Efficient probabilistic model personalization integrating uncertainty on data and parameters: Application to Eikonal-Diffusion models in cardiac electrophysiology

Biophysical models are increasingly used for medical applications at the organ scale. However, model predictions are rarely associated with a confidence measure although there are important sources of uncertainty in computational physiology methods. For instance, the sparsity and noise of the clinical data used to adjust the model parameters (personalization), and the difficulty in modeling accurately soft tissue physiology. The recent theoretical progresses in stochastic models make their use computationally tractable, but there is still a challenge in estimating patient-specific parameters with such models. In this work we propose an efficient Bayesian inference method for model personalization using polynomial chaos and compressed sensing. This method makes Bayesian inference feasible in real 3D modeling problems. We demonstrate our method on cardiac electrophysiology. We first present validation results on synthetic data, then we apply the proposed method to clinical data. We demonstrate how this can help in quantifying the impact of the data characteristics on the personalization (and thus prediction) results. Described method can be beneficial for the clinical use of personalized models as it explicitly takes into account the uncertainties on the data and the model parameters while still enabling simulations that can be used to optimize treatment. Such uncertainty handling can be pivotal for the proper use of modeling as a clinical tool, because there is a crucial requirement to know the confidence one can have in personalized models.     

Sequential identification of boundary support parameters in a fluid-structure vascular model using patient image data

Viscoelastic support has been previously established as a valuable modeling ingredient to represent the effect of surrounding tissues and organs in a fluid-structure vascular model. In this paper, we propose a complete methodological chain for the identification of the corresponding boundary support parameters, using patient image data. We consider distance maps of model to image contours as the discrepancy driving the data assimilation approach, which then relies on a combination of (1) state estimation based on the so-called SDF filtering method, designed within the realm of Luenberger observers and well adapted to handling measurements provided by image sequences, and (2) parameter estimation based on a reduced-order UKF filtering method which has no need for tangent operator computations and features natural parallelism to a high degree. Implementation issues are discussed, and we show that the resulting computational effectiveness of the complete estimation chain is comparable to that of a direct simulation. Furthermore, we demonstrate the use of this framework in a realistic application case involving hemodynamics in the thoracic aorta. The estimation of the boundary support parameters proves successful, in particular in that direct modeling simulations based on the estimated parameters are more accurate than with a previous manual expert calibration. This paves the way for complete patient-specific fluid-structure vascular modeling in which all types of available measurements could be used to estimate additional uncertain parameters of biophysical and clinical relevance.     

Dimensional reductions of a cardiac model for effective validation and calibration

Complex 3D beating heart models are now available, but their complexity makes calibration and validation very difficult tasks. We thus propose a systematic approach of deriving simplified reduced-dimensional models, in “0D”—typically, to represent a cardiac cavity, or several coupled cavities—and in “1D”—to model elongated structures such as muscle samples or myocytes. We apply this approach with an earlier-proposed 3D cardiac model designed to capture length-dependence effects in contraction, which we here complement by an additional modeling component devised to represent length-dependent relaxation. We then present experimental data produced with rat papillary muscle samples when varying preload and afterload conditions, and we achieve some detailed validations of the 1D model with these data, including for the length-dependence effects that are accurately captured. Finally, when running simulations of the 0D model pre-calibrated with the 1D model parameters, we obtain pressure–volume indicators of the left ventricle in good agreement with some important features of cardiac physiology, including the so-called Frank–Starling mechanism, the End-Systolic Pressure–Volume Relationship, as well as varying elastance properties. This integrated multi-dimensional modeling approach thus sheds new light on the relations between the phenomena observed at different scales and at the local versus organ levels.     

Assessing non-linearity in European temperature-sensitive tree-ring data

We test the application of parametric, non-parametric, and semi-parametric calibration models for reconstructing summer (June–August) temperature from a set of tree-ring width and density data on the same dendro samples from 40 sites across Europe. By comparing the performance of the three calibration models on pairs” of tree-ring width (TRW) and maximum density (MXD) or maximum blue intensity (MXBI), we test whether a non-linear temperature response is more prevalent in TRW or MXD (MXBI) data, and whether it is associated with the temperature sensitivity and/or autocorrelation structure of the dendro parameters. We note that MXD (MXBI) data have a significantly stronger temperature response than TRW data as well as a lower autocorrelation that is more similar to that of the instrumental temperature data, whereas TRW exhibits a redder” variability continuum. This study shows that the use of non-parametric calibration models is more suitable for TRW data, while parametric calibration is sufficient for both MXD and MXBI data – that is, we show that TRW is by far the more non-linear proxy.     

