Simulation of left ventricular function during dyskinetic or akinetic aneurysm

Th e purpose of our study was to simulate the hemodynamics of left ventricular function after left ventricular aneurysm (LVA) of various sizes and to validate the results of this computer based simulation with patient data. We developed an equivalent electronic circuit (EEC) that refl ects the hemodynamic conditions of LVA (after acute myocardial infraction) while taking into consideration the resetting of the sympathetic nervous tone in the heart and systemic circuit, the fl uctuating intrathoracic pressure during respiration and passive relaxation of the ventricle during diastole. Th e key feature of the EEC was a subcircuit representing the LVA, with a subcircuit to measure ventricular blood volume (i.e. intraventricular “shunting” of blood fl ow during systole and diastole) between the unaff ected section of the left ventricle and its aneurysm. Th is EEC model can simulate akinetic or dyskinetic LVAs of diff erent sizes and provides realistic beat-to-beat ventricular blood fl ow and pressure tracings that were validated by pressure-volume loop diagrams and by published patient data. In agreement with published data, simulated dyskinetic LVAs have a considerably greater impact on ventricular function than akinetic LVAs. Th e hemodynamic eff ects of ventricular systolic dysfunction following LVA were also evaluated. We conclude that the EEC model qualitatively and to a signifi cant degree quantitatively represents conditions in patients with a dyskinetic or an akinetic LVA and provides realistic beat-to-beat ventricular blood fl ow and pressure tracings. ©  Association of Basic Medical Sciences of FBIH. All rights reserved


INTRODUCTION
Left ventricular aneurysm (LVA) is a mechanical complication of an acute transmural myocardial infarction (AMI).It can follow AMI immediately or can develop weeks to months later and has a reported frequency of  - [].LVA can be defi ned as any localised area of left ventricular akinesia or dyskinesia that reduces left ventricular ejection fraction [].Ventricular akinesia denotes a segment of the ventricular wall that has essentially no contractile function (does not contract during systole).Ventricular dyskinesia denotes a segment of the ventricle wall that exhibits a paradoxical, outward movement during systole causing intraventricular "shunting" of blood fl ow between systole and diastole.After AMI, a portion of the aff ected ventricular wall loses the ability to contract during systole, but retains its elastic properties during diastole, and, provided the aff ected area is of suffi cient size, manifests a ventricular dyskinesia.For the same cardiac out-put, a heart with a dyskinetic LVA needs to increase "stroke" volume (to infl ate the dyskinetic aneurysm during systole and still maintain a normal cardiac output) and has consequently an increased oxygen demand and a lower work reserve.Patient data suggests, that the acute compensatory mechanisms, the Frank-Starling mechanism and the baroreceptor refl ex mediated increase in chronotropic and inotropic activity of the heart, can maintain stroke volume if the noncontractile region involves less than  of the left ventricular circumference [].Over time, the section of the ventricle that experienced AMI undergoes tissue remodelling from muscle tissue to granulation tissue and fi nally to fi brous tissue []; the noninfarcted ventricular region can develop systolic or diastolic dysfunction [,].Th e wall of a chronic LVA has essentially a low compliance and no contractile properties, qualities of an akinetic aneurysm.Compared to a dyskinetic LVA, that aff ects an equal portion of the ventricular wall, an akynetic LVA represents a lesser work load for the healthy segment of the ventricle and a relative improvement of heart function.Cardiovascular physiology can be simulated using either an analog or a digital approach [] but only a few studies have modelled left ventricular function or hemodynamic changes after LVA [-].In this paper we present an equivalent electronic circuit (EEC) computer model that simulates the hemodynamic changes of akinetic or dyskinetic LVA while taking into consideration the resetting of the sympathetic nervous tone in the heart and systemic circuit, the fluctuating intrathoracic pressure during respiration and passive relaxation of the ventricle during diastole.

