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INNOVATIVE METHODOLOGY
1Institute of Immunology and Physiology, Ural Branch of the Russian Academy of Sciences; and 2Ural State University, Ekaterinburg, Russia; and 3Cardiac Mechano-Electric Feedback, University Laboratory of Physiology, Oxford, United Kingdom
Submitted 29 March 2005 ; accepted in final form 15 July 2005
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ABSTRACT |
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 TOP ABSTRACT HYBRID DUPLEX METHODRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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heart muscle; muscle mechanics; mathematical model; real-time control; stretch
The mechanisms underlying cardiac mechanical heterogeneity and their relevance for (patho)physiological function of the heart are poorly understood, partially because there are no simple, yet representative, experimental models. Because of the complex geometry of the heart, the transmural and longitudinal disparity of deformation and tension fields (1, 3, 34), inhomogeneous innervation and blood supply (50), and the complex sequence of electrical activation of the myocardium, it is difficult to adequately control experimental conditions in native tissue models. To resolve this problem, it is necessary to develop simple, yet physiologically relevant, experimental and theoretical models of the inhomogeneous myocardium.
The most fundamental model of cardiac heterogeneity involves two interacting myocardial elements: a duplex. Mechanical interaction of duplex elements can be in series or in parallel. The first findings from experiments using such mechanically connected pairs of cardiac muscle were published 35 years ago (44, 47). In these and a limited number of later studies (37), the effects of mechanical heterogeneity were studied using two normal papillary muscles connected in series or one hypoxic and one normal papillary muscle.
We developed the duplex method further by creating six principally different duplex configurations, each with its own advantages and limitations (for reviews see Refs. 26 and 27). The first two duplex configurations consist of two papillary muscles, or trabeculae, connected in series or in parallel: biological duplexes. Replacing both biological samples (in the 2 configurations described above) with interacting computational models yields two further duplex configurations: serial and parallel virtual duplexes. The remaining two configurations are based on the real-time interaction between one biological muscle and one virtual muscle: serial and parallel hybrid duplexes. Using the duplex method, we previously observed a range of fundamentally new results (22–27, 39, 40) which confirm that biomechanical characteristics of inhomogeneous myocardium are principally different from those of isolated muscle or homogeneous myocardium.
The technically most challenging, but analytically most rewarding, implementation of the above-described configurations is the hybrid duplex. It requires the laboratory expertise and techniques required for in vitro biological work, together with reliable in silico implementations of cardiac electromechanics used in theoretical studies, plus a solution to the instrumentation and control challenges imposed by the need to make biological and virtual duplex elements interact in real time. The present report provides a detailed description of methodological solutions for serial and parallel hybrid duplexes and identifies their advantages for exploring basic phenomena of heterogeneous myocardial activity. This study is focused on describing the hybrid duplex method, as it is a relatively new concept and a fair amount of detail is required for its clarification. [Further experimental data obtained by the duplex method are available elsewhere (24, 26, 39).]
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HYBRID DUPLEX METHOD |
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TOPABSTRACTHYBRID DUPLEX METHODRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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The hybrid duplex approach allows one to analyze and compare the mechanical activity of duplex elements in isolation and after mechanical coupling, in series or in parallel, under isometric, isotonic, and auxotonic modes of contraction.
The virtual muscle is represented by an extensively tested mathematical model of active myocardial mechanics (13, 16, 39). The model provides access to a wide range of parameter variations to simulate physiological or pathological properties of cardiac muscle. This allows one to monitor and alter mechanical heterogeneity of the hybrid duplex. An advantage, unique to the hybrid duplex approach, is that one and the same biological muscle can be coupled to a range of different virtual samples. This can be used to simulate the interaction of muscle segments in various positions relative to each other, different environmental conditions, or pathophysiological states. Another advantage of the hybrid duplex approach lies in the ability to observe any intracellular process simulated in the virtual representation of myocardium. This is in addition to experimentally recorded parameters and provides for more detailed analysis options (with the necessary proviso, of course, that mathematical model insight remains theoretical until the model is experimentally validated).
In addition to simulating functional heterogeneities, the temporal relation of duplex element activation is under experimental control. Varying the time delay in electrical activation of elements allows representation of the activation delays between distant myocardial segments in vivo.
The main difference in controlling the activity of a single muscle and a muscle coupled in silico to another muscle is that the control signal for each muscle may not be defined before their joined contraction. In a duplex, the mechanical activity of each element depends on the state of the other, which poses significant challenges for control and feedback routines. Thus the hybrid approach implements the interaction of a real muscle with a virtual muscle (i.e., computer model) by means of signal exchange in real time (Fig. 1).
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Real muscle. In a hybrid duplex, one of the elements is a biological preparation, such as papillary muscle or trabecula. We used rabbit and rat right ventricular preparations to obtain samples representative of fast (rat) and slow (rabbit) contractions to illustrate the feasibility of our control routine in the hybrid duplex setting.
Because isolated myocardial preparations differ in their individual mechanical characteristics, all preparations are initially extensively characterized for parameters such as passive stiffness, rate of isometric tension development, time to peak isometric force, and amplitude of isometric force or isotonic shortening. Because each of these characteristics can be freely adjusted in a virtual muscle, one can expose the biological sample in a hybrid duplex to a partner element of closely corresponding mechanical properties (homogeneous duplex) or an element with mechanical characteristics that differ in a defined way (heterogeneous duplex), and one can go back and forth between settings (for preparation of biological samples, see Experimental Protocols).
