Parameter estimation and sensitivity analysis of a nonlinearly elastic static lung model

A model for the static pressure-volume behavior of the lung parenchyma based on a pseudo-elastic strain energy function was tested. Values of the model parameters and their variances were estimated by an optimal least-squares fit of the model-predicted pressures to the corresponding data from excised, saline-filled dog lungs. Although the model fit data from twelve lungs very well, the coefficients of variation for parameter values differed greatly. To analyze the sensitivity of the model output to its parameters, we examined an approximate Hessian, H, of the least-squares objective function. Based on the determinant and condition number of H, we were able to set formal criteria for choosing the most reliable estimates of parameter values and their variances. This in turn allowed us to specify a normal range of parameter values for these dog lungs. Thus the model not only describes static pressure-volume data, but also uses the data to estimate parameters from a fundamental constitutive equation. The optimal parameter estimation and sensitivity analysis developed here can be widely applied to other physiologic systems.     

胶州湾生物-物理耦合模型参数灵敏度分析

参数灵敏度分析旨在评价模型中各参数对模拟结果的影响程度,是参数优化和模型校正的基础步骤,也是认识模型行为的重要工具。所建的胶州湾生物-物理耦合模型包括浮游植物、浮游动物、营养盐、碎屑和溶解氧5类状态变量,对其涉及的50个参数进行灵敏度分析,得到3个非常灵敏性参数、2个灵敏性参数、11个比较灵敏性参数和34个不太灵敏性参数。非常灵敏及灵敏性参数包括浮游植物生长速率(μPRPC)、暗反应修正因子(FAC)、光饱和强度(α)、浮游植物死亡率(μDEPC)和水体消光系数(bla),主要影响浮游植物生长和死亡过程,反映了浮游植物在生态系统中的基础性和重要性作用。这5个参数显著地影响碳和营养盐循环,是整个胶州湾生态系统最主要的影响参数,应优先进行优化。比较灵敏性参数的影响主要表现在营养盐对浮游植物生长或死亡的限制以及温度对光饱和量的限制,浮游动物生长、牧食和死亡过程以及浮游植物生物量对牧食的限制,叶绿素a的生产,缺氧条件下沉积物释放磷以及浮游植物对磷的摄取等过程,这些参数对于各状态变量的灵敏性存在不同程度的差异,从而表征不同的特点。与不太灵敏性参数相关的过程主要为叶绿素a和碎屑消光作用,温度对浮游植物生长、浮游动物牧食、碎屑和沉积物矿化的限制,碎屑和沉积物矿化与沉降,与无机氮相关的大部分过程,溶解氧浓度变化等,这些过程除了受模型内部参数影响外,还在很大程度上受水深、海水温度和陆源污染等外部因素影响。比较灵敏及不太灵敏性参数影响模型局部过程,是模型校正的重要依据,除了非常灵敏及灵敏性参数以外,叶绿素a、浮游动物、碎屑和无机磷四种状态变量可分别根据叶绿素a最大生产系数(K CHmax)、浮游动物一级死亡率(μDEZC1)、有机碎屑矿化率(μREDC)和浮游植物磷摄取的半饱和常数(h UPPP)进行校正。与营养盐相关参数的灵敏度分析表明,胶州湾浮游植物处于磷限制,无机氮主要受陆源排污影响。因此,对无机氮的校正主要通过合理设置沿岸河流径流量或陆源污染物浓度与比例以及无机氮初始场。溶解氧对各参数均不太灵敏。     

Uncertainty quantification and sensitivity analysis of an arterial wall mechanics model for evaluation of vascular drug therapies

Quantification of the uncertainty in constitutive model predictions describing arterial wall mechanics is vital towards non-invasive assessment of vascular drug therapies. Therefore, we perform uncertainty quantification to determine uncertainty in mechanical characteristics describing the vessel wall response upon loading. Furthermore, a global variance-based sensitivity analysis is performed to pinpoint measurements that are most rewarding to be measured more precisely. We used previously published carotid diameter–pressure and intima–media thickness (IMT) data (measured in triplicate), and Holzapfel–Gasser–Ogden models. A virtual data set containing 5000 diastolic and systolic diameter–pressure points, and IMT values was generated by adding measurement error to the average of the measured data. The model was fitted to single-exponential curves calculated from the data, obtaining distributions of constitutive parameters and constituent load bearing parameters. Additionally, we (1) simulated vascular drug treatment to assess the relevance of model uncertainty and (2) evaluated how increasing the number of measurement repetitions influences model uncertainty. We found substantial uncertainty in constitutive parameters. Simulating vascular drug treatment predicted a 6% point reduction in collagen load bearing (\(L_\mathrm {coll}\)), approximately 50% of its uncertainty. Sensitivity analysis indicated that the uncertainty in \(L_{\mathrm {coll}}\) was primarily caused by noise in distension and IMT measurements. Spread in \(L_{\mathrm {coll}}\) could be decreased by 50% when increasing the number of measurement repetitions from 3 to 10. Model uncertainty, notably that in \(L_{\mathrm {coll}}\), could conceal effects of vascular drug therapy. However, this uncertainty could be reduced by increasing the number of measurement repetitions of distension and wall thickness measurements used for model parameterisation.     

