Numerical models: realization and applications. Chapter 4 CIRCULATORY SYSTEM MODELS ABSTRACT

Transcription

1 18 Numerical models: realization and applications Chapter 4 CIRCULATORY SYSTEM MODELS Gianfranco Ferrari, Claudio De Lazzari, Krzysztof Zieliński, Libera Fresiello, Krzysztof Jakub Palko ABSTRACT This chapter illustrates the basic structure, the organization and some examples of digital computer circulatory models applications. A special attention is given to the realization of graphical user interfaces and to choice of software platforms. Mechanical circulatory assistance is treated giving two examples where it is represented with two different approaches: representing the physical device or its functional aspects. The parallel LVAD assistance is simulated modeling the pneumatic ventricle and its main components. IABP assistance is simulated modeling the IABP as a zero average flow generator. In both cases the assistance is connected to the comprehensive basic circulatory model that permits to analyze the mutual influence between the device and the circulatory conditions INTRODUCTION A digital computer model consists basically of a set of equations to be implemented, represented and solved to obtain information about the phenomena of interest. Computers are a very important tool to get information on the phenomena represented by any model. As it will be shown in the next Chapters 7 and 8, computers are not the only way to implement a model even if digital computer models are undoubtedly cheap, fast and reliable tools. In the case of biological system models, as the circulatory system (or the respiratory system that will be discussed in the next Chapters 5 and 6), the models (independently of their structure that can be numerical, as it is discussed in this chapter, physical or hybrid, as it will be discussed in chapters 7 and 8) can necessarily reproduce only a part of their complex behavior as it was also remarked in Chapters 1-2 that defined also the general principles for comprehensive circulatory and respiratory system modeling. Coming to digital computer models, the problem is not only to solve the equations but also to present the results in an adequate and complete form. To perform these tasks, it is necessary to choose the high level software language or platform that will be used to transform the equation system and its solutions into information that can be transferred to the end user. The end user

2 19 requirements and the environment where the model will be used imply different technical solutions in the model development. A digital computer model contains in general an equation solving block, a user interface giving access to data presentation, data storing and model control blocks. The equation solving block, if the model is aimed at reproducing periodic phenomena as in the case of comprehensive circulatory and respiratory system modeling, should include mechanisms that permit the reproduction of such phenomena. Definition of the biological system (circulatory or respiratory) properties to be represented Model development: mathematical equations Model verification Software development Choice of the modeling software language or platform Model environment (numerical or hybrid) and data to be presented (user interface) Parameter (or their range) identification Model development Model exploitation Feedbacks from end-users Figure 1 : digital computer model development main steps What follows attaches to digital computer models even if it could be extended to models based on different structures (physical or hybrid). In general, the model should be able to: reproduce the phenomena of interest and give the information necessary to understand the phenomena and meet the end-user requirements. present information in a form complete and sufficiently clear to the end-users. make the technical aspects transparent, although the end user must be aware of the limits of the model, i.e. what the model is able to represent and in which conditions. In general, the model development is an iterative process as it is schematically sketched in the block diagram in Figure 1. The bi-directional flow of information with end-users is fundamental to optimize the model exploitation. Digital computers are probably the most widespread tool for circulatory system model implementation. The number of existing models is enormous and it is difficult to mention the most

3 20 important as many of them are aimed at reproducing very specific circulatory phenomena. A similar situation exists for comprehensive circulatory system models: some of them were developed for External WHILE loop Cardiac cycle counter Internal WHILE loop Cardiac cycle step counter Equation solving (Euler method) Cardiac Cycle Complete? N Y Stop simulation Figure 2 : block diagram showing the software organization. It is based on two nested WHILE loops that permit to solve the equations with Euler s method within the internal loop. The step within the internal loop is equal to the time interval chosen for Euler s method. specific applications as the study of environmental conditions effects [1],[2] or to evaluate the effects of mechanical circulatory assistance on the whole circulatory system [3],[4]. However, it is indubitable that, among circulatory system models, comprehensive models have a huge number of potential applications due to the ease to adapt them to different applications provided that they are modularly designed to heighten their flexibility. The comprehensive circulatory system models discussed in this book were designed (see Chapters 1-2) to exploit at best their implicit flexibility. They are in general designed to be transformed into hybrid [5], if necessary. However, the family of digital computer models developed so far includes also models that were originally designed to be used in their numerical form only [6]. The next paragraphs will therefore discuss models that, while sharing the same basic structure illustrated in Figure 15, Chapter 2, were developed for numerical-hybrid and numerical only applications. The next sections of this chapter will be devoted to analyze the model features and the user interfaces, the software platforms, the general principles of MCA modeling and two different