Virtual muscle.
The virtual element of hybrid duplexes is represented by the Ekaterinburg model of cardiac mechanical activity (16, 39). The model comprises a system of ordinary differential equations to describe mechanical processes in cardiomyocytes, including the kinetics of force-generating cross bridges based on the kinetics of intracellular Ca2+. In the framework of the model, we describe dynamic changes in muscle length (L) and force (F) under various contraction conditions. For example, the model can calculate the change in F on changes in L, in particular, in isometric mode with L 
const or the change in L on changes in F, e.g., in isotonic mode with F const.
The rheological scheme of the model represents a force-generating contractile unit, reproducing the properties of activated cardiac sarcomeres, and two nonlinear elastic units (one in parallel and one in series with the contractile unit). The force generated by a sarcomere is assumed to be proportional to an averaged force of a force-generating cross bridge multiplied by the fraction of force-generating cross bridges per sarcomere. The former is assumed to depend explicitly on the shortening/lengthening velocity of the sarcomere, and the latter is calculated from a kinetic equation of cross-bridge attachment/detachment that depends on sarcomere length, velocity of sarcomere shortening, and concentration of Ca2+ bound to troponin C (TnC).
Three types of cooperativity of the contractile proteins, observed in biochemical experiments, have been included in the model (13): the affinity of TnC for Ca2+ tends to increase with the concentration of strongly bound cross bridges, with an increasing concentration of Ca2+-TnC complexes, and with the amount of free actin sites increases as a result of end-to-end interactions between adjacent tropomyosins. Thus it was possible to take into account the feedback effects of the mechanical characteristics of contraction on Ca2+-TnC kinetics. These feedback mechanisms reveal themselves in a number of experimentally observed effects, including load-dependent relaxation, and these effects have been successfully simulated within the framework of the model (16, 39).
The kinetics of free intracellular Ca2+ concentration ([Ca2+]i), which is the "link" between electrical, mechanical, and chemical processes in a cardiac cell, are also described in detail elsewhere (39). The model accounts for the Ca2+ exchange with extra- and intracellular sources, including the sarcoplasmic reticulum, and intracellular Ca2+ buffers (including TnC), which play an important role in shaping [Ca2+]i kinetics. The well-developed mathematical description of Ca2+ handling has allowed us to successfully simulate a number of experimentally observed effects of mechanical conditions on muscle contraction and on the Ca2+ transient (16, 24, 39).
The model of cardiac mechanical activity is a system of 10 ordinary differential equations for muscle and sarcomere shortening/lengthening and concentration of force-generating cross bridges. The "chemical" block of the system comprises equations for [Ca2+]i, the Ca2+-TnC complex, Ca2+ bound to other ligands, and, finally, Ca2+ kinetics in net sarcoplasmic reticulum and its junctional compartments. The input signal for the model is a command change in muscle length or load, which determines the equation for muscle deformation.
Combination of the above-described mechanical model with appropriate models of cellular electrophysiology advances the hybrid duplex approach to studying effects of mechanical coupling on electromechanical activity of the interacting elements and mechanoelectrical feedback in heterogeneous myocardial systems. Such a model of myocardial electromechanical activity, combining the Ekaterinburg mechanical module with a cardiac electrophysiology module (31) from the OxSoft heart family, has been developed by our groups (40, 41).
The combined model contains differential equations to describe changes in membrane potential and transmembrane ionic currents, along with the above-mentioned mechanical processes and the kinetics of [Ca2+]i. The total number of state variables is 25, where the electrophysiological part comprises membrane potential, gating parameters of ionic channels, and intracellular Na+ and extracellular K+ concentrations. The actual model can be downloaded from the following website: http://www.physiome.org.nz/publications/PBMB-2003-82/Markhasin/. A pseudocode description of the computational procedure used to calculate one integration step of the system using an Euler numerical integrator is presented in the APPENDIX.
By varying model parameter values, one can simulate variations in muscle-specific behavior. For the proof-of-principle work discussed here, parameters of the mechanical model were initially fitted to mimic the time course of contractions of fast- or slow-contracting biological muscle samples. Then parameters of the virtual muscle element in the hybrid duplex were adjusted to create controlled heterogeneous pairs with defined differences in their velocity of contraction, passive stiffness, diastolic length, or peak force (for details see Ref. 40 and the above-mentioned web site). All parameter ranges were selected to mimic variations in biomechanical properties of ventricular myocardium that are plausible in normal conditions or in pathology.
Computer-Based Control System
General consideration of controlling a hybrid duplex. Afterloaded contractions of a hybrid duplex occur at constant afterload or during (more physiologically) changing external loads. Thus any contractile cycle requires switching between isometric and isotonic (or auxotonic) modes of contraction and relaxation. Specified kinetic conditions should be continuously maintained between virtually interacting elements. Two cases pose special demands in this context.
IN-SERIES DUPLEX.
