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Ridder BJ, Leishman DJ, Bridgland-Taylor M, Samieegohar M, Han X, Wu WW, Randolph A, Tran P, Sheng J, Danker T, Lindqvist A, Konrad D, Hebeisen S, Polonchuk L, Gissinger E, Renganathan M, Koci B, Wei H, Fan J, Levesque P, Kwagh J, Imredy J, Zhai J, Rogers M, Humphries E, Kirby R, Stoelzle-Feix S, Brinkwirth N, Rotordam MG, Becker N, Friis S, Rapedius M, Goetze TA, Strassmaier T, Okeyo G, Kramer J, Kuryshev Y, Wu C, Himmel H, Mirams GR, Strauss DG, Bardenet R, Li Z. A systematic strategy for estimating hERG block potency and its implications in a new cardiac safety paradigm. Toxicol Appl Pharmacol 2020; 394:114961. [PMID: 32209365 PMCID: PMC7166077 DOI: 10.1016/j.taap.2020.114961] [Citation(s) in RCA: 21] [Impact Index Per Article: 5.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2019] [Revised: 02/14/2020] [Accepted: 03/19/2020] [Indexed: 12/13/2022]
Abstract
Introduction hERG block potency is widely used to calculate a drug's safety margin against its torsadogenic potential. Previous studies are confounded by use of different patch clamp electrophysiology protocols and a lack of statistical quantification of experimental variability. Since the new cardiac safety paradigm being discussed by the International Council for Harmonisation promotes a tighter integration of nonclinical and clinical data for torsadogenic risk assessment, a more systematic approach to estimate the hERG block potency and safety margin is needed. Methods A cross-industry study was performed to collect hERG data on 28 drugs with known torsadogenic risk using a standardized experimental protocol. A Bayesian hierarchical modeling (BHM) approach was used to assess the hERG block potency of these drugs by quantifying both the inter-site and intra-site variability. A modeling and simulation study was also done to evaluate protocol-dependent changes in hERG potency estimates. Results A systematic approach to estimate hERG block potency is established. The impact of choosing a safety margin threshold on torsadogenic risk evaluation is explored based on the posterior distributions of hERG potency estimated by this method. The modeling and simulation results suggest any potency estimate is specific to the protocol used. Discussion This methodology can estimate hERG block potency specific to a given voltage protocol. The relationship between safety margin thresholds and torsadogenic risk predictivity suggests the threshold should be tailored to each specific context of use, and safety margin evaluation may need to be integrated with other information to form a more comprehensive risk assessment. hERG potency/safety margin is a widely used nonclinical cardiac safety strategy. A new regulatory paradigm promotes the integration of nonclinical and clinical data. Lack of uncertainty quantification hindered using hERG potency in the new paradigm. A systematic method was established to address this limitation. Analysis supports using different safety margin thresholds in different context.
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Affiliation(s)
- Bradley J Ridder
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Derek J Leishman
- Department of Toxicology and Pathology, Eli Lilly and Company, Indianapolis, IN, USA
| | | | - Mohammadreza Samieegohar
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Xiaomei Han
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Wendy W Wu
