Model to describe the degree of twitch potentiation during neuromuscular monitoring

D. J. Eleveld*,1, A. F. Kopman2, J. H. Proost1 and J. M. K. H. Wierda1

1 Research Group for Experimental Anesthesiology and Clinical Pharmacology, PO Box 30001, 9700 RB Groningen, The Netherlands. 2 Department of Anesthesiology, St Vincent’s Hospital and Medical Center, 170 West 12th Street, New York, NY 10011, USA

*Corresponding author. E-mail: d.j.eleveld@anest.azg.nl

Accepted for publication: October 6, 2003


    Abstract
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Background. Neuromuscular block is estimated by comparing the evoked peak twitch with a control value measured in the absence of neuromuscular block. In practice, this control value is often difficult to determine because repeated motor nerve stimulation enhances the evoked mechanical response of the corresponding muscle, resulting in an increased twitch response. This is known as twitch potentiation or the staircase phenomenon. It is probably the result of myosin light chain phosphorylation creating an increased twitch force for a given amount of Ca2+ released at each action potential. Modelling of potentiation may improve studies of neuromuscular blocking agents using mechanomyography or accelerometry.

Methods. We used one- and two-exponential models to describe the degree of myosin light chain phosphorylation and associated twitch potentiation. These models were fitted to accelerographic twitch force measurements for various stimulation patterns and frequencies used in neuromuscular monitoring.

Results. Fitting a two-exponential model to twitch data for various stimulation rates and patterns provides better prediction than a one-exponential model. A one-exponential model performs poorly when the stimulation rate varies during measurement.

Conclusions. We conclude that a two-exponential model can predict the degree of twitch potentiation for the stimulation patterns and frequencies tested more accurately than a one-exponential model. However, if only one stimulation frequency is used, a one-exponential model can provide good accuracy. We illustrate that such a potentiation model can improve the ability of pharmacodynamic-pharmacokinetic neuromuscular block models to predict twitch response in the presence of a neuromuscular blocking agent.

Br J Anaesth 2004; 92: 373–80

Keywords: monitoring, neuromuscular function; muscle, contractility; muscle, skeletal; pharmacodynamics, models


    Introduction
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Neuromuscular block is often estimated by comparing an evoked mechanomyographic or accelerographic twitch response with a control value measured in the absence of neuromuscular blocking agents. The control twitch value is assumed to be the ‘maximal’ twitch, and is usually determined after induction of anaesthesia, before administration of a neuromuscular blocking agent. The degree of neuromuscular block is expressed as one minus the ratio of an evoked twitch to the control value.

As all neuromuscular block measurements are related to the control value, it is important that it be determined accurately. However, this can be difficult because the repeated motor nerve stimulation necessary for neuromuscular monitoring enhances the evoked mechanical response of the corresponding muscle, resulting in an increased twitch response.1 2 This is known as twitch potentiation or the staircase phenomenon. When subjected to repetitive muscle stimulation, the evoked twitch response increases to a plateau. A change in stimulation frequency alters the twitch response,3 and the twitch height gradually approaches a new plateau. Plateau height increases for increasing stimulation frequencies. With the stimulation patterns and frequencies commonly used in neuromuscular monitoring, potentiation can cause up to a 100% increase in twitch response, and can take up to 25 min to stabilize to within 5% of its plateau value.4

Anaesthetists should be familiar with twitch potentiation because it can lead to incorrect neuromuscular recovery estimations when monitoring with mechanomyography or accelerometry. A common problem is that twitch responses during and after recovery from a dose of neuromuscular blocking agent can be larger than twitch responses before administration of the neuromuscular blocking agent (pre-neuromuscular block). Anaesthetists unfamiliar with potentiation might assume that when the twitch response reaches pre-block levels, recovery is complete. However, the twitch response may not yet have reached its maximum levels. Twitch potentiation is also troublesome for neuromuscular research because accepted pharmacodynamic models, such as those described by Hull and colleagues5 and Sheiner and colleagues,6 assume that pre- and post-neuromuscular block twitch responses are the same.

We developed a model accounting for twitch potentiation, allowing the prediction of twitch response for different stimulation frequencies and patterns.


