US 6,263,239 B1Grant

Method and apparatus for determining the second phase of defibrillator devices

Issue Date:2001-07-17

•39 Claims

•7 Drawing Sheets

Abstract

A method for determining an optimal transchest external defibrillation waveform which, when applied through a plurality of electrodes positioned on a patient's torso will produce a desired response in the patient's cardiac cell membranes. The method includes the steps of providing a quantitative model of a defibrillator circuit for producing external defibrillation waveforms, the quantitative model of a patient includes a chest component, a heart component, a cell membrane component and a quantitative description of the desired cardiac membrane response function. Finally, a quantitative description of a transchest external defibrillation waveform that will produce the desired cardiac membrane response function is computed. The computation is made as a function of the desired cardiac membrane response function, the patient model and the defibrillator circuit model.

Metadata

Assignee

SurVivaLink Corporation

Inventors

James E. Brewer
Gary B. Stendahl
Kenneth F. Olson

Application Information

Application Number:US 09/383,561

Filing Date:1999-08-26

Priority Date:1996-07-01

Art Unit:7

Classifications

IPC:

A61N 139

Field of Search:

607 5607 6607 7607 8

Patent Drawings (7 sheets)

Description

[0002] This application claims priority to copending non-provisional application Ser. No. 08/886,736, filed Jul. 1, 1997, entitled METHOD AND APPARATUS FOR DETERMINING THE SECOND PHASE OF EXTERNAL DEFIBRILLATOR DEVICES, which is based on provisional patent application Ser. No. 60/021,161, filed Jul. 1, 1996 entitled DYNAMIC SECOND PHASE (φ₂) WITH SELF-CORRECTING CHARGE BURPING FOR EXTERNAL DEFIBRILLATOR DEVICES, the contents of both applications of which are herein incorporated by reference.

Field of the Invention

[0003] This invention relates generally to an electrotherapy method and apparatus for delivering an electrical pulse to a patient's heart. In particular, this invention relates to a method and apparatus for tailoring a second phase of biphasic waveform delivered by an external defibrillator, to random patients, by performing intelligent calculations and analysis to the results of a first phase segment of a biphasic defibrillation waveform and other parameters pertaining thereto based on theory and practice as disclosed herein.

Background of the Invention

[0004] Devices for defibrillating a heart have been known for sometime now. Implantable defibrillators are well accepted by the medical community as effective tools to combat fibrillation for an identified segment of the population. A substantial amount of research in fibrillation and the therapy of defibrillation has been done. Much of the most recent research has concentrated on understanding the effects that a defibrillation shock pulse has on fibrillation to terminate such a condition.

[0005] A monophasic waveform is defined to be a single phase, capacitive-discharge, time-truncated, waveform with exponential decay. A biphasic waveform is defined to comprise two monophasic waveforms, separated by time and of opposite polarity. The first phase is designated φ₁and the second phase is designated φ₂. The delivery of φ₁is completed before the delivery of φ₂is begun.

[0006] After extensive testing, it has been determined that biphasic waveforms are more efficacious than monophasic waveforms. There is a wide debate regarding the reasons for the increased efficacy of biphasic waveforms over that of a monophasic waveforms. One hypothesis holds that φ₁defibrillates the heart and φ₂performs a stabilizing action that keeps the heart from refibrillating.

[0007] Biphasic defibrillation waveforms are now the standard of care in clinical use for defibrillation with implantable cardioverter-defibrillators (ICDs), due to the superior performance demonstrated over that of comparable monophasic waveforms. To better understand these significantly different outcomes, ICD research has developed cardiac cell response models to defibrillation. Waveform design criteria have been derived from these first principles and have been applied to monophasic and biphasic waveforms to optimize their parameters. These principles-based design criteria have produced significant improvements over the current art of waveforms.

[0008] In a two paper set, Blair developed a model for the optimal design of a monophasic waveform when used for electrical stimulation. (1) Blair, H. A., “On the Intensity-time relations for stimulation by electric currents.” I. J. Gen. Physiol. 1932; 15: 709-729. (2) Blair, H. A., “On the Intensity-time Relations for stimulation by electric currents. II. J. Gen. Physiol. 1932; 15: 731-755. Blair proposed and demonstrated that the optimal duration of a monophasic waveform is equal to the point in time at which the cell response to the stimulus is maximal. Duplicating Blair's model, Walcott extended Blair's analysis to defibrillation, where they obtained supporting experimental results. Walcott, et al., “Choosing the optimal monophasic and biphasic waveforms for ventricular defibrillation.” J. Cardiovasc Electrophysiol. 1995; 6: 737-750.

[0009] Independently, Kroll developed a biphasic model for the optimal design of φ₂for a biphasic defibrillation waveform. Kroll, M. W., “A minimal model of the single capacitor biphasic defibrillation waveform.” PACE 1994; 17: 1782-1792. Kroll proposed that the φ₂stabilizing action removed the charge deposited by φ₁from those cells not stimulated by φ₁. This has come to be known as “charge burping”. Kroll supported his hypothesis with retrospective analysis of studies by Dixon, et al., Tang, et al., and Freese, et al. regarding single capacitor, biphasic waveform studies. Dixon, et al., “Improved defibrillation thresholds with large contoured epicardial electrodes and biphasic waveforms.” Circulation 1987; 76: 1176-1184; Tang, et al. “Ventricular defibrillation using biphasic waveforms: The Importance of Phasic duration.” J. Am. Coll. Cardiol. 1989; 13: 207-214; and Feeser, S. A., et al. “Strength-duration and probability of success curves for defibrillation with biphasic waveforms.” Circulation 1990; 82: 2128-2141. Again, the Walcott group retrospectively evaluated their extension of Blair's model to φ₂using the Tang and Feeser data sets. Their finding further supported Kroll's hypothesis regarding biphasic defibrillation waveforms. For further discussions on the development of these models, reference may be made to PCT publications WO 95/32020 and WO 95/09673 and to U.S. Pat. No. 5,431,686.

[0010] The charge burping hypothesis can be used to develop equations that describe the time course of a cell's membrane potential during a biphasic shock pulse. At the end of φ₁, those cells that were not stimulated by φ₁have a residual charge due to the action of φ₁on the cell. The charge burping model hypothesizes that an optimal pulse duration for φ₂is that duration that removes as much of the φ₁residual charge from the cell as possible. Ideally, these unstimulated cells are set back to “relative ground.” The charge burping model proposed by Kroll is based on the circuit model shown in FIG. 2bwhich is adapted from the general model of a defibrillator illustrated in FIG. 2a.

[0011] The charge burping model also accounts for removing the residual cell membrane potential at the end of a φ₁pulse that is independent of a φ₂. That is, φ₂is delivered by a set of capacitors separate from the set of capacitors used to deliver φ₁. This charge burping model is constructed by adding a second set of capacitors, as illustrated in FIG. 3. In this figure, C₁represents the φ₁capacitor set, C₂represents the φ₂capacitor set R_Hrepresents the resistance of the heart, and the pair C_Mand R_Mrepresent membrane series capacitance and resistance of a single cell. The node V_Srepresents the voltage between the electrodes, while V_Mdenotes the voltage across the cell membrane.

[0012] External defibrillators send electrical pulses to the patient's heart through electrodes applied to the patient's torso. External defibrillators are useful in any situation where there may be an unanticipated need to provide electrotherapy to a patient on short notice. The advantage of external defibrillators is that they may be used on a patient as needed, then subsequently moved to be used with another patient.

[0013] However, this important advantage has two fundamental limitations. First, external defibrillators do not have direct contact with the patient's heart. External defibrillators have traditionally delivered their electrotherapeutic pulses to the patient's heart from the surface of the patient's chest. This is known as the transthoracic defibrillation problem. Second, external defibrillators must be able to be used on patients having a variety of physiological differences. External defibrillators have traditionally operated according to pulse amplitude and duration parameters that can be effective in all patients. This is known as the patient variability problem.

[0014] The prior art described above effectively models implantable defibrillators, however it does not fully addressed the transthoracic defibrillation problem nor the patient variability problem. In fact, these two limitations to external defibrillators are not fully appreciated by those in the art. For example, prior art disclosures of the use of truncated exponential monophasic or biphasic shock pulses in implantable or external defibrillators have provided little guidance for the design of an external defibrillator that will successfully defibrillate across a large, heterogeneous population of patients. In particular, an implantable defibrillator and an external defibrillator can deliver a shock pulse of similar form, and yet the actual implementation of the waveform delivery system is radically different.

[0015] In the past five years, new research in ICD therapy has developed and demonstrated defibrillation models that provide waveform design rules from first principles. These defibrillation models and their associated design rules for the development of defibrillation waveforms and their characteristics were first developed by Kroll and Irnich for monophasic waveforms using effective and rheobase current concepts. (1) Kroll, M. W., “A minimal model of the monophasic defibrillation pulse.” PACE 1993; 15: 769. (2) Irnich, W., “Optimal truncation of defibrillation pulses.” PACE 1995; 18: 673. Subsequently, Kroll, Walcott, Cleland and others developed the passive cardiac cell membrane response model for monophasic and biphasic waveforms, herein called the cell response model. (1) Kroll, M. W., “A minimal model of the single capacitor biphasic defibrillation waveform.” PACE 1994; 17: 1782. (2) Walcott, G. P., Walker, R. G., Cates. A. W., Krassowska, W., Smith, W. M, Ideker R E. “Choosing the optimal monophasic and biphasic waveforms for ventricular defibrillation.” J Cardiovasc Electrophysiol 1995; 6: 737; and Cleland B G. “A conceptual basis for defibrillation waveforms.” PACE 1996; 19: 1186).