Procedures
Simulations of cardiovascular variables during left ventricular aneurysm (LVA) were performed by developing an equivalent electronic circuit (EEC) with Electronics Workbench (EWB) Personal version . [].Left ventricular acute myocardial infarction (LVAMI) and the resulting LVA are simulated by modifying a previously developed model of the cardiovascular system for a healthy, young adult under resting conditons (i.e.heart frequency rate  beats/min) in a recumbent position [-].In this model venous tone, heart rate and contractility of both ventricles is modulated by a negative feedback mechanism and the infl uence of respiration on cardiac fi lling and output is also taken into account.The EEC of left ventricle was modified to simulate LVAMI and subsequent LVA (Figure ).It consists of two, electronically almost identical, sections S (simulating an initially normal ventricular section) and S (simulating the ventricle section that developed LVAMI and LVA), with two capacitors, C and C.The main components of sections S and S are (a) the "gain stage" (simulating contractility) of the ventricle and (b) the capacitor, simulating part of the left ventricle.Total left ventricular mass, and its total capacitance should be constant, i.e.  mL/mm Hg.Therefore the sum of both capacitors, C in S and C in S, should be always  μF (Figure ).Gain (contractility) in sections S and S is modulated by homeostatic negative feedback.In resting conditions the modulation factor is .If e.g.arterial pressure is decreased, the modulation factor can be -via the myocardial contractility circuit -increased to  [-].To simulate a "mild" LVAMI and a subsequent small LVA (involving  of left ventricular mass), the capacitance of C (section S) is set to . μF and the capacitance of C (section S) is set to . μF.A "severe" LVAMI and a large LVA (involving  of left ventricular mass), is simulated by adjusting the capacitance of C (section S) to . μF and the capacitance of C (section S) to . μF.Th e voltage (pressure) output from capacitors C and C (in sections S and S), representing the complete ventricle, is fed directly between both diodes D (representing the mitral and the aortic valve; SW) to simulate the time course of LVA formation and its consequences on ventricular hemodinamics..LVA formation (defined in our model as an acute, regional ventricular failure with strongly decreased contractility, but conserved diastolic distensibility) is induced by operating Sw at . s.The section S is switched off from the myocardial contractility circuit, thereby decreasing the modulation factor from  to ..Contrary to that, gain is normal in section S and modulation factor is controlled by homeostatic negative feedback..The setting of additional switches (Sw  to ), shown in the table in Figure , simulates two possible outcomes that follow an acute LVA: a decreased diastolic distensibility and scarring of the ventricular mass forming the LVA (in ventricle section S) and a systolic dysfunction (a reduced contractiliy) of the prevoiusly unaffected portion of the left ventricle (in ventricle section S).. Simulation of a decreased diastolic distensibility and scarring of the LVA (while the rest of the myocardium, S, is functioning normally) is achieved by operating Sw at . s and Sw at . s to simulate a progressive, two-step decrease in distensibility (step I and step II in Figure ).In this way either capacitor Cx ( nF) or capacitor Cy ( nF) is connected to capacitor C.In this condition switches Sw and Sw are not activated.Th erefore in this condition the contractility of section S, functioning normally, is modulated by negative feedback (Figure ).. Simulation of reduced contractility of S (the initially normal ventricular section, unaff ected by LVA) is achieved by operating Sw at . s and Sw at . s to simulate a progressive, two-step (mild and severe) decrease in contractility of section S (step I and step II in Figure ).In this way the modulation circuit is switched-off completely and either battery . V or battery . V is connected to the contractility circuit of S.Th us the contractility of S is decreased, inducing a mild or severe S systolic dysfunction (Figure ).We studied how the size of LVA aff ects not only the enddiastolic pressure and the end-diastolic volume of the whole left ventricle, but also the end-diastolic volume of section S (that developed the LVA).To meet this end the ratio between the section of the LV unaffected by LVA-MI (section S) and the affected portion of the LV that developed a LVA (section S), the S/S ratio, is varied in  increments to affect , to  of total left ventricular mass.In Figures. and  .

Simulation of a large LVA and its subsequent decreased distensibility
Th e simulation of a large LVA and its subsequent, two-step decreased distensibility is presented in

Analysis of left ventricle P-V loop diagrams after LVA and its subsequent decreased distensibility or after left ventricle systolic dysfunction
In a relatively small aneurysm (-; S is  of ventricular mass) the P-V loop diagram is shifted to the right and to slightly higher left ventricle (LV) pressures.If in this condition the diastolic distensibility of section S is decreased (step II), the P-V loop diagram is shifted to the left and to slightly lower

Eff ect of aneurysm size on left ventricle EDP and EDV
By increasing the size of the LVA, from  to  of the left ventricle, the end-diastolic volume of the aneurysm (EDVS) is increased.At the same time also the EDVLV is increased.However, the EDVS/EDVLV relation is shifted to lower EDVLV levels, if S distensibil-ity is decreased.And, contrary to that, this relation is shifted to even higher EDVLV levels, if S is affected by systolic dysfunction (

