Sensitivity analysis of respiratory parameter uncertainties: impact of criterion function form and constraints

A sensitivity analysis based on weighted least-squares regression is presented to evaluate alternative methods for fitting lumped-parameter models to respiratory impedance data. The goal is to maintain parameter accuracy simultaneously with practical experiment design. The analysis focuses on predicting parameter uncertainties using a linearized approximation for joint confidence regions. Applications are with four-element parallel and viscoelastic models for 0.125- to 4-Hz data and a six-element model with separate tissue and airway properties for input and transfer impedance data from 2-64 Hz. The criterion function form was evaluated by comparing parameter uncertainties when data are fit as magnitude and phase, dynamic resistance and compliance, or real and imaginary parts of input impedance. The proper choice of weighting can make all three criterion variables comparable. For the six-element model, parameter uncertainties were predicted when both input impedance and transfer impedance are acquired and fit simultaneously. A fit to both data sets from 4 to 64 Hz could reduce parameter estimate uncertainties considerably from those achievable by fitting either alone. For the four-element models, use of an independent, but noisy, measure of static compliance was assessed as a constraint on model parameters. This may allow acceptable parameter uncertainties for a minimum frequency of 0.275-0.375 Hz rather than 0.125 Hz. This reduces data acquisition requirements from a 16- to a 5.33- to 8-s breath holding period. These results are approximations, and the impact of using the linearized approximation for the confidence regions is discussed.     

Elasticity Properties of Lung Parenchyma Derived from Experimental Distortion Data

Distortion data for dog lungs obtained experimentally by Hoppin et al. (1975) are used to arrive at a general mathematical model (strain energy function) which describes finite deformation of lung parenchyma. The strain energy function is correlated with average alveolus model proposed by Frankus and Lee (1974). The latter predicts parenchyma distortion properties from uniformly ventilated pressure-volume data only.     

Comparison of balloon and transducer catheters for estimating lung elasticity.

Pleural pressure is usually estimated with a balloon catheter (BC) positioned in the middle third of the esophagus. An alternate method, which avoids potential inaccuracies associated with changes in balloon volume, is a catheter-mounted transducer (CMT) system. To assess the accuracy of a CMT system in defining the elastic properties of the lungs, we compared the static pressure-volume (PV) properties of the lungs measured sequentially with CMT and BC systems in six healthy subjects each on two occasions, relating static transpulmonary pressure (Pst,L) to lung volume during interrupted exhalations from total lung capacity (TLC). PV data were fitted with an exponential function (least-squares method), and the exponent (k) was used to define the shape of the PV curve; position was defined by Pst,L at TLC and at 90 and 60% TLC. These data were examined for agreement (paired t test) and repeatability (coefficient of repeatability). No significant differences were demonstrated: k was 0.10 +/- 0.02 and 0.11 +/- 0.03 (SD) and Pst,L at 60% TLC was 8.27 +/- 2.09 and 8.37 +/- 1.63 cmH2O for the CMT and BC systems, respectively. The coefficient of repeatability for each parameter was not significantly different but was consistently less with the BC, suggesting better repeatability. We conclude that a CMT system is an acceptable alternative to a BC system for defining the elastic properties of lungs.     

Testing the fit of a quantal model of neurotransmission

Many studies of synaptic transmission have assumed a parametric model to estimate the mean quantal content and size or the effect upon them of manipulations such as the induction of long-term potentiation. Classical tests of fit usually assume that model parameters have been selected independently of the data. Therefore, their use is problematic after parameters have been estimated. We hypothesized that Monte Carlo (MC) simulations of a quantal model could provide a table of parameter-independent critical values with which to test the fit after parameter estimation, emulating Lilliefors's tests. However, when we tested this hypothesis within a conventional quantal model, the empirical distributions of two conventional goodness-of-fit statistics were affected by the values of the quantal parameters, falsifying the hypothesis. Notably, the tests' critical values increased when the combined variances of the noise and quantal-size distributions were reduced, increasing the distinctness of quantal peaks. Our results support two conclusions. First, tests that use a predetermined critical value to assess the fit of a quantal model after parameter estimation may operate at a differing unknown level of significance for each experiment. Second, a MC test enables a valid assessment of the fit of a quantal model after parameter estimation.     