During externally isometric contractions (Ld = const), duplex element interaction is governed by Eq. 1, in which virtual muscle force is matched to force generated by the biological sample
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where Ld/Fd, Lmus/Fmus, and Lmod/Fmod represent the ratio of length to force of duplex (d), biological muscle (mus), and virtual muscle model (mod), respectively. During afterloaded contractions, after isometric mode, where duplex force reaches a predetermined threshold, the control mode is switched to the isotonic (Fd = const) or the auxotonic condition, where each muscle is allowed to shorten independently, but under the same predefined external load on the duplex.
IN-PARALLEL DUPLEX.
During externally isometric contractions, duplex elements develop force independently until their sum reaches a predefined external afterload, imposed on the duplex. During the subsequent isotonic phase of duplex contraction (if any), duplex element interaction is governed by Eq. 2, where changes of the virtual element length must be equal to changes of the biological sample
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Attaining the control conditions identified in Eqs. 1 and 2 allows the hybrid combination to mimic collective behavioral aspects of mechanically interacting muscle segments in heterogeneous myocardium.
Recurrent control strategy.
IN-SERIES DUPLEX CONTROL.
To explain the operation of the control algorithm in more detail, let us consider the recurrent control strategy for both elements of an in-series hybrid duplex. In isometric conditions, total duplex length is fixed to Ld, and the forces developed by duplex elements are matched by changing their individual lengths (by matching amounts, but in opposite directions) to maintain their sum constant and equal to Ld (in accordance with Eq. 1). Let us use tk to denote a discrete time point of duplex control with a time step h, where h = tk + 1 – tk. The following events occur during each interval between tk and tk + 1 (where numbers used to identify events are 1–4 in Fig. 2; see also a pseudocode of the computational cycle in the APPENDIX).
At t = tk, signals of force [Fmus(tk)] and length [Lmus(tk)] of the biological muscle are sampled and transferred to the control routine.
Digitized values of Fmus(tk) and Lmus(tk), together with the virtual muscle force and length for tk [Fmod(tk) and Lmod(tk)], which were calculated during a previous interval [tk – 1,tk], are processed within the controller block of the computer program, where the control errors are calculated as follows: 
1 = Fmus(tk) – Fmod(tk) and 2 = Ld – Lmod(tk) – Lmus(tk). The controller output is as follows. The load on the virtual muscle during the interval [tk,tk + 1] is adjusted as Fmod(tk + s) = Fmod(tk) + K11 (0 
s h). The command signal for muscle length during the interval [tk,tk + 1] is set as Lmus(tk + s) = Lmus(tk) + K22. Control parameters K1 and K2 are specified below.
The control signal Fmod(tk + s) is applied to the model computation unit, where an integration step of the model is calculated using an Euler integrator (for computational procedure see the APPENDIX). Here, Lmod(tk + 1) is calculated as the procedure output. Lmod(tk + 1) and Fmod(tk + 1) = Fmod(tk + h), as well as other model state variables, are stored for further use during the next control cycle.
The control signal Lmus(tk + s) is transmitted to the servomotor to adjust muscle length before the next sampling cycle.
The routine is then repeated at tk + 1, where the actual values of force [Fmus(tk + 1)] corresponding to length [Lmus(tk + 1)] of the real muscle are recorded, and so on.
The parameters K1 and K2 are analogous to "gain" parameters used in conventional proportional controllers. They allow one to account for the time course of the original signal and the discrepancy in the connection conditions. The choice of specific values for these coefficients is based on preliminary numerical experiments with two virtual muscles, the interaction of which was simulated using the above-described algorithm. K1 and K2 were fitted to ensure small differences between the results obtained using the above-described algorithm and a numerical solution of the virtual duplex model. K1 and K2 values depend on biological muscle stiffness and should be decreased with increasing stiffness (in this case, the discrepancy in control conditions may increase). Special procedures using gain coefficients similar to those of standard proportional-integral-derivative control systems are utilized to stabilize the system and minimize the errors. In particular, the control signals [Lmus(tk + s) and Fmod(tk + s)] can be predicted not only on the basis of instantaneous values of the control errors, but also on the basis of information from previous integration steps, which increase control accuracy. Furthermore, command signals may be defined not as discrete, piecewise-constant values (as described above), but via smooth extrapolation between subsequent start and end levels over the duration of the control step. We have successfully implemented such options, for example, by using the previous values of biological muscle force to adjust the current load signal to the virtual muscle during in-series connection. Here, we describe the simplest variant of the algorithm to illustrate the principle of signal exchange between interacting muscles in a more straightforward fashion.
Another key feature of the control procedure is its sensitivity to the selection of elements for length or load control. Numerical experiments on in-series virtual duplexes showed that, during isometric duplex contraction, the mechanically slower muscle would preferably be controlled by length and the faster muscle by load. Otherwise, recurrent control procedures may become unstable (error increase to indefinite). Intuitively, this can be explained as follows. It is clear that less-stable control is provided if the reference signal changes rapidly (because this may result in overcorrection of the control signal). Because at the same control step a force increment would be greater for the fast- than for the slow-contracting muscle, use of the fast-changing force as a reference signal to control the second muscle might cause a deterioration in control reliability of the pair. Thus a muscle with potentially slower force generation should be chosen to transmit its force signals to a faster counterpart. In turn, the slow-contracting muscle would be controlled by the length change recorded from the fast-contracting muscle, where an average rate of shortening/lengthening in the isometric mode of duplex contractions is several times smaller than the velocity of force increase/decrease. The model predictions were confirmed in experiments on the hybrid duplex system, where swapping the controlled variables for the faster and slower elements caused oscillations with increasing amplitude in the output signals followed by breakdown of the control procedure. Different combinations of controlled variables for hybrid duplex elements (e.g., load-length, length-load, load-load, or length-length) were successfully implemented (in addition to that described above). The particular, control pattern should be chosen depending not only on the basis of the individual properties (e.g., fast-slow) of the elements, but also on the basis of the mode of contraction (isometric or isotonic) and the element connection pattern (in series or in parallel) to optimize the control routine.