[0016] A significant increase in the understanding of waveform design has occurred and substantial improvements have been made by using these newly developed design principles. Block et al. has recently written a comprehensive survey of the new principles-based theories and their impact on optimizing internal defibrillation through improved waveforms. Block M, Breithardt G., “Optimizing defibrillation through improved waveforms.” PACE 1995; 18: 526.

[0017] There have not been significant developments in external defibrillation waveforms beyond the two basic monophasic waveforms: the damped sine or the truncated exponential. To date, their design for transthoracic defibrillation has been based almost entirely on empirically derived data. It seems that the design of monophasic and biphasic waveforms for external defibrillation has not yet been generally influenced by the important developments in ICD research.

[0018] Recently there has been reported research on the development and validation of a biphasic truncated exponential waveform in which it was compared clinically to a damped sine waveform. For additional background, reference may be made to U.S. Pat. Nos. 5,593,427, 5,601,612 and 5,607,454. See also: Gliner B E, Lyster T E, Dillon S M, Bardy G H, “Transthoracic defibrillation of swine with monophasic and biphasic waveforms.” Circulation 1995; 92: 1634-1643; Bardy G H, Gliner B E, Kudenchuk P J, Poole J E, Dolack G L, Jones G K, Anderson J, Troutman C, Johnson G.; “Truncated biphasic pulses for transthoracic defibrillation.” Circulation 1995; 91: 1768-1774; and Bardy G H et al, “For the Transthoracic Investigators. Multicenter comparison of truncated biphasic shocks and standard damped sine wave monophasic shocks for transthoracic ventricular defibrillation.” Circulation 1996; 94: 2507-2514. Although the research determined a usable biphasic waveform, there was no new theoretical understanding determined for external waveform design. It appears that external waveform research may develop a “rules-of-thumb by trial and error” design approach much like that established in the early stages of theoretical ICD research. The noted limitations of the transthoracic biphasic waveform may be due in part to a lack of principles-based design rules to determine its waveform characteristics.

[0019] There is a continued need for a device designed to perform a quick and automatic adjustment of phase 2 relative to phase 1 if AED's are to be advantageously applied to random patients according to a cell response model. Further, the model and the device must be adaptable to patient variance and be able to provide automatic adjustment in a dynamic environment.

Summary of the Invention

[0020] The present invention relates to an external defibrillation method and apparatus that addresses the limitations in the prior art. The present invention incorporates three singular practices that distinguish the practice of designing external defibrillators from the practice of designing implantable defibrillators. These practices are 1) designing multiphasic transthoracic shock pulse waveforms from principles based on cardiac electrophysiology, 2) designing multiphasic transthoracic shock pulse waveforms in which each phase of the waveform can be designed without implementation limitations placed on its charging and delivery means by such means for prior waveform phases, and 3) designing multiphasic transthoracic shock pulse waveforms to operate across a wide range of parameters determined by a large, heterogeneous population of patients.

[0021] In particular, the present invention provides for a method and apparatus for tailoring and reforming a second phase (φ₂) of a biphasic defibrillation waveform relative to a first phase (φ₁) of the waveform based on intelligent calculations. The method includes the steps of determining and providing a quantitative description of the desired cardiac membrane response function. A quantitative model of a defibrillator circuit for producing external defibrillation waveforms is then provided. Also provided is a quantitative model of a patient which includes a chest component, a heart component and a cell membrane component. A quantitative description of a transchest external defibrillation waveform that will produce the desired cardiac membrane response function is then computed. Intelligent calculations based on the phase 1 cell response is then computed to determine the desired phase 2 waveform. The computation is made as a function of the desired cardiac membrane response function, the patient model and the defibrillator circuit model.

Brief Description of the Drawings

[0022] FIGS. 1aand 1bare perspective views of an AED according to the present invention.

[0023] FIG. 2ais a very simplified defibrillator model.

[0024] FIG. 2bis a known monophasic defibrillation model.

[0025] FIG. 3 is a known biphasic defibrillation model.

[0026] FIG. 4 represents a monophasic or biphasic capacitive-discharge external defibrillation model according to the present invention.

[0027] FIG. 5arepresents a monophasic capacitor-inductor external defibrillator model according to the present invention.

[0028] FIG. 5brepresents an alternative embodiment of a biphasic capacitor-inductor external defibrillator model according to the present invention.

[0029] FIG. 6 illustrates a biphasic waveform generator utilizing the present invention.

[0030] FIG. 7 is a schematic diagram of a circuit which enables the implementation of the present invention.

[0031] FIG. 8 illustrates a biphasic waveform in relation to a cellular response curve.

Description of the Preferred Embodiment

[0032] The present invention provides a method and apparatus for tailoring a second phase (φ₂) of a biphasic waveform delivered by an external defibrillator, to random patients, by performing intelligent calculations and analysis to the results of a first phase (φ₁) segment of a biphasic defibrillation waveform and other parameters pertaining thereto. Prior to describing the present invention, a discussion of the development of an external defibrillation model will be given.

[0033] External Defibrillator Model

[0034] The apparatus of the present invention is an automated external defibrillator (AED) illustrated in FIGS. 1aand 1b. FIG. 1aillustrates an AED 10, including a plastic case 12 with a carrying handle 14. A lid 16 is provided which covers an electrode compartment 18. An electrode connector 20, a speaker 22 and a diagnostic panel (not shown) are located on case 12 within electrode compartment 18. FIG. 1billustrates AED 10 having a pair of electrodes 24 connected thereto. Electrodes 24 can be pre-connected to connector 20 and stored in compartment 18.

[0035] The operation of AED 10 is described briefly below. A rescue mode of AED 10 is initiated when lid 16 is opened to access electrodes 24. The opening of lid 16 is detected by AED 10 to effectively turn on the device. AED 10 then quickly runs a short test routine. After electrodes 24 have been placed on the patient, AED 10 senses patient specific parameters, such as impedance, voltage, current, charge or other measurable parameters of the patient. The patient specific parameters are then utilized in the design of optimal waveforms as will be described below.

[0036] If a shockable condition is detected through electrodes 24, a plurality of capacitors inside of AED 10 are charged from an energy source, typically a detachable battery pack. Based upon the patient specific parameters sensed, the duration and other characteristics of a discharge waveform are then calculated. The energy stored in AED 10 is then discharged to the patient through electrodes 24.

[0037] For a more detailed description of the physical structure of AED 10 or the process involved in sensing, charging, shocking and testing, reference should be made to applicants co-pending application Ser. No. 08/512,441, filed Aug. 8, 1995 entitled AUTOMATED EXTERNAL DEFIBRILLATOR WITH SELF-TEST SYSTEM which is assigned to the assignee of the present invention, the disclosure of which is herein incorporated by reference.

[0038] In the present invention it is not assumed that both phases of a biphasic waveform are delivered using the same set of capacitors or that both phases of a biphasic waveform are delivered using the capacitor set in the same electrical configuration, although such an embodiment is considered within the spirit and scope of the present invention.

[0039] Transthoracic defibrillation is generally performed by placing electrodes on the apex and anterior positions of the chest wall. With this electrode arrangement, nearly all current passing through the heart is conducted by the lungs and the equipotential surfaces pass through the myocardium normal to the electrode axis. The present invention uses the transthoracic charge burping model to develop design equations that describe the time course of a cell's membrane potential during a transthoracic biphasic shock pulse. These equations are then used to create equations that describe the design of monophasic and biphasic shock pulses for transchest defibrillation to optimize the design of φ₁for defibrillating and the design φ₂for stabilizing. These optimizing shock pulse design equations are called design rules.

[0040] According to the present invention, the main series pathway for current is to pass through the chest wall, the lungs, and the heart. Additionally, there are two important shunting pathways in parallel with the current pathway through the heart. These shunting pathways must be taken into consideration. The lungs shunt current around the heart through a parallel pathway. The second shunting pathway is provided by the thoracic cage. The resistivity of the thoracic cage and the skeletal muscle structure is low when compared to lungs. The high resistivity of the lungs and the shunting pathways are characterizing elements of external defibrillation that distinguish the art from intracardiac defibrillation and implantable defibrillation technologies.

[0041] Therefore, in the transthoracic defibrillation model of the present invention illustrated in FIG. 4, there are several resistances in addition to those discussed for the charge burping model above. R_Srepresents the resistance of the defibrillation system, including the resistance of the defibrillation electrodes. R_CWand R_LSrepresent the resistances of the chest wall and the lungs, respectively, in series with resistance of the heart, R_H. R_TCand R_LPrepresent the resistances of the thoracic cage and the lungs, respectively, in parallel with the resistance of the heart.

