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Kimmig F, Caruel M, Chapelle D. Varying thin filament activation in the framework of the Huxley'57 model. INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING 2022; 38:e3655. [PMID: 36210493 DOI: 10.1002/cnm.3655] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/22/2021] [Revised: 04/29/2022] [Accepted: 09/24/2022] [Indexed: 06/16/2023]
Abstract
Muscle contraction is triggered by the activation of the actin sites of the thin filament by calcium ions. It results that the thin filament activation level varies over time. Moreover, this activation process is also used as a regulation mechanism of the developed force. Our objective is to build a model of varying actin site activation level within the classical Huxley'57 two-state framework. This new model is obtained as an enhancement of a previously proposed formulation of the varying thick filament activation within the same framework. We assume that the state of an actin site depends on whether it is activated and whether it forms a cross-bridge with the associated myosin head, which results in four possible states. The transitions between the actin site states are controlled by the global actin sites activation level and the dynamics of these transitions is coupled with the attachment-detachment process. A preliminary calibration of the model with experimental twitch contraction data obtained at varying sarcomere lengths is performed.
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Affiliation(s)
- François Kimmig
- LMS, École Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France
- Inria, Palaiseau, France
| | - Matthieu Caruel
- CNRS, UMR 8208, MSME, Univ Paris Est Creteil, Univ Gustave Eiffel, Créteil, France
| | - Dominique Chapelle
- LMS, École Polytechnique, CNRS, Institut Polytechnique de Paris, Palaiseau, France
- Inria, Palaiseau, France
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2
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Ortes F, Jinha A, Herzog W, Ziya Arslan Y. Sensitivity of muscle force response of a two-state cross-bridge model to variations in model parameters. Proc Inst Mech Eng H 2022; 236:1513-1520. [DOI: 10.1177/09544119221122062] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
Muscle models based on the cross-bridge theory (Huxley-type models) are frequently used to calculate muscle forces for different contractile conditions. Dynamic and nonlinear characteristics of muscle forces produced during isometric, concentric, and eccentric contractions can be represented to a limited extent by using cross-bridge models. Cross-bridge models use various parameters to simulate force responses. However, there remains uncertainty as to the effect of changes in model parameters on force responses in Huxley-type models. In this study, we aimed to analyze the sensitivity of force response to changes in model parameters in Huxley-type models. A two-state Huxley model was used to determine the cross-bridge attachment distributions and forces for shortening and lengthening contractions. Sensitivity of muscle force to changes in attachment rate, detachment rate, and cross-bridge binding distance was examined within a range of ±20% of the nominal value using Monte Carlo simulations. Changes in the detachment rate influenced the predicted muscle forces the most for lengthening contractions, while changes in attachment rate and binding distance affected forces the most for shortening contractions. These results show once more the asymmetry between shortening and lengthening contractions and the difficulty in using a single cross-bridge model to predict forces during shortening and elongation accurately.