4 21 applications of MCA based on Visual Basic and LabVIEW software platforms. The model developed in LabVIEW is arranged to be transformed into hybrid if and when necessary MODEL FEATURES AND USER INTERFACES Gianfranco Ferrari, Krzysztof Zieliński (A) Ncyc=0 (B) Next step External loop Run Y Pv>P at N Solve Equations P v >P as Y N Time=Time+dt Compute Ejection Pressure N Time> Tcyc Compute Q out Compute V v Internal loop Y Compute Filling Pressure Time=0 Ncyc=Ncyc+1 Compute Q in Figure 3 : : flow chart of the main tasks in the model software. Panel (A) shows the flow chart corresponding to the structure illustrated in Figure 2. Panel (B) illustrates the flow chart in the case of ventricular models using separate equations for ventricular filling and ejection. Q in, Q out : ventricular input and output flows P v, V v : ventricular pressure and volume Pas, Pat: arterial and atrial pressures The first important notation is that circulatory system models should take into account the periodicity of the cardiovascular phenomena: they are governed by HR that ranges approximately between 60 and 200 bts min-1 (corresponding to a cardiac cycle duration T c ranging from 1 s to 300 ms). The second important notation regards the algorithms to solve differential equations. The system of equations, algebraic and differential, representing the circulatory system model shown in Figure 15, Chapter 2.6, must be solved with one of the existing numerical methods to solve differential equations [7],[8]. The choice of the algorithm depends partially on the software platform that will be used to implement the model. What can be said in general is that software

5 22 External WHILE loop Main functions: Data analysis Simulation settings Data storing Tuning algorithm Non-time critical processes Cardiac cycle counter Average values Ventricular en. Max and Min Parameters setting Parameters Average Instantaneous Input variables Tuning controls Start ejection trigger Internal WHILE loop Main functions: Equation solving Process control (In the case of hybrid applications) Data presentation Error management Cardiac cycle step counter Euler method Formula node vi DACQ board AO single upd. Vi Digital I/O GPIB board p-v loops Time course Average values Time critical processes Figure 4 : the block diagram shows the same structure of Figure 2 where the main functions of the software were added. The block diagram includes some notations that are valid for hybrid application and will be discussed later on. Data presentation can be performed at the end of the cardiac cycle (and therefore in the external WHILE loop) in some fully numerical versions of the software. platforms oriented towards mathematical problem solving have optimized built in equation solving packages but usually do not permit to construct efficient personalized user interfaces. If the model, as in cases discussed here, has to be a user friendly tool, that moreover has to comply with specific requirement as velocity of execution, it is better to choose less accurate but fast methods for equation solving. We adopted the Euler method, a good compromise between, in general contradictory requirements of accuracy, stability and velocity of execution. It simplifies programming and gets the software ready for further applications such as hybrid modeling (see Chapter 8). The reason to choose Euler s method is evident looking at Figure 2. The structure shown there permits first of all to take into account the periodicity of cardiovascular phenomena. The use of two nested WHILE structures (logic structures that are executed until a certain condition becomes false or true) permits to separate equation solving (internal WHILE loop) from all other actions (for example execution control, data storing or, in some cases, data presentation) that can be performed in the external loop. Practically, the internal loop is divided into steps corresponding to one time interval ( t) of Euler s method adopted to solve the equations. The number of steps is equal to T c divided by t: this number is used for the logic condition governing the execution of the internal WHILE loop. The further advantage of this solution consists in its ease application to