Modeling the Carotid Sinus Baroreceptor

A mathematical model that describes the relationship between sinus pressure and nerve discharge frequency of the carotid sinus baroreceptor is presented. It is partly based upon the single-fiber data obtained by Clarke from the sinus nerve of a dog. The model takes into account what is currently known about the physiology of the baroreceptor. It consists of two nonlinear ordinary differential equations and eight free parameters. With one set of values for these eight parameters, the model reproduces well the experimental results reported by Clarke for positive ramp pressure inputs. Only three parameters needed to be adjusted in order to fit the dynamic data. The remaining five were obtained from static and steady-state data.     

Structure-function relationships in the pulmonary arterial tree

Knowledge of the relationship between structure and function ofthe normal pulmonary arterial tree is necessary for understanding normal pulmonary hemodynamics and the functional consequences of thevascular remodeling that accompanies pulmonary vascular diseases. In aneffort to provide a means for relating the measurable vascular geometryand vessel mechanics data to the mean pressure-flow relationship andlongitudinal pressure profile, we present a mathematical model of thepulmonary arterial tree. The model is based on the observation that thenormal pulmonary arterial tree is a bifurcating tree in which theparent-to-daughter diameter ratios at a bifurcation and vesseldistensibility are independent of vessel diameter, and although theactual arterial tree is quite heterogeneous, the diameter of eachroute, through which the blood flows, tapers from the arterial inlet toessentially the same terminal arteriolar diameter. In the model theaverage route is represented as a tapered tube through which the bloodflow decreases with distance from the inlet because of the diversion offlow at the many bifurcations along the route. The taper and flowdiversion are expressed in terms of morphometric parameters obtainedusing various methods for summarizing morphometric data. To help putthe model parameter values in perspective, we applied one such methodto morphometric data obtained from perfused dog lungs. Modelsimulations demonstrate the sensitivity of model pressure-flowrelationships to variations in the morphometric parameters. Comparisonsof simulations with experimental data also raise questions as to the"hemodynamically" appropriate ways to summarize morphometric data.     

The brownian ratchet and power stroke models for posttranslational protein translocation into the endoplasmic reticulum

A quantitative analysis of experimental data for posttranslational translocation into the endoplasmic reticulum is performed. This analysis reveals that translocation involves a single rate-limiting step, which is postulated to be the release of the signal sequence from the translocation channel. Next, the Brownian ratchet and power stroke models of translocation are compared against the data. The data sets are simultaneously fit using a least-squares criterion, and both models are found to accurately reproduce the experimental results. A likelihood-ratio test reveals that the optimal fit of the Brownian ratchet model, which contains one fewer free parameter, does not differ significantly from that of the power stroke model. Therefore, the data considered here cannot be used to reject this import mechanism. The models are further analyzed using the estimated parameters to make experimentally testable predictions.     

Determination of the average shape of flagellar bends: a gradient curvature model

Data obtained by manual digitization of photographs of flagellar bending waves have been analyzed by determining size parameters for the bends by least-squares fitting of a model waveform. These parameters were then used to normalize the data so that the average shape of the bends could be determined. Best fits were obtained with a model waveform derived from the constant curvature waveforms used previously but with provision for a linear change in curvature across the central region of the bend-the gradient curvature model (GCM). The central regions of the GCM bending waves are separated by transition regions with length determined by a parameter called the truncation factor (FT). Fitting the GCM to sine-generated bending waves give optimal fit when FT = 0.34. Fitting the GCM to four different samples of flagellar bending waves gave best fits with values of FT ranging from 0.17 for ATP-reactivated Lytechinus spermatozoa beating at approximately 10 Hz to 0.32 for live spermatozoa of Arbacia. The difference between the Arbacia waveforms and a sine-generated waveform is therefore very small, but a sine-generated waveform lacks the degree of freedom represented by FT that is required to fit other waveforms optimally. The residual differences between the waveform data and optimal GCM waveforms were averaged and found to be small. In most cases, the curvature in the central region of the optimal GCM decreased in magnitude towards the tip of the flagellum; however, this slope was highly variable and sometimes positive. Significant variations in both this slope and FT were found in individual bends as they propagated along a flagellum.     