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Affiliation(s)
- Faruk Ortes
- Department of Mechanical Engineering, Faculty of Engineering, Istanbul University-Cerrahpasa, Istanbul, Turkey
| | - Azim Jinha
- Human Performance Lab, Faculty of Kinesiology, University of Calgary, Calgary, Canada
| | - Walter Herzog
- Human Performance Lab, Faculty of Kinesiology, University of Calgary, Calgary, Canada
| | - Yunus Ziya Arslan
- Institute of Graduate Studies in Science and Engineering, Department of Robotics and Intelligent Systems, Turkish-German University, Istanbul, Turkey
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3
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Morris CJ, Zawieja DC, Moore JE. A multiscale sliding filament model of lymphatic muscle pumping. Biomech Model Mechanobiol 2021; 20:2179-2202. [PMID: 34476656 PMCID: PMC8595193 DOI: 10.1007/s10237-021-01501-0] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/17/2021] [Accepted: 08/01/2021] [Indexed: 11/30/2022]
Abstract
The lymphatics maintain fluid balance by returning interstitial fluid to veins via contraction/compression of vessel segments with check valves. Disruption of lymphatic pumping can result in a condition called lymphedema with interstitial fluid accumulation. Lymphedema treatments are often ineffective, which is partially attributable to insufficient understanding of specialized lymphatic muscle lining the vessels. This muscle exhibits cardiac-like phasic contractions and smooth muscle-like tonic contractions to generate and regulate flow. To understand the relationship between this sub-cellular contractile machinery and organ-level pumping, we have developed a multiscale computational model of phasic and tonic contractions in lymphatic muscle and coupled it to a lymphangion pumping model. Our model uses the sliding filament model (Huxley in Prog Biophys Biophys Chem 7:255-318, 1957) and its adaptation for smooth muscle (Mijailovich in Biophys J 79(5):2667-2681, 2000). Multiple structural arrangements of contractile components and viscoelastic elements were trialed but only one provided physiologic results. We then coupled this model with our previous lumped parameter model of the lymphangion to relate results to experiments. We show that the model produces similar pressure, diameter, and flow tracings to experiments on rat mesenteric lymphatics. This model provides the first estimates of lymphatic muscle contraction energetics and the ability to assess the potential effects of sub-cellular level phenomena such as calcium oscillations on lymphangion outflow. The maximum efficiency value predicted (40%) is at the upper end of estimates for other muscle types. Spontaneous calcium oscillations during diastole were found to increase outflow up to approximately 50% in the range of frequencies and amplitudes tested.
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Affiliation(s)
- Christopher J Morris
- Department of Bioengineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK
| | - David C Zawieja
- College of Medicine Faculty, Texas A&M University, Texas, USA
| | - James E Moore
- Department of Bioengineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, UK.
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Schroeder RT, Kuo AD. Elastic energy savings and active energy cost in a simple model of running. PLoS Comput Biol 2021; 17:e1009608. [PMID: 34813593 PMCID: PMC8651147 DOI: 10.1371/journal.pcbi.1009608] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2021] [Revised: 12/07/2021] [Accepted: 11/02/2021] [Indexed: 11/18/2022] Open
Abstract
The energetic economy of running benefits from tendon and other tissues that store and return elastic energy, thus saving muscles from costly mechanical work. The classic "Spring-mass" computational model successfully explains the forces, displacements and mechanical power of running, as the outcome of dynamical interactions between the body center of mass and a purely elastic spring for the leg. However, the Spring-mass model does not include active muscles and cannot explain the metabolic energy cost of running, whether on level ground or on a slope. Here we add explicit actuation and dissipation to the Spring-mass model, and show how they explain substantial active (and thus costly) work during human running, and much of the associated energetic cost. Dissipation is modeled as modest energy losses (5% of total mechanical energy for running at 3 m s-1) from hysteresis and foot-ground collisions, that must be restored by active work each step. Even with substantial elastic energy return (59% of positive work, comparable to empirical observations), the active work could account for most of the metabolic cost of human running (about 68%, assuming human-like muscle efficiency). We also introduce a previously unappreciated energetic cost for rapid production of force, that helps explain the relatively smooth ground reaction forces of running, and why muscles might also actively perform negative work. With both work and rapid force costs, the model reproduces the energetics of human running at a range of speeds on level ground and on slopes. Although elastic return is key to energy savings, there are still losses that require restorative muscle work, which can cost substantial energy during running.
