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J Thorac Cardiovasc Surg 2003;125:699-710
© 2003 The American Association for Thoracic Surgery

Cardiopulmonary Support and Physiology

Pressure-flow loops and instantaneous input impedance in the thoracic aorta: Another way to assess the effect of aortic bypass graft implantation on myocardial, brain, and subdiaphragmatic perfusion

From the Hemodynamics and Cardiovascular Mechanics Laboratory,^a School of Medicine; Department of Cardiac Surgery,^b La Timone Hospital; and Physics Laboratory,^c Saint-Jerome University of Sciences, Marseilles, France.

Received for publication Jan 15, 2002. Revisions requested April 10, 2002; revisions received May 8, 2002. Accepted for publication July 8, 2002. Address for reprints: Professor T. G. Mesana, Ottawa Heart Institute, 40 Ruskin St, Ottawa, Ontario K1Y 4W7, Canada (E-mail: tmesana{at}ottawaheart.ca).

	Abstract

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
Background: The serious disturbances in ventriculoarterial coupling after thoracic aorta bypass grafting are addressed through aortic entry impedance in the frequency domain from flow-pressure waves. We designed a method for synthesizing pressure and flow waves to evaluate opposal to aortic flow along the cardiac cycle, addressing myocardial, brain, and visceral tissue perfusions from pressure-flow hysteresis loops and forward-backward aortic entry impedance in the ascending aorta, transverse aortic arch, and distal descending aorta, respectively, before and after extra-anatomic grafting of the descending aorta in the swine.
Methods: Twelve pigs underwent extra-anatomic grafting (woven double-velour prosthesis, 18-mm diameter), bypassing the descending aorta. Periarterial flow and endovascular pressure signals were mathematically synthesized (error minimization) to yield continuous functions of flow, pressure along the cardiac cycle before treatment for mean hemodynamics, pressure-flow hysteresis loops, and aortic entry impedance.
Results: Grafting of the descending aorta overshadowed pressure-flow hysteresis loops in the ascending aorta by shortening maximum pressure delay on maximum flow and diastolic flow reversal. Clamping of the descending aorta substantially restored hemodynamics in the ascending aorta, although the diastolic flow decrease was accelerated. Identical processes developed in the transverse aorta. Subdiaphragmatic descending aortic flow was flattened after grafting and restored, although thickened, after clamping of the descending aorta. Flow wave peak was framed by a diastolic aortic entry impedance peak, which was damped along the transverse aortic arch (aortic entry impedance peak in the ascending aorta, 1700 ± 102 kN · s · m^-5; aortic entry impedance peak in the descending aorta, 292 ± 45 kN · s · m^-5; P < .05). After grafting, the aortic entry impedance peak was transferred to early systole (aortic entry impedance peak in the transverse aortic arch, 2104 ± 94 kN · s · m^-5; aortic entry impedance peak in the descending aorta, 450 ± 75 kN · s · m^-5; P < .05). Clamping of the descending aorta attenuated the early systolic aortic entry impedance peak (aortic entry impedance peak in the transverse aortic arch, 1269 ± 104 kN · s · m^-5; aortic entry impedance peak in the descending aorta, 491 ± 75 kN · s · m^-5; P < .05), although aortic entry impedance in the descending aorta remained higher than before grafting (P < .05). Specifically, the backward flow ascending aorta to coronary trunks generated a backward aortic entry impedance peak (2234 ± 350 kN · s · m^-5) superimposed onto the forward aortic entry impedance peak with asymptotic boundaries that diminished after grafting and further enlarged after clamping of the descending aorta.
Conclusions: Hemodynamic opposition of grafting of the descending aorta are specific to the aortic site and cardiac cycle and are dependent on clamping of the descending aorta. Our approach to thoracic aorta hemodynamics could enable optimization of bypass grafting.

