the following description is of the best mode currently contemplated for practicing the invention . the basic concept of the invention relating to forming an efficient defibrillation waveform can be practiced with two or more capacitors within the icd . a preferred number of capacitors is three . however , the basic concept will first be explained in the context of a two - capacitor icd . in accordance with one aspect of the invention , then a biphasic pulse or waveform is generated by an icd device having two capacitors that includes a positive phase of duration t 1 ms and a negative phase of duration t 2 ms , as shown in fig1 . first and second capacitors , c a and c b , within the icd device are initially charged to a voltage v 1 and are connected in parallel . the biphasic defibrillation pulse begins by discharging the charged parallel capacitors through the cardiac tissue by way of defibrillation electrodes in contact with the cardiac tissue . thus , a leading edge of the biphasic pulse starts at a first peak voltage of approximately v 1 volts ( the charge on the first and second capacitors when first connected to the electrodes ). during a first portion of the positive phase of the biphasic pulse , the amplitude of the biphasic pulse decays from the first peak voltage v 1 to a voltage v 2 in accordance with a first time constant τ 1 . the first time constant τ 1 varies as a function of ( c a + c b ) r , where c a is the value of the first capacitor , c b is the value of the second capacitor , and r is an effective resistance associated with the discharge through the first and second electrodes . a second portion of the positive phase begins by connecting the first and second capacitors in series . this sudden series connection increases the defibrillation pulse to a second peak voltage of approximately 2 ( v 2 ) volts ( the sum of the voltages on each of the first and second capacitors at the time the series connection is made ), as illustrated in fig1 . the amplitude of the biphasic pulse decays during the second portion of the positive phase from the second peak voltage 2 ( v 2 ) to a voltage v 3 in accordance with a second time constant τ 2 . the second time constant τ 2 varies as a function of ( c a c b / c a c b ) ) r . advantageously , the voltage at the trailing edge of the positive phase , v 3 , occurs at a time that is near the maximum cell membrane response . the negative phase of the biphasic waveform begins by inverting the polarity of the series - connected first and second capacitors . such negative phase thus commences at a third peak voltage of approximately − v3 volts , and decays thereafter towards zero in accordance with the second time constant τ 2 . after a prescribed time period t 2 , the negative phase ends . the biphasic waveform produced in accordance with the two - capacitor icd is illustrated in fig1 . the first portion of the positive phase may terminate when either : ( 1 ) the voltage decreases below a threshold voltage v 3 ; or ( 2 ) a prescribed time period t a has elapsed . the tissue membrane voltage that results when the waveform of fig1 is applied to excitable cardiac tissue membranes is as shown in fig2 . this membrane voltage is obtained by modeling the tissue membranes as taught in the blair reference , previously cited . as shown in fig1 - 20 , the optimum duration for t a will be described in more detail . a functional block diagram of the pulse generation circuitry used to generate the biphasic waveform of the two - capacitor icd is shown in fig3 . as seen in fig3 a cardiac tissue - stimulating device 10 includes a power source 12 , e . g ., at least one battery , a timing and control circuit 14 , a charging circuit 16 , an isolation switch network sw 1 , a series parallel switch network sw 2 , at least two capacitors c a and c b , an output switch network sw 3 , and at least two electrodes 20 and 22 . the electrodes 20 and 22 are adapted to be positioned within or on the heart . the electrodes 20 and 22 are connected