Incorrect calculation of total electrical energy delivered by a deep brain stimulator
Mark D. McAuley
Abstract
Modern deep brain stimulation (DBS) devices typically enable the programmer of the device to specify the pulse amplitude, the number of pulses delivered per second and the duration of each pulse – i.e. current, frequency and pulse width, respectively. Some devices also enable considerable control over the administration of the electric charge, via accurately-powered segmented electrodes. A DBS programmer may also consider the total electrical energy delivered (TEED) per second by such a device. TEED (1s) was used in 2002 by Moro et al. [[1]Moro E. Esselink R.J. Xie J. Hommel M. Benabid A.L. Pollak P. The impact on Parkinson’s disease of electrical parameter settings in STN stimulation.Neurology. 2002; 59: 706-713https://doi: 10.1212/wnl.59.5.706Crossref PubMed Scopus (379) Google Scholar], and subsequently modified in a 2005 letter by Koss et al. [[2]Koss A.M. Alterman R.L. Tagliati M. Shils J.L. Calculating total electrical energy delivered by deep brain stimulation systems.Ann Neurol. 2005; 58: 168https://doi: 10.1002/ana.20525. (author reply 168–9)Crossref PubMed Scopus (125) Google Scholar]. Koss and colleagues provided a mathematical derivation concluding:TEED (1s) = (voltage2 × frequency × pulsewidth) / impedance (1s)(1) Although some programming guidelines discourage the use of TEED (1s) to censor or edit the combinations of stimulation parameters [[3]Marks Jr., W.J. Deep brain stimulation management.2ed. Cambridge University Press, 2016: 62Google Scholar], TEED (1s) is considered by some DBS programmers in current clinical practice. Imbalzano and colleagues provide an interesting case report in which they varied the frequency of the pulse delivered to the subthalamic nucleus by a DBS device and observed the resultant clinical response [[4]Imbalzano G. Artusi C.A. Montanaro E. Romagnolo A. Rizzone M.G. Lopiano L. Zibetti Maurizio Tuning deep brain stimulation related depression by frequency modulation: a case report.Brain Stimul. 2020; 13: 1265-1267https://doi: 10.1016/j.brs.2020.06.006Abstract Full Text Full Text PDF PubMed Scopus (1) Google Scholar]. Within the legend of their figure, Imbalzano et al. state: “∗ Expected threshold calculated by TEED (1s); TEED (1s) = (voltage2 x frequency x amplitude)/impedance; TEED (1s): total electrical energy delivered in 1 second”. Imbalzano and colleagues referenced Koss et al., but mistakenly replaced ‘pulsewidth’ with ‘amplitude’ relative to Equation (1). Furthermore, Koss et al. had provided a voltage-based equation for TEED (1s), whilst Imbalzano and colleagues used a current-based device. Taking a similar approach to Koss et al., and using standard terms, one can perform a relevant mathematical derivation commencing with Ohm’s law:V(V) = I(A) × R(Ω)(2) and the non-alternating current equation for power:P(W) = V(V) × I(A)(3) where the physical quantities in Equations ((2), (3)) are P = power, V = voltage, I = current and R = resistance, and the appropriate units are respectively (W) = watts, (V) = volts, (A) = amps and (Ω) = ohms. Combining Equations ((2), (3)):P(W) = I(A)I(A) × R(Ω)(4) A square pulse of width pw, measured in seconds, repeated at a frequency f, measured in Hz (pulses per second), requires the power equation to be modified to account for the time that energy is not flowing (noting this is true when pw(sec)×f(Hz)<1):P(W)=I(A)2×pw(sec)×f(Hz)×R(Ω)(5) Equation (5) provides a current-based equation to approximate the power delivered by a DBS device. Note that the equation is approximate because DBS devices do not output a perfectly square pulse and the nature of the pulse generated may change over time due to battery performance. The relationship between frequency and current in Equation (5) implies that 130 Hz and 2.0 mA delivers the same power (energy per second) as 80 Hz and 2.5 mA [= (130/80)0.5 × 2.0 mA] and as 60 Hz and 2.9 mA [= (130/60)0.5 × 2.0 mA]. The above-mentioned figure incorrectly indicates that the equivalent energy per second to 130 Hz and 2 mA is 80 Hz and 4 mA, and, 60 Hz and 5 mA. Before this recent article by Imbalzano and colleagues, Ricchi et al. also incorrectly state that TEED (1s) is a function of voltage squared and amplitude [[5]Ricchi V. Zibetti M. Angrisano S. Merola A. Arduino N. Artusi C.A. et al.Transient effects of 80 Hz stimulation on gait in STN DBS treated PD patients: a 15 months follow-up study.Brain Stimul. 