	Introduction

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 

View larger version (67K): [in this window] [in a new window]   Mesana, Mekkaoui, and Rolland

Pioneering works in the creation of long bypasses from the AA ³ or ventricular outflow tract ^1,2 to the abdominal aorta, as well as later studies ^4-6 and simulations, ^7,8 showed that bypasses markedly increased characteristic impedance and lowered aortic total compliance, whereas they slightly changed vascular resistance, which in turn impaired energy transfer in the ventriculoarterial coupling. Pressure- or volume-dependent molecular mechanisms subsequently contribute to the progression of abnormal myocardial energetics and systolic dysfunction characteristics of ventricular failure. ^9-12 Clinical observations documented left ventricular hypertrophy after bypass procedures as a result of increasing aortic input impedance. ^6,13 Conversely, actions on aortic input impedance are of potential therapeutic interest because they improve ventricular function in patients with dilated cardiomyopathy. ^11,13

Myocardial consequences after thoracic aortic surgery usually are identified by calculating input impedance from blood pressure and flow measurements by using frequency domain techniques. ^9,12,14-16 Impedance then is commonly displayed as amplitude and phase shifts across a frequency spectrum, and finally, the amplitude of the impedance at a given frequency represents the relative resistance to blood flow at that frequency. ^{1-4,6-9,11-16} In this way the aortic input impedance offers an appropriate description of the total hydraulic load imposed on the left ventricle because matching heart rate to frequencies that minimize input impedance results in more efficient blood flow by diminishing energy requirements. ^1,2,12,15-17 Nevertheless, even when the frequency analysis provides an appropriate description of the total hydraulic load of the left ventricle, instantaneous measurements of the flow-pressure relationships fail to describe aortic entry impedance (Ze) changes along the cardiac cycle, especially Ze backward and forward variations. From a therapeutic standpoint, we further need to assess the effects that noncompliant conduits, such as bypass Dacron grafts, have on hemodynamics and on upstream and downstream tissue perfusion. In this study we decided to use an extra-anatomic bypass graft of the long DA segment, which reproduces clinical situations, such as coarctation or thromboexclusion of the aorta. Then, from data acquisition methods previously developed in our laboratory for hemodynamic investigations of atherosclerosis development and stent implantation in swine, ^18-20 we applied a new mathematic synthesizing method to convert the pressure and flow measurement points into mathematic equations enabling adequate assessment of pressure-flow (P-Q) relationships as P-Q hysteresis loops and profiles, all a cardiac cycle along.

	Materials and methods

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
Twelve swine (Pietrain/Duroc bred, 10 months old, 73 ± 3 kg) were handled in accordance with the Marseilles School of Medicine's Animal Care and Use Committee, which granted approval of the protocol. As previously described, ^18-20 the swine were anesthetized (intramuscular ketamine, 18 mg/kg, and atropine, 20 µg/kg), and ventilated with endotracheal tubing (oxygen-air controlled mixture) and intravenous (ear and vena cava) and intra-arterial (superior femoral artery, 6F introducer sheath; BSC) lines for pressure monitoring. Six ECG leads and an SaO₂/plethysmographic tail probe (HP78354C) were placed, and then animals were anesthetized (chloralose/urethane 5:25%; w/v, 1.25 mL/kg) and perfused with Ringer lactate (15 mL · kg^-1 · h^-2) solution. Aseptic techniques were used with prophylactic antibiotics (amoxicillin/clavulanic acid 1 g:200 mg, 20 mL, intravenous). A left posterior thoracotomy (fourth to sixth intercostal spaces) was performed before controlling the thoracic aorta at the measurement and grafting sites. Before aortic clamping, anticoagulation (clotting time >500 seconds) was performed by means of central administration of heparin (350 IU/kg heparin; Choay) and, to attenuate aortic crossclamping-induced changes in hemodynamics and heart rate, the animals received esmolol-chloridrate (Brevibloc, 100 mg/10 mL; Sigma-Tau) as 500 µg · kg^-1 · min^-1 within 1 minute before clamping and 50 µg · kg^-1 · min^-1 for the next 10 minutes to return arterial pressure and heart rate to basal values; the effects of esmolol disappeared within 15 to 20 minutes. The aortic sites to be investigated (Figure 1) were 2 cm below the AA pericardial reflexion for the AA, 2 cm after the left subclavian artery embranchment for the TA, and 2 cm above the diaphragm for the TA. Side aortic crossclamping for graft implantation was performed in the distal aortic arch and 4 cm into the distal DA. A woven double-velour vascular graft (Hemashield, Meadox Medicals Inc) extended from the upper DA (between the anterior bronchial arteries and the first posterior intercostal arteries embranchment) to the DA. Measurements of aortic diameter yielded values of 20.2, 18.4, and 15.4 mm for the AA, TA and DA, respectively. The vascular graft was 18 mm in diameter, and length was determined so that the graft paralleled the DA, realizing 45° aorta-to-graft angles. Animals were killed with 15 mL of 20% KCl administered intravenously.

