Optimal experiment selection for parameter estimation in biological differential equation models

ABSTRACT: BACKGROUND: Parameter estimation in biological models is a common yet challenging problem. In this work we explore the problem for gene regulatory networks modeled by differential equations with unknown parameters, such as decay rates, reaction rates, Michaelis-Menten constants, and Hill coefficients. We explore the question to what extent parameters can be efficiently estimated by appropriate experimental selection. RESULTS: A minimization formulation is used to find the parameter values that best fit the experiment data. When the data is insufficient, the minimization problem often has many local minima that fit the data reasonably well. We show that selecting a new experiment based on the local Fisher Information of one local minimum generates additional data that allows one to successfully discriminate among the many local minima. The parameters can be estimated to high accuracy by iteratively performing minimization and experiment selection. We show that the experiment choices are roughly independent of which local minima is used to calculate the local Fisher Information. CONCLUSIONS: We show that by an appropriate choice of experiments, one can, in principle, efficiently and accurately estimate all the parameters of gene regulatory network. In addition, we demonstrate that appropriate experiment selection can also allow one to restrict model predictions without constraining the parameters using many fewer experiments. We suggest that predicting model behaviors and inferring parameters represent two different approaches to model calibration with different requirements on data and experimental cost.     

Sensitivity analysis of an energetic muscle model applied at whole body level in recumbent pedalling

Musculoskeletal models are used in order to describe and analyse the mechanics of human movement. In order to get a complete evaluation of the human movement, energetic muscle models were developed and were shown to be promising.

The aim of this work is to determine the sensitivity of muscle mechanical and energetic model estimates to changes in parameters during recumbent pedalling.

Inputs of the model were electromyography and joint angles, collected experimentally on one participant. The sensitivity analysis was performed on muscle-specific tension, physiological cross-sectional area, muscle maximal force, tendon rest length and percentage of fast-twitch fibres using an integrated sensitivity ratio. Soleus, gastrocnemius, vasti, gluteus and medial hamstrings were selected for the analyses.

The energetic model was found to be always less sensitive to parameter changes than the mechanical model. Tendon slack length was found to be the most critical parameter for both energetic and mechanical models even if the effect on the energetic output was smaller than on muscle force and joint moments.     

Characterization of arterial wall mechanical behavior and stresses from human clinical data

This paper demonstrates the feasibility of material identification and wall stress computation for human common carotid arteries based on non-invasive in vivo clinical data: dynamical intraluminal pressure measured by applanation tonometry, and medial diameter and intimal-medial thickness measured by high-resolution ultrasound echotracking. The mechanical behavior was quantified assuming an axially pre-stretched, thick-walled, cylindrical artery subjected to dynamical blood pressure and perivascular constraints. The wall was further assumed to be three-dimensional and to consist of a nonlinear, hyperelastic, anisotropic, incompressible material with smooth muscle activity and residual stresses. Mechanical contributions by individual constituents-an elastin-dominated matrix, collagen fibers, and vascular smooth muscle-were accounted for using a previously proposed microstructurally motivated constitutive relation. The in vivo boundary value problem was solved semi-analytically to compute the inner pressure during a mean cardiac cycle. Using a nonlinear least-squares method, optimal model parameters were determined by minimizing differences between computed and measured inner pressures over a mean cardiac cycle. The fit-to-data from two healthy patients was very good and the predicted radial, circumferential, and axial stretch and stress fields were sensible. Hence, the proposed approach was able to identify complex geometric and material parameters directly from non-invasive in vivo human data.     

Fast and accurate fitting of relaxation dispersion data using the flexible software package GLOVE

Relaxation dispersion spectroscopy is one of the most widely used techniques for the analysis of protein dynamics. To obtain a detailed understanding of the protein function from the view point of dynamics, it is essential to fit relaxation dispersion data accurately. The grid search method is commonly used for relaxation dispersion curve fits, but it does not always find the global minimum that provides the best-fit parameter set. Also, the fitting quality does not always improve with increase of the grid size although the computational time becomes longer. This is because relaxation dispersion curve fitting suffers from a local minimum problem, which is a general problem in non-linear least squares curve fitting. Therefore, in order to fit relaxation dispersion data rapidly and accurately, we developed a new fitting program called GLOVE that minimizes global and local parameters alternately, and incorporates a Monte-Carlo minimization method that enables fitting parameters to pass through local minima with low computational cost. GLOVE also implements a random search method, which sets up initial parameter values randomly within user-defined ranges. We demonstrate here that the combined use of the three methods can find the global minimum more rapidly and more accurately than grid search alone.     