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Aaron Randolph
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Phu Tran
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Jiansong Sheng
- CiPA LAB, 900 Clopper Rd, Suite 130, Gaithersburg, MD 20878, USA
| | - Timm Danker
- NMI-TT GmbH, Markwiesenstr. 55, 72770 Reutlingen, Germany
| | | | - Daniel Konrad
- B'SYS GmbH, The Ion Channel Company, Benkenstrasse 254, CH-4108, Witterswil, Switzerland
| | - Simon Hebeisen
- B'SYS GmbH, The Ion Channel Company, Benkenstrasse 254, CH-4108, Witterswil, Switzerland
| | - Liudmila Polonchuk
- F. Hoffmann-La Roche AG, F. Hoffmann-La Roche Ltd Bldg. 73/R. 103b Grenzacherstrasse, 124, CH-4070 Basel, Switzerland
| | - Evgenia Gissinger
- F. Hoffmann-La Roche AG, F. Hoffmann-La Roche Ltd Bldg. 73/R. 103b Grenzacherstrasse, 124, CH-4070 Basel, Switzerland
| | | | - Bryan Koci
- Eurofins Scientific, Eurofins Discovery, 6 Research Park Drive, St. Charles, MO 63304, USA
| | - Haiyang Wei
- Eurofins Scientific, Eurofins Discovery, 6 Research Park Drive, St. Charles, MO 63304, USA
| | - Jingsong Fan
- Bristol-Myers Squibb Company, Discovery Toxicology, Bristol-Myers Squibb, 3551 Lawrenceville, Princeton Rd, Lawrence Township, NJ 08648, USA
| | - Paul Levesque
- Bristol-Myers Squibb Company, Discovery Toxicology, Bristol-Myers Squibb, 3551 Lawrenceville, Princeton Rd, Lawrence Township, NJ 08648, USA
| | - Jae Kwagh
- Bristol-Myers Squibb Company, Discovery Toxicology, Bristol-Myers Squibb, 3551 Lawrenceville, Princeton Rd, Lawrence Township, NJ 08648, USA
| | | | - Jin Zhai
- Merck & Co., Inc, Kenilworth, NJ, USA
| | - Marc Rogers
- Metrion Biosciences Limited, Riverside 3, Suite 1, Granta Park, Great Abington, Cambridge CB21, 6AD, United Kingdom
| | - Edward Humphries
- Metrion Biosciences Limited, Riverside 3, Suite 1, Granta Park, Great Abington, Cambridge CB21, 6AD, United Kingdom
| | - Robert Kirby
- Metrion Biosciences Limited, Riverside 3, Suite 1, Granta Park, Great Abington, Cambridge CB21, 6AD, United Kingdom
| | | | - Nina Brinkwirth
- Nanion Technologies Munich, Ganghoferstrasse 70A, 80339 Munich, Germany
| | | | - Nadine Becker
- Nanion Technologies Munich, Ganghoferstrasse 70A, 80339 Munich, Germany
| | - Søren Friis
- Nanion Technologies Munich, Ganghoferstrasse 70A, 80339 Munich, Germany
| | - Markus Rapedius
- Nanion Technologies Munich, Ganghoferstrasse 70A, 80339 Munich, Germany
| | - Tom A Goetze
- Nanion Technologies Munich, Ganghoferstrasse 70A, 80339 Munich, Germany
| | - Tim Strassmaier
- Nanion Technologies, USA, 1 Naylon Place, Suite C, Livingston, NJ 07039, USA
| | - George Okeyo
- Nanion Technologies, USA, 1 Naylon Place, Suite C, Livingston, NJ 07039, USA
| | - James Kramer
- Charles River Laboratories, 14656 Neo Parkway, Cleveland, OH 44128, USA
| | - Yuri Kuryshev
- Charles River Laboratories, 14656 Neo Parkway, Cleveland, OH 44128, USA
| | - Caiyun Wu
- Charles River Laboratories, 14656 Neo Parkway, Cleveland, OH 44128, USA
| | - Herbert Himmel
- Bayer AG, RD-TS-TOX-SP-SPL1, Aprather Weg 18a, 42096 Wuppertal, Germany
| | - Gary R Mirams
- Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, United Kingdom
| | - David G Strauss
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA
| | - Rémi Bardenet
- Université de Lille, CNRS, Centrale Lille, UMR 9189 - CRIStAL, Villeneuve d'Ascq, France
| | - Zhihua Li
- Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, 10903 New Hampshire Ave, Silver Spring, MD 20993, USA.