    Methods
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
The staircase effect is well documented, but the underlying mechanism7 8 is not well known. Calcium ions released from the sarcoplasmic reticulum in response to an action potential bind to calmodulin. Ca2+/calmodulin binds to myosin light chain kinase and the enzyme is converted from an inactive to an active form. The activated kinase phosphorylates a specific serine residue in the amino-terminal portion of the regulatory light chain of myosin, leading to an increase in the rate by which myosin cross bridges move into a force-producing state. This creates an increased twitch force for a given amount of Ca2+ released at each action potential. This mechanism is activated in vertebrate striated muscle and therefore applicable to current neuromuscular monitoring techniques used in anaesthesia. Force production is increased per evoked twitch, both for mechanomyography and accelerometry, although not necessarily to the same degree.

We used exponential models to predict the degree of twitch potentiation on accelerographic twitch data for various stimulation frequencies and patterns. As there are several steps between Ca2+ release from evoked twitches and increased twitch force production, and because these steps are not observable during standard neuromuscular monitoring, we used a simplified exponential model. The degree of twitch potentiation (P1) increases as the result of Ca2+ release from an evoked twitch because of Ca2+/calmodulin complex formation and subsequent myosin light chain phosphorylation. Myosin light chain phosphorylation decays in the inter-stimulus period, and therefore so does the degree of twitch potentiation.

The equation describing twitch potentiation decay for a one-exponential model is:

{delta}P1/{delta}t = –k10·P1

where k10 is a rate constant describing the decay of potentiation in the time period between stimulations.

A two-exponential potentiation model introduces a second exponential term (P2). The equations for decay of potentiation in a two-exponential model are:

{delta}P1/{delta}t = k21·P2–(k12+k10P1

{delta}P2/{delta}t = k12·P1–k21·P2

where k10, k12 and k21 are rate constants describing the decay of potentiation in the time period between stimulations.

After a twitch takes place, the degree of potentiation increases by a constant (r) related to the amount of calcium release per twitch. For the purposes of this model, we assume that potentiation takes place immediately after an evoked twitch. The underlying physiological process is not instantaneous; it is likely to occur much faster than the stimulation intervals commonly used in neuromuscular monitoring, however.

During neuromuscular block some, or all, of the muscles motor fibres are blocked, and do not release calcium when stimulated. For an entire muscle, the total amount of calcium released per twitch is reduced and therefore the increase in potentiation per stimulus is similarly reduced. We defined the underlying twitch response (utr) as a relative measure of calcium release per stimulus, and the output of the pharmacodynamic model. No neuromuscular block corresponds with a utr of 1 and complete neuromuscular block corresponds with utr of 0. This factor ensures that the degree of twitch potentiation will decay during complete neuromuscular block as a result of calcium release suppression.

Increase in P1 from stimulation = r·utr

As the degree of potentiation is affected by evoked twitches, only the very first evoked twitch of neuromuscular monitoring can be assumed to be unpotentiated, and can therefore be a measurable control value. The evoked potentiated twitch response is calculated relative to this value. In our models, we also included a factor Kscale to compensate for inaccuracies in determining this control value. The use of Kscale also ensures that the model fitting process does not require the researcher to ‘choose’ a normalization value thus making the model fitting process more objective.

Twitch (%) = 100·Kscale·(1+P1)·utr

The train-of-four stimulation pattern consists of four separate twitches and was modelled as such with an increase in P1 from stimulation for each twitch and decay of potentiation in the short period between twitches.

Parameter estimation
We estimated model parameters for 10 data sets of accelerographic data published by one of the authors (AK) to demonstrate the magnitude of the staircase phenomenon in man using various stimulation patterns and frequencies.4 This study collected supramaximal thumb accelerographic data after ulnar nerve stimulation during anaesthesia. The Human Subject Review Committee (St Vincent’s Hospital and Medical Center, New York, NY, USA) approved the procedure and informed consent was obtained. Anaesthesia was induced with alfentanil (15–40 µg kg–1) plus propofol (2.0–2.5 mg kg–1). Laryngeal mask placement or tracheal intubation was accomplished without the use of neuromuscular blocking agents. Anaesthesia was maintained with nitrous oxide (65–75% inspired), propofol (50–75 µg kg–1 min–1) and intermittent doses of fentanyl if required. Ventilation was controlled, and end-tidal carbon dioxide tension (PCO2) was maintained between 4.5 and 5.3 kPa. After induction of anaesthesia, the evoked supramaximal response (pulse width 200 µs) of the adductor pollicis was recorded in three groups each of 10 patients, each group having a different stimulation pattern. Evoked twitch data were measured by the TOF-Guard® acceleromyograph.