IN-PARALLEL DUPLEX CONTROL.
In contrast to in-series hybrid duplexes, the mechanical environment of the biological muscle in-parallel hybrid duplexes is determined by setting muscle load via a servomotor. This depends on the current force generated by the virtual muscle. Shortening of the biological muscle at a given load is recorded by a length transducer and imposed as a command length change on the virtual muscle. Any change in force developed by the latter is calculated and used to generate the updated command load signal imposed on the biological sample. Thus the difference of the algorithm for controlling parallel duplexes during isotonic contraction consists of 1) the purpose of control, which is to ensure equal shortening of elements (
Lmus = Lmod), and 2) the equality of the sum of element forces to the overall load applied to the duplex (Fd = Fmus + Fmod; see Eq. 2). In this case, control errors are determined as follows: 1 = Fd – Fmus(tk) – Fmod(tk) and 2 = Lmus(tk) – Lmod(tk). Because of the necessity to control the biological muscle via servomotor-applied loads, command signals are generated for real muscle load and virtual muscle length.
Technical Arrangement of Experiments
The hardware used to implement the hybrid duplex method is represented schematically in Fig. 1. The distinctive feature of our setup, compared with the conventional configurations of mechanographic installations, is the possibility of using two separate servomotors: a "load (F) motor" and a "length (L) motor." In particular, this approach is distinct from the setup we implemented for biological duplexes, composed of two myocardial preparations, where one length motor with negative feedback from the force transducers is used to control common shortening of coupled muscle pairs during afterloaded contractions (24, 26). The use of two motors allows us to swap easily different algorithms for controlling the length of a biological preparation via the L motor during isometric contractions of the hybrid duplex (where the rod of the F motor is firmly pressed to the rod of the L motor and, thus, follows its movement) or the load on the preparation via the F motor during afterloaded shortening. This facilitates muscle control and improves ease of operation and stability of control procedures. Therefore, we have opted for the more-complicated configuration of the setup as our general approach.
We designed the F motor to set muscle load using a small electrodynamic loudspeaker coil (0.1 W) and an optical length transducer consisting of two optoelectronic pairs. The transmissive lever system has been implemented to maintain constant interaction of the magnetic field with the current coil up to the maximum displacement of the lever. This allows us to obtain an almost linear dependence of the force on the control current (drift 5%). The F motor set is provided with clock jewels and long current-contact jaws to make the system highly compliant and, thus, suitable for very small load applications (lever end compliance 
50 m/N). The lever is made of bamboo (inertial mass 3 mg), maximum force output is 50 mN, maximum rod displacement is 2 mm, and peak frequency is 0.5 kHz. The mean square magnitude of signal noise of transducer displacement within the working range of frequencies (0–0.5 kHz) is 1 µm, nonlinearity is 5%, and temperature drift is 2 µm/K.
The L motor is based on a commercially available, higher-powered electrodynamic speaker (10 W). The electromechanical system of the L motor can develop a higher force output (up to 10 N) and is equipped with a very rigid suspension system. The proportional plus integroderivative analogous system implements feedback control of the current in the coil from an optical displacement sensor. This feedback loop supports the same high operating speeds (up to 0.5 kHz), while maximum displacement supported by the motor is 2 mm (nonlinearity 5%), and feedback-regulated compliance is 10–5 m/N. The noise of the length transducer in the frequency band of 0–0.5 kHz is 1 µm, and nonlinearity is 5%. The temperature shift of the transducer is <1 µm/K.
Load and length of the preparation are set by feeding a signal into the L or F motor controller via a digital-to-analog (D/A) converter. The coefficient of correspondence between the control signal and the load/rod travel is predetermined experimentally and recorded in the control program for signal calibration and conditioning.
For precision measurement of the force developed by a muscle preparation, we use a force transducer based on a silicone strain-gauge bridge (model C-03, METRAN-SENSOR, Ekaterinburg, Russian Federation) and a conventional measuring amplifier. The operating range is 50 mN, with a resonance frequency at 1 kHz, noise of 2 x 10–2 mN, and compliance of 0.2 mm/N.
The analog-to-digital (A/D) and D/A data conversions are performed via an ACL-8316/12 (Adlink Technology), with an A/D and D/A resolution of 16/12-bit, A/D conversion time of 10 µs, and D/A settling time of 6 µs.
The program for controlling the dynamic interaction between elements of the hybrid duplex and for computing the output signals of the virtual muscle is run on an IBM-compatible personal computer (at least Pentium III, 800 MHz) with Windows 2000 and the HyperKernel (NEMATRON, Ann Arbor, MI) real-time integrated environment. HyperKernel is implemented as an extension (subsystem) for Windows. It has a task scheduler, its own set of services, and its own kernel. HyperKernel and Windows 2000 are operated in turns at strictly determined time intervals that can be set between 25 and 250 µs.