View larger version (95K): [in this window] [in a new window]   Fig. 1. The 3 sites for hemodynamic measurement were the AA, the TA, and the DA. The extra-anatomic bypass graft (EAG; double-headed arrow) was proximally inserted between the bronchial arteries and the first intercostal arteries embranchment. The native DA after grafting finally was clamped (C).

Mathematic analysis and synthesis of flow and pressure waveforms
The study aimed at determining mean hemodynamics, P-Q relationships as P-Q hysteresis loops, and profile from mathematic synthesis of P-Q waves along a cardiac cycle. Cycle-to-cycle wave variability was overcome by generating a steady-state cycle (256 points) from time-averaged P-Q cardiac cycles calculated by using linear interpolation (mean period): , where N_j is defined as the number of points within each cardiac cycle, N_c is defined as the number of cycles, and f is defined as the sampling frequency. ¹⁹

The discrete pressure and flow signals were the resulting waves carrying energy and quantity of movement along the thoracic aorta. The following schedule was followed to obtain the pressure and flow analytic functions. First, complex pressure and flow waveforms were fractionated into successive segments of curves separated by 2 opposite extreme points, which were scaled (Figure 2 and Appendix 1). A mathematic modeling of the segment has been developed that applied to each segment after scaling of the 2 extreme points to measured values, offsetting in 2-dimensional coordinate axis, and settling into the increasing or decreasing nature of the waveform segment (Appendix 2). The initial waveform was synthesized by summing individual segments and fitting the curve to the original pulse wave to obtain the error, which was treated in the same manner. The final function was the sum of the serial functions. The model applied to both pressure and flow measured waves. The quadratic error normalized to the original waveforms was found to be very low, irrespective of the experimental conditions (Figure 3 and Appendix 3).

View larger version (17K): [in this window] [in a new window]   Fig. 2. The first 2 steps in mathematic synthesizing of the measured flow waveform in the native AA. A, The flow wave was fractioned into successive segments of curves separated by 2 opposite extreme points according to an algorithm described in Appendix 1. B, Each elementary segment of curve, after scaling the 2 extreme points to measured values, was synthesized into the mathematic function and further considered as part of the second derivative of a sigmoidal function.

View larger version (29K): [in this window] [in a new window]   Fig. 3. The measured flow waveform was synthesized by summing the individual successive segments and fitting the curve to the original pulse wave to obtain the first model m0 and error 0. Error 0 was iteratively treated in the same manner to give m1 and error 1 and m2 and error 2, subsequently. The synthesized function finally was the sum of serial functions. The model applied to both pressure and flow measured waves, with a quadratic error normalized to original waveforms ranging from 10-7 to 10-8, irrespective of the experimental conditions.

Statistical analysis
Data treatment and statistical analyses were performed by using Systat 9.0 software (SASS Inc). Results are given as mean ± SD. Statistical analysis was performed with the Kruskal-Wallis test (ie, nonparametric analog of 1-way analysis of variance), which was reduced to the Mann-Whitney test (ie, nonparametric analog of 2-sample t test) when there were 2 groups.