to the output switch sw 3 through conventional leads 21 and 23 , respectively . a voltage sense amplifier 24 senses the voltage held on the capacitor c b ( which will be the same voltage as capacitor c a when c a and c b are connected in parallel ). in some embodiments of the invention , a current sense amplifier 26 may also be used to sense the current flowing to or returning from one of the electrodes 20 or 22 . in fig3 such current is sensed by differentially measuring the voltage across a small current - sense resistor r s connected in series with electrode 22 . the outputs of the voltage sense amplifier 24 and the current sense amplifier 26 are directed to the timing and control circuit 14 . a suitable cardiac activity sensor 28 is also employed within the device 10 in order to detect cardiac activity . the function of the sensor 28 is to sense cardiac activity so that an assessment can be made by the timing and control circuitry whether a defibrillation pulse needs to be generated and delivered to the cardiac tissue . such sensor 28 may take many forms , e . g , a simple r - wave sense amplifier of the type commonly employed in implantable pacemakers . the details of the sensor 28 are not important for purposes of the present invention . the power source 12 is connected to provide operating power to all components and circuitry within the device 10 . the power source 12 also provides the energy needed to generate the biphasic defibrillation pulse . that is , energy stored within the power source 12 is used to charge capacitors c a and c b , through the charging circuit 18 , up to the desired initial defibrillation starting pulse voltage v 1 . such charging is carried out under control of the timing and control circuit 14 . typically , v 1 may be a relatively high voltage , e . g ., 350 volts , even though the power source 12 may only be able to provide a relatively low voltage , e . g ., 3 - 6 volts . the charging circuit 16 takes the relatively low voltage from the power source 12 and steps it up to the desired high voltage v 1 , using conventional voltage step - up techniques as are known in the art . this stepped - up voltage v 1 is then applied through the isolation switch sw 1 to both capacitors c a and c b at a time when c a and c b are connected in parallel , i . e ., when sw 2 is in its “ p ” position , and at a time when the output switch is in its open , or off , position . as the capacitors c a and c b are being charged , the voltage sense amplifier 24 monitors the voltage level on the capacitors . when the desired voltage v 1 has been reached , the timing and control circuitry 14 turns off the charging circuit 16 and opens the isolation switch sw 1 , thereby holding the voltage v 1 on capacitors c a and c b until such time as a defibrillation pulse is needed . when a defibrillation pulse is called for by the timing and control circuit 14 , the output switch sw 3 is placed in its positive phase position , pos , thereby connecting the parallel connected capacitors c a and c b ( on which the starting voltage v 1 resides ) to the cardiac tissue through the electrodes 20 and 22 . such connection starts the discharge of capacitors c a and c b through the cardiac tissue in accordance with the first time constant τ 1 as described above in connection in fig1 . after a period of time t a , or as soon as the voltage across the parallel - connected capacitors c a and c b has decreased to the threshold value v 2 ( as sensed by the voltage sense amplifier 24 ), the timing and control circuit switches sw 2 to its series - connected or “ s ” position , thereby connecting the capacitors c a and c b in series across the electrodes 20 and 22 . such series connection doubles the voltage across the electrodes 20 and 22 to a value of 2 ( v 2 ) thereafter , the discharge of the series - connected capacitors c a and c b continues through the cardiac tissue in accordance with the second time constant τ2 as described above . this discharge continues until the end of