In Kopman’s study,4 only one of the groups received multiple stimulation frequencies and patterns and data from this group was used in the present study. The other groups only received one stimulation frequency and were not suitable for our purposes of modelling potentiation over various stimulation rates and patterns. The group used consisted of single twitch (ST) stimulation every second (ST1) for at least 10 min followed by train-of-four (TOF) stimulation every 15 s (TOF15) for at least 10 min. Some data sets also contained one or two additional periods of other stimulation rates, usually longer than 10 min. These included periods of ST stimulation every 10 s (ST10) and TOF stimulation every 5 min (TOF300).

As the time periods have different stimulation frequencies and therefore a different number of stimulations, squared prediction error (difference between measured and predicted twitch responses) was weighted by the time interval between stimulations. This weighting is necessary to avoid the great amount of information in the 1 Hz sampling periods from overwhelming the information in the lower frequency sampling periods and resulting in very poor prediction accuracy for low frequency sampling periods. Parameter estimations were done using a maximum likelihood method and a modified Powell’s method for optimization. As the fastest stimulation frequency (1 Hz) is much lower than the twitch fusion frequency, we assumed that all twitches are independent. Transducer and amplifier gains remained constant throughout each experiment and therefore we assumed a constant measurement error variance. These data sets are obtained in the absence of neuromuscular block and therefore the utr parameter is 1 and constant. We compared the goodness of fit of the one- and two-exponential model fittings by calculating the Akaike Information Criteria (AIC) for each of the model fits.

We fitted one- and two-exponential potentiation models to the twitch data. We also fitted a one-exponential model with separate r and k10 values for each stimulation frequency/pattern. When using this model, no twitch weighting was used because separate r and k10 values for each stimulation frequency/pattern prevent the prediction error in the 1 Hz data sections from overwhelming the prediction error in the lower frequency data sections.

To illustrate the effect that a potentiation model can have on neuromuscular block modelling, we fitted a pharmacokinetic–pharmacodynamic (PK–PD) single effect compartment model with a sigmoidal concentration response (pharmacodynamic) relationship,6 with and without a potentiation model to mechanomyographic neuromuscular block data on file (single subject) in our department. The patient was anaesthetized with propofol and sufentanil and a mixture of nitrous oxide/oxygen. Mechanomyographical monitoring of the adductor pollicis was applied. The stimulation mode was ST10 with a pulse width of 200 µs. Rocuronium was infused at a rate of 116.7 µg kg–1 min–1 for 3.7 min and was discontinued at 70% twitch depression. Arterial blood samples were obtained during onset and offset of the block and for 4 h after rocuronium administration. Plasma concentrations were determined by high performance liquid chromatography. When fitting using standard PK–PD modelling practice, twitches measured during twitch stabilization (before the start of infusion) are not included in the model fitting. Twitches were normalized to the post-recovery value.

When combining a PK–PD model and a potentiation model, the pharmacodynamic ‘response’ is considered to be the utr, relating relative calcium release per evoked twitch. In the absence of muscle relaxation utr has a value of 1.0, but the presence of neuromuscular block lowers this value according to a sigmoidal concentration response (pharmacodynamic) relationship. The potentiation model transforms utr to evoked twitch force. We used a one-exponential potentiation model because the data contain only one stimulation pattern (ST10). Twitches measured during twitch stabilization (before the start of the infusion) were included in the combined model fitting.


    Results
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Table 1 shows the results of the one- and two-exponential model fittings with the mean and standard deviation (SD) of the fitted model parameters.