Biphasic Regulation of Myosin Light Chain Phosphorylation by p21-activated Kinase Modulates Intestinal Smooth Muscle Contractility

Supraphysiological mechanical stretching in smooth muscle results in decreased contractile activity. However, the mechanism is unclear. Previous studies indicated that intestinal motility dysfunction after edema development is associated with increased smooth muscle stress and decreased myosin light chain (MLC) phosphorylation in vivo, providing an ideal model for studying mechanical stress-mediated decrease in smooth muscle contraction. Primary human intestinal smooth muscle cells (hISMCs) were subjected to either control cyclical stretch (CCS) or edema (increasing) cyclical stretch (ECS), mimicking the biophysical forces in non-edematous and edematous intestinal smooth muscle in vivo. ECS induced significant decreases in phosphorylation of MLC and MLC phosphatase targeting subunit (MYPT1) and a significant increase in p21-activated kinase (PAK) activity compared with CCS. PAK regulated MLC phosphorylation in an activity-dependent biphasic manner. PAK activation increased MLC and MYPT1 phosphorylation in CCS but decreased MLC and MYPT1 phosphorylation in hISMCs subjected to ECS. PAK inhibition had the opposite results. siRNA studies showed that PAK1 plays a critical role in regulating MLC phosphorylation in hISMCs. PAK1 enhanced MLC phosphorylation via phosphorylating MYPT1 on Thr-696, whereas PAK1 inhibited MLC phosphorylation via decreasing MYPT1 on both Thr-696 and Thr-853. Importantly, in vivo data indicated that PAK activity increased in edematous tissue, and inhibition of PAK in edematous intestine improved intestinal motility. We conclude that PAK1 positively regulates MLC phosphorylation in intestinal smooth muscle through increasing inhibitory phosphorylation of MYPT1 under physiologic conditions, whereas PAK1 negatively regulates MLC phosphorylation via inhibiting MYPT1 phosphorylation when PAK activity is increased under pathologic conditions.     

Finite-element Analysis of a Stenotic Artery Revascularization Through a Stent Insertion

Short-term and long-term clinical follow-up data clearly indicate the superiority of stenting techniques within the family of mechanical treatments for percutaneous coronary revascularizations. However, restenosis phenomena are in general still present, representing the major drawback for this innovative non-invasive approach.

Experimental evidence indicates the mechanical interaction between the stent and the artery as a significant cause for the activation of stent-related restenosis. At the same time, the literature shows a significant lack of computational investigations within this field, possibly as consequence of the complexity of the problem.

According to these considerations, the aim of the present work is to study the bio-mechanical interaction between a balloon-expandable stent and a stenotic artery, highlighting considerations able to improve the general understanding of the problem.

In particular, given an initial stent design (J&J Palmaz-Schatz like), we show the presence of possible areas of artery injury during the stent deployment and areas of non-uniform contact pressure after the stent apposition, due to a non-uniform stent expansion. Since these concentrated mechanical actions can play an important role in the activation of restenosis mechanisms, we propose a modified stent design, which shows a more uniform expansion and for which typical stenting parameters (i.e., residual stenosis, elastic recoil, foreshortening) are computed and presented.     

Model of geometrical and smooth muscle tone adaptation of carotid artery subject to step change in pressure

Recent experimental studies have shown significant alterations of the vascular smooth muscle (VSM) tone when an artery is subjected to an elevation in pressure. Therefore, the VSM participates in the adaptation process not only by means of its synthetic activity (fibronectins and collagen) or proliferative activity (hypertrophy and hyperplasia) but also by adjusting its contractile properties and its tone level. In previous theoretical models describing the time evolution of the arterial wall adaptation in response to induced hypertension, the contribution of VSM tone has been neglected. In this study, we propose a new biomechanical model for the wall adaptation to induced hypertension, including changes in VSM tone. On the basis of Hill's model, total circumferential stress is separated into its passive and active components, the active part being the stress developed by the VSM. Adaptation rate equations describe the geometrical adaptation (wall thickening) and the adaptation of active stress (VSM tone). The evolution curves that are derived from the theoretical model fit well the experimental data describing the adaptation of the rat common carotid subjected to a step increase in pressure. This leads to the identification of the model parameters and time constants by characterizing the rapidity of the adaptation processes. The agreement between the results of this simple theoretical model and the experimental data suggests that the theoretical approach used here may appropriately account for the biomechanics underlying the arterial wall adaptation.     