the positive phase . the positive or first phase ends at a time t 1 from the beginning of the positive phase ( as measured by timing circuits within the timing and control circuit 14 ), or when the voltage has decayed to a value v 3 ( as sensed by voltage sense amplifier 24 ). alternatively , the positive phase may end as a function of the sensed current ( as sensed by the current sense amplifier 26 ), e . g ., at a time when the sensed current has decreased from a peak value by a prescribed amount or percentage . as soon as the positive phase ends , the timing and control circuit 14 switches the output switch sw 3 to the negative phase position , neg , thereby reversing the polarity of the discharge of the series - connected capacitors c a and c b through the cardiac tissue . the negative phase lasts thereafter for a time period t 2 determined by the timing and control circuitry . the functions represented by the functional block diagram of fig3 may be implemented by those of skill in the art using a wide variety of circuit elements and components . it is not intended that the present invention be directed to a specific circuit , device or method ; but rather that any circuit , device or method which implements the functions described above in connection with fig3 to produce a defibrillation waveform of the general type shown in fig1 be covered by the invention . turning next to fig4 there is shown a simplified schematic diagram of an icd having three 120 μf capacitors c 1 , c 2 and c 3 . the manner of charging the capacitors while they are connected in parallel is the same or similar to that shown in fig3 . when the capacitors c 1 , c 2 and c 3 have been charged to a high voltage , e . g ., 370 v , a stored energy of approximately 25 joules is realized . once the capacitors have been charged by the icd , the capacitors are configured for a parallel discharge . this is accomplished by closing switches s 1 , s 2 , s 3 and s 4 , while maintaining switches s 5 and s 6 open . the parallel discharge takes place from time t = 0 until a time d 1 . once d 1 elapses , one of two options may be used to discharge the remaining charge . in accordance with a first option , or option 1 , after d 1 has elapsed ( i . e ., after the capacitors are discharged in parallel until time d 1 ), all of the capacitors are discharged in series for the remainder of the pulse . this is accomplished by opening s 1 , s 2 , s 3 and s 4 and closing s 5 and s 6 . at a later time , d 2 , the “ h bridge ” circuit 40 ( fig4 ) is used to reverse the polarity of the output . at yet a later time , d , the output pulse is truncated . the waveform generated in accordance with option 1 is illustrated in fig5 . the tissue membrane voltage associated with the waveform of fig5 is modeled and computed , using the blair model , as shown in fig6 . for the example shown in fig5 and 6 , the optimum value of d 1 is nominally about 3 . 5 ms . the optimum choice of d 2 is when the elapsed time at d 2 is about 1 . 5 times the elapsed time at d 1 , or when the elapsed time at d 2 ( from t = 0 ) is about 5 . 25 ms . in accordance with a second option , or option 2 , the capacitors c 1 and c 2 remain in parallel and are in series with c 3 until time d 2 . this is accomplished by opening s 3 and s 4 and closing s 6 . after d 2 all the capacitors are in series ( s 1 and s 2 also open , 55 closed ) until c 3 runs out of charge at a time d 4 . after d 4 , the diode d 1 bypasses the depleted capacitor and the time constant of discharge is of c 1 and c 2 in series . at a time d 3 , where d 2 & lt ; d 3 & lt ; d 4 , the polarity of the output is reversed using the h bridge 40 . the pulse is truncated at time d . the resulting waveform is shown in fig7 . the resulting membrane voltage is modeled and computed and shown in fig8 . for the example shown in fig7 and 8 , the optimum values of d , is 2 . 7 ms , d 2 is 1 . 5 times d 1 ( or about 4 ms ) , d 3 is d 2 + 1 . 25 ms . the value of d 4 is computed to be about 7 . 