Numerical Modeling of Interstitial Fluid Flow Coupled with Blood Flow through a Remodeled Solid Tumor Microvascular Network

Modeling of interstitial fluid flow involves processes such as fluid diffusion, convective transport in extracellular matrix, and extravasation from blood vessels. To date, majority of microvascular flow modeling has been done at different levels and scales mostly on simple tumor shapes with their capillaries. However, with our proposed numerical model, more complex and realistic tumor shapes and capillary networks can be studied. Both blood flow through a capillary network, which is induced by a solid tumor, and fluid flow in tumor’s surrounding tissue are formulated. First, governing equations of angiogenesis are implemented to specify the different domains for the network and interstitium. Then, governing equations for flow modeling are introduced for different domains. The conservation laws for mass and momentum (including continuity equation, Darcy’s law for tissue, and simplified Navier–Stokes equation for blood flow through capillaries) are used for simulating interstitial and intravascular flows and Starling’s law is used for closing this system of equations and coupling the intravascular and extravascular flows. This is the first study of flow modeling in solid tumors to naturalistically couple intravascular and extravascular flow through a network. This network is generated by sprouting angiogenesis and consisting of one parent vessel connected to the network while taking into account the non-continuous behavior of blood, adaptability of capillary diameter to hemodynamics and metabolic stimuli, non-Newtonian blood flow, and phase separation of blood flow in capillary bifurcation. The incorporation of the outlined components beyond the previous models provides a more realistic prediction of interstitial fluid flow pattern in solid tumors and surrounding tissues. Results predict higher interstitial pressure, almost two times, for realistic model compared to the simplified model.     

Parameter estimation and sensitivity analysis in an agent-based model of Leishmania major infection

Computer models of disease take a systems biology approach toward understanding host-pathogen interactions. In particular, data driven computer model calibration is the basis for inference of immunological and pathogen parameters, assessment of model validity, and comparison between alternative models of immune or pathogen behavior. In this paper we describe the calibration and analysis of an agent-based model of Leishmania major infection. A model of macrophage loss following uptake of necrotic tissue is proposed to explain macrophage depletion following peak infection. Using Gaussian processes to approximate the computer code, we perform a sensitivity analysis to identify important parameters and to characterize their influence on the simulated infection. The analysis indicates that increasing growth rate can favor or suppress pathogen loads, depending on the infection stage and the pathogen's ability to avoid detection. Subsequent calibration of the model against previously published biological observations suggests that L. major has a relatively slow growth rate and can replicate for an extended period of time before damaging the host cell.     

Symmetry and the stationary state behavior of enzyme membranes

An analysis of the stationary state behavior of model enzyme (or catalytic) membranes is considered. In particular, membrane functional symmetry is shown to be of critical importance in deriving a unique set of global linear phenomenological relations from their local counterparts. Indeed the appropriate transformation of the dissipation function, based on the correct identification of the “transport plane” within the membrane, leads to a global analog of the Curie Principle. By extending the argument from the near-equilibrium regime to the pseudo-first-order kinetic regime a set of practical equations can be derived. These make it possible to obtain local transport and reaction parameters from global measurements. The facilitation of transport by reaction is discussed.     

Identifiability and retrievability of unique parameters describing intrinsic Andrews kinetics

A key factor contributing to the variability in the microbial kinetic parameters reported from batch assays is parameter identifiability, i.e., the ability of the mathematical routine used for parameter estimation to provide unique estimates of the individual parameter values. This work encompassed a three-part evaluation of the parameter identifiability of intrinsic kinetic parameters describing the Andrews growth model that are obtained from batch assays. First, a parameter identifiability analysis was conducted by visually inspecting the sensitivity equations for the Andrews growth model. Second, the practical retrievability of the parameters in the presence of experimental error was evaluated for the parameter estimation routine used. Third, the results of these analyses were tested using an example data set from the literature for a self-inhibitory substrate. The general trends from these analyses were consistent and indicated that it is very difficult, if not impossible, to simultaneously obtain a unique set of estimates of intrinsic kinetic parameters for the Andrews growth model using data from a single batch experiment.     