6 23 hybrid configurations as it will be shown later in this paragraph. The external WHILE loop plays also a role of cardiac cycle counter. In the case of models where ventricular filling and ejection phases are separated (see Paragraphs 2.4 and 2.6) the solution of ventricular equations (within the general structure depicted in Figure 2) is realized on the basis of the flow chart reported in Figure 3 panel (B) that shows as well the flow chart (panel (A)) corresponding to the structure reproduced in Figure 2. The conditions on P as and P at imply the presence of ventricular valves When a condition on pressure is not met, the corresponding flow is set to zero. This is the behavior of an ideal purely Equation solving: calculated variables A/D converter Measured variables A/D converter Equation solving: calculated variables D/A converter Interface Physical process Interface D/A converter Numerical Physical Numerical Figure 5 : general schematics of the relationship between physical and numerical parts of the model in the case of hybrid applications. The block Physical process can be either a hydraulic, electrical and pneumatic model. The interface block corresponds to the impedance transformer described in Chapter 8. Data measured in the physical section are sent to the numerical section via A/D converters. Equations are solved in the numerical section and calculated variables are input into the physical section via D/A conventers. resistive valve with no backflow (resistance close to zero when the valve is open upstream pressure>downstream pressure, infinite when the valve is closed upstream pressure<downstream pressure). The models include the possibility to simulate non-ideal valves with a finite backflow resistance. The other important functions performed by the model are illustrated in Figure 4. They include data presentation and storing, model parameters control and some algorithms developed for special applications [4],[9]. The block diagram of Figure 4 is valid for all the discussed circulatory system models except for data presentation that is performed at the end of the cardiac cycle for fully numerical models. Hybrid models considerations and specific features Hybrid models will be discussed in detail in Chapter 8. However, as far as the software is concerned, it is important to evidence some important issues. The main feature of a hybrid model is that it is based on a real time process. Equations are solved using the variables calculated by the model and the variables acquired in the physical environment as it is schematically depicted in the

7 24 block diagram in Figure 5. This implies that for each single step inside the internal WHILE loop in Figure 2 some data must be sampled, acquired and transferred via the A/D converter to the numerical model, that in turn, after solving the equations, calculates the variables that, via the D/A converter, are transferred to the physical section [5]. All these actions must be performed within the sampling time (set equal to the time interval t used for equation solving with Euler s method). Sampling time is set for most applications to 1-2 ms [5],[10]. As one can imaging timing is a critical issue in the real time hybrid process. From this point of view, looking at Figure 1 and Figure 3, it is evident why, in the case of hybrid applications, data presentation stays inside the internal loop. In this way, it is possible to display data point by point within the sampling time without introducing any delay into the external WHILE loop before starting a new cardiac cycle. Some further details about this specific problem will be given in the next paragraph 4.3. User interfaces Main User Interface Ventricular Ventricular and Aortic P-V loops Pressures Time Courses Hemodynamics Variables (avgd. in the cardiac cycle) Main Controls & Data Presentation panel Control of Main Circulatory Parameters Flows and Ventricular Volumes Time Courses Secondary Controls panel Simulation Parameters: Time Interval Graphics Starting Values Running Mode (numerical/hybrid) Ventricular Ejection and Application Filling Parameters Parameters Figure 6 : circulatory system model GUI structure. GUI is based on a 3-layer main structure. Each layer has different tasks corresponding at different complexity levels. The first layer (Main User Interface) contains the basic data presentations only. When circulatory system models or models in general are used by specialists for research purposes, this is not in general a critical issue. If on the contrary models have to be used as tools in environments where simplicity of use and fast responses are important, a friendly and simple graphical user interface (GUI) becomes of the utmost importance. It is not possible to give general and universal rules. However data presentation should be limited to what is necessary to understand the model behavior (in relation to the circulatory phenomena that it can simulate) and is, at the same time, familiar for the user. The same rule should be applied for the model run controls that should be limited to what is necessary for the basic operations with the model. All other data and controls should be available but not normally present on the main user interface. The circulatory system