Glc-6-PD and PKG contribute to hypoxia-induced decrease in smooth muscle cell contractile phenotype proteins in pulmonary artery

Persistent hypoxic pulmonary vasoconstriction (HPV) plays a significant role in the pathogenesis of pulmonary hypertension, which is an emerging clinical problem around the world. We recently showed that hypoxia-induced activation of glucose-6-phosphate dehydrogenase (Glc-6-PD) in pulmonary artery smooth muscle links metabolic changes within smooth muscle cells to HPV and that inhibition of Glc-6PD reduces acute HPV. Here, we demonstrate that exposing pulmonary arterial rings to hypoxia (20-30 Torr) for 12 h in vitro significantly (P < 0.05) reduces (by 30-50%) SM22α and smooth muscle myosin heavy chain expression and evokes HPV. Glc-6-PD activity was also elevated in hypoxic pulmonary arteries. Inhibition of Glc-6-PD activity prevented the hypoxia-induced reduction in SM22α expression and inhibited HPV by 80-90% (P < 0.05). Furthermore, Glc-6-PD and protein kinase G (PKG) formed a complex in pulmonary artery, and Glc-6-PD inhibition increased PKG-mediated phosphorylation of VASP (p-VASP). In turn, increasing PKG activity upregulated SM22α expression and attenuated HPV evoked by Glc-6-PD inhibition. Increasing passive tension (from 0.8 to 3.0 g) in hypoxic arteries for 12 h reduced Glc-6-PD, increased p-VASP and SM22α levels, and inhibited HPV. The present findings indicate that increases in Glc-6-PD activity influence PKG activity and smooth muscle cell phenotype proteins, all of which affect pulmonary artery contractility and remodeling.     

Towards in vivo aorta material identification and stress estimation

This paper addresses the problem of constructing a mechanical model for the abdominal aorta and calibrating its parameters to in vivo measurable data. The aorta is modeled as a pseudoelastic, thick-walled, orthotropic, residually stressed cylindrical tube, subjected to an internal pressure. The model parameters are determined by stating a minimization problem for the model pressure and computing the optimal solution by a minimization algorithm. The data used in this study is in vivo pressure–diameter data for the abdominal aorta of a 24-year-old man. The results show that the axial, circumferential and radial stresses have magnitudes in the span 0 to 180 kPa. Furthermore, the results show that it is possible to determine model parameters directly from in vivo measurable data. In particular, the parameters describing the residual stress distribution can be obtained without interventional procedures.     

Pulsatile Flow Inside Moderately Elastic Arteries,Its Modelling and Effects of Elasticity

Pulsatile flow inside a moderately elastic circular conduit with a smooth expansion is studied as a model to understand the influence of wall elasticity in artery flow. The solution of the simultaneous fluid-wall evolution is evaluated by a perturbative method, where the zeroth order solution is represented by the flow in a rigid vessel; the first order correction gives the wall motion and induced flow modification without the need to solve the difficult coupled problem. Such an approach essentially assumes a locally infinite celerity, therefore it represent a good approximation for the fluid-wall interaction in sites of limited extent (branches, stenosis, aneurism, etc.), which include typical situations associated with vascular diseases. The problem is solved numerically in the axisymmetric approximation; the influence of wall elasticity on the flow and on the unsteady wall shear stress is studied in correspondence of parameters taken from realistic artery flow. Attention is posed to the role of phase difference between the incoming pressure and flow pulses.     

Adaptation of active tone in the mouse descending thoracic aorta under acute changes in loading

Arteries can adapt to sustained changes in blood pressure and flow, and it is thought that these adaptive processes often begin with an altered smooth muscle cell activity that precedes any detectable changes in the passive wall components. Yet, due to the intrinsic coupling between the active and passive properties of the arterial wall, it has been difficult to delineate the adaptive contributions of active smooth muscle. To address this need, we used a novel experimental–computational approach to quantify adaptive functions of active smooth muscle in arterial rings excised from the proximal descending thoracic aorta of mice and subjected to short-term sustained circumferential stretches while stimulated with various agonists. A new mathematical model of the adaptive processes was derived and fit to data to describe and predict the effects of active tone adaptation. It was found that active tone was maintained when the artery was adapted close to the optimal stretch for maximal active force production, but it was reduced when adapted below the optimal stretch; there was no significant change in passive behavior in either case. Such active adaptations occurred only upon smooth muscle stimulation with phenylephrine, however, not stimulation with KCl or angiotensin II. Numerical simulations using the proposed model suggested further that active tone adaptation in vascular smooth muscle could play a stabilizing role for wall stress in large elastic arteries.     