Bayesian calibration of a stochastic,multiscale agent-based model for predicting in vitro tumor growth

Hybrid multiscale agent-based models (ABMs) are unique in their ability to simulate individual cell interactions and microenvironmental dynamics. Unfortunately, the high computational cost of modeling individual cells, the inherent stochasticity of cell dynamics, and numerous model parameters are fundamental limitations of applying such models to predict tumor dynamics. To overcome these challenges, we have developed a coarse-grained two-scale ABM (cgABM) with a reduced parameter space that allows for an accurate and efficient calibration using a set of time-resolved microscopy measurements of cancer cells grown with different initial conditions. The multiscale model consists of a reaction-diffusion type model capturing the spatio-temporal evolution of glucose and growth factors in the tumor microenvironment (at tissue scale), coupled with a lattice-free ABM to simulate individual cell dynamics (at cellular scale). The experimental data consists of BT474 human breast carcinoma cells initialized with different glucose concentrations and tumor cell confluences. The confluence of live and dead cells was measured every three hours over four days. Given this model, we perform a time-dependent global sensitivity analysis to identify the relative importance of the model parameters. The subsequent cgABM is calibrated within a Bayesian framework to the experimental data to estimate model parameters, which are then used to predict the temporal evolution of the living and dead cell populations. To this end, a moment-based Bayesian inference is proposed to account for the stochasticity of the cgABM while quantifying uncertainties due to limited temporal observational data. The cgABM reduces the computational time of ABM simulations by 93% to 97% while staying within a 3% difference in prediction compared to ABM. Additionally, the cgABM can reliably predict the temporal evolution of breast cancer cells observed by the microscopy data with an average error and standard deviation for live and dead cells being 7.61±2.01 and 5.78±1.13, respectively.     

Fast parameter calibration of a cardiac electromechanical model from medical images based on the unscented transform

Patient-specific cardiac modelling can help in understanding pathophysiology and predict therapy planning. However, it requires to personalize the model geometry, kinematics, electrophysiology and mechanics. Calibration aims at providing proper initial values of parameters before performing the personalization stage which involves solving an inverse problem. We propose a fast automatic calibration method of the mechanical parameters of a complete electromechanical model of the heart based on a sensitivity analysis and the Unscented Transform algorithm. A new implementation of the complete Bestel–Clement–Sorine (BCS) cardiac model is also proposed, in a modular and efficient framework. A complete sensitivity analysis is performed that reveals which observations on the volume evolution are significant to characterize the global behaviour of the myocardium. We show that the calibration method gives satisfying results by optimizing up to 5 parameters of the BCS model in only one iteration. This method was evaluated synthetically as well as on 7 volunteers with a mean relative error from the real data of 10 %. This calibration is designed to replace manual parameter estimation as well as initialization steps that precede automatic personalization algorithms based on images.     

Parameter identification of the calvin photosynthesis cycle

Summary This article is concerned with the determination of kinetic parameters of the Calvin photosynthesis cycle which is described by seventeen nonlinear ordinary differential equations. It is shown that the task requires dynamic data for several sets of initial conditions. The numerical technique is based upon an algorithm for non-linear optimization and Gear's numerical integration scheme for stiff systems of differential equations. The sensitivity of the parameters to noise in the data is tested with a method adapted from Rosenbrook and Storey. A preliminary set of parameters has been obtained from a preliminary set of experimental data. The numerical methods are then tested with synthetic data derived from these parameters. The mathematical model and the results obtained in the simulation are used as an aid in designing new experiments.     

Requirements for Glucose, Nitrate, and Phosphate and Their Variations in the Analysis of the Mixed Culture of Pseudomonas aeruginosa Variants

The calibration data and sensitivity analysis for the variational model of Pseudomonas aeruginosa mixed culture containing R, S, and M phenotypic variants are presented. The model with calibrated parameters (requirements of the variants for glucose, nitrates, and phosphates) adequately describes the experiment. The sensitivity indices (the species abundances and total community size) changed least under the influence of the parameter variations in the one-factor limitation areas. The variant proportions are more sensitive to variation of the requirements than the community size.     

Extravascular fluid dynamics of the embryonic chick wing bud

While a number of models of positional information in the chick wing bud have involved the diffusion of morphogens to establish chemical gradients of morphogenetic activity, only recently have there been attempts to characterize the milieu in which such diffusion must take place. We report an analysis of the fluid dynamics of the extravascular (interstitial) spaces of stage 22-25 chick wing buds, into which aqueous aniline blue dye was microinjected as a visible, unreactive tracer. Six sites along the antero-posterior (A-P) and proximo-distal (P-D) axes were chosen for study. Injections of dye into the posterior half of the wing bud exhibited marked directionality of extravascular transport (mean of all posterior sites = 68%), while anterior injections showed little or no directionality (mean of all anterior sites = 13%). All instances of directed dye movement were disto-proximal, the same direction as the blood flow through the marginal veins. In embryos gassed in situ with CO2, which reversibly stopped the heartbeat and vascular flow, directionality was abolished, yet diffusion rates were unaffected. Posterior disto-proximal extravascular dye movement was correlated with the greater diameter, flow velocity, and volume flow rate of the posterior marginal vein, compared to the anterior marginal vein. Radial diffusion rates were measured, and posterior disto-proximal rates were corrected for measured disto-proximal directionality by the use of a simple diffusion-translation model. Three-way analysis of variance showed that directionality-uncorrected disto-proximal rates in posterior sites were not significantly different from anterior radial rates, but that directionality-corrected posterior rates did differ significantly (P less than 0.0001). A significant stage effect (P less than 0.005) and a significant interaction between the A-P axis and stage (P less than 0.05) were also found. We hypothesize that the spatial arrangement and flow patterns of the vasculature physically determine the fluid dynamics of the interstitium. Based on these observations, we also suggest that disto-proximal extravascular fluid movement in the posterior wing bud presents a barrier to the free diffusion of aqueous molecules, including morphogens originating in the "zone of polarizing activity."     