6 ms . the choice of d can be in the range of 1 . 5 to 2 . 0 times that of d 3 . with either option 1 or option 2 , the choice of the values d 1 , d 2 and d 3 are primarily functions of the icd &# 39 ; s capacitance value , the discharge pathway impedance , and the tissue time constant ( τ m ). the advantage of option 2 is that the peak waveform voltage is lower than option 1 yet a minute increase in membrane voltage over option 1 is achieved . however , option 1 is simpler to implement and diode d 1 is not needed since all the capacitors are discharged equally . the advantages of either option 1 or option 2 are better appreciated by comparing the results of such discharge , as presented in fig5 , 7 and 8 , with the corresponding discharge achieved with a two - capacitor icd series discharge , as is commonly used in a conventional icd of the prior art . the discharge waveform achieved with a conventional two - capacitor icd using series discharge , and the resulting membrane voltage , is shown in fig9 and 10 , respectively . note , that to store equal energy to the three capacitor icd , each capacitor of the two - capacitor icd must have 1 . 5 times the capacitance value , or two capacitors each with c = 180 μf . as can be seen from a comparison of fig9 and 10 with fig5 and 6 ( option 1 ), and 5 a and 5 b ( option 2 ), for equal stored energy , the value of the peak membrane voltage for option 2 is 1 . 18 times higher than the membrane voltage realized using the conventional waveform . similarly , option 1 yields a membrane voltage that is 1 . 17 times higher than is realized using the conventional waveform . in other words , a 25 joule icd with three 120μf capacitors and a switching network as in option 2 performs equally to a 34 . 4 joule conventional icd with two 180μf capacitors . this represents a remarkable improvement in performance . as shown in fig1 , the two - step waveform has been reproduced . although identical in nature to that shown in fig1 the designators have been changed slightly for purposes of the in depth analysis that will follow . as described above in conjunction with fig3 two capacitors , c a & amp ; c b , have been charged to the same initial voltage , v 01 . the system resistance ( as seen by device ) is given by r s . for purposes of this discussion , the myocardium has been modeled as a parallel - rc circuit with myocardial tissue time constant , τ m . the amplitude of each step of the positive portion of the defibrillation waveform , shown in fig1 , can be characterized with the following basic equations : v s1 ( t 1 )= v 01 · exp [− t 1 / τ s1 ] 0 ≦ t 1 ≦ d 1 v s2 ( t 2 )= v 02 · exp [− t 2 / τ 2 ] 0 ≦ t 2 ≦ d 2 v s1 is the exponential decay during the first period , t 1 , ( i . e ., step 1 ); v s2 is the exponential decay during the second period , t 2 , ( i . e ., step 2 ); τ s1 is the time constant of c a and c b in parallel ; τ s2 is the time constant of c a and c b in series ; v 01 is the initial voltage during step 1 on the capacitors c a and c b once fully charged to the source voltage , v 01 ; and v 02 is the initial voltage during step 2 remaining on the capacitors c a and c b now configured in series . the analysis that follows directly will explain how to determine the absolute and approximate solutions for the optimal durations , d 1 and d 2 , to maximize induced myocardial potential , v m ( t ), when the two capacitors are arranged in a parallel - series , two - step arrangement . consider the myocardial responses to v s1 ( t 1 ) [ step 1 ] and v s2 ( t 2 ) [ step 2 ] separately . note that the following derivations ( equations 1 - 4 ) make absolutely no assumptions regarding any specific relationships between the characteristics of step 1 and step 2 . the “ step 1 ” myocardial response , v m1 , to the step 1 waveform , v s1 , is described by :  v m1  ( t 1 )  t 1 + v m1  ( t 1 ) τ m ∝ v s1  ( t 1 ) τ m ( eq . 1 ) v m1  ( t 1 ) = { v 01 α 1 · ( exp  [ - t 1 τ s1 ] - exp  [ - t 1 τ m ] ) τ s1 ≠ τ m v 01 τ s1 · ( t 1 · exp  [ - t 1 τ s1 ] ) τ s1 = τ m   where   α 1 = 1 - ( τ m / τ s1 ) . ( eq . 