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Affiliation(s)
| | - Arthur D. Kuo
- Faculty of Kinesiology, University of Calgary, Alberta, Canada
- Biomedical Engineering Program, University of Calgary, Alberta, Canada
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Lemaire KK, Baan GC, Jaspers RT, van Soest AJK. Comparison of the validity of Hill and Huxley muscle tendon complex models using experimental data obtained from rat m. soleus in situ. J Exp Biol 2016; 219:977-87. [DOI: 10.1242/jeb.128280] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/10/2015] [Accepted: 01/15/2016] [Indexed: 11/20/2022]
Abstract
The relationship between mechanical and metabolic behaviour in the widely used Hill muscle-tendon complex (MTC) model is not straightforward, while this is an integral part of the Huxley model. In this study we assessed to what extent Huxley and Hill type MTC models yield adequate predictions of mechanical muscle behaviour during stretch-shortening cycles (SSC). In fully anaesthetized male Wistar rats (N=3), m. soleus was dissected completely free, except for the insertion. Cuff electrodes were placed over the n. ischiadicus. The distal end of the tendon was connected to a servo motor, via a force transducer. The setup allowed for full control over muscle stimulation and length, while force was measured. Quick release and isovelocity contractions (part 1), and SSC (part 2) were imposed. Simulations of part 2 were made with both a Hill and a Huxley MTC model, using parameter values determined from part 1. A modification to the classic two-state Huxley model was made to incorporate series elasticity, activation dynamics and active and passive force-length relations. Results were similar for all rats. Fitting of the free parameters to data of part 1 was near perfect (R2 > .97). During SSC, predicted peak force and force during relaxation deviated from the experimental data, for both models. Overall, both models yielded similarly adequate predictions of the experimental data. We conclude that Huxley and Hill MTC models are equally valid with respect to mechanical behaviour.
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Affiliation(s)
- Koen K. Lemaire
- MOVE Research Institute Amsterdam, Department of Human Movement Sciences, VU University Amsterdam, Van Der Boechorststraat 9, 1081 Amsterdam, The Netherlands
| | - Guus C. Baan
- Laboratory for Myology, MOVE Research Institute Amsterdam, Department of Human Movement Sciences, VU University Amsterdam, Van Der Boechorststraat 9, 1081 Amsterdam, The Netherlands
| | - Richard T. Jaspers
- Laboratory for Myology, MOVE Research Institute Amsterdam, Department of Human Movement Sciences, VU University Amsterdam, Van Der Boechorststraat 9, 1081 Amsterdam, The Netherlands
| | - A. J. Knoek van Soest
- MOVE Research Institute Amsterdam, Department of Human Movement Sciences, VU University Amsterdam, Van Der Boechorststraat 9, 1081 Amsterdam, The Netherlands
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Studies on biomechanics of skeletal muscle based on the working mechanism of myosin motors: An overview. CHINESE SCIENCE BULLETIN-CHINESE 2012. [DOI: 10.1007/s11434-012-5438-y] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Mavritsaki E, Lepora N, Porrill J, Yeo CH, Dean P. Response linearity determined by recruitment strategy in detailed model of nictitating membrane control. BIOLOGICAL CYBERNETICS 2007; 96:39-57. [PMID: 17021829 DOI: 10.1007/s00422-006-0105-5] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2006] [Accepted: 07/18/2006] [Indexed: 05/12/2023]
Abstract
Many models of eyeblink conditioning assume that there is a simple linear relationship between the firing patterns of neurons in the interpositus nucleus and the time course of the conditioned response (CR). However, the complexities of muscle behaviour and plant dynamics call this assumption into question. We investigated the issue by implementing the most detailed model available of the rabbit nictitating membrane response (Bartha and Thompson in Biol Cybern 68:135-143, 1992a and in Biol Cybern 68:145-154, 1992b), in which each motor unit of the retractor bulbi muscle is represented by a Hill-type model, driven by a non-linear activation mechanism designed to reproduce the isometric force measurements of Lennerstrand (J Physiol 236:43-55, 1974). Globe retraction and NM extension are modelled as linked second order systems. We derived versions of the model that used a consistent set of SI units, were based on a physically realisable version of calcium kinetics, and used simulated muscle cross-bridges to produce force. All versions showed similar non-linear responses to two basic control strategies. (1) Rate-coding with no recruitment gave a sigmoidal relation between control signal and amplitude of CR, reflecting the measured relation between isometric muscle force and stimulation frequency. (2) Recruitment of similar strength motor units with no rate coding gave a sublinear relation between control signal and amplitude of CR, reflecting the increase in muscle stiffness produced by recruitment. However, the system response could be linearised by either a suitable combination of rate-coding and recruitment, or by simple recruitment of motor units in order of (exponentially) increasing strength. These plausible control strategies, either alone or in combination, would in effect present the cerebellum with the simplified virtual plant that is assumed in many models of eyeblink conditioning. Future work is therefore needed to determine the extent to which motor neuron firing is in fact linearly related to the nictitating membrane response.