2012; 5: 388-392https://doi: 10.1016/j.brs.2011.07.001Abstract Full Text Full Text PDF PubMed Scopus (82) Google Scholar]. A different mistake relating to TEED can be seen in Schor & Nelson, who also reference Koss et al., but then incorrectly state: “TEED is calculated as follows: [current2× frequency × pulse width]/impedance.” [[6]Schor J.S. Nelson A.B. Multiple stimulation parameters influence efficacy of deep brain stimulation in Parkinsonian mice.J Clin Invest. 2019; 129: 3833-3838https://doi: 10.1172/JCI122390Crossref PubMed Scopus (18) Google Scholar] Schor & Nelson have replaced ‘voltage’ with ‘current’ relative to Equation (1), rather than use Ohm’s Law to appropriately transform Equation (1) for a current-based device. In addition to the above inconsistent equations, statements in the published literature have the potential to perpetuate misunderstandings of the energy delivered by DBS devices per second. For example, Conway et al. have published a meta-analysis in which they reference Koss et al. and then state: “… the total electrical energy delivered (TEED) for the patient is calculated by multiplying the values for amplitude, frequency, pulse width, and biological impedance” and, later, “[g]iven that the TEED represents the product of stimulation frequency, amplitude, pulse width, and biological impedance, any changes that are made to one of these parameters (e.g., lowering frequency) ultimately changes the TEED, unless a compensatory change is made to one of the other parameters (e.g., by increasing amplitude)” [[7]Conway Z.J. Silburn P.A. Thevathasan W. O’Maley K. Naughton G.A. Cole M.H. Alternate subthalamic nucleus deep brain stimulation parameters to manage motor symptoms of Parkinson’s Disease: systematic review and meta-analysis.Mov Disord Clin Pract. 2018; 6: 17-26https://doi: 10.1002/mdc3.12681Crossref PubMed Scopus (5) Google Scholar]. Such statements could leave a reader with the incorrect impression that TEED (1s) is a function of current, rather than current squared, and therefore that halving the frequency would require doubling of the current to keep the TEED constant. Further, some of the terminology used when referring to TEED is potentially confusing. Conway et al. refer to ‘amplitude’, which could mean either voltage or current, but in their case clearly refers to the current. The TEED (1s) derived by Koss et al. is the total energy delivered in one second – i.e. the energy per second, which is normally referred to as the power. Also, the use of the term impedance could be simplified by referring to the resistance, which is the real part of the complex impedance. The examples provided in this letter illustrate the publication of incorrect equations, potentially confusing statements, and the use of unclear terminology for TEED (1s) in several journals. While the examples provided do not necessarily indicate flaws in the conclusions drawn in those papers, such inexact statements risk causing misunderstandings regarding how to correctly calculate TEED (1s). On a personal note, as a patient who had a DBS device implanted in May 2020, I would encourage DBS programmers to consider Equation (5) – specifically, the electrical energy delivered by a DBS device every second (i.e. the power) is a function of current squared, pulse width, frequency and resistance. MDM: Conception, execution and writing. Ethics clearance was not required for this letter to the editor. Informed patient consent was not necessary for this work. I confirm that I have read the Journal’s position on issues involved in ethical publication and affirm that this work is consistent with those guidelines. This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.