[0042] The design rules for external defibrillation waveforms are determined in three steps. In the first step, the transchest forcing function is determined. The transchest forcing function is the name that is given to the voltage that is applied across each cardiac cell during an external defibrillation shock. In the second step, the design equations for φ₁of a shock pulse are determined. The design equations are the equations describing the cell's response to the φ₁transchest forcing function, the equation describing the optimal φ₁pulse duration, and the equation describing the optimal φ₁capacitor. Therefore, step two relates the cell response to the action of a monophasic shock pulse or the first phase of a biphasic shock pulse. This relation is used to determine the optimal design rules and thereby design parameters for the implementation of this phase in an external defibrillator. It will be clear to those in the art that step two is not restricted to capacitor discharge shock pulses and their associated transchest forcing function. Another common implementation of an external defibrillator incorporates a damped sine wave for a shock pulse and can be either a monophasic or biphasic waveform. This type of external defibrillator is modeled by the circuits shown in FIGS. 5aand 5b. In the third step, the design equations for φ₂of a shock pulse are determined. The design equations are the equations describing the cell's response to the φ₂transchest forcing function, the equation describing the optimal φ₂pulse duration and the equation describing the optimal φ₂capacitor. These design equations are employed to determine the optimal design rules and thereby design parameters of φ₂of a biphasic shock pulse with respect to how the cell responds to the shock pulse. An important element of this invention is to provide shock pulse waveforms that are designed from a cardiac cell response model developed from first principles and that correctly determines the effects of the chest and its components on the ability of a shock pulse to defibrillate.

[0043] The transchest forcing function is determined by solving for the voltage found at node V₃in FIG. 4. The transchest forcing function is derived by solving for V₃using the following three nodal equations:

\begin{matrix} \frac{V_{1} - V_{S}}{R_{S}} + \frac{V_{1}}{R_{TC}} + \frac{V_{1} - V_{2}}{R_{CW}} = 0, & (1) \\ \frac{V_{2} - V_{1}}{R_{CW}} + \frac{V_{2}}{R_{LP}} + \frac{V_{2} - V_{3}}{R_{LS}} = 0, and & (2) \\ \frac{V_{3} - V_{2}}{R_{LS}} + \frac{V_{3}}{R_{H}} + \frac{V_{3} - V_{M}}{R_{M}} = 0. & (3) \end{matrix}

[0044] Equation 1 can be rewritten as

\begin{matrix} V_{1} (\frac{1}{R_{S}} + \frac{1}{R_{BC}} + \frac{1}{R_{CW}}) = \frac{V_{S}}{R_{S}} + \frac{V_{2}}{R_{CW}} . & (4A) \\ V_{1} = \frac{V_{S}}{R_{S} Ω_{1}} + \frac{V_{2}}{R_{CW} Ω_{1}}, where Ω_{1} = \frac{1}{R_{S}} + \frac{1}{R_{TC}} + \frac{1}{R_{CW}} . & (4B) \end{matrix}

[0045] Rewriting equation 2, we have

\begin{matrix} V_{2} (\frac{1}{R_{CW}} + \frac{1}{R_{LP}} + \frac{1}{R_{LS}}) = \frac{V_{1}}{R_{CW}} + \frac{V_{3}}{R_{LS}} . & (4C) \end{matrix}

[0046] By substituting equation 4B for V₁into equation 4C, we can solve for V₂as an expression of V_Sand V₃:

\begin{matrix} V_{2} = \frac{V_{S}}{R_{S} R_{CW} Ω_{1} Ω_{2} Ω_{22}} + \frac{V_{3}}{R_{LS} Ω_{2} Ω_{22}}, where Ω_{2} = \frac{1}{R_{LS}} + \frac{1}{R_{LP}} + \frac{1}{R_{CW}}, and Ω_{22} = 1 - \frac{1}{R_{CW}^{2} Ω_{1} Ω_{2}} . & (5) \end{matrix}

[0047] Now solving for V₃as an expression of V_Sand V_M, equation 3 may be re-arranged as

\begin{matrix} V_{3} (\frac{1}{R_{LS}} + \frac{1}{R_{H}} + \frac{1}{R_{M}}) = \frac{V_{2}}{R_{LS}} + \frac{V_{M}}{R_{M}} so that & (6) \\ V_{3} = \frac{V_{2}}{R_{LS} Ω_{3}} + \frac{V_{M}}{R_{M} Ω_{3}} where Ω_{3} = \frac{1}{R_{LS}} + \frac{1}{R_{H}} + \frac{1}{R_{M}} . & (7) \end{matrix}

[0048] Substituting equation 5 for V₂into equation 7, we can solve for V₃as an expression of V_Sand V_M:

\begin{matrix} V_{3} = \frac{V_{S}}{R_{S} R_{CW} R_{LS} Ω_{1} Ω_{2} Ω_{22} Ω_{3} Ω_{33}} + \frac{V_{M}}{R_{M} Ω_{3} Ω_{33}} where & (8) \\ Ω_{33} = 1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})} & (9) \end{matrix}

[0049] From equation 8 we define Ω_Mto be:

\begin{matrix} Ω_{M} = R_{M} Ω_{3} Ω_{33} = R_{M} Ω_{3} (1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})}) Ω_{M} = R_{M} (Ω_{3} - \frac{1}{R_{LS}^{2} (Ω_{2} - \frac{1}{R_{CW}^{2} Ω_{1}})}) . & (10) \end{matrix}

[0050] From equation 8 we also define Ω_Sto be:

[0051] Ω_S=R_SR_CWR_LSΩ₁Ω₂Ω₃Ω₂₂Ω₃₃ (11)

[0052]

\begin{matrix} Ω_{S} = R_{S} R_{CW} R_{LS} Ω_{1} Ω_{2} (1 - \frac{1}{(R_{CW}^{2} Ω_{1} Ω_{2})}) Ω_{3} (1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})}) & (12) \\ Ω_{S} = R_{S} R_{CW} R_{LS} (Ω_{1} Ω_{2} - \frac{1}{R_{CW}^{2}}) (Ω_{3} - \frac{1}{R_{LS}^{2} (Ω_{2} - \frac{1}{R_{CW}^{2} Ω_{1}})}) & (13) \\ so that V_{3} = \frac{V_{S}}{Ω_{S}} + \frac{V_{M}}{Ω_{M}} & (14) \end{matrix}

[0053] is the general transchest transfer function as shown in FIG. 4 or FIGS. 5aand 5b. Equation 14 encapsulates the transchest elements and their association between the forcing function V_S(which models a defibrillation circuit and the shock pulse) and the cell membrane voltage V_M. Therefore, this completes the first step.

[0054] The variable V_Smay now be replaced with a more specific description of the defibrillation circuitry that implements a shock pulse. For a first example, a monophasic time-truncated, capacitive-discharge circuit may be represented by V_S=V₁e^-t/τ₁, where V₁is the leading-edge voltage for the shock pulse and τ₁=RC₁, with R determined below.

[0055] As shown in FIGS. 5aand 5b, a second example would be a monophasic damped sine wave circuit, represented by

\begin{matrix} V_{S} = V_{1} (\frac{τ_{C1}}{τ_{C1} - τ_{L1}}) (ⅇ^{- t / τ_{C1}} - ⅇ^{- t / τ_{L1}}) & (14B) \end{matrix}

[0056] where V₁is the voltage on the charged capacitor C₁τ_C1=RC₁and τ_L1=L₁/R. Every step illustrated below may be performed with this and other similar transchest forcing functions which represent defibrillator circuitry.

[0057] To proceed with step two, from FIG. 4, nodal analysis provides an equation for V_M:

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M} - V_{3}}{R_{M}} = 0. & (15) \end{matrix}

[0058] Rearranging equation 15, we have

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} = \frac{V_{3}}{R_{M}} . & (16) \end{matrix}

[0059] Next, substituting equation 14 as an expression for V₃into equation 16, the cell membrane response is now calculated as follows:

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} = \frac{1}{R_{M}} (\frac{V_{S}}{Ω_{S}} + \frac{V_{M}}{Ω_{M}}) & (17) \\ C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} - \frac{V_{M}}{R_{M} Ω_{M}} = \frac{V_{S}}{R_{M} Ω_{S}} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{S}}{R_{M} Ω_{S}} & (18) \end{matrix}

[0060] Dividing through by C_M, and setting τ_M=R_MC_M, then equation 18 becomes

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{τ_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{S}}{τ_{M}} (\frac{1}{Ω_{S}}) . & (19) \end{matrix}

[0061] Equation 19 is a general ordinary differential equation (ODE) that models the effects of any general forcing function V_Sthat represents a phase of a shock pulse waveform applied across the chest. The general ODE equation 19 models the effects of a general shock pulse phase V_Son the myocardium, determining cardiac cell response to such a shock pulse phase.

[0062] In the equations given below:

[0063] C₁equals the capacitance of the first capacitor bank and V_S=V₁e^-t/τ;

[0064] C₂equals the capacitance of the second capacitor bank and V_S=V₂e^-t/τ;

[0065] R=R_S+R_B, where R_S=System impedance (device and electrodes);

[0066] R_B=body impedance (thoracic cage, chest wall, lungs (series, parallel), heart).