Simulating uterine contraction by using an electro-chemo-mechanical model

Contractions of uterine smooth muscle cells consist of a chain of physiological processes. These contractions provide the required force to expel the fetus from the uterus. The inclusion of these physiological processes is, therefore, imperative when studying uterine contractions. In this study, an electro-chemo-mechanical model to replicate the excitation, activation, and contraction of uterine smooth muscle cells is developed. The presented modeling strategy enables efficient integration of knowledge about physiological processes at the cellular level to the organ level. The model is implemented in a three-dimensional finite element setting to simulate uterus contraction during labor in response to electrical discharges generated by pacemaker cells and propagated within the myometrium via gap junctions. Important clinical factors, such as uterine electrical activity and intrauterine pressure, are predicted using this simulation. The predictions are in agreement with clinically measured data reported in the literature. A parameter study is also carried out to investigate the impact of physiologically related parameters on the uterine contractility.     

Method for characterizing viscoelasticity of human gluteal tissue

Characterizing compressive transient large deformation properties of biological tissue is becoming increasingly important in impact biomechanics and rehabilitation engineering, which includes devices interfacing with the human body and virtual surgical guidance simulation. Individual mechanical in vivo behaviour, specifically of human gluteal adipose and passive skeletal muscle tissue compressed with finite strain, has, however, been sparsely characterised. Employing a combined experimental and numerical approach, a method is presented to investigate the time-dependent properties of in vivo gluteal adipose and passive skeletal muscle tissue. Specifically, displacement-controlled ramp-and-hold indentation relaxation tests were performed and documented with magnetic resonance imaging. A time domain quasi-linear viscoelasticity (QLV) formulation with Prony series valid for finite strains was used in conjunction with a hyperelastic model formulation for soft tissue constitutive model parameter identification and calibration of the relaxation test data. A finite element model of the indentation region was employed. Strong non-linear elastic but linear viscoelastic tissue material behaviour at finite strains was apparent for both adipose and passive skeletal muscle mechanical properties with orthogonal skin and transversal muscle fibre loading. Using a force-equilibrium assumption, the employed material model was well suited to fit the experimental data and derive viscoelastic model parameters by inverse finite element parameter estimation. An individual characterisation of in vivo gluteal adipose and muscle tissue could thus be established. Initial shear moduli were calculated from the long-term parameters for human gluteal skin/fat: G(∞,S/F)=1850 Pa and for cross-fibre gluteal muscle tissue: G(∞,M)=881 Pa. Instantaneous shear moduli were found at the employed ramp speed: G(0,S/F)=1920 Pa and G(0,M)=1032 Pa.     

A minimum area discrepancy method (MADM) for force displacement response correlation

With the increasing use of Computer Aided Engineering, it has become vital to be able to evaluate the accuracy of numerical models. Specific methods such as CORA were developed to objectively evaluate the correlation between a physical test and a numerical simulation results in terms of parameter vs time. However, no metric has so far been developed for Force Vs Deflection (FvD) signals often used in crashworthiness and biomechanics. A unique method called the Minimum Area Discrepancy Method, or MADM, is proposed to address this deficiency. This new method initially calculates a parameter ‘R’ which represents the area between numerical model and the average physical test response and then divides it by the average area generated by the upper and lower test corridors, based on the same standard deviation. The parameter ‘R’ is then normalized between 0 (no correlation) and 1 (perfect correlation) to become the MADM correlation rating. The MADM method was then validated by comparing a one dimensional Finite Element (FE) model of a chest model, under 2 impact velocities, against reference Post Mortem Human Subject (PMHS) data. The MADM method was further used to improve the correlation of this thorax model, by varying model parameters and generating 81 model variations. Based on the MADM ratings, a set parameter values leading to the best fit was identified. The best fit exhibits a response significantly better than the original chest model. MADM is novel, unique, easy to use and fulfills an important gap in objectively evaluating FVD correlation responses. Abbreviations MADM Correlation rating value (Minimum Area Discrepancy Method)