The installation control software consists of two programs interacting through shared memory: one operates in the extension kernel (interchanging signals with equipment and computing the mathematical model), and the other is a Windows 2000 application that uses, among other resources, a real-time extension interface. To ensure that the requirements of a real-time system are met, a procedure for data exchange with the hardware and for computation of the mathematical model is called in the program kernel with the help of the system's real-time timer at a clock frequency of 10 kHz. The procedure effectively fulfills the following tasks (which are particularly critical for in-series hybrid combinations): 1) a signal for stimulating the real muscle is produced, 2) the forces and lengths of the real muscle are recorded, 3) the load on the virtual muscle is shaped (with coupling equations for the hybrid duplex taken into account), 4) the next step of computation of the mathematical model is performed, 5) the buffer memory for exchanging data with the user program is filled, 6) a new length of the preparation is developed, and 7) signals are transmitted to the hardware for real muscle control.
The program features a user-friendly interactive interface that allows the experimenter to vary model coefficients "on the fly," to switch between in-series and in-parallel muscle connections, to set initial conditions (preparation lengths and afterloads), to display and store data, and to perform other tasks. In addition to these features, the program allows control algorithms to be adjusted (e.g., types of algorithms and feedback coefficients).
Experimental Protocols
Biological muscle preparation.
Animals were treated according to the Principles of Laboratory Animal Care of the National Society for Medical Research and the National Institutes of Health Guide for the Care and Use of Laboratory Animals and experiments had full institutional approval. Animals (rabbits or rats) were instantaneously killed by cervical translocation, and hearts were quickly excised. Isolated hearts were placed in a solution containing (in mM) 120 NaCl, 4.7 KCl, 1.2 MgSO4, 25 NaHCO3, 2.5 CaCl2, and 5.5 glucose, buffered to pH 7.35 (Trizma, Sigma-Aldrich), and bubbled with carbogen (95% O2-5% CO2). 2,3-Butanedione monoxime (30 mM) was used to prevent myocardial damage when small muscle preparations were cut from large hearts (18). Papillary muscles or trabeculae were excised from the right ventricle within 5–10 min and placed in a temperature-controlled (30°C) bath. The preparations were 0.5 mm in diameter, and average length was 3 ± 0.5 mm. One end of the muscle strip was tied to the rod of the force transducer and the other end to the lever of the F motor. The lever of the F motor rests against the rod of the L motor (Fig. 1, inset). The preparations were stimulated with 1.5 threshold amplitude electrical pulses at 20 beats/min.
On reaching a steady state in mechanical activity of the preparation, the muscle length was systematically decreased to near slack length (Lmin), defined as the minimum length of muscle for which an active force can be recorded. After the initial length was measured, the preparation was stretched to working length (Lw), defined as the length at which passive tension does not exceed 10–15% of active isometric force. The preparation was allowed to reach steady state at Lw before experimental determination of the passive properties of the preparation commenced. Then stimulation was terminated, and preparation length was changed linearly from Lw to Lmin and returned to Lw (over 5 min to prevent a hysteresis loop). By approximating the "deformation-passive tension" relation, the program calculates the coefficients of the exponential part of the dependence of passive tension on muscle length. These coefficients guide parameter selection in the rheological equations of the virtual muscle to set the mechanical characteristics of the virtual muscle in a defined relation to those of the biological muscle.
Virtual muscle definition. A number of the mechanical parameters of the virtual muscle, such as simulated slack and working length, isometric force, and passive stiffness, were defined, depending on measurements of real muscle mechanical properties, by model coefficient identification. Force and length change signals are scaled with the help of calibration factors. By matching slack lengths of both duplex elements, the program establishes the number of serial sarcomeres in the virtual muscle, with the assumption that the length of an unstretched sarcomere in ventricular muscle is 1.78 µm (17). The interconvertible recalculation of sarcomere/muscle length to each other is necessary, because the length of a sarcomere is a model phase variable, and it is necessary to specify the length of each sarcomere in the virtual muscle in proportion to the working length of the real muscle.
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RESULTS |
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TOPABSTRACTHYBRID DUPLEX METHODRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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Within the framework of a hybrid duplex, it is necessary to compute virtual muscle behavior faster than in real time to allow sufficient time for the interaction with the biological sample (i.e., the model calculation during a control step must be shorter than the step duration). Even though an Euler integrator does not necessarily offer the highest numerical precision, it provides very fast computation of model equations (<30 µs per integration step) at sufficient accuracy. It was chosen after careful assessment of several integration methods (including a 4th-order Runge-Kutta method) revealing negligible (<0.2%) divergence of results at integration steps <100 µs. This integration error is small compared with background noise levels inherent to the technical setup.