[0067] To determine body impedance, R_B, we see that the series combination of R_Hand R_LSyields R_H+R_LS. (FIG. 4). The parallel combination of R_H+R_LSand R_LPyields:

\begin{matrix} \frac{R_{LP} (R_{LS} + R_{H})}{R_{LP} + R_{LS} + R_{H}} . & (20) \end{matrix}

[0068] The series combination of equation 20 and R_CWyields:

\begin{matrix} R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})} . & (21) \end{matrix}

[0069] The parallel combination of equation 21 and R_TCyields:

\begin{matrix} R_{B} = [\frac{R_{TC} [R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})}}{R_{TC} + R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})}}] & (22) \end{matrix}

[0070] where R_Bis the impedance of the body for this model.

[0071] The discharge of a single capacitor is modeled by V_S=V₁e^-t/τfor an initial C₁capacitor voltage of V₁. Placing V_Sinto equation 19 gives:

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{τ_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{1} ⅇ^{- t / τ_{1}}}{τ_{M} Ω_{S}} & (23) \end{matrix}

[0072] where τ_M=R_MC_Mrepresents the time constant of the myocardial cell in the circuit model, and τ₁, which equals R_SC₁, represents the time constant of φ₁. Such a standard linear ODE as equation 23 has the form

\frac{ⅆ y}{ⅆ x} + P (X) Y = Q (x) .

[0073] These linear ODEs have an integration factor that equals e^∫pdx. The general solution to such equations is:

[0074] Y=e^-∫pdx[∫e^∫pdxQdx+c].

[0075] The ODE in equation 23 models the effects of each phase of a time-truncated, capacitor-discharged shock pulse waveform. Equation 23 is a first-order linear ODE, and may be solved using the method of integration factors, to get:

\begin{matrix} V_{M1} (t) = k ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} + (\frac{V_{1}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{2}{Ω_{M}}) - τ_{M}}) ⅇ^{- t / τ_{1}} . & (24) \end{matrix}

[0076] Equation 24 is an expression of cell membrane potential during φ₁of a shock pulse. To determine the constant of integration k, the initial value of V_M1is assumed to be V_M1(0)=V_G(“cell ground”). Applying this initial condition to equation 24, k is found to be

\begin{matrix} k = V_{G} - (\frac{V_{o}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (25) \end{matrix}

[0077] Assuming τ₁=R_C1, where R=R_S+R_B, then the solution to the initial-value problem for φ₁is:

\begin{matrix} V_{M1} (t) = V_{G} ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} + (\frac{V_{1}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- t / τ_{1}} - ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})}) & (26) \end{matrix}

[0078] Equation 26 describes the residual voltage found on a cell at the end of φ₁.

[0079] Assuming V_G=0 and V₁=1, the solution for cell response to an external shock pulse is

\begin{matrix} V_{M1} (t) = (\frac{1}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- \frac{t}{τ_{1}}} - ⅇ^{- (\frac{t}{τ_{M}}) (1 - \frac{1}{Ω_{M}})}) . & (27) \end{matrix}

[0080] We may now determine optimal durations for φ₁according to criteria for desired cell response. One such design role or criterion is that the φ₁duration is equal to the time required for the external defibrillator shock pulse to bring the cell response to its maximum possible level. To determine this duration, equation 27 is differentiated and the resulting equation 27B is set to zero. Equation 27B is then solved for the time t, which represents shock pulse duration required to maximize cardiac cell response.

\begin{matrix} (\frac{AB}{τ_{M}}) ⅇ^{- Bt / τ_{M}} - (\frac{A}{τ_{1}}) ⅇ^{- t / τ_{1}} = 0, where A = (\frac{1}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) and B = 1 - \frac{1}{Ω_{M}} . & (27B) \end{matrix}

[0081] Solving for t, the optimal duration dφ₁for a monophasic shock pulse or φ₁of a biphasic shock pulse is found to be

\begin{matrix} d φ_{1} = (\frac{τ_{1} τ_{M}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) \ln (\frac{τ_{1} (1 - \frac{1}{Ω_{M}})}{τ_{M}}), & (27C) \end{matrix}

[0082] where “1n” represents the logarithm to the base e, the natural logarithm.

[0083] For φ₂, an analysis almost identical to equations 20 through 27 above is derived. The differences are two-fold. First, a biphasic waveform reverses the flow of current through the myocardium during φ₂. Reversing the flow of current in the circuit model changes the sign on the current. The sign changes on the right hand side of equation 23.

[0084] The second difference is the step taken to incorporate an independent φ₂into the charge burping model. Therefore, the φ₂ODE incorporates the C₂capacitor set and their associated leading-edge voltage, V₂, for the φ₂portion of the pulse. Then τ₂represents the φ₂time constant; τ₂=RC₂, and V_S=−V₂e^-t/τ. Equation 23 now becomes:

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + (\frac{V_{M}}{τ_{M}}) (1 - \frac{1}{Ω_{M}}) = \frac{- V_{2} ⅇ^{- t / τ_{2}}}{τ_{M} Ω_{S}} . & (29) \end{matrix}

[0085] Equation 29 is again a first-order linear ODE. In a similar manner, its general solution is determined to be:

\begin{matrix} V_{M2} (t) = k ⅇ^{(- t / τ_{M}) (1 - \frac{1}{Ω_{M}})} - (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (30) \end{matrix}

[0086] To determine the constant of integration k, the value of V_M2at the end of φ₁is

[0087] V_M2(0)=V_M1(d_φ1)=V_φ1′ (31)

[0088] where d_φ1is the overall time of discharge for φ₁and V_φ1is the voltage left on the cell at the end of φ₁. Applying the initial condition to equation 30 and solving for k:

\begin{matrix} k = V_{φ 1} + (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (32) \end{matrix}

[0089] The solution to the initial-value problem for φ₂is

\begin{matrix} V_{M2} (t) = (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} - ⅇ^{- t / τ_{2}}) + V_{φ 1} ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} . & (33) \end{matrix}

[0090] Equation 33 provides a means to calculate the residual membrane potential at the end of φ₂for the cells that were not stimulated by φ₁. Setting Equation 33 equal to zero, we solve for t, thereby determining the duration of φ₂, denoted dφ₂, such that V_M2(dφ₂)=0. By designing φ₂with a duration dφ₂, the biphasic shock pulse removes the residual change placed on a cell by φ₁. We determine dφ₂to be:

\begin{matrix} d_{φ2} = (\frac{τ_{2} τ_{M}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) \cdot \ln (1 + (\frac{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}{τ_{2}}) (\frac{Ω_{S} V_{φ 1}}{V_{2}})) . & (34) \end{matrix}

[0091] From the equations above, an optimal monophasic or biphasic defibrillation waveform may be calculated for an external defibrillator.

[0092] As an example, an external defibrillator may be designed as set forth below. Assume a monophasic truncated exponential shock pulse, a 200 μF capacitor, so that τ₁=R·(200 μF). Suppose also that the external defibrillator is designed to apply the maximal cardiac cell response design rule (equation 27C) to determine the duration of the discharge. Suppose further that the human cardiac cell time constant is estimated to be 3±1 ms. Further assume that the external defibrillator energy source comprises five 1000 μF capacitors in series to implement a 200 μF capacitor bank. If each capacitor is charged to 400V, for a total of 2000V for the leading-edge voltage, this represents 400J of stored energy. The transchest elements are estimated at: 82% current through the thoracic cage; 14% through the chest wall and lungs in parallel; and 4% of applied current through the lung in series with the heart. Then the membrane resistance coefficient Ω_M=5.9, and the system resistance coefficient Ω_S=2.3. Then the table below illustrates the application of the design rule as the overall chest resistance ranges from 25Ω to 200Ω:

[0093] It should be noted and understood that the design of φ₂is independent from φ₁. To design φ₂, the only information necessary from φ₁is where the cell response was left when φ₁was truncated. Additionally, φ₂need not use the same or similar circuitry as that used for φ₁. For example, φ may use circuitry as illustrated in FIG. 4 where φ₂may use circuitry illustrated in FIG. 5a, or vice-versa. The corresponding design rules for a φ₁circuitry may be used in conjunction with the design rules for a φ₂circuitry, regardless of the specific circuitry used to implement each phase of a monophasic or biphasic shock pulse.

[0094] Present Invention

[0095] The present invention is based on the charge burping model hypothesis which postulates and defines an optimal pulse duration for φ₂as a duration that removes as much of the φ₁residual charge from the cell as possible. Ideally, the objective is to maintain unstimulated cells with no charge or set them back to relative ground.

[0096] A further objective of the present invention is to formulate a measurement by which the optimal duration of τ_s(cell time constant) and τ_m(membrane time constant) can be measured. Although one can choose a proper φ₂(fixed) for a given cell response φ₁, in transthoracic shock pulse applications, τ_mis not known and it varies across patients, waveforms and time. For a fixed φ₂, therefore, the error in τ_mcould be substantial. Realizing this, the present invention is designed to correct for “range” of candidate τ_mvalues to fit an optimal duration for a fixed φ₂. In other words, φ₂is selected so that the capacitance in the model is matched with measured R_Hto get a “soft landing” to thereby minimize error due to τ_m±E in charge burping ability of φ₂involving patient variability.