DISCUSSION
The presented EEC model, adapted to simulate LVA, is very similar to that already described [-].Qualitatively, its cardiovascular variables resemble quite well to those in vivo.Quantitatively, however, there are some minor differences which have already been discussed [-].Thus, also in the presented EEC the ejection fraction of the left ventricle is slightly lower (about ) than in vivo (above ).However, the simulated acute compensatory mechanisms can increase the EF to about  [-].
The simulated hemodynamics during LVA cannot be perfectly matched to individual patient data due to the lack of relevant, patient information and also due to the inherent limitations of the model.Relevant patient information, that is usually not available, is (a) the degree of right and left heart failure, (b) the degree of left ventricular tissue remodelling in the area of LVA and in the surrounding ventricular muscle region and (c) the effi ciency of short, medium and  long term compensatory mechanisms that can be modifi ed by the individuals life-style, age, previous or ongoing diseases or medication.Limitations specifi c to the presented model are the assumption of a normal short-term compensatory mechanism and the exclusion of medium (hormonal) and long-term (water and sodium retention) compensatory mechanisms at the time of a LVA.Also the model does not simulate the eff ect of ventricular arrhythmias that can occur after LVA [] and can only partially model the ventricular functional changes due to heart tissue remodelling and alterations of ventricular geometry after a LVA [].
Well documented hemodynamic changes in patients with LVA are, for example, a reduced ejection fraction and an increased left ventricular end diastolic pressure (LVEDP) and volume (LVEDV) [-].The relationship between the size of aneurysm and LVEDV or LVEDP was first studied by Klein et al. [].Small aneurysms, with a LVAto-chamber surface ratio below , did not cause a signifi cant change in LVEDV or LVEDP.However, when the size of LVA exceeded , both the LVEDV and LVEDP increase proportionally with the volume of the aneurysm.Th e results of the EEC computer simulation are qualitatively and to a signifi cant degree quantitatively consistent with the observed hemodynamic changes after LVA in clinical studies [-].Also, the model highlighted the infl uence of LVA on the overall ventricular contractile properties.Systolic ventricular function can be evaluated by EF.In normal conditions, in the EEC model, EF is ..In a  LVA, the EF is decreased to ..However, after LVA distensibility is strongly decreased (step II), EF is improved to ..Contrary to that, when in the same simulation conditions, severe ventricular systolic dysfunction occurs, EF is decreased to ..In our model we recorded a minimal increase of LVEDV and LVEDP when the size of LVA was below  of ventricular mass and a proportional increase in LVEDV and LVEDP when the size of LVA was  or more.After LVA, the changes in LVEDV and LVEDP are reduced by decreased distensibility of the aneurysm and increased by systolic dysfunction (Figure ).The pressure-volume loop diagrams, constructed with the EEC model, correctly reflected the hemodynamic changes after LVA, reduced aneurysmal distensibility or ventricular systolic dysfunction (Figure ).This is consistent with the improvement in left ventricular function after LVA surgery [-, ].
As reported, the acute compensatory mechanisms can maintain stroke volume if the noncontractile region involves less than  of the left ventricular circumference [].Our computer model assumed a normal acute compensatory mechanisms, thus LVAs involving more that  of left ventricular mass could theoretically be compensated.Excluding the possibility of a model generated artefact, we suggest that patients with LVA could have a less efficient acute compensatory mechanism than healthy persons.For example, the acute compensatory response could be suboptimal due to right heart failure or due to a reduced right ventricular filling pressure.Cardiovascular drugs, such as nitro-glycerine, reduce both systemic arterial pressure and venous return, thus reducing the right ventricular fi lling pressure and indirectly the left ventricular fi lling pressure [].Evaluation of LVA is important for understanding the hemodynamic impact of an aneurysm and for planning optimal surgical treatment [, , , ].Th e hemodynamic properties of LVA were simulated with a computer-driven biomechanical model of the left ventricle using a pressure wave chamber system to simulate ventricular and aortic pressures and volumes [].However, this mechanical model cannot simulate passive relaxation of the ventricle during diastole, nor the resetting of the sympathetic nervous tone or the changes in intrathoratic pressure during respiration; it can quantify the diff erent eff ects of dyskinetic and akinetic aneurysms of different sizes on ventricular function.Th erefore, an additional advantage of the present EEC model of LVA over previous models is a realistic beat-to-beat ventricular blood fl ow and pressure tracing during LVA.Realistic beat-to-beat tracings (Figures , 

CONCLUSION
The model enables realistic beat-to-beat ventricular blood flow and pressure tracings that facilitate the understanding of the effects of akinetic or dyskinetic aneurysms of various sizes on ventricular function and correctly predicts the effect of systolic dysfunction or decreased aneurysmal distensibility on ventricular hemodinamics.