We assessed the effects of control step duration on duplex control quality and found that oscillation amplitudes of force and length of duplex elements increase with increasing control step duration. This depends on the individual ratios of passive to active force of the biological preparation. Particularly, during isotonic contraction of parallel elements in a hybrid duplex, force oscillations were prominent when control step duration was increased (Fig. 3A). For the tested control steps, we found that the maximal amplitude of duplex force oscillations did not exceed 0.5, 1.5, and 3.5% of maximal isometric force at step lengths of 70, 100, and 200 µs, respectively (Fig. 3B). At the same time, maximal length oscillation amplitudes were 
0.1% of Lw (not shown). The damping time constant was <60 ms. The prevailing frequency of output force signal oscillations was 100 Hz, which is two orders below the frequency of input control signals, indicating that the output error is determined by noise sources external to the control circuit. On the basis of the analysis described above, we selected a control step of 100 µs for our hybrid setting, inasmuch as this allowed appropriate time for signal exchange and model calculations (including a "time reserve" for more complex model calculations) and small errors of control.
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Contractions of In-Series Duplexes
Before connection of a virtual and a biological muscle in series, it is necessary to match their passive tension to prevent sudden changes in element lengths on duplex formation (see Virtual muscle definition under Experimental Protocols).
In the isometric mode of contraction (Fig. 4, A and B), application of an electrical stimulus to the muscle preparation is followed by monitoring of the increasing isometric tension in both elements, which, together with length values, is fed into the control program. In accordance with the control algorithm, the forces of the real and virtual muscles are compared and, if different, aligned by adjustment of element lengths in opposite directions to maintain constant external length (Fig. 4B). The time course of overall duplex force development differs significantly from that in the individual muscles before their mechanical interaction, and duplex force is between the forces of individual elements in isolation (Fig. 4A). Duplex elements achieve their individual peak force at different lengths (Fig. 4B) because of opposite variation in their lengths during mechanical interaction.
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Contractions of Parallel Duplexes
The isometric phase of afterloaded contractions in parallel hybrid duplexes takes place when the load servomotor lever is pressed to the length motor lever (using the F motor's maximum force, 50 mN). Both muscles therefore contract at a preset and constant length. In each control step, muscle force is sampled, the force of the virtual muscle is calculated, and the two are summed to calculate overall isometric force of the duplex. For isotonic contractions (Fig. 4, C and D), duplex afterload is the sum of active isometric forces of biological and virtual muscles, and the preload is equal to the sum of the passive forces generated by both duplex elements.
During the isometric phase of afterloaded duplex contractions, muscles are allowed to generate isometric forces until their sum reaches the predefined afterload level. Then the duplex control mode is switched to the isotonic mode, where the lever of the load servomotor is freed from the restriction imposed by the rod of the length servomotor, thus allowing the muscle to shorten. The biological muscle length signal is used as model input (see IN-PARALLEL DUPLEX CONTROL under Computer-Based Control System). Duplex elements contract auxotonically, shortening under a common constant load imposed on the duplex (Fig. 4D). On reaching end-systolic length, the duplex begins to relengthen until the working length (from which shortening began) is reached. Then the lever of the length servomotor acts again to restrict force lever motion, and relaxation proceeds isometrically (Fig. 4D).
The experimental recording in Fig. 4D shows two curves illustrating the development of forces by the biological and the virtual muscle in the hybrid duplex. Auxotonic contractions of duplex elements have a polyphasic character, caused by their dynamic interaction. The time of peak velocity of duplex shortening under a given load (Fd) is highlighted in Fig. 4D by a vertical line. The trajectory of muscle shortening during duplex contraction is significantly different from that obtained when individual muscles contract under the same relative afterload, but in the absence of mechanical interaction (Fig. 4C).
The use of two different servomotors allows exposure of a duplex to more physiological loading conditions (12, 33, 42), which mimic certain aspects of the mechanical environment experienced by individual muscle segments in the ventricular wall. The isometric phase of tension development by duplex elements during afterloaded contractions, for example, may be compared with the isovolumic phase of ventricular contraction. Duplex shortening under a constant (or increasing, yet predetermined, load) may be taken to correspond to ventricular ejection. This can be followed by a phase of isometric muscle relaxation, imitating the isovolumic relaxation of the heart. Finally, passive lengthening to the original starting length may be programmed to occur over a time course and with the dynamics typical of fast ventricular filling, slow filling, and, finally, additional filling during atrial systole, as in an intact ventricle.
In this "physiological mode of duplex contraction," after reaching end-systolic dimensions, the length servomotor is used to limit lever movement of the load servomotor, thus preventing isotonic elongation of the myocardial preparation. Thus the control mode is switched from isotonic to isometric, and the elements of the duplex relax at constant lengths. The forces and lengths of the real and virtual muscles are measured during the contraction cycle, and the dynamics of intracellular processes computed in the model (e.g., Ca2+ concentration and formation of Ca2+-TnC complexes) are stored for later analysis.
Data Processing
To assess the effects of mechanical interaction on the mechanical function of duplex elements, we compared a range of characteristic properties, such as the force-velocity and length-force relations, work, power, and time constants of contraction and relaxation, all of which were obtained in individual muscles before and after mechanical coupling.
The force-velocity relation is widely used in muscle physiology to characterize the peak shortening velocity developed by a muscle, depending on the applied load. In the parallel duplex setting, we developed a routine to assess this relation, which is described in detail elsewhere (24, 39). By comparing the forces of each muscle when overall duplex shortening reaches its peak velocity at any given afterload (Fig. 4D), we determined that there are opposite shifts in force development in each individual element, as well as in their relative contribution to overall force development, after duplex formation. The force-velocity relation of individual muscles is distinctly affected by the mechanical interaction in heterogeneous systems, as shown in Fig. 5A: the contraction curves for duplex elements are shifted relative to the force-velocity behavior of duplex elements in isolation.