A mixture theory model of fluid and solute transport in the microvasculature of normal and malignant tissues. II: Factor sensitivity analysis, calibration, and validation

The treatment of cancerous tumors is dependent upon the delivery of therapeutics through the blood by means of the microcirculation. Differences in the vasculature of normal and malignant tissues have been recognized, but it is not fully understood how these differences affect transport and the applicability of existing mathematical models has been questioned at the microscale due to the complex rheology of blood and fluid exchange with the tissue. In addition to determining an appropriate set of governing equations it is necessary to specify appropriate model parameters based on physiological data. To this end, a two stage sensitivity analysis is described which makes it possible to determine the set of parameters most important to the model’s calibration. In the first stage, the fluid flow equations are examined and a sensitivity analysis is used to evaluate the importance of 11 different model parameters. Of these, only four substantially influence the intravascular axial flow providing a tractable set that could be calibrated using red blood cell velocity data from the literature. The second stage also utilizes a sensitivity analysis to evaluate the importance of 14 model parameters on extravascular flux. Of these, six exhibit high sensitivity and are integrated into the model calibration using a response surface methodology and experimental intra- and extravascular accumulation data from the literature (Dreher et al. in J Natl Cancer Inst 98(5):335–344, 2006). The model exhibits good agreement with the experimental results for both the mean extravascular concentration and the penetration depth as a function of time for inert dextran over a wide range of molecular weights.     

Assessing the sensitivity of divergence time estimates to locus sampling,calibration points,and model priors in a RAD-seq phylogeny of Carex section Schoenoxiphium

Restriction site-associated DNA sequencing (RAD-seq) and related methods have become relatively common approaches to resolve species-level phylogeny. It is not clear, however, whether RAD-seq data matrices are well suited to relaxed clock inference of divergence times, given the size of the matrices and the abundance of missing data. We investigated the sensitivity of Bayesian relaxed clock estimates of divergence times to alternative analytical decisions on an empirical RAD-seq phylogenetic matrix. We explored the relative contribution of secondary calibration strategies, amount of missing data, and the data partition analyzed to overall variance in divergence times inferred using BEAST MCMC analyses of Carex section Schoenoxiphium (Cyperaceae)—a recent radiation for which we have nearly complete species sampling of RAD-seq data. The crown node for Schoenoxiphium was estimated to be 15.22 (9.56–21.18) Ma using a single calibration point and low missing data, 11.93 (8.07–16.03) Ma using multiple calibration points and low missing data, and 8.34 (5.41–11.22) using multiple calibrations but high missing data. We found that using matrices with more than half of the individuals with missing data inferred younger mean ages for all nodes. Moreover, we have found that our molecular clock estimates are sensitive to the positions of the calibration(s) in our phylogenetic tree (using matrices with low missing data), especially when only a single calibration was applied to estimate divergence times. These results argue for sensitivity analyses and caution in interpreting divergence time estimates from RAD-seq data.     

Effects of body condition score on direct and indirect measurements of insulin sensitivity in periparturient dairy cows

Reductions in insulin sensitivity in periparturient dairy cows develop as a means to support lactation; however, excessive mobilization of fatty acids (FA) increases the risk for peripartal metabolic disorders. Our objectives were to investigate the effect of prepartum body condition score (BCS) on systemic glucose and insulin tolerance, and to compare direct and indirect measurements of insulin sensitivity in peripartal lean and overweight dairy cows. Fourteen multiparous Holstein cows were allocated into two groups according to their BCS at day −28 prepartum: lean (n = 7; BCS ≤ 3.0) or overweight; (n = 7; BCS ≥ 4.0). Liver biopsies were performed on day −27, −14 and 4, relative to expected parturition. Intravenous insulin or glucose tolerances tests were performed following each liver biopsy. Relative to lean cows, overweight cows exhibited lower dry matter intake, lost more BCS and displayed increased plasma FA and β-hydroxybutyrate concentrations and elevated liver lipid content during peripartum. Glucose clearance rate was lower for all cows postpartum. Prepartum BCS had minimal effects on insulin and glucose tolerance; however, the ability of the cow to restore blood glucose levels following an insulin challenge was suppressed by increased BCS. Glucose-dependent parameters of insulin and glucose tolerance were not correlated with surrogate indices of insulin sensitivity. We conclude that prepartum BCS had minimal effect on systemic insulin sensitivity following parturition. The observed inconsistency between surrogate indices of insulin sensitivity and direct measurements of insulin and glucose tolerance adds support to growing concerns regarding their usefulness as tools to estimate systemic insulin action in periparturient cows.     