View this table:
[in this window]
[in a new window]
 
Table 1 Fitted potentiation model parameters (n=10)
 
Figure 1 shows measured and predicted twitch values for a representative fitting of a one- and a two-exponential potentiation model (Subject 2). The one-exponential model shows a poor ability to predict twitch potentiation for the four different stimulation patterns and frequencies used. The patterns were: ST1 (12 min); followed by TOF15 (22 min); followed by ST10 (15 min); and TOF300 (15 min).



View larger version (16K):
[in this window]
[in a new window]
 
Fig 1 Measured and predicted twitch data for one-exponential and two-exponential potentiation model (Subject 2). Four stimulation patterns are used: ST1 and ST10 (single twitch at intervals of 1 s and 10 s), and TOF15 and TOF300 (train-of-four at intervals of 15 s and 300 s, respectively). For the one-exponential model, prediction accuracy is poor when stimulation rate changes from ST1 to TOF15.

 
The one-exponential model did not predict twitch response with good accuracy. In particular, the rapid decrease in twitch response when changing stimulation pattern from ST1 to TOF15 is poorly modelled. The data set shown in Figure 1 is typical of the one-exponential model fits, in that twitch response was consistently overestimated in this region. The two-exponential model predicts twitch response more accurately.

Figure 2 shows the worst case (largest average squared residual) fit for the two-exponential model (Subject 1). This subject had the standard stimulation periods followed by two supplemental stimulation periods consisting of 15 min TOF300 stimulation, followed by 13 min ST10 stimulation. This fit appears unusual because the steady state potentiation level for ST10 stimulation appears to be very similar to the steady state potentiation level for TOF15 stimulation and this was the only data set to exhibit this characteristic. In all other data sets with similar stimulation patterns, ST10 stimulation had a lower steady state potentiation level than TOF15 stimulation. This is clearly seen in the data for Subject 2 (Fig. 1). A slight change in thumb position could have occurred in Subject 1 (Fig. 2) between 20 and 35 min. This resulted in an unusual value for the potentiation decay constant, which approached zero for this subject when fitting a two-exponential model.



View larger version (12K):
[in this window]
[in a new window]
 
Fig 2 Worst case (largest average residual) of two-exponential potentiation model fitting (Subject 1).

 
To investigate whether a one-exponential model may be reasonably accurate if a single stimulation pattern is applied at a constant frequency, we performed model fitting using separate r and k10 values for each stimulation frequency/pattern with a one-exponential model. Figure 3 shows representative observed and predicted twitch values. Here, prediction error is low.



View larger version (12K):
[in this window]
[in a new window]
 
Fig 3 Measured and predicted twitch data for one-exponential potentiation model (Subject 2) using separate r and k10 values for each stimulation frequency/pattern. Prediction accuracy is good for all stimulation patterns/frequencies.

 
AIC values were calculated for all model fittings. The AIC values for the two-exponential models are much lower (mean 4253, SD 3727) than for the corresponding one-exponential models. This indicates that the two-exponential model calculates twitch enhancement for various stimulation patterns and frequencies more accurately than the one-exponential potentiation model. Allowing for separate r and k10 values for each stimulation frequency/pattern and using a one-exponential model further reduces AIC values (mean 900, SD 626).

Figure 4 shows the observed and predicted time course of rocuronium using standard PK–PD modelling practice compared with a combined PK–PD-potentiation model. The standard PK–PD model assumes equal pre- and post-neuromuscular block twitch values and this example shows that this may not be the case for measured data. The standard PK–PD model ignores twitches measured before neuromuscular blocking agent administration (during twitch stabilization) because the model cannot predict the changes in twitch response that occur. Prediction accuracy is poor in the pre-onset phase.



View larger version (15K):
[in this window]
[in a new window]
 
Fig 4 Measured and predicted neuromuscular block data using standard PK–PD model and a combined PK–PD-potentiation (one-exponential) model. Stimulation pattern was ST10. A fast infusion of rocuronium was started at time 0 and stopped 3.7 min later. The combined PK–PD-potentiation model predicts twitches before antagonist administration accurately while standard PK–PD modelling ignores these data.

 
The combined PK–PD-potentiation model allows accurate prediction of twitches measured during twitch stabilization (before neuromuscular blocking agent administration) and during the pre-onset phase. Twitch modelling accuracy is better than without a potentiation model especially before giving the bolus dose. The average residual is smaller than without a potentiation model even though a larger amount of data is predicted.

