In this study we modeled the quantitative relationship between Vim DBS stimulation parameters and the tremor response in essential tremor. Unlike our earlier patient-specific models (Kuncel et al., 2006
), this model can predict the tremor response to varying stimulation parameters: It incorporated a “competing processes” hypothesis, and made more accurate predictions, with fewer free model parameters, than a conventional model lacking that hypothesis. Our model accounted for 46% of the variance in the data, which is far from perfect: however our goal was not perfect prediction but, rather, to test the “competing processes” hypothesis. Unlike other models (Ushe et al., 2004
), our model accounts for both frequency and voltage-dependence, as well as inter-thalamus variation related to electrode position via the thalamus-specific parameter β1.
1. Competing Processes Model of DBS Effects
We found that, for therapeutically effective stimulation frequencies, the relation of tremor to stimulation voltage was U-shaped with an optimum voltage that differed between subjects and thalami. As voltage increased above the optimum, tremor suppression was decreased, and at higher voltages, stimulation aggravated tremor.
This is easily overlooked in clinical observations limited to patients with well-placed electrodes and moderate stimulation voltages. We detected it because we applied a quantitative analysis to voltage varied across a wide range, in patients with a wide range of electrode placements. We did not detect any sharp division into categories of well- and poorly-placed electrodes. Rather, the U-shaped tremor-voltage relationship was seen in all patients, with only quantitative differences according to electrode location.
We propose a ‘competing processes’ model of Vim DBS to account for tremor suppression and both types of tremor-aggravation. Under this model, DBS exerts a tremor-suppressing effect due to activation of a region near the electrode (presumably Vim nucleus, or, possibly, Vop or prelemniscal radiations), and a tremor-aggravating effect due to current spread to a more distant site (perhaps internal capsule). At voltages less than the optimal voltage Vopt, increasing voltage activates a larger volume of the tremor-suppressing region. At voltages greater than Vopt, increasing voltage results in current spread to the more distant tremor-aggravating region. The two processes have qualitatively similar dependence on frequency, but quantitative differences such that, at low frequency, tremor-aggravation dominates.
The competing processes model arose as a way to account for our observations while being economical in the number of assumptions; in fact, the model makes only one novel assumption (see Appendix
), namely, the notion that Vim DBS has simultaneous competing effects on tremor. We tested that hypothesis by incorporating it into our mathematical model of Vim DBS effects on tremor, then tested the model in two ways. First, we demonstrated that our model predicted Vim DBS effects on tremor more accurately, with fewer free parameters, than a multiple polynomial regression model. The point is not that our model was more accurate than the multiple regression model (although it was). The point is that building certain assumptions into our model increased its accuracy, without sacrificing parsimony, implying that the assumptions were correct. Second, we validated the model by fitting it to a subset of the data (low-frequency stimulation) and demonstrating that it could then accurately predict a separate subset of the data (high-frequency stimulation). The success of the model by both tests supports the competing processes hypothesis on which it was built.
The physical mechanisms of tremor-aggravation and -suppression is not directly addressed by our data. We have elsewhere proposed a mechanism for frequency-dependence of tremor-suppression (Grill et al., 2004
). One possible physical mechanism by which tremor-aggravation might occur is by producing uncomfortable side effects, such as paresthesias. Discomfort would then aggravate tremor because it was stressful or anxiogenic. In its simplest form, this hypothesis is incompatible with our data, because tremor aggravation was prominent at low stimulation frequencies, which did not produce uncomfortable paresthesias. This hypothesis could still apply, however, to high-frequency / high voltage stimulation, if we postulate a second, separate mechanism of tremor aggravation for low-frequency stimulation. When tremor increased with low-frequency stimulation, in agreement with others (Ohye and Narabayashi, 1979
; Bejjani et al., 2000
) we observed a jerky quality apparently different from “natural” tremor,. However, we saw the same with tremor-aggravation with high-frequency high-voltage stimulation; moreover we found that with stimulation-aggravated tremor, the power spectral peak did not broaden or shift () indicating that the movement was still a rhythmical oscillation at the same frequency. A separate mechanism of tremor aggravation by low-stimulation frequency could be incorporated into our model by splitting the tremor-aggravation term into two terms, one operative at low stimulation frequency, and the other at high frequency. At present, however, this seems a needless complication because the increased model complexity is not required to fit the data.
Another possible mechanism of tremor aggravation would be ‘driving’ the essential tremor oscillator. The amplitude of an oscillation can be increased by supplying energy to the oscillator at a susceptible frequency. For a linear oscillator, the forcing function must oscillate at the natural frequency of the oscillator, or a harmonic, or subharmonic thereof: for example, 2 Hz stimulation might aggravate a 4 Hz tremor, and 5 Hz stimulation a 5 Hz tremor, but not vice versa. In its simplest form, this hypothesis is incompatible with our data and that of Ushe et al., 2004
because the relationship of tremor to stimulation frequency is smooth, without sharp peaks and valleys corresponding to resonant frequencies. However, a nonlinear oscillator may exhibit more complex behavior: it is possible that a nonlinear essential tremor oscillator could exhibit a susceptibility to driving corresponding to the smooth frequency-dependence in our model ().