Lung structure parameters estimated from modeling aerosol deposition in isolated dog lungs

A three-compartment model predicting the recovery of aerosol boli (i.e., the ratio of the number of particles expired to the number inspired) as a function of breath-holding time and bolus penetration was fitted to experimental data measured in nine isolated dog lungs. For each lung, the diameters of alveoli and alveolar ducts, as well as the volume fractions of alveoli, alveolar ducts, and airways, were determined as parameters providing the best fit. Parameter values were alveolar diameter = 0.116 +/- 0.007 (SE) mm, alveolar duct diameter = 0.284 +/- 0.015 mm, total alveolar volume/total lung capacity (TLC) = 0.68 +/- 0.02, total alveolar duct volume/TLC = 0.24 +/- 0.02, and total airway volume/TLC = 0.09 +/- 0.01. These values agreed with published values for linear dimensions and volumetric fractions in the canine lung. The mean alveolar diameter determined by the model in the nine lungs agreed closely with a mean value of 0.115 +/- 0.002 mm determined by morphometric analysis of photographs of the subpleural alveoli in the same lungs. The procedure of fitting the model to experimental data appears to have promise as a noninvasive probe of the lung periphery. However, aerosol-derived dimensions were more variable than morphometric ones, possibly because of interlung differences in aerosol distribution not accounted for in the model.     

Parameter estimation in models combining signal transduction and metabolic pathways: the dependent input approach

Biological complexity and limited quantitative measurements pose severe challenges to standard engineering methodologies for modelling and simulation of genes and gene products integrated in a functional network. In particular, parameter quantification is a bottleneck, and therefore parameter estimation, identifiability, and optimal experiment design are important research topics in systems biology. An approach is presented in which unmodelled dynamics are replaced by fictitious 'dependent inputs'. The dependent input approach is particularly useful in validation experiments, because it allows one to fit model parameters to experimental data generated by a reference cell type ('wild-type') and then test this model on data generated by a variation ('mutant'), so long as the mutations only affect the unmodelled dynamics that produce the dependent inputs. Another novel feature of the approach is in the inclusion of a priori information in a multi-objective identification criterion, making it possible to obtain estimates of parameter values and their variances from a relatively limited experimental data set. The pathways that control the nitrogen uptake fluxes in baker's yeast (Saccharomyces cerevisiae) have been studied. Well-defined perturbation experiments were performed on cells growing in steady-state. Time-series data of extracellular and intracellular metabolites were obtained, as well as mRNA levels. A nonlinear model was proposed and was shown to be structurally identifiable given data of its inputs and outputs. The identified model is a reliable representation of the metabolic system, as it could correctly describe the responses of mutant cells and different perturbations.     

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.     

Why are there so many threat displays?

The morphology of branching trees in general, and of the bronchial tree in particular, can be described in terms of three parameters, the diameter, length and branching ratios. These are the factors by which mean diameter and mean length increase in successive orders towards the trachea, and by which the number of branches increases in successive orders away from the trachea. Orders of branching are counted from the periphery towards the trachea, according to the method of Strahler. A model of from two to nine orders, and of constant total length and volume, was used to investigate the effect of varying the above parameters on the calculated pressure difference across the model during flow. In particular, the branching ratio was set at known values for dog and human lungs, and diameter and length ratios were independently varied. Known data from dog and human lungs were found to be close to the points predicted by the model where the lines of minimal resistance and minimal entropy production crossed. Other factors which may affect the values of these parameters are discussed.     