2 ) the “ step 2 ” myocardial response , v m2 , to the step 2 waveform , v s2 , is governed by :  v m2  ( d 1 , t 2 )  t 2 + v m2  ( d 1 , t 2 ) τ m ∝ v s2  ( t 2 ) τ m ( eq . 3 ) with the initial condition : v m2 ( d 1 , 0 )= v m1 ( d 1 ), where d 1 represents the final duration of step 1 . this initial condition ensures that there is a continuity of myocardial voltage when transitioning from the end of step 1 into the start of step 2 . the solution to this differential equation is : v m2  ( d 1 , t 2 ) = v m1  ( d 1 ) · exp  [ - t 2 τ m ] + { v 02  ( d 1 ) α 2 · ( exp  [ - t 2 τ s2 ] - exp  [ - t 2 τ m ] ) τ s2 ≠ τ m v 02  ( d 1 ) τ s2 · ( t 1 · exp  [ - t 2 τ s2 ] ) τ s2 = τ m ( eq . 4 ) where α 2 = 1 −( τ m / τ s2 ), and v 02 is proportional to v s2 ( 0 ) equation ( 4 ) describes a curve with a single maximum value . the step durations , d 1 = d 1 opt and d 2 = d 2 opt , that maximize this shock - induced myocardial voltage , v m2 ( t 1 , t 2 ) can be determined by solving the simultaneous equations given by : ∂ v m2  ( d 1 opt , d 2 opt ) ∂ d 1 opt = 0   ∂ v m2  ( d 1 opt , d 2 opt ) ∂ d 2 opt = 0  ( eq . 5 ) from equation ( 5 ), two equations that describe d 2 opt as a function of d 1 opt can be found ( the following derivations assume τ s1 ≢ τ m and τ s2 ≢ τ m ): d 2 opt = τ m α 2 · ln   { 1 +  ( α 2 α 1 · v 01 ∂ v 02 / ∂ d 1 opt ) · ( 1 τ s1  exp  [ - d 1 opt τ s1 ] - 1 τ m  exp  [ - d 1 opt τ m ] ) } ( eq . 6 ) d 2 opt = τ m α 2 · ln   { t 2 τ m  [ 1 -  ( α 2 α 1 · v 01 v 02  ( d 1 opt ) ) · ( exp  [ - d 1 opt τ s1 ] - exp  [ - d 1 opt τ m ] ) ] } ( eq . 7 ) setting equations ( 6 ) and ( 7 ) equal to each other and simplifying produces the following implicit equation for d 1 opt : ( τ m τ s2 · α 1 v 01 ) = ( 1 / τ s1 ∂ v 02 / ∂ d 1 opt + τ s2 / τ m v 02  ( d 1 opt ) )  exp  [ - d 1 opt τ s1 ] - ( 1 / τ m ∂ v 02 / ∂ d 1 opt + τ s2 / τ m v 02  ( d 1 opt ) )  exp  [ - d 1 opt τ m ] ( eq . 8 ) further simplifications of equation ( 8 ) require that v 02 ( d 1 ) be explicitly defined . when the two system capacitors ( c a & amp ; c b ) are configured into a parallel arrangement during step 1 and then reconfigured into a series arrangement during step 2 , the system time constants can be explicitly defined as : τ s1 = r s ·( c a + c b ) τ s2 = r s ·( c a c b )/( c a + c b ) ( eq . 9 ) v 02 ( d 1 )= 2 · v s1 ( d 1 ) = 2 · v 01 · exp [− d 1 / τ s1 ] ( eq . 10 ) where equation ( 10 ) codifies the notion that , in a parallel - series arrangement , the leading edge voltage of step 2 equals twice the trailing edge voltage of step 1 . substituting equation ( 10 ) into equation ( 8 ) and solving explicitly for d 1 opt and subsequently d 2 opt [ via equation ( 6 ) or ( 7 )] yields : d 1 opt = - τ m α 1 · ln  { ( τ m τ s1 )   ( 2  α 1 - α 2 α 1 - α 2 ) } ( eq . 11 ) d 2 opt = + τ m α 1 · ln  { ( 1 2 )   ( 2  α 1 - α 2 α 1 - α 2 ) } ( eq . 12 ) the maximum myocardial voltage attained using these optimal parallel - series step durations can then be determined by substituting equations ( 10 )-( 12 ) into equation ( 4 ) and simplifying : v m2  ( d 1 opt , d 2 opt ) = v 01  ( 1 2 ) - 1 α 2  ( τ m τ s1 ) 1 α 1 - 1   ( 2  α 1 - α 2 α 1 - α 2 ) 1 α 1 - 1 α 2 ( eq . 13 ) note that equations ( 11 )-( 13 ) are valid for any independent values of c a and c b . according to this simple rc model of defibrillation , successful defibrillation is achieved when the myocardial voltage ( as embodied herein by v m1 and v m2 ) is “ depolarized ” to its threshold value , v th . an equation that describes the minimum relative magnitude for v 0 ( i . e ., the voltage to which each of the capacitors is charged in preparation for the defibrillation shock ) that successfully drives v m2 to v th can be obtained from equation ( 13 ) by setting v m2 = v th and solving for v 01 ( which , for these parallel - series shocks , is equivalent to v 0 ). since the total stored energy in capacitors c a and c b is given by : e stored = 1 2  ( c a + c b ) · v 0 2 ( eq . 