Tremor aggravation and suppression components of the model are dependent on stimulation frequency and voltage. Parameters used to generate these plots were taken from and (thalamus D).
We did not investigate the effects of stimulation on tremor frequency. Inspection of individual power spectra did not reveal a frequency difference between spontaneous and stimulation-aggravated tremor (e.g. ). However, our experiments were not designed to detect such an effect. Experiments concentrating on tremor-frequency effects might shed light on the mechanism by which activation of the tremor-aggravating region aggravates tremor.
2. Clinical Aspects
The present study was primarily intended to elucidate DBS mechanisms, rather than producing data of direct utility to clinicians. Our model could, in principle, be used to aid clinical stimulator adjustments: clinicians could test the effects of low-frequency stimulation, and use the results to predict the optimal voltage for high frequency stimulation, thereby avoiding extensive testing at high frequency. This might be advantageous in some patients, since low frequency stimulation is less apt to cause uncomfortable paresthesias. In practice, however, clinical Vim DBS adjustment for ET is rarely difficult enough to justify the weight of mathematical apparatus developed here.
We suggest that in practice the most important clinical implications derive from the qualitative shape of the response surface. We found that the frequency-voltage response surface was a ridge system (Box and Draper, 1987
) oriented approximately parallel to the frequency axis. This means that, in the part of the frequency-voltage regime where net tremor-suppression takes place, variations in voltage produced a much larger change in tremor suppression than variations in frequency. This suggests that adjustments should begin at a fixed frequency, with more attention to fine-tuning voltage than frequency. If tremor suppression at the optimal voltage is inadequate, further voltage increase is likely to be counterproductive. Increasing frequency may gain additional benefit, although we (Kuncel et al., 2006
) and others (Ushe et al., 2004
)(O'Suilleabhain et al., 2003
) have not found this to yield large improvements in tremor. Increased pulsewidth might also gain additional benefit, but we found the effect of pulsewidth to be quite small, in agreement with O'Suilleabhain et al., 2003
and our own earlier work (Kuncel et al., 2006
This study examined a population of tremor patients that was heterogeneous with respect to electrode placement and clinical outcome. The findings of tremor aggravation by low-frequency stimulation and a U-shaped relationship between tremor and voltage with high-frequency stimulation were consistent across all patients. Additionally, we detected a correlation of a patient-specific characteristic (electrode location) with a model parameter (beta1). The ability to detect correlation is greater when there is more variance in the independent variable, so the clinical heterogeneity helped us detect the correlation, and restricting the population to patients with optimal electrode placement would reduce the strength of the correlation. For the clinician faced with a more heterogeneous patient population, the correlation may be useful for clinical decision-making, e.g. knowing how much improvement in symptoms can be achieved by optimizing stimulation parameters helps to decide whether to reposition the electrode.
3. Limitations of the Model
Our model captures much of the variation in our data in a mathematically tractable way. However, it has limitations. First it is still not a complete model of physical processes. The values of parameters have been fit to achieve a match to the experimental data, not to directly measured physical quantities like distance, electrical impedance, etc. Nonetheless, the present model is a step towards the goal of a complete structural model. For example, we believe a physical model will have to incorporate two different sites where stimulation acts, at different distances from the electrode.
Second, our model does not address effects of varying the active contact on the electrode used for stimulation. We have conducted a separate series of experiments in which we varied this parameter (Kuncel et al., MS in preparation), however, in the experiements reported here, in order to keep the number of stimulator setting combinations to be tested within manageable limits, we conducted all experiements using the electrode contacts previously established, by the treating physician, to be the most effective.
Third, in order to predict Vopt for an individual thalamus, the model needs an estimate of β1 for that thalamus. This requires, at a minimum, tremor measurements made at low stimulation frequency (though in future, it might be possible to estimate β1 from information about the electrode location). This can be regarded as either a defect or a virtue of the model: On the one hand, for maximum accuracy, one (though only one) of the model parameters needs to be fit individually. On the other hand, inter-thalamus variation in the response to stimulation is a real (and clinically important) phenomenon, and any model that fails to take it into account is, in some sense, incomplete.
Finally, our model accounts for less than 100% of the variance in our data. Some of the remaining variance may be truly random, but some may reflect as yet unidentified factors such as, clinical & demographic variables. We believe the present model will aid in uncovering such factors. By modeling, predicting, and then subtracting out dominant effects, subtler effects of other factors may be uncovered.