Parameter estimation of a monod-type model. Part I: Theoretical identifiability and sensitivity analysis

Summary Parameter estimation of a Monod-type model based on the study of the theoretical identifiability of the model followed by the sensitivity analysis of the state variables with respect to parameters is presented. Theorerical identifiability allows to establish the unicity of the solution. On the other hand, sensitivity analysis throws light on the conditions that make parameters identifiable. Thus, the introduction of additional parameters, especially substrate maintenance and death constant, increases the estimation difficulty.     

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.     

A semi-empirical nonlinear viscoelastic model of the arterial wall

A mathematical model was developed for the characterization of quasistatic nonlinear viscoelastic behaviour of large arteries with activated smooth muscle. Arteries were considered to belong to the class of viscoelastic materials with fading memory and the first order term of the integral polynomial constitutive equations of Pipkin and Rogers satisfying the nonlinear superposition principle was modified to handle responses to increasing and decreasing loads independently. The two creep functions contained by the obtained one dimensional constitutive equation were determined experimentally from the series of creep and recovery tests of increasing amplitude performed on isolated canine iliac arteries following the activation of the vascular smooth muscle by normal dose of norepinephrine in vitro. Utilizing the constitutive equation of the arterial wall and the tabulated values of creep functions successive stress-strain hysteresis loops of various constant stress rates were simulated by digital computer. The computed hysteresis loops demonstrated the main characteristics, such as the weak and asymmetric rate-sensitivity of the experimentally observed hysteresis of arteries qualitatively well, thus allowing certain conclusions on the mechanism of quasistatic viscoelastic behaviour of vascular smooth muscle.     

Multiphysics simulation of blood flow and LDL transport in a porohyperelastic arterial wall model

Atherosclerosis localizes at a bend andor bifurcation of an artery, and low density lipoproteins (LDL) accumulate in the intima. Hemodynamic factors are known to affect this localization and LDL accumulation, but the details of the process remain unknown. It is thought that the LDL concentration will be affected by the filtration flow, and that the velocity of this flow will be affected by deformation of the arterial wall. Thus, a coupled model of a blood flow and a deformable arterial wall with filtration flow would be invaluable for simulation of the flow field and concentration field in sequence. However, this type of highly coupled interaction analysis has not yet been attempted. Therefore, we performed a coupled analysis of an artery with multiple bends in sequence. First, based on the theory of porous media, we modeled a deformable arterial wall using a porohyperelastic model (PHEM) that was able to express both the filtration flow and the viscoelastic behavior of the living tissue, and simulated a blood flow field in the arterial lumen, a filtration flow field and a displacement field in the arterial wall using a fluid-structure interaction (FSI) program code by the finite element method (FEM). Next, based on the obtained results, we further simulated LDL transport using a mass transfer analysis code by the FEM. We analyzed the PHEM in comparison with a rigid model. For the blood flow, stagnation was observed downward of the bends. The direction of the filtration flow was only from the lumen to the wall for the rigid model, while filtration flows from both the wall to the lumen and the lumen to the wall were observed for the PHEM. The LDL concentration was high at the lumenwall interface for both the PHEM and rigid model, and reached its maximum value at the stagnation area. For the PHEM, the maximum LDL concentration in the wall in the radial direction was observed at the position of 3% wall thickness from the lumenwall interface, while for the rigid model, it was observed just at the lumenwall interface. In addition, the peak LDL accumulation area of the PHEM moved about according to the pulsatile flow. These results demonstrate that the blood flow, arterial wall deformation, and filtration flow all affect the LDL concentration, and that LDL accumulation is due to stagnation and the presence of filtration flow. Thus, FSI analysis is indispensable.     