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Beattie KA, Hill AP, Bardenet R, Cui Y, Vandenberg JI, Gavaghan DJ, de Boer TP, Mirams GR. Sinusoidal voltage protocols for rapid characterisation of ion channel kinetics. J Physiol 2018; 596:1813-1828. [PMID: 29573276 PMCID: PMC5978315 DOI: 10.1113/jp275733] [Citation(s) in RCA: 41] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Accepted: 02/19/2018] [Indexed: 12/21/2022] Open
Abstract
Key points Ion current kinetics are commonly represented by current–voltage relationships, time constant–voltage relationships and subsequently mathematical models fitted to these. These experiments take substantial time, which means they are rarely performed in the same cell. Rather than traditional square‐wave voltage clamps, we fitted a model to the current evoked by a novel sum‐of‐sinusoids voltage clamp that was only 8 s long. Short protocols that can be performed multiple times within a single cell will offer many new opportunities to measure how ion current kinetics are affected by changing conditions. The new model predicts the current under traditional square‐wave protocols well, with better predictions of underlying currents than literature models. The current under a novel physiologically relevant series of action potential clamps is predicted extremely well. The short sinusoidal protocols allow a model to be fully fitted to individual cells, allowing us to examine cell–cell variability in current kinetics for the first time.
Abstract Understanding the roles of ion currents is crucial to predict the action of pharmaceuticals and mutations in different scenarios, and thereby to guide clinical interventions in the heart, brain and other electrophysiological systems. Our ability to predict how ion currents contribute to cellular electrophysiology is in turn critically dependent on our characterisation of ion channel kinetics – the voltage‐dependent rates of transition between open, closed and inactivated channel states. We present a new method for rapidly exploring and characterising ion channel kinetics, applying it to the hERG potassium channel as an example, with the aim of generating a quantitatively predictive representation of the ion current. We fitted a mathematical model to currents evoked by a novel 8 second sinusoidal voltage clamp in CHO cells overexpressing hERG1a. The model was then used to predict over 5 minutes of recordings in the same cell in response to further protocols: a series of traditional square step voltage clamps, and also a novel voltage clamp comprising a collection of physiologically relevant action potentials. We demonstrate that we can make predictive cell‐specific models that outperform the use of averaged data from a number of different cells, and thereby examine which changes in gating are responsible for cell–cell variability in current kinetics. Our technique allows rapid collection of consistent and high quality data, from single cells, and produces more predictive mathematical ion channel models than traditional approaches. Ion current kinetics are commonly represented by current–voltage relationships, time constant–voltage relationships and subsequently mathematical models fitted to these. These experiments take substantial time, which means they are rarely performed in the same cell. Rather than traditional square‐wave voltage clamps, we fitted a model to the current evoked by a novel sum‐of‐sinusoids voltage clamp that was only 8 s long. Short protocols that can be performed multiple times within a single cell will offer many new opportunities to measure how ion current kinetics are affected by changing conditions. The new model predicts the current under traditional square‐wave protocols well, with better predictions of underlying currents than literature models. The current under a novel physiologically relevant series of action potential clamps is predicted extremely well. The short sinusoidal protocols allow a model to be fully fitted to individual cells, allowing us to examine cell–cell variability in current kinetics for the first time.
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Affiliation(s)
- Kylie A Beattie
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK.,Division of Applied Regulatory Science, Office of Clinical Pharmacology, Office of Translational Sciences, Center for Drug Evaluation and Research, Food and Drug Administration, Silver Spring, MD, USA
| | - Adam P Hill
- Department of Molecular Cardiology and Biophysics, Victor Chang Cardiac Research Institute, Sydney, NSW, 2010, Australia.,St Vincent's Clinical School, UNSW Sydney, Darlinghurst, NSW, 2010, Australia
| | - Rémi Bardenet
- CNRS & CRIStAL, Université de Lille, 59651 Villeneuve d'Ascq, Lille, France
| | - Yi Cui
- Safety Evaluation and Risk Management, Global Clinical Safety and Pharmacovigilance, GlaxoSmithKline, Uxbridge, UB11 1BS, UK
| | - Jamie I Vandenberg
- Department of Molecular Cardiology and Biophysics, Victor Chang Cardiac Research Institute, Sydney, NSW, 2010, Australia.,St Vincent's Clinical School, UNSW Sydney, Darlinghurst, NSW, 2010, Australia
| | - David J Gavaghan
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, OX1 3QD, UK
| | - Teun P de Boer
- Department of Medical Physiology, Division of Heart & Lungs, University Medical Center Utrecht, Utrecht, The Netherlands
| | - Gary R Mirams
- Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, NG7 2RD, UK
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Johnstone R, Bardenet R, Polonchuk L, Davies M, Gavaghan D, Mirams G. Hierarchical Bayesian Fitting of Concentration-effect Models to Ion Channel Screening Data. J Pharmacol Toxicol Methods 2017. [DOI: 10.1016/j.vascn.2017.09.098] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Abstract
Dose-response (or 'concentration-effect') relationships commonly occur in biological and pharmacological systems and are well characterised by Hill curves. These curves are described by an equation with two parameters: the inhibitory concentration 50% (IC50); and the Hill coefficient. Typically just the 'best fit' parameter values are reported in the literature. Here we introduce a Python-based software tool, PyHillFit , and describe the underlying Bayesian inference methods that it uses, to infer probability distributions for these parameters as well as the level of experimental observation noise. The tool also allows for hierarchical fitting, characterising the effect of inter-experiment variability. We demonstrate the use of the tool on a recently published dataset on multiple ion channel inhibition by multiple drug compounds. We compare the maximum likelihood, Bayesian and hierarchical Bayesian approaches. We then show how uncertainty in dose-response inputs can be characterised and propagated into a cardiac action potential simulation to give a probability distribution on model outputs.
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Affiliation(s)
- Ross H Johnstone
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK
| | | | - David J Gavaghan
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK
| | - Gary R Mirams
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK.,Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
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Abstract
Dose-response (or ‘concentration-effect’) relationships commonly occur in biological and pharmacological systems and are well characterised by Hill curves. These curves are described by an equation with two parameters: the inhibitory concentration 50% (IC50); and the Hill coefficient. Typically just the ‘best fit’ parameter values are reported in the literature. Here we introduce a Python-based software tool,
PyHillFit , and describe the underlying Bayesian inference methods that it uses, to infer probability distributions for these parameters as well as the level of experimental observation noise. The tool also allows for hierarchical fitting, characterising the effect of inter-experiment variability. We demonstrate the use of the tool on a recently published dataset on multiple ion channel inhibition by multiple drug compounds. We compare the maximum likelihood, Bayesian and hierarchical Bayesian approaches. We then show how uncertainty in dose-response inputs can be characterised and propagated into a cardiac action potential simulation to give a probability distribution on model outputs.
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Affiliation(s)
- Ross H Johnstone
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK
| | | | - David J Gavaghan
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK
| | - Gary R Mirams
- Computational Biology, Department of Computer Science, University of Oxford, Oxford, UK.,Centre for Mathematical Medicine & Biology, School of Mathematical Sciences, University of Nottingham, Nottingham, UK
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Kursawe J, Bardenet R, Zartman JJ, Baker RE, Fletcher AG. Robust cell tracking in epithelial tissues through identification of maximum common subgraphs. J R Soc Interface 2016; 13:20160725. [PMID: 28334699 PMCID: PMC5134023 DOI: 10.1098/rsif.2016.0725] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/07/2016] [Accepted: 10/17/2016] [Indexed: 11/30/2022] Open
Abstract
Tracking of cells in live-imaging microscopy videos of epithelial sheets is a powerful tool for investigating fundamental processes in embryonic development. Characterizing cell growth, proliferation, intercalation and apoptosis in epithelia helps us to understand how morphogenetic processes such as tissue invagination and extension are locally regulated and controlled. Accurate cell tracking requires correctly resolving cells entering or leaving the field of view between frames, cell neighbour exchanges, cell removals and cell divisions. However, current tracking methods for epithelial sheets are not robust to large morphogenetic deformations and require significant manual interventions. Here, we present a novel algorithm for epithelial cell tracking, exploiting the graph-theoretic concept of a 'maximum common subgraph' to track cells between frames of a video. Our algorithm does not require the adjustment of tissue-specific parameters, and scales in sub-quadratic time with tissue size. It does not rely on precise positional information, permitting large cell movements between frames and enabling tracking in datasets acquired at low temporal resolution due to experimental constraints such as phototoxicity. To demonstrate the method, we perform tracking on the Drosophila embryonic epidermis and compare cell-cell rearrangements to previous studies in other tissues. Our implementation is open source and generally applicable to epithelial tissues.