[0097] The technique of “soft landing” advanced by the present invention limits the error in τ_mand sets φ₂to dynamically adjust within a range of possible τ_mvalues. As discussed hereinbelow, optimizing solutions are used to determine parameters on which intelligent calculations could be made so that autonomous φ₂adjustments for variable R_Hare possible.

[0098] The charge burping model also accounts for removing the residual charge at the end of φ₁based on φ₂delivered by a separate set of capacitors other than those used to deliver φ₁. Referring now to FIG. 3, C₁represents the φ₁capacitor set and C₂represents the φ₂capacitor, R_Hrepresents the resistance of the heart, and the pair C_mand R_mrepresent the membrane series capacitance and resistance of a single cell. The node V_srepresents the voltage between the electrodes, while V_mdenotes the voltage across the cell membrane.

[0099] Accordingly, one of the advantages that AEDs have over ICDs, is that the implementation of a φ₂waveform may be completely independent of the implementation of φ₁. Specifically, the charging and discharging circuits for φ₁and φ₂do not need to be the same circuitry. Unlike ICDs, AEDs are not strictly constrained by space and volume requirements. Within practical limits, in AEDs the capacitance and voltage which characterize φ₂need not depend on the circuitry and the values of φ₁.

[0100] The Lerman-Deale model for AED's define the main series for current to pass through the chest wall, the lungs and the heart. Further, two shunting pathways in parallel with current pathway through the heart are defined. Another shunting pathway is provided by the thoracic cage. However, when compared to the resistivity of the lungs, the thoracic cage resistance is rather negligible.

[0101] Thus, considering the transthoracic defibrillation model of FIG. 4, there are several other resistances in addition to those discussed for the charge burping model hereinabove. R_srepresents the resistance of the defibrillation system, including the resistance of the electrodes. R_CWand R_LSrepresent the resistances of the chest wall and the lungs, respectively, in series with resistance of the heart, R_H. R_TCand R_LPrepresent the resistances of the thoracic cage and the lungs, respectively, in parallel with the resistance of the heart.

[0102] As discussed hereinabove, developing design equations which enable adjustments for variable resistances encountered in the transthoracic defibrillation model of FIG. 4 is one of the advances of the present invention. In order to adjust for variable R_Hboth φ₁and φ₂are assumed fixed. Then φ₂is selected to have a range of capacitance values which permit to optimize the slope of the voltage curve at time t. In other words, C_S2for φ₂is chosen such that

\frac{ⅆ v}{ⅆ t} = 0

[0103] The design parameters of the present invention are derived from equation 35, as follows:

[0104] V_M(t)=V_O(1−e^-). (35)

[0105] Equation 35 can be rewritten as:

\begin{matrix} V_{M2} (t) = (V_{φ1} + {\frac{τ_{2}}{τ_{2} - τ_{M}}} V_{2}) e \frac{- τ}{τ_{M}} - (\frac{τ_{2}}{τ_{2} - τ_{M}}) V_{2} ⅇ^{- t / τ_{2}} = 0. Letting A = V φ_{1} + B and B = \frac{τ_{2}}{τ_{2} - τ_{M}} V_{2}, & (36) \end{matrix}

[0106] equation 36 can be written as:

\begin{matrix} V (t) = Ae \frac{- t}{τ_{M}} - Be \frac{- t}{τ_{2}} . & (37) \end{matrix}

[0107] Differentiating equation 37 with respect to t, we have the following:

\begin{matrix} \frac{ⅆ v}{ⅆ t} = - \frac{A ⅇ^{- t / τ_{M}}}{τ_{M}} + \frac{B}{τ_{2}} ⅇ^{- t / τ_{2}} = 0. & (38) \end{matrix}

[0108] Equation 38 is the profile of φ₁waveform and at

\frac{ⅆ v}{ⅆ t} = 0,

[0109] the slope of the curve is zero, which means the terminal value of the time constant is determinable at this point.

[0110] Thus, solving equation 38 for the value of t, we have:

\begin{matrix} t = \frac{t_{2}}{τ_{2} - τ_{M}} τ_{M} \cdot \ln {\frac{τ_{2}}{τ_{M}} \frac{(V_{φ 1} + \frac{τ_{2}}{τ_{2} - τ_{M}} V_{2})}{(\frac{τ_{2}}{τ_{2} - τ_{M}}) V_{2}}} where & (39) \\ t_{1} = (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}}) \ln {1 + (\frac{τ_{2} - τ_{M}}{τ_{2}}) (\frac{V_{φ 1}}{V_{2}})} & (40) \\ t_{1} = t_{1} + (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \ln {\frac{τ_{2}}{τ_{M}}}) . & (41) \end{matrix}

[0111] For biphasic defibrillation waveforms, it is generally accepted that the ratio of φ₁, duration (τ_m) to φ₂duration (τ₂) should be ≧1. Charge burping theory postulates that the beneficial effects of φ₂are maximal when it completely removes the charge deposited on myocardial cell by φ₁. This theory predicts that φ₁/φ₂should be >1 when τ_sis >3 ms and <1 when τ_s<3 ms. τ_sis defined as the product of the pathway resistance and capacitance. (See NASPE ABSTRACTS, Section 361 entitled Charge Burping Predicts Optimal Ratios of Phase Duration for Biphasic Defibrillation, by Charles D. Swerdlow, M.D., Wei Fan, M.D., James E. Brewer, M.S., Cedar-Sinai Medical Center, Los Angeles, Calif.

[0112] In light of the proposed duration ratio of φ₁and φ₂, wherein the optimal solution is indicated to be at t₁=t₂where t is the duration of φ₁and t₂is the duration of φ₂and superimposing this condition on equation 41 hereinabove, we have:

\begin{matrix} t_{2} = t_{1} + (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \cdot \ln {\frac{τ_{2}}{τ_{M}}}) setting t_{2} = t_{1}, remanaging terms we have: & (42) \\ t_{2} - t_{1} = 0 = \frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \cdot \ln {\frac{τ_{2}}{τ_{M}}} or τ_{2} τ_{M} \cdot \ln (\frac{τ_{2}}{τ_{M}}) = 0 or \ln \frac{τ_{2}}{τ_{M}} = 0 or \frac{τ_{2}}{τ_{M}} = 1. \end{matrix}

[0113] From the result of equation 42 we make the final conclusion that the optimal charge burping is obtained when τ₂=τ_m. From prior definition, we have established that τ_m=R_H·C_S. Thus, in accordance with equation 42, τ₂=τ_m=R_H·C_S.

[0114] Referring now to FIG. 6, a biphasic defibrillation waveform generated using the equations 35-42 is shown. At V_m=0 and dV_m/dt=0, φ₁and φ₂are equal to zero.

[0115] FIG. 7 is a schematic of a circuit which enables the implementation of the theory developed in the present invention. The circuit shows a plurality of double throw switches connecting a plurality of capacitors. The capacitors and the switches are connected to a charge or potential source. The voltage is discharged via electrodes. One aspect of implementing the “soft landing” charge burping technique developed in the present invention is to fix C_Sfor φ₁and fix C_Sfor φ₂. Further, τ_mis fixed. Then a range of resistance values representing R_Hare selected. The τ_mand R_Hranges represent the patient variability problem. The objective is to enable corrective action such that C_Svalues could range between 40 mf-200 mf and dv/dt=0 for τ₂. As indicated hereinabove the error in changing burping is minimized for τ₂when dv/dt=0.

[0116] The implementation of the present invention requires that capacitor bank values be determined for φ₁and φ₂. Specifically, the capacitor values for φ₁should be designed to realize dv/dt=0 and V=0 for φ₂to minimize charge burping error due to R_Hand τ_m. Where a variable resistor is used to set R_Hthus providing a known but variable value and τ_mcan be set within these practical ranges.

[0117] FIG. 8 depicts a biphasic defibrillation waveform 300, generated using equations 35-42 above, in relation to a predicted patient's cellular response curve 304; the cellular response, as explained earlier is based on a patient's measured impedance. As shown, the residual charge left on the cardiac cells after delivery of φ₁has been brought back to zero charge, i.e. charge balanced, after the delivery of φ₂. This charge balance has been achieved because the energy delivered has been allowed to vary.

[0118] Traditionally, all defibrillation waveforms have delivered a fixed energy. This has been primarily because therapy has been measured in joules. These fixed energy waveforms, which include monophasic damped sine, biphasic truncated exponential, and monophasic truncated exponential waveforms, only passively responded to patient and/or system impedance.

[0119] This passive response was by charging a capacitor to a fixed voltage, wherein the voltage was adjusted depending on amount of fixed energy to be delivered that was desired, and discharging it across the patient, whom acts as the load. The energy delivered is controlled by the simple equation:

\begin{matrix} E = \frac{1}{2} {CV}^{2} & (43) \end{matrix}

[0120] where: E is the energy, C is the capacitance of the charging capacitor, and V is the voltage. In the case of the truncated exponential waveform, equation 43 is extended to:

\begin{matrix} E = \frac{1}{2} C (V_{i}^{2} - V_{f}^{2}) & (44) \end{matrix}

[0121] where: V_iis the initial voltage and V_fis the final voltage. Significantly, no form of patient impedance or system impedance appears in either of the above equations.