Impact of imperfect test sensitivity on determining risk factors: the case of bovine tuberculosis

Background

Imperfect diagnostic testing reduces the power to detect significant predictors in classical cross-sectional studies. Assuming that the misclassification in diagnosis is random this can be dealt with by increasing the sample size of a study. However, the effects of imperfect tests in longitudinal data analyses are not as straightforward to anticipate, especially if the outcome of the test influences behaviour. The aim of this paper is to investigate the impact of imperfect test sensitivity on the determination of predictor variables in a longitudinal study.

Methodology/Principal Findings

To deal with imperfect test sensitivity affecting the response variable, we transformed the observed response variable into a set of possible temporal patterns of true disease status, whose prior probability was a function of the test sensitivity. We fitted a Bayesian discrete time survival model using an MCMC algorithm that treats the true response patterns as unknown parameters in the model. We applied our approach to epidemiological data of bovine tuberculosis outbreaks in England and investigated the effect of reduced test sensitivity in the determination of risk factors for the disease. We found that reduced test sensitivity led to changes to the collection of risk factors associated with the probability of an outbreak that were chosen in the ‘best’ model and to an increase in the uncertainty surrounding the parameter estimates for a model with a fixed set of risk factors that were associated with the response variable.

Conclusions/Significance

We propose a novel algorithm to fit discrete survival models for longitudinal data where values of the response variable are uncertain. When analysing longitudinal data, uncertainty surrounding the response variable will affect the significance of the predictors and should therefore be accounted for either at the design stage by increasing the sample size or at the post analysis stage by conducting appropriate sensitivity analyses.     

Application of global sensitivity analysis to determine goals for design of experiments: an example study on antibody-producing cell cultures

Global sensitivity analysis (GSA) can be used to quantify the importance of model parameters and their interactions with respect to model output. In this study, the Sobol' method for GSA is applied to a dynamic model of monoclonal antibody-producing mammalian cell cultures in order to identify the parameters that need to be accurately determined experimentally. Our results show that most parameters have low sensitivity indices and exhibit strong interactions with one another. These parameters can be set at their nominal values and unnecessary experimentation can therefore be avoided. In contrast, certain parameters are identified as sensitive, necessitating their estimation given sufficiently rich experimental data. Moreover, parameter sensitivity varies during culture time in a biologically meaningful manner. In conclusion, GSA can serve as an excellent precursor to optimal experiment design.     

Yield-SAFE: A parameter-sparse, process-based dynamic model for predicting resource capture, growth, and production in agroforestry systems

1. Silvoarable agroforestry (SAF) is the cultivation of trees and arable crops on the same parcel of land. SAF may contribute to modern diversified land use objectives in Europe, such as enhanced biodiversity and productivity, reduced leaching of nitrogen, protection against flooding and erosion, and attractiveness of the landscape. Long-term yield predictions are needed to assess long-term economic profitability of SAF.

2. A model for growth, resource sharing and productivity in agroforestry systems was developed to act as a tool in forecasts of yield, economic optimization of farming enterprises and exploration of policy options for land use in Europe. The model is called Yield-SAFE; from “YIeld Estimator for Long term Design of Silvoarable AgroForestry in Europe”. The model was developed with as few equations and parameters as possible to allow model parameterization under constrained availability of data from long-term experiments.

3. The model consists of seven state equations expressing the temporal dynamics of: (1) tree biomass; (2) tree leaf area; (3) number of shoots per tree; (4) crop biomass; (5) crop leaf area index; (6) heat sum; and (7) soil water content. The main outputs of the model are the growth dynamics and final yields of trees and crops. Daily inputs are temperature, radiation and precipitation. Planting densities, initial biomasses of tree and crop species, and soil parameters must be specified.

