed by simple ratios.

Morphogenetic Implications of Peristalsis-Driven Fluid Flow in the Embryonic Lung

Epithelial organs are almost universally secretory. The lung secretes mucus of extremely variable consistency. In the early prenatal period, the secretions are of largely unknown composition, consistency, and flow rates. In addition to net outflow from secretion, the embryonic lung exhibits transient reversing flows from peristalsis. Airway peristalsis (AP) begins as soon as the smooth muscle forms, and persists until birth. Since the prenatal lung is liquid-filled, smooth muscle action can transport fluid far from the immediately adjacent tissues. The sensation of internal fluid flows has been shown to have potent morphogenetic effects, as has the transport of morphogens. We hypothesize that these effects play an important role in lung morphogenesis. To test these hypotheses in a quantitative framework, we analyzed the fluid-structure interactions between embryonic tissues and lumen fluid resulting from peristaltic waves that partially occlude the airway. We found that if the airway is closed, fluid transport is minimal; by contrast, if the trachea is open, shear rates can be very high, particularly at the stenosis. We performed a parametric analysis of flow characteristics' dependence on tissue stiffnesses, smooth muscle force, geometry, and fluid viscosity, and found that most of these relationships are governed by simple ratios. We measured the viscosity of prenatal lung fluid with passive bead microrheology. This paper reports the first measurements of the viscosity of embryonic lung lumen fluid. In the range tested, lumen fluid can be considered Newtonian, with a viscosity of 0.016 ± 0.008 Pa-s. We analyzed the interaction between the internal flows and diffusion and conclude that AP has a strong effect on flow sensing away from the tip and on transport of morphogens. These effects may be the intermediate mechanisms for the enhancement of branching seen in occluded embryonic lungs.

AN AUGMENTED IMMERSED INTERFACE METHOD FOR MOVING STRUCTURES WITH MASS

We present an augmented immersed interface method for simulating the dynamics of a deformable structure with mass in an incompressible fluid. The fluid is modeled by the Navier-Stokes equations in two dimensions. The acceleration of the structure due to mass is coupled with the flow velocity and the pressure. The surface tension of the structure is assumed to be a constant for simplicity. In our method, we treat the unknown acceleration as the only augmented variable so that the augmented immersed interface method can be applied. We use a modified projection method that can enforce the pressure jump conditions corresponding to the unknown acceleration. The acceleration must match the flow acceleration along the interface. The proposed augmented method is tested against an exact solution with a stationary interface. It shows that the augmented method has a second order of convergence in space. The dynamics of a deformable circular structure with mass is also investigated. It shows that the fluid-structure system has bi-stability: a stationary state for a smaller Reynolds number and an oscillatory state for a larger Reynolds number. The observation agrees with those in the literature.

Mechanical control of spheroid growth: distinct morphogenetic regimes

We develop a model of transport and growth in epithelio-mesenchymal interactions. Analysis of the growth of an avascular solid spheroid inside a passive mesenchyme or gel shows that sustained volumetric growth requires four generic mechanisms: (1) growth factor, (2) protease, (3) control of cellularity, and (4) swelling. The model reveals a bifurcation delineating two distinct morphogenetic regimes: (A) steady growth, (B) growth arrested by capsule formation in the mesenchyme. In both morphogenetic regimes, growth velocity is constant unless and until a complete capsule forms. Comprehensive exploration of the large parameter space reveals that the bifurcation is determined by just two ratios representing the relative strengths of growth and proteolytic activity. Growth velocity is determined only by the ratio governing growth, independent of proteolytic activity. There is a continuum of interior versus surface growth, with fastest growth at the surface. The model provides a theoretical basis for explaining observations of growth arrest despite proteolysis of surrounding tissue, and gives a quantitative framework for the design and interpretation of experiments involving spheroids, and tissues which are locally equivalent to spheroids.