Artery buckling analysis using a four-fiber wall model

Artery bent buckling has been suggested as a possible mechanism that leads to artery tortuosity, which is associated with aging, hypertension, atherosclerosis, and other pathological conditions. It is necessary to understand the relationship between microscopic wall structural changes and macroscopic artery buckling behavior. To this end, the objectives of this study were to develop arterial buckling equations using a microstructure-based 4-fiber reinforced wall model, and to simulate the effects of vessel wall microstructural changes on artery buckling. Our results showed that the critical pressure increased nonlinearly with the axial stretch ratio, and the 4-fiber model predicted higher critical buckling pressures than what the Fung model predicted. The buckling equation using the 4-fiber model captured the experimentally observed reduction of critical pressure induced by elastin degradation and collagen fiber orientation changes in the arterial wall. These results improve our understanding of arterial stability and its relationship to microscopic wall remodeling, and the model provides a useful tool for further studies.     

Model-based analysis of arterial pulse signals for tracking changes in arterial wall parameters: a pilot study

Biomechanics and Modeling in Mechanobiology - Arterial wall parameters (i.e., radius and viscoelasticity) are prognostic markers for cardiovascular diseases (CVD), but their current monitoring...     

Measurement of wall motion and wall shear in a compliant arterial cast

Initial measurements of the time-varying wall shear rate at two sites in a compliant cast of a human aortic bifurcation are presented. The shear rates were derived from flow velocities measured by laser Doppler velocimetry (LDV) near the moving walls of the cast. To derive these shear rate values, the distance from the velocimeter sampling volume to the cast wall must be known. The time variation of this distance was obtained from LDV measurements of the velocity of the wall itself.     

Dynamic Bayesian sensitivity analysis of a myocardial metabolic model

Dynamic compartmentalized metabolic models are identified by a large number of parameters, several of which are either non-physical or extremely difficult to measure. Typically, the available data and prior information is insufficient to fully identify the system. Since the models are used to predict the behavior of unobserved quantities, it is important to understand how sensitive the output of the system is to perturbations in the poorly identifiable parameters. Classically, it is the goal of sensitivity analysis to asses how much the output changes as a function of the parameters. In the case of dynamic models, the output is a function of time and therefore its sensitivity is a time dependent function. If the output is a differentiable function of the parameters, the sensitivity at one time instance can be computed from its partial derivatives with respect to the parameters. The time course of these partial derivatives describes how the sensitivity varies in time.When the model is not uniquely identifiable, or if the solution of the parameter identification problem is known only approximately, we may have not one, but a distribution of possible parameter values. This is always the case when the parameter identification problem is solved in a statistical framework. In that setting, the proper way to perform sensitivity analysis is to not rely on the values of the sensitivity functions corresponding to a single model, but to consider the distributed nature of the sensitivity functions, inherited from the distribution of the vector of the model parameters.In this paper we propose a methodology for analyzing the sensitivity of dynamic metabolic models which takes into account the variability of the sensitivity over time and across a sample. More specifically, we draw a representative sample from the posterior density of the vector of model parameters, viewed as a random variable. To interpret the output of this doubly varying sensitivity analysis, we propose visualization modalities particularly effective at displaying simultaneously variations over time and across a sample. We perform an analysis of the sensitivity of the concentrations of lactate and glycogen in cytosol, and of ATP, ADP, NAD⁺ and NADH in cytosol and mitochondria, to the parameters identifying a three compartment model for myocardial metabolism during ischemia.     

Parameter estimation and sensitivity analysis for batch fermentation of recombinant cells

Summary An interactive scheme for estimating parameters in an unstructured model of a recombinant fermentation process is presented. Sensitivity analysis is simultaneously evaluated in this approach so that the instantaneous influence of parameters on state variables can be inspected. The predicted profiles of fermentation by both the model and the sensitivity analysis based on ±50% variations of the initial concentration of glucose fit the experimental observations.     

Parameter optimization and sensitivity analysis for large kinetic models using a real-coded genetic algorithm

Dynamic modeling is a powerful tool for predicting changes in metabolic regulation. However, a large number of input parameters, including kinetic constants and initial metabolite concentrations, are required to construct a kinetic model. Therefore, it is important not only to optimize the kinetic parameters, but also to investigate the effects of their perturbations on the overall system. We investigated the efficiency of the use of a real-coded genetic algorithm (RCGA) for parameter optimization and sensitivity analysis in the case of a large kinetic model involving glycolysis and the pentose phosphate pathway in Escherichia coli K-12. Sensitivity analysis of the kinetic model using an RCGA demonstrated that the input parameter values had different effects on model outputs. The results showed highly influential parameters in the model and their allowable ranges for maintaining metabolite-level stability. Furthermore, it was revealed that changes in these influential parameters may complement one another. This study presents an efficient approach based on the use of an RCGA for optimizing and analyzing parameters in large kinetic models.     