4. A parameterization of Yield-SAFE is generated, using published yield tables for tree growth and output from the comprehensive crop simulation model STICS. Analysis of tree and crop growth data from two poplar agroforestry stands in the United Kingdom demonstrates the validity of the modelling concept and calibration philosophy of Yield-SAFE. A sensitivity analysis is presented to elucidate which biological parameters most influence short and long-term productivity and land equivalent ratio.

5. The conceptual model, elaborated in Yield-SAFE, in combination with the outlined procedure for model calibration, offers a valid tool for exploratory land use studies.

EQUIL: Simulation and data analysis of binding reactions with arbitrary chemical models

We have developed an algorithm for simulation and analysis of arbitrary chemical systems in equilibrium, with emphasis on ligand binding reactions. The program EQUIL can treat reactions involving multiple ligands, multiple binding sites, ternary complex models, allosteric effectors, competitive and noncompetitive binding, conformational changes, cooperativity, and generally any scheme that can be represented as a set of chemical equations. EQUIL is based on a general thermodynamic model of chemical equilibria; it does not involve nonlinear transformation of experimental data, but it does require the user to define the model of interaction between ligands and receptors by writing down the appropriate chemical reactions. EQUIL contains features of particular importance to ligand binding experiments: variable binding capacities, nonspecific binding, and the ability to simultaneously analyze data from different types of experiments. Furthermore, the simulation feature of EQUIL allows the user to investigate the feasibility of experiments that could possibly distinguish between different reaction models. We illustrate the use of this program on personal computers to analyze and simulate simple and complicated interactions between ligands and receptors.     

Importance of Initial Concentration of Factor VIII in a Mechanistic Model of In Vitro Coagulation

This computational study generates a hypothesis for the coagulation protein whose initial concentration greatly influences the course of coagulation. Many clinical malignancies of blood coagulation arise due to abnormal initial concentrations of coagulation factors. Sensitivity analysis of mechanistic models of blood coagulation is a convenient method to assess the effect of such abnormalities. Accordingly, the study presents sensitivity analysis, with respect to initial concentrations, of a recently developed mechanistic model of blood coagulation. Both the model and parameters to which model sensitivity is being analyzed provide newer insights into blood coagulation: the model incorporates distinct equations for plasma-phase and platelet membrane-bound species, and sensitivity to initial concentrations is a new dimension in sensitivity analysis. The results show that model predictions are most uncertain with respect to changes in initial concentration of factor VIII, and this hypothesis is supported by results from other models developed independently.     

Implementation of chemometrics,design of experiments,and neural network analysis for prior process knowledge assessment,failure modes and effect analysis,scale-down model development,and process characterization for a chromatographic purification of Teriparatide

Process understanding and characterization forms the foundation, ensuring consistent and robust biologics manufacturing process. Using appropriate modeling tools and machine learning approaches, the process data can be monitored in real time to avoid manufacturing risks. In this article, we have outlined an approach toward implementation of chemometrics and machine learning tools (neural network analysis) to model and predict the behavior of a mixed-mode chromatography step for a biosimilar (Teriparatide) as a case study. The process development data and process knowledge was assimilated into a prior process knowledge assessment using chemometrics tools to derive important parameters critical to performance indicators (i.e., potential quality and process attributes) and to establish the severity ranking for the FMEA analysis. The characterization data of the chromatographic operation are presented alongwith the determination of the critical, key and non- key process parameters, set points, operating, process acceptance and characterized ranges. The scale-down model establishment was assessed using traditional approaches and novel approaches like batch evolution model and neural network analysis. The batch evolution model was further used to demonstrate batch monitoring through direct chromatographic data, thus demonstrating its application for continuos process verification. Assimilation of process knowledge through a structured data acquisition approach, built-in from process development to continuous process verification was demonstrated to result in a data analytics driven model that can be coupled with machine learning tools for real time process monitoring. We recommend application of these approaches with the FDA guidance on stage wise process development and validation to reduce manufacturing risks.     

Analysis of factors that influence rates of carbon monoxide uptake, distribution, and washout from blood and extravascular tissues using a multicompartment model.