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Affiliation(s)
- Jochen Kursawe
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Rémi Bardenet
- CNRS and CRIStAL, Université de Lille, 59651 Villeneuve d'Ascq, France
| | - Jeremiah J Zartman
- Department of Chemical and Biomolecular Engineering, University of Notre Dame, 205D McCourtney Hall of Molecular Science and Engineering, Notre Dame, IN 46556, USA
| | - Ruth E Baker
- Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK
| | - Alexander G Fletcher
- School of Mathematics and Statistics, University of Sheffield, Hicks Building, Hounsfield Road, Sheffield S3 7RH, UK
- Bateson Centre, University of Sheffield, Sheffield S10 2TN, UK
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Johnstone R, Bardenet R, de Boer T, Gavaghan D, Davies M, Polonchuk L, Mirams G. Cell-specific mathematical models of cardiac electrophysiology. J Pharmacol Toxicol Methods 2016. [DOI: 10.1016/j.vascn.2016.02.029] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Johnstone RH, Bardenet R, Gavaghan DJ, Polonchuk L, Davies MR, Mirams GR. Hierarchical Bayesian Modelling of Variability and Uncertainty in Synthetic Action Potential Traces. Comput Cardiol (2010) 2016; 43:1089-1092. [PMID: 37551270 PMCID: PMC7614890 DOI: 10.22489/cinc.2016.313-458] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 08/09/2023]
Abstract
There are many sources of uncertainty in the recording and modelling of membrane action potentials (APs) from cardiomyocytes. For example, there are measurement, parameter, and model uncertainties. There is also extrinsic variability between cells, and intrinsic beat-to-beat variability within a single cell. These combined uncertainties and variability make it very difficult to extrapolate predictions from these models, since current AP models have single parameter values and thus produce a single AP trace. We aim to re-parameterise existing AP models to fit experimental data, and to quantify uncertainty associated with ion current densities when measuring and modelling these APs. We then wish to propagate this uncertainty into model predictions, such as ion channel block effected by a pharmaceutical compound. We perform an in silico study using synthetic data generated from different sets of parameters. We then 'forget' these parameter values and re-infer their distributions using hierarchical Markov chain Monte Carlo methods. We find that we can successfully infer the 'correct' distributions for most ion current densities for each AP trace, along with an approximation to the higher-level distribution from which these parameter values were sampled. There is, however, some level of unidentifiability amongst some of the current densities.
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Affiliation(s)
- Ross H Johnstone
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford, UK
| | | | - David J Gavaghan
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford, UK
| | | | | | - Gary R Mirams
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford, UK
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Johnstone RH, Chang ETY, Bardenet R, de Boer TP, Gavaghan DJ, Pathmanathan P, Clayton RH, Mirams GR. Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models? J Mol Cell Cardiol 2015; 96:49-62. [PMID: 26611884 PMCID: PMC4915860 DOI: 10.1016/j.yjmcc.2015.11.018] [Citation(s) in RCA: 84] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Submit a Manuscript] [Subscribe] [Scholar Register] [Received: 05/31/2015] [Revised: 10/13/2015] [Accepted: 11/17/2015] [Indexed: 01/07/2023]
Abstract
Cardiac electrophysiology models have been developed for over 50 years, and now include detailed descriptions of individual ion currents and sub-cellular calcium handling. It is commonly accepted that there are many uncertainties in these systems, with quantities such as ion channel kinetics or expression levels being difficult to measure or variable between samples. Until recently, the original approach of describing model parameters using single values has been retained, and consequently the majority of mathematical models in use today provide point predictions, with no associated uncertainty. In recent years, statistical techniques have been developed and applied in many scientific areas to capture uncertainties in the quantities that determine model behaviour, and to provide a distribution of predictions which accounts for this uncertainty. In this paper we discuss this concept, which is termed uncertainty quantification, and consider how it might be applied to cardiac electrophysiology models. We present two case studies in which probability distributions, instead of individual numbers, are inferred from data to describe quantities such as maximal current densities. Then we show how these probabilistic representations of model parameters enable probabilities to be placed on predicted behaviours. We demonstrate how changes in these probability distributions across data sets offer insight into which currents cause beat-to-beat variability in canine APs. We conclude with a discussion of the challenges that this approach entails, and how it provides opportunities to improve our understanding of electrophysiology. Uncertainty and variability in action potential models can be quantified. A probabilistic method for inferring maximal current densities is developed and applied. We use this to infer the currents responsible for canine beat-to-beat variability. Emulation of mathematical models provides rich information at low computational cost. The importance of considering uncertainty and variability in future is discussed.