[0122] With respect to system impedance, the above equations make the assumption that the internal impedance of the defibrillator is 0 ohms. This is a good approximation for truncated exponential waveforms, but it is not a good approximation for damped sine waveforms. Damped sine waveforms typically have 10-13 ohms internal impedance. This internal impedance effects the delivered energy. The internal impedance of the defibrillator will absorb a portion of the stored energy in the capacitor, thus reducing the delivered energy. For low patient impedances, the absorbed energy can become quite significant, with on the order of 40% of the stored energy being absorbed by the internal resistances.

[0123] With respect to patient impedance, the patient acts as the load and, as such, the peak current is simply a function of the peak voltage via Ohm's law:

\begin{matrix} I = \frac{V}{R} & (45) \end{matrix}

[0124] where: I is the current and R represents the patient's impedance. As equation 45 indicates, the current is inversely proportional to the impedance. This means that there is lower current flow for high impedance patients which, in turn, means that it takes longer to deliver the energy to a high impedance patient. This fact is further exemplified when considering the equation for a truncated exponential voltage at any point in time:

\begin{matrix} V (t) = \frac{1}{2} {CV}_{i} ⅇ^{- \frac{t}{RC}} & (46) \end{matrix}

[0125] Equation 46 shows that patient impedance, R, is reflected within in the voltage equation which forms a part of the energy equation. Equation 46 shows that the duration of a defibrillation waveform extends passively with impedance, since it takes longer to reach the truncate voltage.

[0126] To actively respond to system and patient impedances, as the present invention does with its charge burping model, the energy must be allowed to vary. To explain further, in the charge balance waveform of the present invention, there is a desire to exactly terminate the defibrillation waveform when the cellular response curve returns to zero charge, see again FIG. 8 Failing to terminate the defibrillation waveform when the cellular response curve returns to zero charge, e.g., overshooting or undershooting the neutral condition, can promote refibrillation.

[0127] In achieving this charge balanced waveform, as explained in detail in the specification above: (1) the charging capacitor is charged to a specific charge voltage, i.e., this specific charge voltage is not a system variable; (2) the current is controlled by the patient impedance

(I = \frac{V}{R}),

[0128] i.e., the current is not a system variable; and (3) the duration of the defibrillation pulse is controlled by the expected cellular response curve, i.e., duration is not a system variable. Because items 1-3 are not system variables but rather are preset or set in accordance to the patient at hand, the energy must be allowed to vary or the defibrillation system is over constrained and charge balancing cannot be achieved over the range of patient impedances. To deliver fixed energy in a charge balanced system would require the ability to vary the charge voltage. Varying the charge voltage is typically not done since this requires knowing the exact impedance in advance of the shock delivery. Many defibrillators measure the impedance during a high voltage charge delivery since this is more accurate than low voltage measurements done prior to shock delivery. Varying charge voltage also requires a more expensive defibrillation circuitry since capacitor costs increase dramatically with voltage.

[0129] It should be noted that due to the characteristics of the cellular response curve, the duration of the defibrillation pulse does increase slightly for increases in patient impedance. This increased duration is of a much lesser effect than in a traditional truncated waveform.

[0130] Although the present invention has been described with reference to preferred embodiments, workers skilled in the art will recognize that changes may be made in form and detail without departing from the spirit or scope of the present invention.

Claims

What is claimed:

1. A method for determining the second phase of a biphasic defibrillation shock pulse, said second phase having variable energy, wherein upon application of said second phase of said biphasic defibrillation shock pulse a desired response is produced in a patient's cardiac cell membrane, comprising:

providing a quantitative model of a defibrillator circuit for producing said biphasic defibrillation shock pulse;

providing a quantitative model of a patient that includes a variable heart component;

providing a quantitative description of a predetermined response of said cardiac cell membrane to said shock pulse;

determining a quantitative description of a first phase of said biphasic defibrillation shock pulse that will produce said predetermined response of said cardiac cell membrane, wherein the determination is made as a function of said predetermined response of said cardiac cell membrane, said quantitative model of a defibrillator circuit, and said quantitative model of a patient; and

determining a quantitative description of a second phase of said biphasic defibrillation shock pulse based on said first phase, wherein said quantitative description provides for setting a time duration for said second phase based on said variable heart component whereby an amount of energy to be delivered by said second phase varies according to said time duration that is set.

2. The method of claim 1, wherein said predetermined response has a first instance of zero charge and a second instance of zero charge, and wherein said quantitative description of said first phase provides for initiating said first phase at a time of said first instance of zero charge and said quantitative description of said second phase provides for setting said time duration to end at a time of said second instance of zero charge.

3. The method of claim 1, wherein said quantitative model of a patient further includes a chest component, and a cardiac cell membrane component.

4. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having at least one defibrillation system resistance component.

5. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having at least one chest wall resistance component.

6. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a thoracic cage resistance component.

7. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a one lung series resistance component.

8. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a one lung parallel resistance.

9. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a chest wall resistance component, a thoracic cage resistance component, and a lung resistance component.

10. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a lung parallel resistance component connected in parallel with said variable heart component.

11. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a thoracic cage resistance component connected in parallel with said variable heart component.

12. The method of claim 1, wherein the step of providing a quantitative model of a defibrillator circuit is carried out with a circuit having first and second capacitors.

13. The method of claim 1, wherein the step of providing a quantitative model of a patient is carried out with a circuit having a lung parallel resistance component and a thoracic cage resistance component connected in parallel with said variable heart component.

14. A defibrillation apparatus for delivering a biphasic defibrillation shock pulse to produce a desired response in a patient's cardiac cell membrane, wherein a second phase of said biphasic defibrillation shock pulse has a variable energy, comprising:

a defibrillator circuit; and

a pair of electrodes operably coupled to said defibrillator circuit, wherein said pair of electrodes delivers said biphasic defibrillation shock pulse to said patient's cardiac cell membrane,

wherein said defibrillator circuit has been configured according to:

a quantitative model of a defibrillator circuit for producing said biphasic defibrillation shock pulse;

a quantitative model of a patient that includes a variable heart component; and

a quantitative description of a predetermined response of said cardiac cell membrane to said shock pulse;

a quantitative description of a first phase of said biphasic defibrillation shock pulse that will produce said predetermined response of said cardiac cell membrane, wherein said quantitative description of said first phase has been determined from said predetermined response of said cardiac cell membrane, said quantitative model of said defibrillator circuit, and said quantitative model of said patient; and

a quantitative description of a second phase of said biphasic defibrillation shock pulse wherein said quantitative description of said second phase has been determined based on said first phase and wherein said quantitative description of said second phase includes a time duration setting that has been based on said variable heart component whereby an amount of energy to be delivered by said second phase varies according to said time duration setting.

15. The apparatus of claim 14, wherein said predetermined response has a first instance of zero charge and a second instance of zero charge, and wherein said quantitative description of said first phase includes initiating said first phase at a time of said first instance of zero charge and said quantitative description of said second phase includes said time duration setting that ends said second phase at a time of said second instance of zero charge.

16. The apparatus of claim 14, wherein said quantitative model of a patient further includes a chest component and a cardiac cell membrane component.

17. The apparatus of claim 14, wherein said defibrillator circuit includes a defibrillation system resistance component.

18. The apparatus of claim 14, wherein said defibrillator circuit includes a chest wall resistance component.

19. The apparatus of claim 14, wherein said defibrillator circuit includes a thoracic cage resistance component.

20. The apparatus of claim 14, wherein said defibrillator circuit has a lung series resistance component.

21. The apparatus of claim 14, wherein said defibrillator circuit has a lung parallel resistance component.

22. The apparatus of claim 14, wherein said defibrillator circuit has a chest wall resistance component, a thoracic cage resistance component, and a lung resistance component.

23. The apparatus of claim 14, wherein said defibrillator circuit has a lung parallel resistance component connected with parallel with a variable heart resistance component.

24. The apparatus of claim 14, wherein said defibrillator circuit has a thoracic cage resistance component connected in parallel with a variable heart resistance component.

25. The apparatus of claim 14, wherein said defibrillator circuit has a first and second capacitor.

26. The apparatus of claim 14, wherein said defibrillator circuit has a lung parallel resistance component and a thoracic cage resistance component connected in parallel with a variable heart resistance component.

27. A defibrillation apparatus for delivering a biphasic defibrillation shock pulse to produce a desired response in a patient's cardiac cell membrane, wherein a second phase of said biphasic defibrillation shock pulse has a variable energy, comprising:

an electronic circuit having a defibrillator portion and a patient modeling portion; and

a pair of electrodes operably coupled to said electronic circuit, wherein said pair of electrodes delivers said biphasic defibrillation shock pulse to said patient's cardiac cell membrane,

wherein said electronic circuit has been configured according to:

a quantitative model of a defibrillator circuit for producing said biphasic defibrillation shock pulse;

a quantitative model of a patient that includes a variable heart component; and

a quantitative description of a predetermined response of said cardiac cell membrane to said shock pulse;

28. The apparatus of claim 27, wherein said predetermined response has a first instance of zero charge and a second instance of zero charge, and wherein said quantitative description of said first phase includes initiating said first phase at a time of said first instance of zero charge and said quantitative description of said second phase includes said time duration setting that ends said second phase at a time of said second instance of zero charge.