Identifiability analysis and improvement of a tree water flow and storage model

A recently published tree water flow and storage model (RCGro) for simulating water transport dynamics in trees and related stem diameter variations was improved in order to better describe a data set gathered under mild drought stress conditions. Model improvements were carried out based on the results of a mathematical identifiability analysis. This analysis provided important information with respect to the balance between model complexity and data availability. Using the identifiability analysis results, we were able to (1) highlight weaknesses of the model; (2) obtain information on how the model could be reduced in some places, to improve its identifiability properties, and extended in others, to enhance model performance; (3) identify which measurements are necessary to optimally calibrate the model. The resulting improved model was less complex (contained less unidentifiable parameters), had better dynamic properties and was able to better describe the stress data set.     

Effects of arterial wall models and measurement uncertainties on cardiovascular model predictions

We developed a methodology to assess and compare the prediction quality of cardiovascular models for patient-specific simulations calibrated with uncertainty-hampered measurements. The methodology was applied in a one-dimensional blood flow model to estimate the impact of measurement uncertainty in wall model parameters on the predictions of pressure and flow in an arterial network. We assessed the prediction quality of three wall models that have been widely used in one-dimensional blood flow simulations. A 37-artery network, previously used in one experimental and several simulation studies, was adapted to patient-specific conditions with a set of three clinically measurable inputs: carotid–femoral wave speed, mean arterial pressure and area in the brachial artery. We quantified the uncertainty of the predicted pressure and flow waves in eight locations in the network and assessed the sensitivity of the model prediction with respect to the measurements of wave speed, pressure and cross-sectional area. Furthermore, we developed novel time-averaged sensitivity indices to assess the contribution of model parameters to the uncertainty of time-varying quantities (e.g., pressure and flow). The results from our patient-specific network model demonstrated that our novel indices allowed for a more accurate sensitivity analysis of time-varying quantities compared to conventional Sobol sensitivity indices.     

Multi-dimensional sensitivity analysis and ecological implications of a nutrient uptake model

Mechanistic models of nutrient uptake are essential to the study of plant-soil interactions. In these models, uptake rates depend on the supply of the nutrient through the soil and the uptake capacity of the roots. The behaviour of the models is complex, although only six to ten parameters are used. Our goal was to demonstrate a comprehensive and efficient method of exploring a steady-state uptake model with variation in parameters across a range of values described in the literature. We employed two analytical techniques: the first a statistical analysis of variance, and the second a graphical representation of the simulated response surface. The quantitative statistical technique allows objective comparison of parameter and interaction sensitivity. The graphical technique uses a judicious arrangement of figures to present the shape of the response surface in five dimensions. We found that the most important parameters controlling uptake per unit length of root are the average dissolved nutrient concentration and the maximal rate of nutrient uptake. Root radius is influential if rates are expressed per unit root length; on a surface area basis, this parameter is less important. The next most important parameter is the effective diffusion coefficient, especially in the uptake of phosphorus. The interactions of parameters were extremely important and included three and four dimensional effects. For example, limitation by maximal nutrient influx rate is approached more rapidly with increasing nutrient solution concentration when the effective diffusion coefficient is high. We also note the ecological implications of the response surface. For example, in nutrient-limited conditions, the rate of uptake is best augmented by extending root length; when nutrients are plentiful increasing uptake kinetics will have greater effect.     

Effects of transmural pressure and wall shear stress on LDL accumulation in the arterial wall: a numerical study using a multilayered model

The accumulation of low-density lipoprotein (LDL) is recognized as one of the main contributors in atherogenesis. Mathematical models have been constructed to simulate mass transport in large arteries and the consequent lipid accumulation in the arterial wall. The objective of this study was to investigate the influences of wall shear stress and transmural pressure on LDL accumulation in the arterial wall by a multilayered, coupled lumen-wall model. The model employs the Navier-Stokes equations and Darcy's Law for fluid dynamics, convection-diffusion-reaction equations for mass balance, and Kedem-Katchalsky equations for interfacial coupling. To determine physiologically realistic model parameters, an optimization approach that searches optimal parameters based on experimental data was developed. Two sets of model parameters corresponding to different transmural pressures were found by the optimization approach using experimental data in the literature. Furthermore, a shear-dependent hydraulic conductivity relation reported previously was adopted. The integrated multilayered model was applied to an axisymmetric stenosis simulating an idealized, mildly stenosed coronary artery. The results show that low wall shear stress leads to focal LDL accumulation by weakening the convective clearance effect of transmural flow, whereas high transmural pressure, associated with hypertension, leads to global elevation of LDL concentration in the arterial wall by facilitating the passage of LDL through wall layers.