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Affiliation(s)
- Ross H Johnstone
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK
| | - Eugene T Y Chang
- Insigneo Institute for in-silico Medicine and Department of Computer Science, University of Sheffield, Sheffield S1 4DP, UK
| | - Rémi Bardenet
- CNRS & CRIStAL, Université de Lille, 59651 Villeneuve d'Ascq, France
| | - Teun P de Boer
- Division of Heart & Lungs, Department of Medical Physiology, University Medical Center Utrecht, Utrecht, The Netherlands
| | - David J Gavaghan
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK
| | - Pras Pathmanathan
- U.S. Food and Drug Administration, 10903 New Hampshire Avenue, Silver Spring, MD 20993, USA.
| | - Richard H Clayton
- Insigneo Institute for in-silico Medicine and Department of Computer Science, University of Sheffield, Sheffield S1 4DP, UK.
| | - Gary R Mirams
- Computational Biology, Dept. of Computer Science, University of Oxford, Oxford OX1 3QD, UK.
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Abreu P, Aglietta M, Ahn EJ, Albuquerque IFM, Allard D, Allekotte I, Allen J, Allison P, Almeda A, Alvarez Castillo J, Alvarez-Muñiz J, Ambrosio M, Aminaei A, Anchordoqui L, Andringa S, Antičić T, Aramo C, Arganda E, Arqueros F, Asorey H, Assis P, Aublin J, Ave M, Avenier M, Avila G, Bäcker T, Balzer M, Barber KB, Barbosa AF, Bardenet R, Barroso SLC, Baughman B, Bäuml J, Beatty JJ, Becker BR, Becker KH, Bellétoile A, Bellido JA, Benzvi S, Berat C, Bertou X, Biermann PL, Billoir P, Blanco F, Blanco M, Bleve C, Blümer H, Boháčová M, Boncioli D, Bonifazi C, Bonino R, Borodai N, Brack J, Brogueira P, Brown WC, Bruijn R, Buchholz P, Bueno A, Burton RE, Caballero-Mora KS, Caramete L, Caruso R, Castellina A, Catalano O, Cataldi G, Cazon L, Cester R, Chauvin J, Cheng SH, Chiavassa A, Chinellato JA, Chirinos Diaz J, Chudoba J, Clay RW, Coluccia MR, Conceição R, Contreras F, Cook H, Cooper MJ, Coppens J, Cordier A, Coutu S, Covault CE, Creusot A, Criss A, Cronin J, Curutiu A, Dagoret-Campagne S, Dallier R, Dasso S, Daumiller K, Dawson BR, de Almeida RM, De Domenico M, De Donato C, de Jong SJ, De La Vega G, de Mello Junior WJM, de Mello Neto JRT, De Mitri I, de Souza V, de Vries KD, Decerprit G, del Peral L, del Río M, Deligny O, Dembinski H, Dhital N, Di Giulio C, Díaz Castro ML, Diep PN, Dobrigkeit C, Docters W, D'Olivo JC, Dong PN, Dorofeev A, dos Anjos JC, Dova MT, D'Urso D, Dutan I, Ebr J, Engel R, Erdmann M, Escobar CO, Espadanal J, Etchegoyen A, Facal San Luis P, Fajardo Tapia I, Falcke H, Farrar G, Fauth AC, Fazzini N, Ferguson AP, Ferrero A, Fick B, Filevich A, Filipčič A, Fliescher S, Fracchiolla CE, Fraenkel ED, Fröhlich U, Fuchs B, Gaior R, Gamarra RF, Gambetta S, García B, Garcia-Gamez D, Garcia-Pinto D, Gascon A, Gemmeke H, Gesterling K, Ghia PL, Giaccari U, Giller M, Glass H, Gold MS, Golup G, Gomez Albarracin F, Gómez Berisso