14 ) then the optimal relationship between c a and c b that maximizes myocardial voltage for a given total stored energy can be found by substituting c a = k · c b into equation ( 14 ) and then solving for k in ∂ e stored /∂ k = 9 . the result is : the above result implies that c a should equal c b in order to achieve maximum myocardial impact for any given total energy . the relationship c a = c b is equivalent to τ s1 = 4 · τ s2 [ see equation ( 9 )], from which simplified versions of equations ( 1l )-( 13 ) can be derived : d 1 opt = τ m α 1 · ln  { ( 1 3 )  ( 1 + τ m 2  τ s2 ) } ( eq . 16 ) d 2 opt = + τ m α 2 · ln  { ( 1 3 )   ( 1 + 2  τ s2 τ m ) } ( eq . 17 ) v m2  ( d 1 opt , d 2 opt ) = 2  v 01  ( τ m 2  τ s2 ) 1 α 2 - 1  [ ( 1 3 )  ( 1 + τ m 2  τ s2 ) ] 1 α 1 - 1 α 2 ( eq . 18 ) finally , the optimal capacitance for a given r s and τ m is determined by finding the value of c a that minimizes e stored , that is , solving for c a in ∂ e stored /∂ c a = 0 ( with k = 1 ). the result is : c a = c b = τ m r s ( eq . 19 ) or equivalently , the optimal capacitance ( for a given r s and τ m ) is that which satisfies : 1 2  τ s1 = 2  τ s2 = τ m ( eq .  20 ) d 1 opt =+ 2τ m · 1 n [ 3 / 2 ] ≈ 0 . 811 · τ m ( eq . 21 ) d 2 opt =+ τ m · 1 n [ 3 / 2 ] ≈ 0 . 405 · τ m ( eq . 22 ) further insights into the preceding theoretical calculations can be gleaned from corresponding graphical analyses . the relative stored energy required for defibrillation ( e stored ) for all possible parallel - series two - step waveforms is graphically illustrated in the contour plot of fig1 . in this plot , the x - axis is indexed by the total capacitance ( c a + c b , scaled by τ m / r s ) while the y - axis is indexed by the ratio of the two capacitances ( k = c a / c b ). although perhaps seemingly non - intuitive axis definitions , they efficiently provide complete coverage of the entire parameter space of all possible capacitor combinations for two - step waveforms . as indicated by the horizontal line 100 and the vertical line 102 overlaid on this plot ( and as consistent with the conclusions of equations ( 15 ) and ( 19 )), the most efficient two - step positive portion for the biphasic shock is delivered when : the contours then step out from this optimal point in 1 % increments , thus providing an indication as to the relative sensitivity of the energy efficiency to deviations in either total capacitance or capacitance ratio . in fact , energy efficiency remains quite robust : for example , energy efficiency remains within 1 % of optimal for : ˜ 1 . 5 · τ m / r s & lt ;( c a + c b )& lt ;˜ 2 . 7 · τ m / r s ; and two - dimensional contour plots of optimal step 1 and step 2 durations ( normalized by τ m , i . e ., d 1 opt / τ m and d 2 opt / τ m ) as given by equations ( 11 ) and ( 12 ) are presented in fig1 and 14 , respectively . similar to fig1 , fig1 and 14 have respective horizontal lines 110 , 120 and vertical lines 112 , 122 from have been overlaid on these contour maps as well . their respective intersections 114 , 124 appropriately correspond to the “ 0 . 811 ” and “ 0 . 405 ” coefficients found in equations ( 21 ) and ( 22 ), respectively . since r s and τ m represent patient - specific variables that directly impact the choice of durations used for these stepped waveforms , it is perhaps useful to present example values for d 1 opt and d 2 opt for a representative range of values for r s ( 30 - 90 ω ), τ m ( 2 - 4 ms ), and c a ( 30 - 90 μf ). the tables shown in fig1 - 17 provide such a set of example values , wherein values for d 1 opt and d 2 opt are computed from equations ( 16 ) and ( 17 ), respectively . given the limits of the ranges used for r s , τ m , and c a in the tables shown in fig1 - 17 , d 1 opt and d 2 opt range from lows of 1 . 286 and 0 . 422 ms ( when τ m = 2 ms , c a = 30 μf , and r s = 30 ω ) to highs of 3 . 704 and 2 . 