MARS Approach for Global Sensitivity Analysis of Differential Equation Models with Applications to Dynamics of Influenza Infection

Differential equation models are widely used for the study of natural phenomena in many fields. The study usually involves unknown factors such as initial conditions and/or parameters. It is important to investigate the impact of unknown factors (parameters and initial conditions) on model outputs in order to better understand the system the model represents. Apportioning the uncertainty (variation) of output variables of a model according to the input factors is referred to as sensitivity analysis. In this paper, we focus on the global sensitivity analysis of ordinary differential equation (ODE) models over a time period using the multivariate adaptive regression spline (MARS) as a meta model based on the concept of the variance of conditional expectation (VCE). We suggest to evaluate the VCE analytically using the MARS model structure of univariate tensor-product functions which is more computationally efficient. Our simulation studies show that the MARS model approach performs very well and helps to significantly reduce the computational cost. We present an application example of sensitivity analysis of ODE models for influenza infection to further illustrate the usefulness of the proposed method.     

Theoretical study of the fibrous capsule tissue growth around a disk-shaped implant

We analyze the mathematical properties of the fibrous capsule tissue concentration around a disk-shaped implant. We establish stability estimates as well as monotonicity results that illustrate the sensitivity of this growth to the biocompatibility index parameters of the implant. In addition, we prove that the growth of the tissue increases exponentially in time toward an asymptotic regime. We also study the mathematical properties of the solution of the inverse problem consisting in the determination of the values of the biocompatibility index parameters from the knowledge of some fibrous capsule tissue measurements. We prove that this model calibration problem admits a unique solution, and establish a characterization of the index parameters. Furthermore, we demonstrate analytically that such a solution is not continuous with respect to the data, and therefore the considered inverse problem is ill-posed due to the lack of the stability requirement.     

A Method for Estimating the Predictive Power in a Model of a Biological System with Low Sensitivity to Parameters

The ambiguity of parameter estimates for the model of a biological system may be due to low sensitivity of the model to perturbations of input data (parameters), which mathematically reflects biological mechanisms of robustness. We developed a novel method for estimating the predictive power of a model with the ambiguity of parameter estimates. The predictions are understood as a correct reproduction of the system behavior by the model when changing input data and parameters. The method is based on the relative sensitivity analysis of the fitted model to stiff parameters of the predicted model. The application principles of our approach are demonstrated using a model for the formation of the mRNA expression pattern of the hb gene in the Drosophila embryo and its ability to predict the hb pattern in the Kr null mutant. The nonlinear nature of the system is simulated by a saturating sigmoid function, which is the cause of low sensitivity. Our method allows us to estimate the predictive power of the model and uncover the causes of poor predictions, as well as choose the relevant level of the model detail in terms of predictions.     