29. The apparatus of claim 27, wherein said quantitative model of a patient further includes a chest component and a cardiac cell membrane component.

30. The apparatus of claim 27, wherein said patient modeling portion includes a defibrillation system resistance component.

31. The apparatus of claim 27, wherein said patient modeling portion includes a chest wall resistance component.

32. The apparatus of claim 27, wherein said patient modeling portion includes a thoracic cage resistance component.

33. The apparatus of claim 27, wherein said patient modeling portion has a lung series resistance component.

34. The apparatus of claim 27, wherein said patient modeling portion has a lung parallel resistance component.

35. The apparatus of claim 27, wherein said patient modeling portion has a chest wall resistance component, a thoracic cage resistance component, and a lung resistance component.

36. The apparatus of claim 27, wherein said patient modeling portion has a lung parallel resistance component connected with parallel with a variable heart resistance component.

37. The apparatus of claim 27, wherein said patient modeling portion has a thoracic cage resistance component connected in parallel with a variable heart resistance component.

38. The apparatus of claim 27, wherein said defibrillator portion has a first and second capacitor.

39. The apparatus of claim 27, wherein said patient modeling portion has a lung parallel resistance component and a thoracic cage resistance component connected in parallel with a variable heart resistance component.

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Abstract

Description

Field of the Invention

Background of the Invention

Summary of the Invention

Brief Description of the Drawings

[0022] FIGS. 1aand 1bare perspective views of an AED according to the present invention.

[0023] FIG. 2ais a very simplified defibrillator model.

[0024] FIG. 2bis a known monophasic defibrillation model.

[0025] FIG. 3 is a known biphasic defibrillation model.

[0026] FIG. 4 represents a monophasic or biphasic capacitive-discharge external defibrillation model according to the present invention.

[0027] FIG. 5arepresents a monophasic capacitor-inductor external defibrillator model according to the present invention.

[0028] FIG. 5brepresents an alternative embodiment of a biphasic capacitor-inductor external defibrillator model according to the present invention.

[0029] FIG. 6 illustrates a biphasic waveform generator utilizing the present invention.

[0030] FIG. 7 is a schematic diagram of a circuit which enables the implementation of the present invention.

[0031] FIG. 8 illustrates a biphasic waveform in relation to a cellular response curve.

Description of the Preferred Embodiment

[0033] External Defibrillator Model

\begin{matrix} \frac{V_{1} - V_{S}}{R_{S}} + \frac{V_{1}}{R_{TC}} + \frac{V_{1} - V_{2}}{R_{CW}} = 0, & (1) \\ \frac{V_{2} - V_{1}}{R_{CW}} + \frac{V_{2}}{R_{LP}} + \frac{V_{2} - V_{3}}{R_{LS}} = 0, and & (2) \\ \frac{V_{3} - V_{2}}{R_{LS}} + \frac{V_{3}}{R_{H}} + \frac{V_{3} - V_{M}}{R_{M}} = 0. & (3) \end{matrix}

[0044] Equation 1 can be rewritten as

\begin{matrix} V_{1} (\frac{1}{R_{S}} + \frac{1}{R_{BC}} + \frac{1}{R_{CW}}) = \frac{V_{S}}{R_{S}} + \frac{V_{2}}{R_{CW}} . & (4A) \\ V_{1} = \frac{V_{S}}{R_{S} Ω_{1}} + \frac{V_{2}}{R_{CW} Ω_{1}}, where Ω_{1} = \frac{1}{R_{S}} + \frac{1}{R_{TC}} + \frac{1}{R_{CW}} . & (4B) \end{matrix}

[0045] Rewriting equation 2, we have

\begin{matrix} V_{2} (\frac{1}{R_{CW}} + \frac{1}{R_{LP}} + \frac{1}{R_{LS}}) = \frac{V_{1}}{R_{CW}} + \frac{V_{3}}{R_{LS}} . & (4C) \end{matrix}

[0046] By substituting equation 4B for V₁into equation 4C, we can solve for V₂as an expression of V_Sand V₃:

\begin{matrix} V_{2} = \frac{V_{S}}{R_{S} R_{CW} Ω_{1} Ω_{2} Ω_{22}} + \frac{V_{3}}{R_{LS} Ω_{2} Ω_{22}}, where Ω_{2} = \frac{1}{R_{LS}} + \frac{1}{R_{LP}} + \frac{1}{R_{CW}}, and Ω_{22} = 1 - \frac{1}{R_{CW}^{2} Ω_{1} Ω_{2}} . & (5) \end{matrix}

[0047] Now solving for V₃as an expression of V_Sand V_M, equation 3 may be re-arranged as

\begin{matrix} V_{3} (\frac{1}{R_{LS}} + \frac{1}{R_{H}} + \frac{1}{R_{M}}) = \frac{V_{2}}{R_{LS}} + \frac{V_{M}}{R_{M}} so that & (6) \\ V_{3} = \frac{V_{2}}{R_{LS} Ω_{3}} + \frac{V_{M}}{R_{M} Ω_{3}} where Ω_{3} = \frac{1}{R_{LS}} + \frac{1}{R_{H}} + \frac{1}{R_{M}} . & (7) \end{matrix}

[0048] Substituting equation 5 for V₂into equation 7, we can solve for V₃as an expression of V_Sand V_M:

\begin{matrix} V_{3} = \frac{V_{S}}{R_{S} R_{CW} R_{LS} Ω_{1} Ω_{2} Ω_{22} Ω_{3} Ω_{33}} + \frac{V_{M}}{R_{M} Ω_{3} Ω_{33}} where & (8) \\ Ω_{33} = 1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})} & (9) \end{matrix}

[0049] From equation 8 we define Ω_Mto be:

\begin{matrix} Ω_{M} = R_{M} Ω_{3} Ω_{33} = R_{M} Ω_{3} (1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})}) Ω_{M} = R_{M} (Ω_{3} - \frac{1}{R_{LS}^{2} (Ω_{2} - \frac{1}{R_{CW}^{2} Ω_{1}})}) . & (10) \end{matrix}

[0050] From equation 8 we also define Ω_Sto be:

[0051] Ω_S=R_SR_CWR_LSΩ₁Ω₂Ω₃Ω₂₂Ω₃₃ (11)

[0052]

\begin{matrix} Ω_{S} = R_{S} R_{CW} R_{LS} Ω_{1} Ω_{2} (1 - \frac{1}{(R_{CW}^{2} Ω_{1} Ω_{2})}) Ω_{3} (1 - \frac{1}{(R_{LS}^{2} Ω_{2} Ω_{22} Ω_{3})}) & (12) \\ Ω_{S} = R_{S} R_{CW} R_{LS} (Ω_{1} Ω_{2} - \frac{1}{R_{CW}^{2}}) (Ω_{3} - \frac{1}{R_{LS}^{2} (Ω_{2} - \frac{1}{R_{CW}^{2} Ω_{1}})}) & (13) \\ so that V_{3} = \frac{V_{S}}{Ω_{S}} + \frac{V_{M}}{Ω_{M}} & (14) \end{matrix}

[0055] As shown in FIGS. 5aand 5b, a second example would be a monophasic damped sine wave circuit, represented by

\begin{matrix} V_{S} = V_{1} (\frac{τ_{C1}}{τ_{C1} - τ_{L1}}) (ⅇ^{- t / τ_{C1}} - ⅇ^{- t / τ_{L1}}) & (14B) \end{matrix}

[0057] To proceed with step two, from FIG. 4, nodal analysis provides an equation for V_M:

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M} - V_{3}}{R_{M}} = 0. & (15) \end{matrix}

[0058] Rearranging equation 15, we have

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} = \frac{V_{3}}{R_{M}} . & (16) \end{matrix}

[0059] Next, substituting equation 14 as an expression for V₃into equation 16, the cell membrane response is now calculated as follows:

\begin{matrix} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} = \frac{1}{R_{M}} (\frac{V_{S}}{Ω_{S}} + \frac{V_{M}}{Ω_{M}}) & (17) \\ C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} - \frac{V_{M}}{R_{M} Ω_{M}} = \frac{V_{S}}{R_{M} Ω_{S}} C_{M} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{R_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{S}}{R_{M} Ω_{S}} & (18) \end{matrix}

[0060] Dividing through by C_M, and setting τ_M=R_MC_M, then equation 18 becomes

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{τ_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{S}}{τ_{M}} (\frac{1}{Ω_{S}}) . & (19) \end{matrix}

[0062] In the equations given below:

[0063] C₁equals the capacitance of the first capacitor bank and V_S=V₁e^-t/τ;

[0064] C₂equals the capacitance of the second capacitor bank and V_S=V₂e^-t/τ;

[0065] R=R_S+R_B, where R_S=System impedance (device and electrodes);

[0066] R_B=body impedance (thoracic cage, chest wall, lungs (series, parallel), heart).