	Results

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
Data on mean hemodynamics in the thoracic aorta are shown in Table 1. In the native aorta pulsatile cardiac outflow decreased as diastolic flow increased with position along the overall thoracicaorta, whereas systolic and pulse flows decreased and mean () increased from AA to TA only. Grafting DA decreased diastolic pressure and mean and diastolic flow and increased peripheral resistance and while decreasing the flow in the AA and DA, although not immediately upstream of the graft in the TA. The results indicate that extra-anatomic bypass grafting diminished cardiac outflow while preserving flow input in the TA to the detriment of both coronary and brain flow and decreased flow pulsatility downstream of the graft in the DA. Driving flow solely into the graft by clamping the DA kept mean flow constant while decreasing diastolic and mean blood pressure subsequently and increased diastolic flow with the accompanying decreases in systolic flow and change in flow in the AA. The stiff graft offered additional resistance to blood flow. As in clamped conditions, peripheral resistance and increased in the thoracic aorta. Grafting the DA therefore partially restored the upstream coronary and brain perfusion but failed to restore flow pulsatility downstream in the DA.

View larger version (23K): [in this window] [in a new window]   Fig. 4. Synthesized pressure and flow waves were plotted as parametric equations for P-Q relationships to give the hysteresis loops in the AA, the TA, and the DA in the native (N) and grafted DA, before (EAG) and after (C) clamping of the DA. Arrows indicate the direction of loops.

In the TA P-Q loops exhibited similarities to AA loops: maximum flow preceded maximum pressure, and there existed diastolic backward flow to the supra-aortic trunk, although to a lower extent than in the AA. The processes also were furthered in the grafted TA. In contrast, they were restored after aortic clamping, although they were flattened compared with those in the native situation. The blood volume diminished slightly from 21.94 ± 1.87 mL/cycle in the TA to 16.56 ± 1.43 mL/cycle in the DA and further decreased to 9.78 ± 0.98 mL/cycle after clamping (P < .05).

In the DA hysteresis loops contrasted with those in the AA and TA, with the former presenting the stiff characteristics of the viscoelastic abdominal aorta. (Maximum pressure was reached as systolic flow was still increasing to maximum flow, and the decreasing diastolic flow rate was lower than decreasing pressure rate.) After bypassing the DA, the P-Q loop was drastically flattened downstream of the graft. Clamping the DA substantially restored the flow profile, although the diastolic flow wave decrease was accelerated. In parallel blood volume substantially decreased from 22.72 ± 1.87 mL/cycle in the DA to 12.29 ± 1.21 mL/cycle in the grafted DA (P < .05) and remained identical (10.69 ± 1.01 mL/cycle) after clamping.

The population-averaged entry impedance profile (Figure 5) before grafting shows that P-Q wave peaks were framed in all thoracic aortas by increased episode. peak lengthened most during diastole, cropping up as P-Q systolic peaks returned to minimal values. As the distance from the heart increases, peak and plateau decreased from 1700 ± 102 and 459 ± 34 kN · s · m^-5 in the TA to 292 ± 45 and 231 ± 51 kN · s · m^-5 in the DA, respectively, whereas the valley under P-Q peaks increased from 56 ± 12 kN · s · m^-5 in the TA to 71 ± 18 kN · s · m^-5 in the DA (P < .05). Figure 5 also shows that additional impedance events having asymptotic boundaries superimposed onto the profile but only in the AA. They were mathematically attributable to changes in flow slopes and directions. Specifically, the early diastolic flow was decreasing (ie, it had a negative slope) but was still forward in direction before turning to backward flow (ie, as crossing zero flow value). Thereafter (the time of mitral valve opening), the backward flow lowered (ie, returned to a positive slope), before turning into a forward flow as crossing zero flow value (Figure 5, insert). The forward peak correlated the changes in the flow slope, whereas the superimposed backward peak dropped between the 2 flow zero value, crossing over with asymptotic boundaries. Therefore the backward flow fulfilling the coronary trunks had its own backward peak (2234 ± 350 kN · s · m^-5).