Regulation of hepatic stem/progenitor phenotype by microenvironment stiffness in hydrogel models of the human liver stem cell niche

Human livers have maturational lineages of cells within liver acini, beginning periportally in stem cell niches, the canals of Hering, and ending in polyploid hepatocytes pericentrally and cholangiocytes in bile ducts. Hepatic stem cells (hHpSCs) in vivo are partnered with mesenchymal precursors to endothelia (angioblasts) and stellate cells, and reside in regulated microenvironments, stem cell niches, containing hyaluronans (HA). The in vivo hHpSC niche is modeled in vitro by growing hHpSC in two-dimensional (2D) cultures on plastic. We investigated effects of 3D microenvironments, mimicking the liver's stem cell niche, on these hHpSCs by embedding them in HA-based hydrogels prepared with Kubota's Medium (KM), a serum-free medium tailored for endodermal stem/progenitors. The KM-HA hydrogels mimicked the niches, matched diffusivity of culture medium, exhibited shear thinning and perfect elasticity under mechanical loading, and had predictable stiffness depending on their chemistry. KM-HA hydrogels, which supported cell attachment, survival and expansion of hHpSC colonies, induced transition of hHpSC colonies towards stable heterogeneous populations of hepatic progenitors depending on KM-HA hydrogel stiffness, as shown by both their gene and protein expression profile. These acquired phenotypes did not show morphological evidence of fibrotic responses. In conclusion, this study shows that the mechanical properties of the microenvironment can regulate differentiation in endodermal stem cell populations.

Lung organogenesis

Developmental lung biology is a field that has the potential for significant human impact: lung disease at the extremes of age continues to cause major morbidity and mortality worldwide. Understanding how the lung develops holds the promise that investigators can use this knowledge to aid lung repair and regeneration. In the decade since the "molecular embryology" of the lung was first comprehensively reviewed, new challenges have emerged-and it is on these that we focus the current review. Firstly, there is a critical need to understand the progenitor cell biology of the lung in order to exploit the potential of stem cells for the treatment of lung disease. Secondly, the current familiar descriptions of lung morphogenesis governed by growth and transcription factors need to be elaborated upon with the reinclusion and reconsideration of other factors, such as mechanics, in lung growth. Thirdly, efforts to parse the finer detail of lung bud signaling may need to be combined with broader consideration of overarching mechanisms that may be therapeutically easier to target: in this arena, we advance the proposal that looking at the lung in general (and branching in particular) in terms of clocks may yield unexpected benefits.

Convergent extension by intercalation without mediolaterally fixed cell motion

We construct and implement a stochastic model of convergent extension, using a minimal set of assumptions on cell behavior. In addition to the basic assumptions of volume conservation, random cell motion, and cell-cell and cell-ECM adhesion, and a non-standard assumption that cytoskeletal polymerization generates an internal pressure tending to keep cells convex, we find that we need only two conditions for convergent extension. (1) Each cell type has a particular aspect ratio towards which it regulates its geometry. We do not require that cells align in a specific orientation, e.g. to be oriented mediolaterally. (2) The elongating tissue is composed of cells that prefer to be elongated, and these cells must be accompanied by cells which prefer to be round. The latter effectively provide a boundary to capture. In simulations, our model tissue extends and converges to a stacked arrangement of elongated cells one cell wide, an arrangement which is seen in ascidian notochords, but which has not been observed in other models. This arrangement is achieved without any direct mediolateral bias other than that which is provided by the physical edge of the adjacent tissue.

Mechanics of mesenchymal contribution to clefting force in branching morphogenesis

Branching morphogenesis is ubiquitous and may involve several different mechanisms. Glandular morphogenesis is affected by growth, cell rearrangements, changes in the basal lamina, changes in the stromal ECM, changes in cell-cell and cell-ECM adhesions, mesenchymal contractility, and possibly other mechanisms. We have developed a 3D model of the mechanics of clefting, focusing in this paper solely on the potential role of mesenchyme-generated traction forces. The tissue mechanics are assumed to be those of fluids, and the hypothesized traction forces are modeled as advected by the deformations which they generate. We find that mesenchymal traction forces are sufficient to generate a cleft of the correct size and morphology, in the correct time frame. We find that viscosity of the tissues affects the time course of morphogenesis, and also affects the resulting form of the organ. Morphology is also strongly dependent on the initial distribution of contractility. We suggest an in vitro method of examining the role of mesenchyme in branching morphogenesis.