[0067] To determine body impedance, R_B, we see that the series combination of R_Hand R_LSyields R_H+R_LS. (FIG. 4). The parallel combination of R_H+R_LSand R_LPyields:

\begin{matrix} \frac{R_{LP} (R_{LS} + R_{H})}{R_{LP} + R_{LS} + R_{H}} . & (20) \end{matrix}

[0068] The series combination of equation 20 and R_CWyields:

\begin{matrix} R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})} . & (21) \end{matrix}

[0069] The parallel combination of equation 21 and R_TCyields:

\begin{matrix} R_{B} = [\frac{R_{TC} [R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})}}{R_{TC} + R_{CW} + \frac{R_{LP} (R_{LS} + R_{H})}{(R_{LP} + R_{LS} + R_{H})}}] & (22) \end{matrix}

[0070] where R_Bis the impedance of the body for this model.

[0071] The discharge of a single capacitor is modeled by V_S=V₁e^-t/τfor an initial C₁capacitor voltage of V₁. Placing V_Sinto equation 19 gives:

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + \frac{V_{M}}{τ_{M}} (1 - \frac{1}{Ω_{M}}) = \frac{V_{1} ⅇ^{- t / τ_{1}}}{τ_{M} Ω_{S}} & (23) \end{matrix}

\frac{ⅆ y}{ⅆ x} + P (X) Y = Q (x) .

[0073] These linear ODEs have an integration factor that equals e^∫pdx. The general solution to such equations is:

[0074] Y=e^-∫pdx[∫e^∫pdxQdx+c].

\begin{matrix} V_{M1} (t) = k ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} + (\frac{V_{1}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{2}{Ω_{M}}) - τ_{M}}) ⅇ^{- t / τ_{1}} . & (24) \end{matrix}

\begin{matrix} k = V_{G} - (\frac{V_{o}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (25) \end{matrix}

[0077] Assuming τ₁=R_C1, where R=R_S+R_B, then the solution to the initial-value problem for φ₁is:

\begin{matrix} V_{M1} (t) = V_{G} ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} + (\frac{V_{1}}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- t / τ_{1}} - ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})}) & (26) \end{matrix}

[0078] Equation 26 describes the residual voltage found on a cell at the end of φ₁.

[0079] Assuming V_G=0 and V₁=1, the solution for cell response to an external shock pulse is

\begin{matrix} V_{M1} (t) = (\frac{1}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- \frac{t}{τ_{1}}} - ⅇ^{- (\frac{t}{τ_{M}}) (1 - \frac{1}{Ω_{M}})}) . & (27) \end{matrix}

\begin{matrix} (\frac{AB}{τ_{M}}) ⅇ^{- Bt / τ_{M}} - (\frac{A}{τ_{1}}) ⅇ^{- t / τ_{1}} = 0, where A = (\frac{1}{Ω_{S}}) (\frac{τ_{1}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) and B = 1 - \frac{1}{Ω_{M}} . & (27B) \end{matrix}

[0081] Solving for t, the optimal duration dφ₁for a monophasic shock pulse or φ₁of a biphasic shock pulse is found to be

\begin{matrix} d φ_{1} = (\frac{τ_{1} τ_{M}}{τ_{1} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) \ln (\frac{τ_{1} (1 - \frac{1}{Ω_{M}})}{τ_{M}}), & (27C) \end{matrix}

[0082] where “1n” represents the logarithm to the base e, the natural logarithm.

\begin{matrix} \frac{ⅆ V_{M}}{ⅆ t} + (\frac{V_{M}}{τ_{M}}) (1 - \frac{1}{Ω_{M}}) = \frac{- V_{2} ⅇ^{- t / τ_{2}}}{τ_{M} Ω_{S}} . & (29) \end{matrix}

[0085] Equation 29 is again a first-order linear ODE. In a similar manner, its general solution is determined to be:

\begin{matrix} V_{M2} (t) = k ⅇ^{(- t / τ_{M}) (1 - \frac{1}{Ω_{M}})} - (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (30) \end{matrix}

[0086] To determine the constant of integration k, the value of V_M2at the end of φ₁is

[0087] V_M2(0)=V_M1(d_φ1)=V_φ1′ (31)

[0088] where d_φ1is the overall time of discharge for φ₁and V_φ1is the voltage left on the cell at the end of φ₁. Applying the initial condition to equation 30 and solving for k:

\begin{matrix} k = V_{φ 1} + (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) . & (32) \end{matrix}

[0089] The solution to the initial-value problem for φ₂is

\begin{matrix} V_{M2} (t) = (\frac{V_{2}}{Ω_{S}}) (\frac{τ_{2}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) (ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} - ⅇ^{- t / τ_{2}}) + V_{φ 1} ⅇ^{- (t / τ_{M}) (1 - \frac{1}{Ω_{M}})} . & (33) \end{matrix}

\begin{matrix} d_{φ2} = (\frac{τ_{2} τ_{M}}{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}) \cdot \ln (1 + (\frac{τ_{2} (1 - \frac{1}{Ω_{M}}) - τ_{M}}{τ_{2}}) (\frac{Ω_{S} V_{φ 1}}{V_{2}})) . & (34) \end{matrix}

[0091] From the equations above, an optimal monophasic or biphasic defibrillation waveform may be calculated for an external defibrillator.

[0094] Present Invention

\frac{ⅆ v}{ⅆ t} = 0

[0103] The design parameters of the present invention are derived from equation 35, as follows:

[0104] V_M(t)=V_O(1−e^-). (35)

[0105] Equation 35 can be rewritten as:

\begin{matrix} V_{M2} (t) = (V_{φ1} + {\frac{τ_{2}}{τ_{2} - τ_{M}}} V_{2}) e \frac{- τ}{τ_{M}} - (\frac{τ_{2}}{τ_{2} - τ_{M}}) V_{2} ⅇ^{- t / τ_{2}} = 0. Letting A = V φ_{1} + B and B = \frac{τ_{2}}{τ_{2} - τ_{M}} V_{2}, & (36) \end{matrix}

[0106] equation 36 can be written as:

\begin{matrix} V (t) = Ae \frac{- t}{τ_{M}} - Be \frac{- t}{τ_{2}} . & (37) \end{matrix}

[0107] Differentiating equation 37 with respect to t, we have the following:

\begin{matrix} \frac{ⅆ v}{ⅆ t} = - \frac{A ⅇ^{- t / τ_{M}}}{τ_{M}} + \frac{B}{τ_{2}} ⅇ^{- t / τ_{2}} = 0. & (38) \end{matrix}

[0108] Equation 38 is the profile of φ₁waveform and at

\frac{ⅆ v}{ⅆ t} = 0,

[0109] the slope of the curve is zero, which means the terminal value of the time constant is determinable at this point.

[0110] Thus, solving equation 38 for the value of t, we have:

\begin{matrix} t = \frac{t_{2}}{τ_{2} - τ_{M}} τ_{M} \cdot \ln {\frac{τ_{2}}{τ_{M}} \frac{(V_{φ 1} + \frac{τ_{2}}{τ_{2} - τ_{M}} V_{2})}{(\frac{τ_{2}}{τ_{2} - τ_{M}}) V_{2}}} where & (39) \\ t_{1} = (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}}) \ln {1 + (\frac{τ_{2} - τ_{M}}{τ_{2}}) (\frac{V_{φ 1}}{V_{2}})} & (40) \\ t_{1} = t_{1} + (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \ln {\frac{τ_{2}}{τ_{M}}}) . & (41) \end{matrix}

\begin{matrix} t_{2} = t_{1} + (\frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \cdot \ln {\frac{τ_{2}}{τ_{M}}}) setting t_{2} = t_{1}, remanaging terms we have: & (42) \\ t_{2} - t_{1} = 0 = \frac{τ_{2} τ_{M}}{τ_{2} - τ_{M}} \cdot \ln {\frac{τ_{2}}{τ_{M}}} or τ_{2} τ_{M} \cdot \ln (\frac{τ_{2}}{τ_{M}}) = 0 or \ln \frac{τ_{2}}{τ_{M}} = 0 or \frac{τ_{2}}{τ_{M}} = 1. \end{matrix}

[0114] Referring now to FIG. 6, a biphasic defibrillation waveform generated using the equations 35-42 is shown. At V_m=0 and dV_m/dt=0, φ₁and φ₂are equal to zero.

\begin{matrix} E = \frac{1}{2} {CV}^{2} & (43) \end{matrix}

[0120] where: E is the energy, C is the capacitance of the charging capacitor, and V is the voltage. In the case of the truncated exponential waveform, equation 43 is extended to:

\begin{matrix} E = \frac{1}{2} C (V_{i}^{2} - V_{f}^{2}) & (44) \end{matrix}

[0121] where: V_iis the initial voltage and V_fis the final voltage. Significantly, no form of patient impedance or system impedance appears in either of the above equations.

[0123] With respect to patient impedance, the patient acts as the load and, as such, the peak current is simply a function of the peak voltage via Ohm's law:

\begin{matrix} I = \frac{V}{R} & (45) \end{matrix}

\begin{matrix} V (t) = \frac{1}{2} {CV}_{i} ⅇ^{- \frac{t}{RC}} & (46) \end{matrix}

(I = \frac{V}{R}),