View larger version (19K): [in this window] [in a new window]   Fig. 5. The entry impedance, as calculated from pressure and flow functions along a mean cardiac cycle, is plotted in conjunction with measured flow waves in the native AA (N AA), the native TA (N TA), and the native DA (N DA). Changes in the vicinity of zero values are magnified (insert) in the native AA to highlight the forward-backward changes in impedance and their flow counterparts.

View larger version (21K): [in this window] [in a new window]   Fig. 6. The entry impedance, as calculated from pressure and flow functions along a mean cardiac cycle, is plotted in conjunction with measured flow waves in the extra-anatomically grafted AA (EAG AA), the extra-anatomically grafted TA (EAG TA), and the extra-anatomically grafted DA (EAG DA). Changes in the vicinity of zero values are magnified (insert) in the extra-anatomically grafted AA to highlight the forward-backward changes in impedance and their flow counterparts.

View larger version (20K): [in this window] [in a new window]   Fig. 7. The entry impedance, as calculated from pressure and flow functions along a mean cardiac cycle, is plotted in conjunction with measured flow waves after clamping of the native DA in the extra-anatomically grafted AA (C AA), the extra-anatomically grafted TA (C TA), and the extra-anatomically grafted DA (C DA). Changes in the vicinity of zero values are magnified (insert) in the extra-anatomically grafted AA to highlight the forward-backward changes in impedance and their flow counterparts.

	Discussion

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

Consequences of extra-anatomic grafting
The results show that the acknowledged opposition to blood flow exerted by the extra-anatomic bypass in the DA ^1-9 was in fact accounted for by a presystolic peak upstream of the graft and was cumulative with doubling in magnitude of the diastolic further downstream of the graft. The physiologic architecture of the diastolic flow assistance to upstream embranchments was disadvantaged by erasing the early diastolic phase peak. Forcing the flow to the graft conduit markedly restored the quality of blood flow in the grafted TA by reducing the early systolic peak and enlarging the diastolic peak and by shifting down the diastolic peak from a forward profile toward a backward profile concomitantly. Finally, the results emphasized that the aortic flow to the downstream visceral port was disadvantaged by DA bypassing; either the native aorta was clamped or it was not.

The striking observation in the present study is that the well-known early cardiac compensatory responses to grafting a long stiff conduit from the cardiac outflow tract to the abdominal aorta ^{1-3,5,6,11-13} were not detectable, whereas to the contrary, lowering cardiac output develops after graft-induced relief in AA and TA flows. Distal to the heart (ie, after the major branching of the coronary and supra-aortic trunks), such an opposition to blood flow generated a (moderate) well-known jet stream-like process at the entrance of the prosthesis-DA bifurcation. ²¹ The process partly emptied the AA and TA to pertain blood flow immediately upstream the graft but to the detriment of blood flow drives to coronary and supra-aortic trunks. Clamping the DA returned flow profiles to basic features by restoring diastolic fluxes to coronary and brain territories through different processes than increasing loading in thoracic aortic segments I and II. The responses again were achieved without generating the previously established cardiac contractile compensatory responses. ^{1-3,5,6,11-13} However, these studies also demonstrated that the myocardial response to implanting a stiff conduit in the aorta was found to be more pronounced when the graft was implanted directly onto the aortic root than when implanted downstream in the AA. Our results therefore strongly support the view that implanting the graft in the DA was not capable of inducing the appropriate signals driving the myocardial energetic response. As speculative attempts to explain the absence of compensatory cardiac effects, it should be noted that grafting the DA distally from the heart enables the aortic root and AA to play their forming actions on the blood flow wave, which is obviously not the case when grafting the aortic root directly. Alternately, it is also noticeable that with regard to peripheral resistance and mean values, TA hemodynamically was less affected than the AA and DA were by DA grafting, obviously because of TA localization immediately upstream of the graft. It is therefore conceivable that unidentified sensors for the cardiac compensatory effects are located at this site. Such hypotheses deserve further experimental investigations of the aortic blood flow under pharmacologic and mechanical manipulation of the AA and TA.