689 ms ( when τ m = 4 ms , c a = 90 μf , and r s = 90 ω ), respectively . of course , d 1 opt and / or d 2 opt could move outside of these ranges if any one or more of r s , τ m , and c a exceed the limits used for these tables . in those cases , equations ( 16 ) and ( 17 ) could be used to compute exactly the optimal step durations for any combination of r s , τ m and c a . in another embodiment , the device could also determine d 1 opt and d 2 opt based on measured values for r s , and / or a programmed value for τ m based on a particular value for c a and c b . by way of example , if the capacitance value for c a and c b is set to 60 μf , so that equation 19 is satisfied for a tissue resistance , r s equal to nominally 50 ohms and a tissue time constant , τ m , then for a range for τ m , of 2 ms to 4 ms , and a range for r s of 30 - 90 ohms , then : ( c a + c b )* r s / τ m = 0 . 9 to further assist with interpreting the results embodied in fig1 and 14 and the table shown in fig1 - 17 , fig1 graphs a subset of those data as simple functions of r s and τ m . in particular , fig1 presents a pair of graphs : the left and right halves plot d 1 opt and d 2 opt , respectively , as functions of r s for three representative values of τ m ( 2 , 3 , and 4 ms ). for these graphs , c a = c b = 60 μf ( thus k = 1 . 0 ). consistent with the data in the tables shown in fig1 - 17 both d 1 opt and d 2 opt increase in value with increasing r s or τ m . moreover , this figure helps illustrate how d 1 opt appears significantly more sensitive to relative changes in τ m than in r s , while d 2 opt appears to have the opposite sensitivity . while fig1 - 17 provide a comprehensive overview of all possible parallel - series two - step waveforms , it is also useful to consider some specific examples that can aid in illustrating the relative improvements gained by using such a parallel - series two - step capacitor arrangement over the traditional one - step arrangement . [ 0118 ] fig1 graphically compares the positive portion of the biphasic shock waveform shapes ( v s , top two waveforms , 150 and 160 ) and associated tissue responses ( v m , bottom two waveforms , 152 and 162 ) for one - step , 150 , and parallel - series two - step , 160 , shocks having equal stored energies and leading - edge voltages . τ m = 3 ms , r s = 50 ω , c a = c b = 60 μf the one - step shock is generated by essentially keeping c a and c b in a parallel arrangement for its entire shock duration , for a constant effective capacitance of 120 μf . as is evident from the tissue responses ( i . e ., comparing the one - step response 152 to the two - step response 162 ), two - step the myocardial voltage ( 162 ) reaches a higher higher final cell membrane potential (+ 18 . 6 %) in a shorter total duration ( 3 . 65 vs . 4 . 16 ms 12 . 3 %) as compared to the final cell membrane potential ( 152 ) using the one - step shock . a consequence of this improved tissue response is that this two - step waveform requires a lower effective leading - edge voltage ( and hence a lower stored energy ) to achieve the same defibrillation efficacy as its equivalent one - step waveform . [ 0123 ] fig2 illustrates this scenario by resealing the results presented in fig1 such that the strength of each shock is sufficient to produce tissue responses of equal amplitudes . consistent with the results presented in fig1 , this two - step positive portion of the biphasic shock waveform 164 theoretically requires a 15 . 6 % lower leading - edge voltage than its one - step counterpart 154 , which translates into a 28 . 8 % reduction in required stored energy , and a potentially lower pain waveform for the patient since the leading edge of the shocking pulse is reduced . [ 0124 ] fig2 and 22 illustrate analogous results to those depicted in fig2 , but for relatively extreme combinations of r s and c a . in fig2 , r s = 30 ω and c a = c b = 30 μf , while in fig2 , r s = 90 ω and c a = c b = 90 μf . as is evident in fig2 and 22 , the shape of the optimal parallel - series two - step waveform depends strongly on the magnitudes of r s and c a . furthermore , the relative improvement in energy efficiency also strongly depends on these values . for example , in fig2 , the two - step waveform 166 induced an equivalent final tissue response as its one - step waveform 156 , but with an 8 . 