M, Gonçalves P, Gonzalez D, Gonzalez JG, Gookin B, Góra D, Gorgi A, Gouffon P, Gozzini SR, Grashorn E, Grebe S, Griffith N, Grigat M, Grillo AF, Guardincerri Y, Guarino F, Guedes GP, Guzman A, Hague JD, Hansen P, Harari D, Harmsma S, Harrison TA, Harton JL, Haungs A, Hebbeker T, Heck D, Herve AE, Hojvat C, Hollon N, Holmes VC, Homola P, Hörandel JR, Horneffer A, Horvath P, Hrabovský M, Huege T, Insolia A, Ionita F, Italiano A, Jarne C, Jiraskova S, Josebachuili M, Kadija K, Kampert KH, Karhan P, Kasper P, Kégl B, Keilhauer B, Keivani A, Kelley JL, Kemp E, Kieckhafer RM, Klages HO, Kleifges M, Kleinfeller J, Knapp J, Koang DH, Kotera K, Krohm N, Krömer O, Kruppke-Hansen D, Kuehn F, Kuempel D, Kulbartz JK, Kunka N, La Rosa G, Lachaud C, Lauer R, Lautridou P, Le Coz S, Leão MSAB, Lebrun D, Lebrun P, Leigui de Oliveira MA, Lemiere A, Letessier-Selvon A, Lhenry-Yvon I, Link K, López R, Lopez Agüera A, Louedec K, Lozano Bahilo J, Lu L, Lucero A, Ludwig M, Lyberis H, Macolino C, Maldera S, Mandat D, Mantsch P, Mariazzi AG, Marin J, Marin V, Maris IC, Marquez Falcon HR, Marsella G, Martello D, Martin L, Martinez H, Martínez 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Taşcău O, Tavera Ruiz CG, Tcaciuc R, Tegolo D, Thao NT, Thomas D, Tiffenberg J, Timmermans C, Tiwari DK, Tkaczyk W, Todero Peixoto CJ, Tomé B, Tonachini A, Travnicek P, Tridapalli DB, Tristram G, Trovato E, Tueros M, Ulrich R, Unger M, Urban M, Valdés Galicia JF, Valiño I, Valore L, van den Berg AM, Varela E, Vargas Cárdenas B, Vázquez JR, Vázquez RA, Veberič D, Verzi V, Vicha J, Videla M, Villaseñor L, Wahlberg H, Wahrlich P, Wainberg O, Walz D, Warner D, Watson AA, Weber M, Weidenhaupt K, Weindl A, Westerhoff S, Whelan BJ, Wieczorek G, Wiencke L, Wilczyńska B, Wilczyński H, Will M, Williams C, Winchen T, Winnick MG, Wommer M, Wundheiler B, Yamamoto T, Yapici T, Younk P, Yuan G, Yushkov A, Zamorano B, Zas E, Zavrtanik D, Zavrtanik M, Zaw I, Zepeda A, Zhu Y, Zimbres Silva M, Ziolkowski M. Measurement of the proton-air cross section at √s=57 TeV with the Pierre Auger Observatory. Phys Rev Lett 2012; 109:062002. [PMID: 23006259 DOI: 10.1103/physrevlett.109.062002] [Citation(s) in RCA: 44] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/02/2011] [Indexed: 06/01/2023]
Abstract
We report a measurement of the proton-air cross section for particle production at the center-of-mass energy per nucleon of 57 TeV. This is derived from the distribution of the depths of shower maxima observed with the Pierre Auger Observatory: systematic uncertainties are studied in detail. Analyzing the tail of the distribution of the shower maxima, a proton-air cross section of [505±22(stat)(-36)(+28)(syst)] mb is found.
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Affiliation(s)
- P Abreu
- LIP and Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal
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