To better understand factors that influence carbon monoxide (CO) washout rates, we utilized a multicompartment mathematical model to predict rates of CO uptake, distribution in vascular and extravascular (muscle vs. other soft tissue) compartments, and washout over a range of exposure and washout conditions with varied subject-specific parameters. We fitted this model to experimental data from 15 human subjects, for whom subject-specific parameters were known, multiple washout carboxyhemoglobin (COHb) levels were available, and CO exposure conditions were identical, to investigate the contributions of exposure conditions and individual variability to CO washout from blood. We found that CO washout from venous blood was biphasic and that postexposure times at which COHb samples were obtained significantly influenced the calculated CO half times (P < 0.0001). The first, more rapid, phase of CO washout from the blood reflected the loss of CO to the expired air and to a slow uptake by the muscle compartment, whereas the second, slower washout phase was attributable to CO flow from the muscle compartment back to the blood and removal from blood via the expired air. When the model was used to predict the effects of varying exposure conditions for these subjects, the CO exposure duration, concentration, peak COHb levels, and subject-specific parameters each influenced washout half times. Blood volume divided by ventilation correlated better with half-time predictions than did cardiac output, muscle mass, or ventilation, but it explained only approximately 50% of half-time variability. Thus exposure conditions, COHb sampling times, and individual parameters should be considered when estimating CO washout rates for poisoning victims.     

An effective-diffusivity model of pulmonary capillary exchange: General theory, limiting cases, and sensitivity analysis

A family of possible models of capillary-tissue exchange useful for interpreting multiple-tracer data from the pulmonary circulation was derived from the convective-diffusion equation. The models are simplifications of a uniform-transit-time model with two serial diffusion layers outside of the capillary. Modifications of this four-parameter model were derived, and the importance of simplifying assumptions were compared using moment analysis and transform-domain equivalence. A permeability-diffusion model was derived by assuming that the layer nearer the capillary contributed a constant resistance to tracer movement. Using sensitivity analysis, we found that the three parameters of this permeability-diffusion model could not be determined independently, and that further model simplification was highly desirable. Two distinct paths of further simplification were explored.The Sangren-Sheppard model was considered as one path. An alternative path of simplification led to a new model of tracer behavior which we have called an effective-diffusivity model. Moment matching was used to determine the relationships among these models. Sensitivity analysis of the Sangren-Sheppard and effective-diffusivity models showed that the parameters of both of these models were more easily identified. However, the sensitivity analysis also showed that these two models had quite different sensitivities to their respective volume parameters. The Sangren-Sheppard model prediction was affected at all times by a change in the extravascular volume parameter, while the effective-diffusivity model prediction was affected only at the longest times. We concluded that the effective-diffusivity model may be a better alternative to the Sangren-Sheppard model under some conditions. The parameters of the effective-diffusivity model provide a more reliable index of the physiology of capillary-tissue exchange as small molecules as measured by the multiple tracer method in the pulmonary circulation.     

生态模型的灵敏度分析

灵敏度分析用于定性或定量地评价模型参数误差对模型结果产生的影响,是模型参数化过程和模型校正过程中的有用工具,具有重要的生态学意义．灵敏度分析包括局部灵敏度分析和全局灵敏度分析．局部灵敏度分析只检验单个参数的变化对模型结果的影响程度;全局灵敏度分析则检验多个参数的变化对模型运行结果总的影响,并分析每一个参数及其参数之间相互作用对模型结果的影响．目前,在对生态模型的灵敏度分析中,越来越倾向于使用全局灵敏度分析的方法．但国内仍多采用局部灵敏度分析方法,很少采用全局灵敏度分析方法．文中详细论述了局部灵敏分析和全局灵敏度分析的主要方法(一次变换法、多元回归法、Morris法、Sobol’法、傅里叶幅度灵敏度检验法和傅里叶幅度灵敏度检验扩展法),希望能为国内生态模型的发展提供一个比较完善的灵敏度分析方法库．结合国内外的灵敏度分析发展现状,指出联合灵敏度研究、灵敏度共性研究及空间直观景观模型的灵敏度分析将为生态模型灵敏度分析研究中的热点和难点．     

Bayesian sensitivity analysis of a model of the aortic valve

Understanding the mechanics of the aortic valve has been a focus of attention for many years in the biomechanics literature, with the aim of improving the longevity of prosthetic replacements. Finite element models have been extensively used to investigate stresses and deformations in the valve in considerable detail. However, the effect of uncertainties in loading, material properties and model dimensions has remained uninvestigated. This paper presents a formal statistical consideration of a selected set of uncertainties on a fluid-driven finite element model of the aortic valve and examines the magnitudes of the resulting output uncertainties. Furthermore, the importance of each parameter is investigated by means of a global sensitivity analysis. To reduce computational cost, a Bayesian emulator-based approach is adopted whereby a Gaussian process is fitted to a small set of training data and then used to infer detailed sensitivity analysis information. From the set of uncertain parameters considered, it was found that output standard deviations were as high as 44% of the mean. It was also found that the material properties of the sinus and aorta were considerably more important in determining leaflet stress than the material properties of the leaflets themselves.