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Affiliation(s)
- Eirini Mavritsaki
- Department of Psychology, Sheffield University, Sheffield, S10 2TP, UK
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Oomens CWJ, Maenhout M, van Oijen CH, Drost MR, Baaijens FP. Finite element modelling of contracting skeletal muscle. Philos Trans R Soc Lond B Biol Sci 2003; 358:1453-60. [PMID: 14561336 PMCID: PMC1693246 DOI: 10.1098/rstb.2003.1345] [Citation(s) in RCA: 98] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Abstract] [MESH Headings] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
To describe the mechanical behaviour of biological tissues and transport processes in biological tissues, conservation laws such as conservation of mass, momentum and energy play a central role. Mathematically these are cast into the form of partial differential equations. Because of nonlinear material behaviour, inhomogeneous properties and usually a complex geometry, it is impossible to find closed-form analytical solutions for these sets of equations. The objective of the finite element method is to find approximate solutions for these problems. The concepts of the finite element method are explained on a finite element continuum model of skeletal muscle. In this case, the momentum equations have to be solved with an extra constraint, because the material behaves as nearly incompressible. The material behaviour consists of a highly nonlinear passive part and an active part. The latter is described with a two-state Huxley model. This means that an extra nonlinear partial differential equation has to be solved. The problems and solutions involved with this procedure are explained. The model is used to describe the mechanical behaviour of a tibialis anterior of a rat. The results have been compared with experimentally determined strains at the surface of the muscle. Qualitatively there is good agreement between measured and calculated strains, but the measured strains were higher.
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Affiliation(s)
- C W J Oomens
- Department of Biomedical Engineering, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands.
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Gielen AWJ, Oomens CWJ, Bovendeerd PHM, Arts T, Janssen JD. A Finite Element Approach for Skeletal Muscle using a Distributed Moment Model of Contraction. Comput Methods Biomech Biomed Engin 2001; 3:231-244. [PMID: 11264850 DOI: 10.1080/10255840008915267] [Citation(s) in RCA: 33] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
Abstract
The present paper describes a geometrically and physically nonlinear continuum model to study the mechanical behaviour of passive and active skeletal muscle. The contraction is described with a Huxley type model. A Distributed Moments approach is used to convert the Huxley partial differential equation in a set of ordinary differential equations. An isoparametric brick element is developed to solve the field equations numerically. Special arrangements are made to deal with the combination of highly nonlinear effects and the nearly incompressible behaviour of the muscle. For this a Natural Penalty Method (NPM) and an Enhanced Stiffness Method (ESM) are tested. Finally an example of an analysis of a contracting tibialis anterior muscle of a rat is given. The DM-method proved to be an efficient tool in the numerical solution process. The ESM showed the best performance in describing the incompressible behaviour.
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Zahalak GI. The two-state cross-bridge model of muscle is an asymptotic limit of multi-state models. J Theor Biol 2000; 204:67-82. [PMID: 10772849 DOI: 10.1006/jtbi.2000.1084] [Citation(s) in RCA: 26] [Impact Index Per Article: 1.1] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/22/2022]
Abstract
The relationship between the two-state model of muscle contraction and multi-state models is examined from the perspective of matched asymptotic expansions, under the assumption that transition rates between attached states are fast compared to those between detached and attached states. A detailed formal analysis of a three-state model reveals that the classic Huxley (1957. Prog. Biophys. Biophys. Chem.7, 225-318) rate equation, as modified for thermodynamic self-consistency by Hill et al. (1975. Biophys. J.15, 335-372), governs the "outer" solution of the three-state equations. Thus, the two-state model remains a valid description of muscle dynamics on physiologically relevant time scales, which are slow compared to millisecond-scale transitions between attached states. But the asymptotic analysis reveals also that the cross-bridge force must be considered to be a nonlinear function of the cross-bridge strain, in contrast to the usual assumption of two-state models. This apparent, or effective, force is determined by both the intrinsic stiffness of the cross-bridge and the equilibrium distribution of cross-bridges among attached states. Further, the asymptotic analysis yields an expression for the energy liberation rate that implies a reduced rate in stretch vs. shortening. Some behaviors of multi-state models that are suggested by the three-state analysis are discussed in qualitative terms.