Semi-mechanistic partial buffer approach to modeling pH, the buffer properties, and the distribution of ionic species in complex solutions

In many biological science and food processing applications, it is very important to control or modify pH. However, the complex, unknown composition of biological media and foods often limits the utility of purely theoretical approaches to modeling pH and calculating the distributions of ionizable species. This paper provides general formulas and efficient algorithms for predicting the pH, titration, ionic species concentrations, buffer capacity, and ionic strength of buffer solutions containing both defined and undefined components. A flexible, semi-mechanistic, partial buffering (SMPB) approach is presented that uses local polynomial regression to model the buffering influence of complex or undefined components in a solution, while identified components of known concentration are modeled using expressions based on extensions of the standard acid-base theory. The SMPB method is implemented in a freeware package, (pH)Tools, for use with Matlab. We validated the predictive accuracy of these methods by using strong acid titrations of cucumber slurries to predict the amount of a weak acid required to adjust pH to selected target values.

Optimizing detection of tissue anisotropy by fluorescence recovery after photobleaching

Fluorescence recovery after photobleaching (FRAP) has been widely used to measure fluid flow and diffusion in gels and tissues. It has not been widely used in detection of tissue anisotropy. This may be due to a lack of applicable theory, or due to inherent limitations of the method. We discuss theoretical aspects of the relationship between anisotropy of tissue structure and anisotropy of diffusion coefficients, with special regard to the size of the tracer molecule used. We derive a semi-mechanistic formula relating the fiber volume fraction and ratio of fiber and tracer molecule diameters to the expected anisotropy of the diffusion coefficients. This formula and others are tested on simulated random walks through random simulated and natural media. We determine bounds on the applicability of FRAP for detection of tissue anisotropy, and suggest minimum tracer sizes for detection of anisotropy in tissues of different composition (fiber volume fraction and fiber diameter). We find that it will be easier to detect anisotropy in monodisperse materials than in polydisperse materials. To detect mild anisotropy in a tissue, such as cartilage, which has a low fiber fraction would require a tracer molecule so large that it would be difficult to deliver to the tissue. We conclude that FRAP can be used to detect tissue anisotropy when the tracer molecule is sufficiently large relative to the fiber diameter, volume fraction, and degree of polydispersivity, and when the anisotropy is sufficiently pronounced.

Quantifying the significance of phage attack on starter cultures: a mechanistic model for population dynamics of phage and their hosts isolated from fermenting sauerkraut

We investigated the possibility of using starter cultures in sauerkraut fermentation and thereby reducing the quantity of salt used in the process. This, in turn, would reduce the amount of waste salt that would enter in our water resources. Phage, naturally present in sauerkraut fermentation, could potentially affect the starter cultures introduced. Thus, a mechanistic mathematical model was developed to quantify the growth kinetics of the phage and starter cultures. The model was validated by independent experiments with two Leuconostoc mesenteroides strains isolated from sauerkraut and their corresponding phage. Model simulations and experimental evidence showed the presence of phage-resistant cell populations in starter cultures which replaced phage-sensitive cells, even when the initial phage density (P(0)) and multiplicity of infection (MOI) were low (P(0) < 1 x 10(3) PFU/ml; MOI < 10(-4)) in the MRS media. Based on the results of model simulation and parameter optimization, it was suggested that the kinetic parameters of phage-host interaction, especially the adsorption rate, vary with the initial phage and host densities and with time. The model was validated in MRS broth. Therefore, the effects of heterogeneity and other environmental factors, such as temperature and pH, should be considered to make the model applicable to commercial fermentations.