Possible extension from hemodynamics to biomechanics: clinics and therapeutics
Potential extensions of the present results into vascular biomechanical changes in the profile are limited by the putative relationships lying between P-Q waves and biomechanical properties of the thoracic aorta. Because it has recently been stressed that inferring mechanical properties of an arterial system from measured pressure and flow, now referred to as the hemodynamic inverse problem, is questionable because there exists an infinite number of solutions of different arterial systems describing a measured pressure-flow pair. ¹⁶ Studying the aortic solid mechanics of the thoracic aorta through local measurements of diameters and pressure is required to determine from where the changes in hemodynamics originated.

Although clinical implications and therapeutic perspectives for the results remain in the speculative domain until prospective investigations are carried out, the present results provide potential mechanisms for some well-known acute or late complications to aortic surgery, as well as cardiac consequences. Specifically, it remains to be established to what extent changes in aortic hemodynamics and subsequent brain perfusion contribute to the neurologic outcome after thoracic aortic surgery. ^23,24 Changes in TA hemodynamics could participate in embolic complications by loosening atheromatous plaques, as well as contributing to temporary neurologic dysfunction, long-term deficits in cognitive functions, ^14,22,23 or spinal cord ischemia, ²⁴ depending on compensatory adaptative changes by the brain or intercostal arteries.

Alternately, long-term follow-up with sophisticated imaging methods after surgical intervention (including redo operations) of coarctation of the aorta by means of insertion of synthetic prosthetic grafts (ie, historical series from the 1980s) provide arguments for the relevance of the presently described hemodynamic consequences of prosthetic aortic bypass. Despite the excellent early clinical results, an increased number of patients showed aneurysm formation at the repair site, and the aneurysms that develop after aortic repair are true aneurysms, as opposed to false aneurysms occurring at the suture line. ^25-27 Aortic wall weakness, compliance mismatch between graft and native wall, and abnormal hemodynamics as flow acceleration and turbulence patterns have been advocated as underlying physiopathologic mechanisms. ^21,28,29 The diastolic changes we report here, coupled with increasing presystolic emergence of Ze peaks in the immediate upstream and downstream vicinity of the stiff graft, provide realistic explanations for transient flow accumulation, resulting in the presence of a presystolic transitory blood vortex in both areas where aneurysm and dilation occurred. ^28,29 These are supported by recent findings obtained by investigating changes in geometry, and strain of 3-dimensional reconstructed synthetic patch repair of aortic coarctation substantially increases the wall stress immediately adjacent to the aorta, ²⁶ which is to be accounted for by the hemodynamic changes we observed in the TA.

Finally, the cardiac consequences of the present study remain speculative in the midterm and long-term, although the newly alleviated hemodynamics in the AA and arch are likely to release a chronic compensatory cardiac response in the long term. Such an assumption might answer the unidentified mechanisms underlying the unpredictable pattern of the cardiac consequences of left ventricular remodeling in patients after successful repair of coarctation of the aorta. ³⁰ The mechanisms of increased relative wall thickness and reduced peak and end-systolic wall stresses are still unknown but are potentially accounted for by the changes in hemodynamics we report here.

Conclusions
The originality of our work consisted in a new approach for analysis of P-Q waves, P-Q loops, and profile in the thoracic aorta along a mean cardiac cycle to demonstrate that the natural hemodynamics provided assistance to the flow with an unexpected diastolic peak facilitating tissue perfusion. Our approach highlights that, upstream of the graft, the surgical procedure impaired the diastolic assistance effect to blood flow to the coronary and supra-aortic trunks. On a long-term basis and with regard to the absence of cardiac compensatory mechanisms, it remains to established whether such modifications might lead to cardiac and arterial deterioration and further organ dysfunction. Further studies should involve orthotopic grafting, should compare various types of substitutes, including aortic stent grafts, or both. Alternate type of grafts should be designed along with better design of anastomotic junctions and should be submitted to in vivo experimental quality control assessment using the presently described hemodynamic characteristics before clinical use.