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Another important characteristic of cardiac contractility is the dependence of isometric muscle force on the length of a preparation (equivalent to the Frank-Starling law of the heart). For the elements of a serial hybrid duplex, this curve may be obtained by determining the length of each element at the peak of duplex force generation, obtained at various fixed lengths of the pair. At each duplex length, a steady state must be reached before measurements are obtained. Comparison of the length-force relations for the elements of a serial hybrid duplex in isolation and during their mechanical interaction showed a shift in the slope of the length-force relation (Fig. 5B).
Using the above data, one can calculate several derivative characteristics of myocardial mechanical activity, such as work, power, or time constants of activation and relaxation during contraction, in isolation and in a duplex. The virtual muscle in a hybrid duplex is represented by a mathematical model that describes the kinetics of intracellular processes underlying contraction. This provides access to investigation of a host of subcellular parameters, here during interaction with the real muscle, by comparing findings in isolated and connected duplex elements (Fig. 6).
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DISCUSSION |
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The hybrid duplex we developed has conceptual analogies to the electrotonic cell-coupling approach of Joyner et al. (14), who used real-time electrical interaction between a real cell and a virtual model. The focus of the studies presented here is the mechanical interaction of cardiac muscle elements and their effects on mechanical and electrical functions of the interacting elements. The two approaches require principally different methods, models, apparatus, and algorithms, but they are clearly compatible, and it is interesting to contemplate the possibilities of a combined electromechanical coupling system (although this is beyond the scope of this study).
In the hybrid duplex method presented here, a real myocardial preparation interacts mechanically, and in real time, with a virtual muscle. It is important that the virtual partner is represented by a mathematical model simulating a sufficiently broad range of essential biomechanical characteristics and reactions in cardiac muscle (24, 39). Another important facet of the model is that it must be based on, and verified against, experimentally established molecular and cellular behavior involved in contraction. If these conditions are met, the hybrid duplex allows exposure of the biological preparation to a mechanical environment that mimics essential aspects of in situ cardiac mechanics and investigation of the molecular and cellular mechanisms involved in this feedback-driven control system by assessment of virtual muscle behavior. This is relevant, in particular, because some of these processes may not be easily (or not at all) accessible in biological preparations. By modulating, for instance, the kinetics of Ca2+-TnC complex formation in the virtual muscle, one can conduct experiments on a range of hybrid duplex scenarios to guide hypothesis formation for subsequent experimental validation.
Another interesting facet of the hybrid duplex is that one and the same biological preparation can be exposed to interaction with a large variety of models mimicking different aspects of cardiac heterogeneity and/or disease states. Such combinations of normal and pathologically changed segments in the ventricular wall are frequently encountered in cardiac pathology, for example, in the case of local ischemia or hypertrophy of the myocardium (38).
It is useful to compare the hybrid duplex method with the other configurations of the duplex method (26). Certainly, a duplex consisting of two natural muscles is the most physiological model of the interaction of inhomogeneous myocardium. However, duplexes of this type feature several limitations. Thus it is practically impossible to obtain a homogeneous duplex consisting of two mechanically identical myocardial elements. Homogeneous duplex experiments are, however, essential for understanding the effects of heterogeneity. In experiments with heterogeneous virtual duplexes in series, we found that the effects of delays in activation timing in an inhomogeneous duplex depended crucially on whether the fast- or slow-contracting element was activated first. This delay required element heterogeneity for optimization of duplex mechanical performance. In contrast, homogeneous duplex activity would always be impeded by any activation delay, highlighting the physiological relevance of matching cardiac tissue heterogeneity to the activation sequence (26, 40).
Another shortcoming of biological duplexes is the fact that all muscle preparations differ in parameters such as initial length, time to peak of isometric contraction, peak force, rate of tension development, rate of relaxation, stiffness, and the parameters described by the force-velocity and length-force relations. Because the combinations of element properties featured in biological duplexes are, essentially, random, it is difficult to assess the statistical significance of any particular observation.
Nonetheless and despite the above-mentioned reservations, biological duplexes serve several important functions. They can provide new insight into the effects of cardiac heterogeneity that may subsequently be "dissected" in more detail using the hybrid duplex approach. Thus, in our experiments using heterogeneous biological duplexes consisting of a fast- and a slow-contracting element, we observed reproducible biomechanical effects that were identified to be related to the sequence of activation of slow- and fast-contracting duplex elements (24, 26, 39). Underlying mechanisms were subsequently investigated in virtual duplex models and are now studied using a hybrid approach. Also, biological duplexes can serve as a test bed for assessment and verification of hybrid duplex-driven hypotheses and conclusions. Thus it would seem that the combination of virtual, hybrid, and biological duplexes offers advantages that may not be derived from any individual approach in isolation.