Force and deformation on branching rudiments: cleaving between hypotheses

A mathematical model of the forces and deformations of the tissues involved in branching morphogenesis is developed and solved. The epithelium and mesenchyme are modeled as Stokes fluids separated by an interface. Each fluid is assumed to have constant viscosity. An initially 3-lobed rudiment is deformed by three inwardly directed point forces. Relationships between the physical parameters of the model (tissue viscosity, clefting force, surface tension) and the time course and morphology are explored. We find that the surface tension, clefting force, and viscosity ratio of the two tissues have significant effects on the branching. We conclude that epithelial branching in soft gels is fundamentally different from epithelial branching in mesenchyme, because of the different mechanics. We propose that a complete understanding of branching morphogenesis requires measurements of the mechanical aspects.

Enhanced growth of tumors in SPARC null mice is associated with changes in the ECM

SPARC, a 32-kDa glycoprotein, participates in the regulation of morphogenesis and cellular differentiation through its modulation of cell-matrix interactions. Major functions defined for SPARC in vitro are de-adhesion and antiproliferation. In vivo, SPARC is restricted in its expression to remodeling tissues, including pathologies such as cancer. However, the function of endogenous SPARC in tumor growth and progression is not known. Here, we report that implanted tumors grew more rapidly in mice lacking SPARC. We observed that tumors grown in SPARC null mice showed alterations in the production and organization of ECM components and a decrease in the infiltration of macrophages. However, there was no change in the levels of angiogenic growth factors in comparison to tumors grown in wild-type mice, although there was a statistically significant difference in total vascular area. Whereas SPARC did inhibit the growth of tumor cells in vitro, it did not have a demonstrable effect on the proliferation or apoptosis of tumor cells in vivo. These data indicate that host-derived SPARC is important for the appropriate organization of the ECM in response to implanted tumors and highlight the importance of the ECM in regulating tumor growth.

Energy-based dynamic model for variable temperature batch fermentation by Lactococcus lactis

We developed a mechanistic mathematical model for predicting the progression of batch fermentation of cucumber juice by Lactococcus lactis under variable environmental conditions. In order to overcome the deficiencies of presently available models, we use a dynamic energy budget approach to model the dependence of growth on present as well as past environmental conditions. When parameter estimates from independent experimental data are used, our model is able to predict the outcomes of three different temperature shift scenarios. Sensitivity analyses elucidate how temperature affects the metabolism and growth of cells through all four stages of fermentation and reveal that there is a qualitative reversal in the factors limiting growth between low and high temperatures. Our model has an applied use as a predictive tool in batch culture growth. It has the added advantage of being able to suggest plausible and testable mechanistic assumptions about the interplay between cellular energetics and the modes of inhibition by temperature and end product accumulation.

Multiphase mechanics of capsule formation in tumors

The presence of a capsule around a tumor is known to be correlated with benign status, and the absence of a capsule often has negative implications for patient prognosis. A mechanical description is presented of the growth of a tumor and the resulting deformations of surrounding normal tissue. A mathematical model of the mechanics is analyzed using physical parameters measured in vivo and in vitro. The model has only three dimensionless parameters, and its results are very robust with respect to parameter variation. We show that the presence of contractility in the surrounding tissue, corresponding to a host defense, can make an existing capsule denser and constrain the tumor better, but cannot be responsible for the observed pressure gradients in encapsulated tumors. Some implications for treatment are discussed.