	Appendix 1

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
The measured waves underwent the segmentation process into consecutive segments separated by 2 opposite extreme points defined as the location of the minimum and maximum, which were identified by using the following algorithm. Each maximum was determined in the wave as P_maxi = (t₁,y₁), with y₁ obeying the condition y_1-1 < y_i > y_{i + 1}, and each minimum as P_mini = (t_i,y_i), with y_i obeying the condition y_{i - 1} > y₁ < y_{i + 1}. The point of origin, p0 = (t₀,y₀), initially was found to be a maximum or a minimum entity when y₀ > y_{0 + 1} or y₀ < y_{0 + 1}, respectively. As a result of segmentation of the original pulse wave, the J segment had the extreme coordinates: .

	Appendix 2

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
The basic mathematic model considered that the curve was the second derivative of a sigmoidal function and extracted the elementary segment between the minimum and maximum (shown in Figure 2). It follows that f(t) is the sigmoidal function and (t) is the second derivative. Therefore:

The function (t) is the result of 2 reverse and shifted waves:

The use of a door function, , extracts useful part of the curve only:

The analytic expression of the elementary segment is as follows:
The J segment, obtained as described in Appendix 1, was fitted with the model as follows. From the extreme coordinates of the J segment the model segment was scaled so that _j(t_jM) = y_jM; (y_jm) = y_jm to yield the subsequent function that fit each waveform segment:

Scaling the curve required the introduction of the following parameters. The definition domain of J segment identified A_j along the ordinate scale with A_j = |y_jM - y_jm| and B_j along the abscissa scale with B_j = |t_jM - t_jm|. The segment of curve was scaled with _j and _j as 2 values determined, by using an empiric method, to be _j = ln(5,2A_j) and . Offsets in the function were accounted for by introducing the parameter in the horizontal offset and where _j = y_jm - y_m in the vertical offset for each waveform segment, with as the extreme coordinates of the model segment. The parameter _j accounted for the increasing or decreasing nature of the waveform segment, with .

	Appendix 3

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
The measured wave was reconstructed through a series of initial fits and error functions (Figure 3), with successive models being obtained by treating the measured wave and recurrent errors with equation 1. The first model m0: , although a reasonable approximation of the measured curve, still contained significant error (error 0). Error 0 was then fit with equation 1, yielding m1: and error 1. This error was subsequently fit with equation 1, yielding m2: , where l, m, and n were, respectively, the number of segments that make up m0, m1, and m2. The 3 resulting models were summed to obtain the final synthesized waveform as follows:

The accuracy (physics, geometrics, and statistics) was checked by using a quadratic error normalized to the original pulse wave, according to the following:

where the discrete counterpart is

where S was the measured wave, T was the mean cardiac cycle period value, and t was the time variable. In each case the error was found to be on the order of 10^-7.

	Appendix 4

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 
Mean hemodynamics of pressure and flow waves were determined as the systolic (maximum), diastolic (minimum), pulse ( = |max-min|) and mean values, as computerized from their respective _p(t) and _Q(t) functions and expressed as millimeters of mercury and milliliters per minute, respectively. The algebraic mean pressure () and flow () values are calculated as and respectively, with T:mean cardiac cycle period value.

Parametric equations of hysteresis loops for P-Q relationships (Figure 3) were defined as follows:

Peripheral resistance (PR; mm Hg · mL^-1 · min^-1) and mean entry impedance (, mm Hg · mL^-1 · min^-1) were calculated as , respectively.

The instantaneous Ze (in kN · s · m^-5) during the cardiac cycle is the ratio of the function of the pressure and flow:

	References

Top Abstract Introduction Materials and methods Results Discussion Appendix 1 Appendix 2 Appendix 3 Appendix 4 References

 

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