Two-zone model for stream and river ecosystems

A mechanistic two-zone model is developed to represent the food web dynamics of stream and river ecosystems by considering the benthic and nonbenthic (or water-column) zones as two separate, but interacting biotopes. Flow processes, solar radiation, and temperature are the dynamic external environmental drivers. State variables are defined to represent the hierarchical levels of detritus, limiting nutrient, vegetation, and invertebrates. The fish trophic level is included as a constant input parameter. Model parameters, constants, and boundary conditions are defined based on watershed as well as channel hydrology, stream geomorphology, and biological activities. Recent advances in ecological science and engineering are used in representing important biogeochemical processes. In particular, the turbulent diffusion, as well as sloughing or detachment, processes are defined based on these recent advancements. The two-zone model was evaluated for a gravel bed prealpine Swiss stream named River Necker with data for the study period of January 1992 through December 1994. The model was able to capture the general trends and magnitudes of the food web state variables. A comprehensive relative sensitivity analysis with five moment-based measures found that approximately 5% of the model parameters were important in predicting benthic vegetation. Results of sensitivity analysis guided the model calibration. Simulated benthic vegetation with the calibrated model, which was obtained by adjusting only four parameters, corresponded with observed data. Hydrology-dependent sloughing and detachment were dominant in determining the response of benthic vegetation and invertebrates. The proposed two-zone food web model is a potentially useful research tool for stream and river ecosystems.     

Development and evaluation of the herd dynamic milk model with focus on the individual cow component

The herd dynamic milk (HDM) model is a dynamic model capable of simulating the performance of individual dairy animals (from birth to death), with a daily time step. Within this study, the HDM model is described and evaluated in relation to milk production, body condition score (BCS) and BCS change throughout lactation by comparing model simulations against data from published experimental studies. The model’s response to variation in genetic potential, herbage allowance and concentrate supplementation was tested in a sensitivity analysis. Data from experiments in Ireland and France over a 3-year period (2009–11) were used to complete the evaluation. The aim of the Irish experiment was to determine the impact of different stocking rates (SRs) (SR1: 3.28 cow/ha, SR2: 2.51 cow/ha) on key physical, biological and economic performance. The aim of the French experiment was to evaluate over a prolonged time period, the ability of two breeds of dairy cows (Holstein and Normande) to produce and to reproduce under two feeding strategies (high level and low level) in the context of compact calving. The model evaluation was conducted at the herd level with separate evaluations for the primiparous and multiparous cows. The evaluation included the two extreme SRs for the Irish experiment, and an evaluation at the overall herd and individual animal level for the different breeds and feeding levels for the French data. The comparison of simulation and experimental data for all scenarios resulted in a relative prediction error, which was consistently <15% across experiments for weekly milk production and BCS. In relation to BCS, the highest root mean square error was 0.27 points of BCS, which arose for Holstein cows in the low feeding group in late lactation. The model responded in a realistic fashion to variation in genetic potential for milk production, herbage allowance and concentrate supplementation.     

Pulse pressure variation tracking using sequential Monte Carlo methods

The pulse pressure variation (PPV) is a measure of the respiratory effect on the variation of systemic arterial blood pressure (ABP) in patients receiving full mechanical ventilation. It is a promising predictor of increases in cardiac output due to an infusion of fluid. We describe a novel automatic algorithm to estimate the PPV of ABP signals recorded under full respiratory support. The algorithm utilizes our recently developed sequential Monte Carlo method (SMCM), which is called a maximum a-posteriori adaptive marginalized particle filter (MAM-PF). MAM-PF estimates the state-space model parameters of the ABP signal continuously and its upper and lower envelopes are derived as a combination of those parameter estimates. Then, the continuous PPV values can be easily obtained based on those estimated envelopes. We report the assessment results of the proposed algorithm on real ABP signals.     

Automated determination of glucose in fermentation broths with p-hydroxy-benzoic-acid hydrazide (p-HBAH)

Summary The substrate concentration is usually the most important process variable for automatic control of cultivation and for product formation. Glucose is the most popular substrate. An on-line technique for glucose analysis is described which is based on the p-HBAH method. The influence of medium components and metabolites was determined. Their influence on the measurement can be taken into account based on the calibration curves presented. This method attains the sensitivity of enzymatic glucose analysis and is less expensive and easier to handle.     

Analytical comparison of absolute quantification strategies to investigate the insulin signaling pathway in fat cells

So far, mass spectrometry-based targeted proteomics is the most sensitive approach to answer and address specific biological questions in an accurate and quantitative fashion. However, the data analysis design used for such quantification varies in the field leading to discrepancies in the reported values. In this study, different quantification strategies based on calibration curves were evaluated and compared. The best accuracy and coefficient of variation was achieved by ratio to ratio calibration curves. We applied the ratio to ratio quantification approach to analyze very low abundant insulin signaling proteins such as PIK3RA (0.10–0.93 fmol/μg), AKT1 (0.1–0.39 fmol/μg), and the insulin receptor (0.22–2.62 fmol/μg) in a fat cell model and demonstrated the adaptation of this pathway at different states of insulin sensitivity.     