DECLARATION OF INTEREST
Th e authors report no confl ict of interest.
FIGURE 1. Electronic circuit of the left ventricle subdivided in section S1 (with initial normal function) and S2 (developing infarction and aneurysm).The time setting of switches in S2 determine whether S1 and S2 operate as a single unit, or as two distinctly separate units.
 this is indicated by acronyms -, -, -, - and -, respectively.Results of the simulations are shown graphically as the time course of equivalent variables.Thus electrical variables (voltage, current, resistance, capacitance and charge) correspond to physiological variables (pressure, blood fl ow, resistance, capacitance and volume).The interdependence of pressure and volume of the left ventricle is shown by pressure-volume analysis (P-V loop diagrams) describing the left ventricle work-load during one cardiac cycle (Figure ).Acronyms of variables studied are listed in Table Figure .Changes in cardiovascular variables in "severe" myocardial infarction and a large myocardial aneurysm (involving  of left ventricle mass) are presented in the upper, middle and bottom graphs of Figure .Each graph is subdivided into four time intervals: normal conditions -before development of an LVAMI and LVA ( s -. s); consequences of LVAMI and subsequent LVA -decreased contractility and conserved distensibility in S (. s -.s); loss of contractility in S with its distensibility decreased fi rst mildly (step I; . s -.s) and then strongly (step II; . s - s LV pressures.Contrary to that, in severe S systolic dysfunction the P-V loop diagram is shifted extremely to the right side of the diagram, showing also relatively high LV pressures (Figure ., Top graph).In a relatively large aneurysm (-; S is  of ventricle mass) the P-V loop diagram is qualitatively similar as in the Top graph of Figure , but quantitatively all changes are more pronounced (Figure ., Bottom graph).

Figure  ,
Top graph).EDPLV changes follow the changes in EDVLV (Figure , Bottom graph).

FIGURE 4 .
FIGURE 4. Pressure-Volume loop diagrams of left ventricle (LV) in normal conditions (N) aneurysm formation (Aneurysm), after distensibility of aneurysm is maximally decreased (STEP II) and after severe systolic dysfunction of section S1 (i.e. in the LV section that did not develop an aneurysm).Top graph: aneurysm in 20% of left ventricle mass.Bottom graph: aneurysm in 50% of left ventricle mass.
) enable the construction of pressure-volume loop diagrams (Figure) that facilitate our understanding of the eff ects of dyskinetic or akinetic aneurysms on ventricular function.A LVA shifts the ventricles pressure-volume loop to the right, this shift is partially reversed by a decrease in aneurysmal distensibility or alternatively, further aggravated by systolic dysfunction of the initially intact ventricular section (i.e.section S; Figure).In addition, the EEC model adjusts the shape of the ventricles pressure-volume loop to refl ect the corresponding changes in ventricular contractility (Figure).

TABLE 1 .
).In normal conditions AoP, MAoP, CO, CVV, Sy, LAtP, ITP are in steady state; heart rate is /min.Decreased contractility in S results in a Sy, LAtP and heart rate increase (/ min), and an AoP, MAoP, CVV and CO decrease for about  s.Steady state is established at about  s of simulation time.Th erefore heart rate is /min again.AoP, MAoP, CO are almost normal, CVV decreased while LAtP is strongly and Sy is slightly increased.A decrease in distensibility of S (step I, step II) results in a temporary increase in AoP, slight increase in MAoP and a decrease in Sy (Figure , Top graph).Before myocardial infarction, EDVLV and ESVLV are in a steady state.SVLV is  mL, EF is ..Decreased contractility in S results in a large increase of EDVLV and ESVLV.Steady state is established within about  s.SVLV is  mL Recorded variables (with corresponding units) and acronyms used in text and illustrations.
Simulation of a large LVA with a subsequent LV systolic dysfunction Th e simulation of a large aneurysm with a subsequent LV systolic dysfunction is presented in Figure .If the simulation shown in Figure  is repeated and modified by inducing first a mild (.s-.s)andlater a severe (.s-s)attenuation of S contractility, it results in a increase of EDVLV and ESVLV, maintaining the SVLV, but further decreasing EF (fi rst to  and later to .).The paradoxical increase of S blood volume is maintained.FIGURE 2. The time course of cardiovascular variables showing initially normal conditions when ejection fraction (EF) is 44.4%, the eff ects of reduced contractility in 50% of left ventricle mass (in section S2) due to development of myocardial infarction and aneurysm (EF is 26.8%) and fi nally if in section S2 the distensibility is decreased in two steps (step I and step II) thus improving EF fi rst to 27.7% and later to 29.2%.Acronyms are explained in Table1.