Identification in an anaerobic batch system: global sensitivity analysis, multi-start strategy and optimization criterion selection

Several mathematical models have been developed in anaerobic digestion systems and a variety of methods have been used for parameter estimation and model validation. However, structural and parametric identifiability questions are relatively seldom addressed in the reported AD modeling studies. This paper presents a 3-step procedure for the reliable estimation of a set of kinetic and stoichiometric parameters in a simplified model of the anaerobic digestion process. This procedure includes the application of global sensitivity analysis, which allows to evaluate the interaction among the identified parameters, multi-start strategy that gives a picture of the possible local minima and the selection of optimization criteria or cost functions. This procedure is applied to the experimental data collected from a lab-scale sequencing batch reactor. Two kinetic parameters and two stoichiometric coefficients are estimated and their accuracy was also determined. The classical least-squares cost function appears to be the best choice in this case study.     

生态过程模型敏感参数最优取值的时空异质性分析——以BIOME-BGC模型为例

生态过程模型是当前研究陆地生态系统水循环、碳循环有力的工具,但此类模型参数众多,参数的合理取值对模型模拟结果有重要影响.以往研究对模型参数的敏感性以及参数的优化取值有诸多的分析和讨论,但有关参数最优取值的时空异质性关注较少.本文以BIOME-BGC模型为例,在常绿阔叶林、落叶阔叶林、C3草地3种植被类型下,通过构建敏感性判别指数,筛选出模型的敏感参数,并在每种植被类型下选取两个试验站点,使用模拟退火算法结合实测通量数据构建目标函数,获取各站点敏感参数逐月的最优取值,然后构建时间异质性判别指数、空间异质性判别指数和时空异质性判别指数对模型敏感参数最优取值的时空异质性进行定量分析.结果表明:BIOME-BGC模型在3种植被类型下遴选出的敏感参数大部分一致,少数有差异,但参数的敏感性强弱在不同植被类型下的表现不尽相同;BIOME-BGC模型敏感参数的最优取值,大都具有不同程度的时空异质性,但不同植被类型下,敏感参数最优取值的时空异质性表现各异;敏感参数中与植被生理、生态相关的参数,其时空异质性相对较小,而与环境、物候相关的参数,其时空异质性普遍较大;在3种植被类型下,模型敏感参数最优取值的时间异质性与空间异质性表现出显著的线性相关性;依据其最优取值的时空异质性,可对BIOME-BGC模型敏感参数进行类型划分,以便在实践应用中采取不同的参数率定策略.本研究结论有助于加深对生态过程模型参数特性及最优取值的理解,可为实践应用中模型参数的合理取值提供一种思路和参考.     

Static lung-lung interactions in unilateral emphysema.

Motivated by the introduction of single-lung transplantation into clinical practice, we compared the static mechanical properties of the respiratory system in six supine dogs before (at baseline) with those after the induction of unilateral emphysema. Relaxation volume (Vrel), total lung capacity (TLC), and static compliance of the emphysematous lung increased to 214 +/- 68, 186 +/- 39, and 253 +/- 95% (SD) of baseline, respectively. Vrel of the nonemphysematous lung fell to 81 +/- 28% of baseline, with no significant change in TLC of the nonemphysematous lung or its pressure-volume relationship, indicating that unilateral hyperinflation does not cause dropout of contralateral lung units. After unilateral emphysema, the chest wall shifted to a higher unstressed or neutral volume (when pleural pressure equals atmospheric pressure) in three of six animals, minimizing the anticipated decrease in lung recoil pressure at the higher respiratory system Vrel. The pattern of relative lung emptying in the intact dog and in the excised lungs was similar during stepwise deflations from TLC, suggesting that mean pleural pressure of the hemithoraces is equal. We conclude that in the dog the static volume distribution between emphysematous and nonemphysematous lungs is determined only by differences in lung recoil and compliance.     