Theoretical analysis of conjugate localization in two-step cancer chemotherapy

Considerable research has been aimed at improving the efficacy of chemotherapeutic agents for cancer therapy. A promising two-step approach that is designed to minimize systemic drug toxicity while maximizing activity in tumors employs monoclonal antibody-enzyme conjugates for the activation of anti-cancer prodrugs. A mathematical model based on the biology of human 3677 melanoma xenografts in nude mice is presented, analyzed, and numerically simulated to study the biodistribution, pharmacokinetics, and intratumoral localization properties of L49-beta-lactamase fusion proteins in solid tumor masses. The model predictions were compared with published experimental data and an excellent correlation was found to exist. Analytic expressions for the total concentration of conjugate in the tumor, the time at which the concentration is maximal, and the half life of conjugate in the tissue were derived. From these results, key parameters were isolated; and the effects of the tumor vasculature, binding kinetics, and administration schedule were investigated. The antibody-antigen dissociation ratio, the conjugate permeability, and the inter-capillary half distance within the tumor mass were found to strongly influence localization and retention in the tumor. The model was used to examine various dosing strategies in an attempt to determine which regimen would provide the best biodistribution results. The results of administering a uniform dose of conjugate via bolus injection, multiple injections, and continuous infusion were compared. The model predicts that when saturation of binding sites does not occur, dosing strategy has little effect on the amount of conjugate that localizes in the tumor.

Mathematical and experimental analysis of localization of anti-tumour antibody-enzyme conjugates

Considerable research has been aimed at improving the efficacy of chemotherapeutic agents for cancer therapy. A promising two-step approach that is designed to minimize systemic drug toxicity while maximizing activity in tumours employs monoclonal antibody (mAb)-enzyme conjugates for the activation of anticancer prodrugs. We present, analyse and numerically simulate a mathematical model based on the biology of the system to study the biodistribution, pharmacokinetics and localization properties of mAb-enzyme conjugates in tumour tissue. The model predictions were compared with experimental observations and an excellent correlation was found to exist. In addition, the critical parameters affecting conjugate half-life were determined to be the inter-capillary half-distance and the antibody-antigen binding affinity. An approximation is presented relating the per cent injected dose per gram to inter-capillary half-distance and time. Finally, the model was used to examine various dosing strategies in an attempt to determine which regimen would provide the best biodistribution results. We compared the results of administering a uniform dose of fusion protein via bolus injection, multiple injections and continuous infusion. The model predicts that dosing strategy has little effect on the amount of conjugate that localizes in the tumour.

Model and analysis of chemotactic bacterial patterns in a liquid medium

A variety of spatial patterns are formed chemotactically by the bacteria Escherichia coli and Salmonella typhimurium. We focus in this paper on patterns formed by E. coli and S. typhimurium in liquid medium experiments. The dynamics of the bacteria, nutrient and chemoattractant are modeled mathematically and give rise to a nonlinear partial differential equation system. We present a simple and intuitively revealing analysis of the patterns generated by our model. Patterns arise from disturbances to a spatially uniform solution state. A linear analysis gives rise to a second order ordinary differential equation for the amplitude of each mode present in the initial disturbance. An exact solution to this equation can be obtained, but a more intuitive understanding of the solutions can be obtained by considering the rate of growth of individual modes over small time intervals.

A minimal mechanism for bacterial pattern formation

Colonies of Escherichia coli or Salmonella typhimurium form geometrically complex patterns when exposed to, or feeding on, intermediates of the tricarboxylic acid (TCA) cycle. In response to the TCA cycle intermediate, the bacteria secrete aspartate, a potent chemo-attractant. As a result, the cells form high-density aggregates arranged in striking regular patterns. The simplest are temporary spots formed in a liquid medium by both E. coli and S. typhimurium. In semi-solid medium S. typhimurium forms concentric rings arising from a low-density bacterial lawn, which are either continuous or spotted, whereas E. coli forms complex patterns arising from a dense swarm ring, including interdigitated spots (also called sunflower spirals), radial spots, radial stripes and chevrons. We present a mathematical model that captures all three of the pattern-forming processes experimentally observed in both E. coli and S. typhimurium, using a minimum of assumptions.

A mechanical model for the formation of vascular networks in vitro

Endothelial cells, when cultured on gelled basement membrane matrix exert forces of tension through which they deform the matrix and at the same time they aggregate into clusters. The cells eventually form a network of cord-like structures connecting cell aggregates. In this network, almost all of the matrix has been pulled underneath the cell cords and cell clusters. This phenomenon has been proposed as a possible model for the growth and development of planar vascular systems in vitro. Our hypothesis is that the matrix is reorganized and the cellular networks form as a result of traction forces exerted by the cells on the matrix and the latter's elasticity. We construct and analyze a mathematical model based on this hypothesis and examine conditions necessary for the formation of the pattern. We show cell migration is not necessary for pattern formation and that isotropic, strain-stimulated traction is sufficient to form the observed patterns.