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Affiliation(s)
- G I Zahalak
- Departments of Biomedical Engineering and Mechanical Engineering, Washington University, St. Louis, MO 63130, USA
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Zahalak GI, de Laborderie V, Guccione JM. The effects of cross-fiber deformation on axial fiber stress in myocardium. J Biomech Eng 1999; 121:376-85. [PMID: 10464691 DOI: 10.1115/1.2798334] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/08/2022]
Abstract
We incorporated a three-dimensional generalization of the Huxley cross-bridge theory in a finite element model of ventricular mechanics to examine the effect of nonaxial deformations on active stress in myocardium. According to this new theory, which assumes that macroscopic tissue deformations are transmitted to the myofilament lattice, lateral myofilament spacing affects the axial fiber stress. We calculated stresses and deformations at end-systole under the assumption of strictly isometric conditions. Our results suggest that at the end of ejection, nonaxial deformations may significantly reduce active axial fiber stress in the inner half of the wall of the normal left ventricle (18-35 percent at endocardium, depending on location with respect to apex and base). Moreover, this effect is greater in the case of a compliant ischemic region produced by occlusion of the left anterior descending or circumflex coronary artery (26-54 percent at endocardium). On the other hand, stiffening of the remote and ischemic regions (in the case of a two-week-old infarct) lessens the effect of nonaxial deformation on active stress at all locations (9-32 percent endocardial reductions). These calculated effects are sufficiently large to suggest that the influence of nonaxial deformation on active fiber stress may be important, and should be considered in future studies of cardiac mechanics.
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Affiliation(s)
- G I Zahalak
- Department of Biomedical Engineering, Washington University, St. Louis, MO 63130-4899, USA
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Abstract
A new phenomenological model of activated muscle is presented. The model is based on a combination of a contractile element, an elastic element that engages upon activation, a linear dashpot and a linear spring. Analytical solutions for a few selected experiments are provided. This model is able to reproduce the response of cat soleus muscle to ramp shortening and stretching and, unlike standard Hill-type models, computations are stable on the descending limb of the force-length relation and force enhancement (depression) following stretching (shortening) is predicted correctly. In its linear version, the model is consistent with a linear force-velocity law, which in this model is a consequence rather than a fundamental characteristic of the material. Results show that the mechanical response of activated muscle can be mimicked by a viscoelastic system. Conceptual differences between this model and standard Hill-type models are analyzed and the advantages of the present model are discussed.
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Affiliation(s)
- M Forcinito
- Department of Mechanical Engineering, The University of Calgary, Alberta, Canada
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13
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Abstract
The Distribution Moment (DM) model has simulated experimental data on skeletal muscle, but it has not been used previously to study the mechanics of active contraction in cardiac muscle. In contrast to previous models of striated muscle contraction, all parameters have physical meaning and assumptions concerning biophysical events within the cell are consistent with available data. In order to simulate cardiac muscle deactivation using the DM model it was necessary to make the cross-bridge detachment rates large for large displacements from the neutral equilibrium position of a cross-bridge. To examine the effect of cooperativity on cardiac muscle contraction, we used the DM model's tight coupling scheme with binding of one or two calcium sites regulating contraction. As observed experimentally, our model predicted a reduction of isometric tension development following rapid shortening lengthening transients when contraction is regulated by either one or two calcium binding sites. The predicted deactivating effect increased if the transient was applied late in the twitch when contraction is regulated by two calcium binding sites, but not when it is regulated by one site. This is the first study in which deactivation has been simulated without making any provisions for length-dependent calcium trononin dissociation.
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Affiliation(s)
- J M Guccione
- Department of Biomedical Engineering, Washington University, St. Louis, MO 63130-4899, USA.
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