8 % shorter duration ( 2 . 1 vs . 2 . 3 ms ), a 6 . 5 % lower leading - edge voltage , and a 12 . 6 % reduction in required stored energy . in fig2 , the relative improvements were a 14 . 3 % shorter duration ( 5 . 3 vs . 6 . 3 ms ), a 25 . 9 % lower leading - edge voltage , and a 45 . 0 % reduction in required stored energy . thus , these comparisons suggest that there would be especially great incentive for utilizing two - step waveforms instead of traditional one - step waveforms when the magnitudes of r s and c a are large , while the incentive is relatively minimal when the magnitudes of r s and c a are small . unfortunately , because of the inherent limitations of this theoretical model , it is not possible to directly compare amplitude - based results ( e . g ., leading - edge voltage , required stored energy ) derived for differing r s or τ m . for this reason , the results of fig2 - 22 are all self - normalized ( that is , there is no relationship between the amplitudes in these graphs ). finally , while equations ( 16 ) and ( 17 ) provide exact formulas for determining d 1 opt and d 2 opt when k = 1 ( i . e . , c a = c b ) , it is sometimes helpful and / or practical to also identify various approximations to such solutions . consider the following infinite series expansion of the natural logarithm : ln  [ x ] = 2 · [ ( x - 1 x + 1 ) + 1 3 · ( x - 1 x + 1 ) 3 + 1 5 · ( x - 1 x + 1 ) 5 + …  ] ( 23 ) utilizing just the first term of this expansion , equations ( 16 ) and ( 17 ) can be simplified to : d 1 opt ≈ 2  τ m 3 - α 1 = 2  τ s1 · τ m 2  τ s1 + τ m ⇒ 1 d 1 opt ≈ 1 2  τ s1 + 1 τ m = 1 4  r s  c a + 1 τ m ( 24 ) d 2 opt ≈ 2  τ m 3 - 2  α 2 = τ s2 · 2  τ m τ s2 + 2  τ m ⇒ 1 d 2 opt ≈ 1 τ s2 + 1 2  τ m = 1 2 · ( 4 r s   c a + 1 τ m ) ( 25 ) in words , these relationships suggest that the optimal step durations can be well approximated by computing variously weighted parallel combinations of system and myocardial time constants . and despite using only one term of equation ( 23 ), these approximations are relatively quite accurate over a broad range of τ s1 / τ m and τ s2 / τ m ratios ( only their ratios , not their absolute values , impact their accuracy ). for example , the relative error for d 1 opt is less than 5 % for 0 . 4 & lt ; τ s1 / τ m & lt ; 5 , while the relative error for d 2 opt is less than 5 % for 0 . 2 & lt ; τ s2 / τ m & lt ; 3 . when equation ( 20 ) is also satisfied ( that is , when system and myocardial time constants are ideally matched ), these relative errors are each only 1 . 35 %. in all cases , these approximation calculations underestimate the true values by these respective relative errors . while the invention herein disclosed has been described by means of specific embodiments and applications thereof , numerous modifications and variations could be made thereto by those skilled in the art without departing from the scope of the invention set forth in the claims .
in an icd , a highly efficient biphasic defibrillation pulse is generated by switching at least two charged capacitors from a parallel connection to various combinations of a parallel / series connection or a series connection during the first phase of the defibrillation pulse . such mid - stream parallel / series connection changes of the capacitors and steps up the voltage applied to the cardiac tissue during the first phase . a stepped - up voltage during the first phase , in turn , gives an extra boost to , and thereby forces additional charge into , the cardiac tissue cells , and thereby transfers more charge to the membrane of the excitable cardiac cell than if the capacitors were continuously discharged in series . phase reversal is timed with the cell membrane reaching its maximum value at the end of the first phase .