There are differences in the pharmacodynamic parameters fitted to the rocuronium data presented. For the data set presented, the combined PK–PD-potentiation model produces similar keo values (0.19 min–1), higher EC50 values (1740 µg litre–1 vs 1556 µg litre–1) and very slightly higher gamma values (3.12 vs 3.05). These differences should be viewed with caution until a more detailed investigation of the effect of PK–PD-potentiation modelling on pharmacodynamic parameters has been completed.


    Discussion
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
Our results indicate that a two-exponential model predicts twitch potentiation better than a one-exponential model for the stimulation frequencies and patterns tested here. The major shortcoming of the one-exponential potentiation model is that the rapid decreases in twitch response when changing stimulation pattern from ST1 to TOF15 are poorly modelled. A one-exponential model may still be accurate if a single stimulation pattern is applied at a constant frequency, as is frequently the case in current neuromuscular block monitoring practice.

One might expect some absolute maximum upper limit regardless of stimulation frequency to the degree of myosin light chain phosphorylation and therefore to the degree of twitch potentiation. We did not observe this effect in this experiment, which is likely because of the limited range of stimulation frequencies commonly used in neuromuscular monitoring.

If, during neuromuscular monitoring, neuromuscular block is initiated before the twitch response has stabilized to its stimulation frequency dependent plateau, then the post-block twitch response will not be the same as before onset of block. This makes neuromuscular modelling difficult because existing models assume that pre-neuromuscular block and post-recovery twitch responses are the same. When applying existing models to measured data, it becomes unclear which of the pre- or post-neuromuscular block twitch responses is the real ‘maximal’ twitch, and it shows how it is possible that these two ‘maximal’ twitches could be measurably different. Current neuromuscular studies9 address this problem by attempting to create a stable degree of potentiation before the administration of neuromuscular blocking agent to equalize the pre- and post-block twitch values. This is done with pre-block periods of repetitive stimulation and/or tetanic stimulation.4 10 The degree of potentiation is then assumed constant over the monitoring period. After individual data collection is completed, the investigator chooses a twitch control value, usually the post-recovery twitch response. These techniques do not always work perfectly and pre-block and post-recovery twitch responses are frequently quite different.

Tetanic stimulation has a higher frequency than is used for neuromuscular monitoring (usually ST10 or TOF15) and therefore can potentially create a greater degree of twitch potentiation. Stopping the tetanic stimulation after a short time (usually after 2 or 5 s) before maximum potentiation results in a lower degree of potentiation than a tetanic stimulation of longer duration would have achieved. With careful choice of tetanic stimulation duration and neuromuscular monitoring stimulation frequency, the average difference between the degree of potentiation evoked by these two different stimulation frequencies can be reduced. This technique is often used during twitch stabilization because it produces on average similar pre- and post-block twitch responses.4 10 This is cited as sufficient evidence that tetanic stimulation and 2 min of twitch stabilization is sufficient to obviate the need for prolonged twitch stabilization.10 However, it is important to note that only the average pre- and post-block twitch responses are similar and individuals still exhibit considerable differences in pre- and post-block twitch values. In fact, the variance of post-block to pre-block twitch ratio is unchanged whether or not tetanic stimulation is applied.9 Tetanic stimulation and 2 min of twitch stabilization reduce the apparent effects of potentiation for a group, but do not address the underlying problem of how pre- and post-block twitch values can differ and yet both be ‘maximal’.

The assumption that the degree of twitch potentiation is constant throughout the course of neuromuscular block is also probably incorrect. This is because, during neuromuscular block, calcium release from an action potential is suppressed through the effect of the neuromuscular blocking agent on the action of acetylcholine. This suppression of calcium release is equivalent to the absence of stimulation and without calcium release the degree of myosin light chain phosphorylation will decay, and therefore the degree of twitch potentiation will also decay. Potentiation during twitch stabilization must therefore decay during neuromuscular block. The degree of twitch potentiation is dependent on the evoked twitch response and not constant throughout the course of neuromuscular block. This is in contrast to assumptions made in current neuromuscular block modelling practice.