The validation of a generalized Hooke's law for coronary arteries

The exponential form of constitutive model is widely used in biomechanical studies of blood vessels. There are two main issues, however, with this model: 1) the curve fits of experimental data are not always satisfactory, and 2) the material parameters may be oversensitive. A new type of strain measure in a generalized Hooke's law for blood vessels was recently proposed by our group to address these issues. The new model has one nonlinear parameter and six linear parameters. In this study, the stress-strain equation is validated by fitting the model to experimental data of porcine coronary arteries. Material constants of left anterior descending artery and right coronary artery for the Hooke's law were computed with a separable nonlinear least-squares method with an excellent goodness of fit. A parameter sensitivity analysis shows that the stability of material constants is improved compared with the exponential model and a biphasic model. A boundary value problem was solved to demonstrate that the model prediction can match the measured arterial deformation under experimental loading conditions. The validated constitutive relation will serve as a basis for the solution of various boundary value problems of cardiovascular biomechanics.     

Parameter estimation of a plant uptake model for cyanide: application to hydroponic data

A plant uptake model is applied to describe free cyanide and ferrocyanide transport and fate in willow (Salix eriocephala var. Michaux) grown in hydroponics. The model is applied to experimental data to determine best-fit parameter values, their associated uncertainty, and their relative importance to field-scale phytoremediation applications. The fitted model results, using least-squares optimization of the observed log concentrations, indicate that free cyanide volatilization from leaf tissue and free cyanide cell wall adsorption were negligible. The free cyanide maximum uptake rate and assimilate (noncyanide 15N) first-order leaf loss rate were the only coefficients that significantly affected the model goodness of fit and were concurrently sensitive to data uncertainty in the parameter optimization. Saturation kinetics may be applicable for free cyanide uptake into plants, but not for ferrocyanide uptake, which may occur via preferential protein-mediated or inefficient transpiration stream uptake. Within the free cyanide system, the relative magnitudes of the saturation uptake parameters and the demonstration of an active role for plants in uptake relative to transpiration suggest the potential importance of preferential diffusion through the cell membranes as reported in the literature, rather than protein-mediated uptake. The fitted 13-parameter model matched the observed data well except for the predicted stem and leaf tissue assimilate concentrations, which were significantly underestimated, particularly in the free cyanide system. These low predicted values, combined with the slightly underestimated solution free cyanide removal, suggest that noncyanide 15N redistribution in phloem should be considered.     

Modelling of the heart and pericardium at end-diastole

Herein we present a refined version of Vito's two-sphere static model of the heart with pericardium and discuss its possible applications. The improvements we make on Vito's model are: (i) Vito assumed that the elastic materials which constitute the model 'heart' and 'pericardium' are isotropic; we relax this assumption to that of transverse-isotropy. (ii) Our analysis, which does not assume the existence of stored-energy functions, links the model directly to empirical stress-strain relations of suitable biaxial uniform-extension tests; two such stress-strain relations (one for the pericardium, one for the myocardium, both of which may be described by the same equation except for difference in the values of response parameters) now define the model completely, so we avoid altogether the difficult task of determining full-fledged constitutive equations for the pericardium and myocardium. As for applications, we contend that the concentric spheres in static equilibrium can be taken as a model of the left ventricle and pericardium at end-diastole. We show that the model when equipped with suitable stress-strain relations does give good fit to the pressure-volume data which Spotnitz et al. (1966, Circulation Res., 18, 49-66) obtained from excised canine left ventricles and to the pericardium data which Pegram et al. (1975, Circulation Res., 9, 707-714) obtained from closed chest, anaesthetized dogs. Three different empirical formulae were tried in the data-fitting as the equation that describes the requisite stress-strain relations. The 'exponential law' gave the best results.     

Least-squares estimation of batch culture kinetic parameters

This article concerns the development of a simple and effective least-squares procedure for estimating the kinetic parameters in Monod expressions from batch culture data. The basic approach employed in this work was to translate the problem of parameter estimation to a mathematical model containing a single decision variable. The resulting model was then solved by an efficient one-dimensional search algorithm which can be adapted to any microcomputer or advanced programmable calculator. The procedure was tested on synthetic data (substrate concentrations) with different types and levels of error. The effect of endogeneous respiration on the estimated values of the kinetic parameters was also assessed. From the results of these analyses the least-squares procedure developed was concluded to be very effective.