MADM_n,m MADM correlation rating using a specific scaling value of ‘n’ and power rating ‘m’

FvD Force versus Displacement

FvT Force versus Time

DvT Displacement versus Time

NM Numerical model

PE Physical Experiment

A_model Area under the average signal and the Numerical Model

A_upper Area under the average signal +1 standard deviation

A_lower Area under the average signal -1 standard deviation

R Ratio between A_model and the average of A_upper and A_lower

     

The importance of proper model assumption in bayesian phylogenetics

We studied the importance of proper model assumption in the context of Bayesian phylogenetics by examining >5,000 Bayesian analyses and six nested models of nucleotide substitution. Model misspecification can strongly bias bipartition posterior probability estimates. These biases were most pronounced when rate heterogeneity was ignored. The type of bias seen at a particular bipartition appeared to be strongly influenced by the lengths of the branches surrounding that bipartition. In the Felsenstein zone, posterior probability estimates of bipartitions were biased when the assumed model was underparameterized but were unbiased when the assumed model was overparameterized. For the inverse Felsenstein zone, however, both underparameterization and overparameterization led to biased bipartition posterior probabilities, although the bias caused by overparameterization was less pronounced and disappeared with increased sequence length. Model parameter estimates were also affected by model misspecification. Underparameterization caused a bias in some parameter estimates, such as branch lengths and the gamma shape parameter, whereas overparameterization caused a decrease in the precision of some parameter estimates. We caution researchers to assure that the most appropriate model is assumed by employing both a priori model choice methods and a posteriori model adequacy tests.     

A Model of Human Muscle Energy Expenditure

A model of muscle energy expenditure was developed for predicting thermal, as well as mechanical energy liberation during simulated muscle contractions. The model was designed to yield energy (heat and work) rate predictions appropriate for human skeletal muscle contracting at normal body temperature. The basic form of the present model is similar to many previous models of muscle energy expenditure, but parameter values were based almost entirely on mammalian muscle data, with preference given to human data where possible. Nonlinear phenomena associated with submaximal activation were also incorporated. The muscle energy model was evaluated at varying levels of complexity, ranging from simulated contractions of isolated muscle, to simulations of whole body locomotion. In all cases, acceptable agreement was found between simulated and experimental energy liberation. The present model should be useful in future studies of the energetics of human movement using forward dynamic computer simulation.     

Mechanical strain induces specific changes in the synthesis and organization of proteoglycans by vascular smooth muscle cells

In the mechanically active environment of the artery, cells sense mechanical stimuli and regulate extracellular matrix structure. In this study, we explored the changes in synthesis of proteoglycans by vascular smooth muscle cells in response to precisely controlled mechanical strains. Strain increased mRNA for versican (3.2-fold), biglycan (2.0-fold), and perlecan (2.0-fold), whereas decorin mRNA levels decreased to a third of control levels. Strain also increased versican, biglycan, and perlecan core proteins, with a concomitant decrease in decorin core protein. Deformation did not alter the hydrodynamic size of proteoglycans as evidenced by molecular sieve chromatography but increased sulfate incorporation in both chondroitin/dermatan sulfate proteoglycans and heparan sulfate proteoglycans (p < 0.05 for both). Using DNA microarrays, we also identified the gene for the hyaluronan-linking protein TSG6 as mechanically induced in smooth muscle cells. Northern analysis confirmed a 4.0-fold increase in steady state mRNA for TSG6 following deformation. Size exclusion chromatography under associative conditions showed that versican-hyaluronan aggregation was enhanced following deformation. These data demonstrate that mechanical deformation increases specific vascular smooth muscle cell proteoglycan synthesis and aggregation, indicating a highly coordinated extracellular matrix response to biomechanical stimulation.     

A Symmetric Extended Logistic Model with Applications to Experimental Toxicity Data

The fit of the logit and probit models for quantal response data can be improved by embedding these classical models within a richer parametric family indexed by one or two shape parameters. In this paper, a symmetric extended logistic model indexed by a shape parameter λ is discussed with application to dose response curves. The usual maximum likelihood method is employed to estimate the parameters of the model. The need to include the shape parameter λ is illustrated by analyzing a set of real experimental data and comparing the fit of the extended logistic model to those obtained by the standard logit and probit models.