Current PK–PD neuromuscular block modelling techniques ignore twitches recorded during twitch stabilization because existing models are incapable of predicting the changes in twitch response that occur. The addition of a potentiation model to a PK–PD model as described here allows the degree of twitch potentiation to be estimated throughout the entire monitoring period, including during twitch stabilization. It also allows the post-supramaximal twitch to be used as a potentiation control value, instead of relying on a post-monitoring normalization technique to ‘choose’ a baseline value. Furthermore, it allows twitches recorded during stabilization to contribute to potentiation model fitting and thus possibly improve modelling accuracy.

We could have modelled the degree of potentiation using abstract compartmental models since they are mathematically equivalent to the exponential models discussed here and it remains possible to convert our model to compartmental form. This may allow interpretation of the possible underlying physical processes, but these interpretations should be treated with caution. One possible interpretation is that the first compartment corresponds to the Ca2+/calmodulin complex binding to myosin light chain kinase, and the second compartment would correspond to free Ca2+/calmodulin complex.

We illustrate that a potentiation model in conjunction with a PK–PD model may be useful to model changing degrees of potentiation during neuromuscular monitoring and thereby increase twitch response modelling accuracy and applicability. Further studies are necessary to determine the effect of the use of a potentiation model on pharmacodynamic parameter estimations.

The search for the ‘true maximal twitch’ is often considered when using current PK–PD modelling techniques. In our opinion, all pre-drug twitches and post-recovery twitches are ‘maximal’ twitches, but they differ in magnitude because of differing degrees of potentiation. The potentiation mechanism described here can be used to model the changes in the magnitude of the ‘maximal’ twitch response throughout neuromuscular monitoring.

We conclude that a two-exponential model can predict the degree of twitch potentiation for different stimulation patterns and frequencies more accurately than a one-exponential model. However, a one-exponential model may still be useful if the stimulation pattern is constant throughout the monitoring period.


    References
 Top
 Abstract
 Introduction
 Methods
 Results
 Discussion
 References
 
1 Krarup C. Enhancement and diminution of mechanical tension evoked by staircase and by tetanus in rat muscle. J Physiol 1981; 311: 355–72[ISI][Medline]

2 Ritchie JM, Wilkie DR. The effect of previous stimulation on the active state of muscle. J Physiol 1955; 130: 488–96[ISI][Medline]

3 VanSanten G, Fidler V, Wierda JMKH. Stabilization of twitch force during mechanomyography of the adductor pollicis muscle. J Clin Monit Comput 1998; 14: 457–63[CrossRef][ISI][Medline]

4 Kopman AF, Kunmar S, Klewicka MM, Neuman GG. The staircase phenomenon. Anesthesiology 2001; 95: 403–7[CrossRef][ISI][Medline]

5 Hull CJ, Van Beem HBH, McLeod K, Sibbald A, Watson MJ. A pharmacodynamic model for pancuronium. Br J Anaesth 1978; 50: 1113–23[Abstract]

6 Sheiner LB, Stanski DR, Vozeh S, Miller RD, Ham J. Simultaneous modelling of pharmacokinetics and pharmacodynamics: application to D-tubocurarine. Clin Pharmacol Ther 1979; 25: 358–71[ISI][Medline]

7 Sweeny HL, Bowman BF, Stull JT. Myosin light chain phosphorylation in vertebrate striated muscle: regulation and function. Am J Physiol Cell Physiol 1993; 264: C1085–95[Abstract/Free Full Text]

8 Klug GA, Botterman BR, Stull JT. The effect of low frequency stimulation on myosin light chain phosphorylation in skeletal muscle. J Biol Chem 1982; 257: 4688–90[Abstract/Free Full Text]

9 Viby-Mogensen J, Engbaek J, Eriksson LI, et al. Good clinical research practice (GCRP) in pharmacodynamic studies of neuromuscular blocking agents. Acta Anaesthesiol Scand 1996; 40: 59–74[ISI][Medline]

10 Lee GC, Lyengar S, Szenohradszky J, et al. Improving the design of muscle relaxant studies: Stabilization period and tetanic recruitment. Anesthesiology 1997; 86: 48–54[CrossRef][ISI][Medline]