Inter-model consistency and complementarity: learning from ex-vivo imaging and electrophysiological data towards an integrated understanding of cardiac physiology

Computational models of the heart at various scales and levels of complexity have been independently developed, parameterised and validated using a wide range of experimental data for over four decades. However, despite remarkable progress, the lack of coordinated efforts to compare and combine these computational models has limited their impact on the numerous open questions in cardiac physiology. To address this issue, a comprehensive dataset has previously been made available to the community that contains the cardiac anatomy and fibre orientations from magnetic resonance imaging as well as epicardial transmembrane potentials from optical mapping measured on a perfused ex-vivo porcine heart. This data was used to develop and customize four models of cardiac electrophysiology with different level of details, including a personalized fast conduction Purkinje system, a maximum a posteriori estimation of the 3D distribution of transmembrane potential, the personalization of a simplified reaction-diffusion model, and a detailed biophysical model with generic conduction parameters. This study proposes the integration of these four models into a single modelling and simulation pipeline, after analyzing their common features and discrepancies. The proposed integrated pipeline demonstrates an increase prediction power of depolarization isochrones in different pacing conditions.     

Chemometric modelling based on 2D-fluorescence spectra without a calibration measurement

MOTIVATION: 2D fluorescence spectra provide information from intracellular compounds. Fluorophores like trytophan, tyrosine and phenylalanin as well as NADH and flavins make the corresponding measurement systems very important for bioprocess supervision and control. The evaluation is usually based on chemometric modelling using for their calibration procedure off-line measurements of the desired process variables. Due to the data driven approach lots of off-line measurements are required. Here a methodology is presented, which enables to perform a calibration procedure of chemometric models without any further measurement. RESULTS: The necessary information for the calibration procedure is provided by means of the a priori knowledge about the process, i.e. a mathematical model, whose model parameters are estimated during the calibration procedure, as well as the fact that the substrate should be consumed at the end of the process run. The new methodology for chemometric calibration is applied for a batch cultivation of aerobically grown S. cerevisiae on the glucose Schatzmann medium. As will be presented the chemometric models, which are determined by this method, can be used for prediction during new process runs. AVAILABILITY: The MATHLAB routine is free available on request from the authors.     

Multi-scale lines and edges in V1 and beyond: Brightness,object categorization and recognition,and consciousness

In this paper we present an improved model for line and edge detection in cortical area V1. This model is based on responses of simple and complex cells, and it is multi-scale with no free parameters. We illustrate the use of the multi-scale line/edge representation in different processes: visual reconstruction or brightness perception, automatic scale selection and object segregation. A two-level object categorization scenario is tested in which pre-categorization is based on coarse scales only and final categorization on coarse plus fine scales. We also present a multi-scale object and face recognition model. Processing schemes are discussed in the framework of a complete cortical architecture. The fact that brightness perception and object recognition may be based on the same symbolic image representation is an indication that the entire (visual) cortex is involved in consciousness.     

Regression Calibration in Semiparametric Accelerated Failure Time Models

Summary In large cohort studies, it often happens that some covariates are expensive to measure and hence only measured on a validation set. On the other hand, relatively cheap but error‐prone measurements of the covariates are available for all subjects. Regression calibration (RC) estimation method ( Prentice, 1982 , Biometrika 69 , 331–342) is a popular method for analyzing such data and has been applied to the Cox model by Wang et al. (1997, Biometrics 53 , 131–145) under normal measurement error and rare disease assumptions. In this article, we consider the RC estimation method for the semiparametric accelerated failure time model with covariates subject to measurement error. Asymptotic properties of the proposed method are investigated under a two‐phase sampling scheme for validation data that are selected via stratified random sampling, resulting in neither independent nor identically distributed observations. We show that the estimates converge to some well‐defined parameters. In particular, unbiased estimation is feasible under additive normal measurement error models for normal covariates and under Berkson error models. The proposed method performs well in finite‐sample simulation studies. We also apply the proposed method to a depression mortality study.