Evaluation of feline leukemia virus control measures

A susceptible-infected-recovered-susceptible (SIRS) model of the epidemiology of feline leukemia virus is formulated and analysed. The dynamics of the disease are dramatically different in no-risk, low-risk and high-risk subpopulations of asocial, free roaming, and multiple cat household cats. Among low risk (<1% prevalence) free roaming cats, the model predicts that an effective immunization rate of 4% year-1, or an effective removal rate of 8% year-1 are adequate to control the disease completely. Under higher risk (10% prevalence) conditions, an effective immunization rate of 23-72% year-1 or a removal rate of 69-145% year-1 are required for control. At very high (30%) prevalence rates, even heroic measures may not suffice to substantially reduce disease prevalence: a vaccination rate of 100% year-1 even if attainable, would only slightly reduce disease prevalence from 30% to 29%. We conclude that the current estimated effective feline leukemia virus immunization rate of 11-19% of the general population is inadequate to provide herd immunity in the subpopulation of cats which are genuinely at risk of infection. A substantial increase in the vaccination rate and/or intensification of test and removal efforts in the at risk population would be required to attain an effective level of protection.

Simple versus sophisticated models of breath alcohol exhalation profiles

For medicolegal purposes, breath alcohol content is typically determined from an end-expiratory sample. Measurements obtained by this method necessarily underestimate the alveolar breath alcohol content, and therefore underestimate the blood alcohol content. We suggest and analyse an improved paradigm which uses the entire time-series of breath alcohol measurements during exhalation, not simply the last recorded value. We present two mathematical models for the exhaling lung, and discuss the implications of each for more accurate and therefore more reliable breath alcohol measurement.

A mechanism for early branching in lung morphogenesis

The lung is a highly branched fluid-filled structure, that develops by repeated dichotomous branching of a single bud off the foregut, of epithelium invaginating into mesenchyme. Incorporating the known stress response of developing lung tissues, we model the developing embryonic lung in fluid mechanical terms. We suggest that the repeated branching of the early embryonic lung can be understood as the natural physical consequence of the interactions of two or more plastic substances with surface tension between them. The model makes qualitative and quantitative predictions, as well as suggesting an explanation for such observed phenomena as the asymmetric second branching of the embryonic bronchi.

A thin-shell model of the cornea and its application to corneal surgery

BACKGROUND

We present a thin-shell, analytical model of the vertebrate cornea to study the changes in shape resulting from surgical operations (eg, radial keratotomy).

METHODS

A simple closed-form solution is derived for a thin linearly elastic spherical model of the cornea. We assume that the shell is symmetrical about a central axis and that the modulus of elasticity is the same in all directions. The surgery is modeled by allowing the modulus of elasticity (or equivalently the thickness) of the shell to depend upon position, measured as an angle from the axis of symmetry.

RESULTS

The analytical nature of the solution allows us to compute the principal curvatures of the cornea explicitly. For example, for representative parameters, the model predicts the average corneal curvature changes from about 43 diopters before keratotomy to about 38 D after keratotomy.

CONCLUSIONS

The model is used to estimate Young's modulus from experimental data reported previously by Thomas et al (Invest Ophthalmol Vis Sci 1991;32:1000), as well as to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.

Analytical model of corneal surgery

We present a model of the human cornea in order to study the changes in its shape resulting from surgical operations (e.g., radial keratotomy). A simple closed-form solution is given for a thin linearly elastic spherical shell model of the cornea. We assume axisymmetry and isotropy in the shell surface. The surgery is modeled by permitting Young's modulus and shell thickness to depend on position. The analytical nature of the solution permits principal shell curvatures to be explicitly calculated. The model is used to investigate the effect of surgery on corneal flattening and the associated sensitivity to intraocular pressure changes.