The hybrid duplex combines some of the advantages of biological and virtual duplexes. Thus, in a hybrid duplex, differences in a number of parameters of the biological and virtual objects can be eliminated by selection of model parameters to match, for example, stiffness, length, and forces of the elements. This makes it possible to assess the effects of asynchronous activation in isolation (i.e., not confounded by additional mechanosensitive pathways) (19). Only in a hybrid duplex is it possible to study the response of the real muscle to a "sparring partner," the mechanical identity of which is open to user intervention on the fly. This enables one to explore separately the influence of individual factors on mechanical interaction between heterogeneous duplex elements. This is important, for example, to the understanding of mechanical interaction between subepi- and subendocardial layers of myocardium, which contract asynchronously, have different resting lengths and stiffness, and shorten to a different degree during systole (4, 7). Moreover, many pathologies give rise to severe changes in regional cardiac mechanical properties, and the relevance of such differences and dyssynchronies is only starting to emerge.
The duplex approach described here can be advanced in several ways. Myocardial preparations, even small trabeculae, are heterogeneous in themselves (5). This heterogeneity may be partially assessed using laser diffraction techniques to study biological element structural characteristics (e.g., sarcomere length) in more detail. Our experimental setup is open for such addition. Alternatively, controlled heterogeneity may be introduced into the model description of the virtual element. However, even at the present stage, our approach allows one to assess heterogeneity effects in larger-scale interaction between heterogeneous muscle segments and effects of intramuscular (i.e., intercellular or even intersarcomere) heterogeneity. Techniques to address mechanical interactions at the level of single cells (e.g., using computer-controlled carbon fibers to subject myocytes to predetermined regimens of mechanical activity depending on the behavior of interacting virtual cells) are emerging.
The hybrid duplex approach can be used to study the effects of mechanical interactions not only on mechanical behavior, but also on Ca2+ kinetics and electrophysiological characteristics. This requires corresponding experimental methods to measure Ca2+ transients and action potentials in the biological sample and/or to advance the models used to describe the virtual muscle by including more detailed electrophysiological behavior. A suitable model of cardiac electromechanical activity is available (40, 41) and has been implemented in hybrid settings to examine the principal possibility of using complex integrative models within the constraints of a real-time setup.
We combined a papillary muscle of guinea pig right ventricle with a virtual sample where the mechanical parameters were fitted to simulate guinea pig-like contractions while the electrical characteristics where typical of guinea pig ventricular cells (Fig. 7). This hybrid duplex was used to verify the feasibility of such an extended hybrid duplex approach. From a computational point of view, the combined model is much more time consuming than the mechanics-only model because of the increased dimension of the system and more complex calculations. Moreover, the model system is stiffer and needs a smaller integration step than that used to integrate the mechanical model. In particular, in hybrid duplex settings, we had to use an integration step equal to at least one-half of the control step (h/2), so two integration steps were calculated during each control step to obtain the model solution. Nevertheless, after some optimization of computational procedures, we succeeded in compiling all the processes to control the hybrid duplex in real time. Our pilot experiments showed a significant change in action potential shape and duration in the virtual muscle because of mechanical interaction with the biological preparation (Fig. 7), thus revealing a contribution of mechanoelectrical feedback to heterogeneous myocardium performance.
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APPENDIX |
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TOPABSTRACTHYBRID DUPLEX METHODRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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ModelCalcEuler(t, Load)
Euler integrator of model ordinary differential equations. New state-of-phase variables are calculated as follows: X(t + h) = X(t) + dX(t), where X is a phase vector and dX = f(t,X) * h is an increment vector calculated using the right side of ordinary differential equation X' = f(t,X). One integration step h of the system is calculated using phase variables stored in a global array after previous procedure reference (subroutine call). Load imposed on the virtual muscle is an input parameter (Load) used on the right side of the model equation to calculate virtual muscle length (Lmod). New values of state variables are stored at the same array at procedure exit. Calculation of the most key state variables [e.g., membrane potential (Em), concentrations of free intracellular Ca2+ and Ca2+-TnC complexes (CaTnC), force-generating cross-bridge fraction (N), and sarcomere length (Ls)] of the model system is demonstrated below.
Electrical part of the model.
Membrane potential increment as a sum of ionic currents depending on current membrane potential, concentrations of ions, and probabilities of ionic channels to be in the open state are calculated in the electrical part of the model
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Chemical part of the model.
Ca2+ kinetics in cytosol and sarcoplasmic reticulum are calculated in the chemical part of the model to accomplish feedback between mechanical and chemical variables (see dCaTnC)
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Mechanical part of the model.
Force-generating cross-bridge increment, sarcomere length, and muscle length were calculated as a model output
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Updating state variables.
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SerialDuplexControl(t)
At t = tk, signals of the force and length of the biological muscle (Fmus and Lmus) are sampled and digitized
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Calculation of corrected control signals for load on the virtual muscle and length of the biological preparation (Fmod corr and Lmus corr) is based on measured values of Fmus, Lmus, and calculated values of virtual muscle force and length (Fmod and Lmod) that stored as global variables during the previous procedure
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Model calculation.
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The control signal Lmus corr is transmitted to the muscle length servomotor displacement before the next sample is taken
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GRANTS |
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ACKNOWLEDGMENTS |
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FOOTNOTES |
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The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
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REFERENCES |
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TOPABSTRACTHYBRID DUPLEX METHODRESULTSDISCUSSIONAPPENDIXGRANTSREFERENCES |
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