8 25 models discussed in this chapter [4],[6] share the same GUI structure shown in Figure 6. The GUI aspect is of course different for different software platforms. The Main User Interface presents time courses of ventricular and arterial pressures and ventricular work cycles for both left and right ventricles. A set of panels accessible from each of the layers in Figure 6 completes the GUI and permit to set important parameters such as HR, ventricular E max, data storing and, in the case of models with hybrid capabilities, the model run mode (numerical or hybrid) SOFTWARE PLATFORMS Gianfranco Ferrari, Krzysztof Jakub Palko In general, the choice of the software platform, that is to say the software environment where the model will be developed, is dependent on the use of the model and its complexity. In fact, it is possible to choose a variety of software platforms. They can be roughly divided as follows: Mathematical applications: e.g. Mathcad (Mathsoft Inc.), MATLAB or Simulink (Mathworks Inc.); they permit the simulation of different models and have programming capabilities that can be extended using additional modules including A/D and D/A conversions. Electrical simulators: e.g. SPICE. They can be used to simulate different models using the electro-hydraulic analogy. They have limited programming capabilities. Graphical languages: e.g. LabVIEW (National Instruments). This graphical language is spread worldwide and permits complex model simulations and has powerful graphical and programming capabilities. It is an easy to learn language including A/D and D/A routines permitting the interaction with a physical process. Compartmental model languages: they are useful in several modelling applications and are well suitable for educational applications. Compiled software platforms: e.g. C++ Visual Basic. They are powerful languages that offer wide possibilities in modelling and in data presentation. They require a good experience in programming and any change in the applications requires re-compilation. Interpreted software platforms: e.g. Basic. They are suitable for modelling applications. Their programming possibilities present some limitations. In the applications discussed here we used essentially two platforms: C++ together with Visual Basic and LabVIEW.The latter is more flexible and is adapted to both numerical and hybrid applications. LabVIEW platform deserves a further comment, especially for it use in hybrid applications. LabVIEW for Windows suffers for many limitations potentially dangerous for time

9 26 critical tasks in the case of hybrid models (see Figure 4). The use of LabVIEW Real Time (RT) can overcome these limitations [11]. LabView RT is based on the principle to divide all control processes between time critical (RT Target) and non time critical (Host Computer) functions. The block diagram of Figure 7 illustrates the division of tasks between real time target and host computer. The real time target runs on a data acquisition board including an on board processor BASIC PRICIPLES OF CIRCULATORY ASSISTANCE MODELING Gianfranco Ferrari The simulation of mechanical circulatory assistance is not the main goal of this book. In fact, it is rather aimed at illustrating the process of using comprehensive circulatory system models to built what we defined a modeling platform with hybrid features. The latter imply that a physical device can be directly connected to the modeling platform that is able to react appropriately to the presence External while loop Cardiac cycle counter Main functions: Data analysis Average values Ventricular en. Max and min Main functions: Data acquisition Internal while loop Cardiac cycle step counter DACQ board AI one pt vi LabView Host Computer Data presentation Data storing p-v loops Time course Average values Parameters Average Instantaneous Equation solving Process control Euler method Formula node vi DACQ board AO single upd. Vi Digital I/O GPIB board User Interface Comunication TCP/IP Software structure RT Engine RT Target Figure 7 : division of tasks between RT target and host computer using LabVIEW RT. The functions allocated on the host computer do not interfere with the time-critical processes run on the real time board. of the assist device making its simulation useless in this way. However, it can be useful anyway to simulate a physical device, being aware that any simulation is always an approximation of the reality. Two examples of MCA simulation will be given in the next two paragraphs. The chosen examples are paradigmatic of two approaches to MCA simulation. In fact, one can think to simulate the real device reproducing its main physical components and the relationship among them. Another possibility, is to adopt a comprehensive approach simulating the main functional aspects of the device. Both approaches are valid and the choice depends on the aim of the simulation. If, for example, it is necessary to study the effects of different VAD synchronization, may be not

10 27 necessary to reproduce a specific real device but rather an abstraction simulating the interaction of a pressure generator with a failing natural ventricle using different synchronizations. If, on the contrary, it is necessary to study some specific aspects of a physical device (for example the stress on the diaphragm of a pneumatic ventricle in different driving and loading conditions), then it will be necessary to use an approximation of the physical device closer to the reality. The chosen examples include two different type of assistance and two different ways to model it. The first example is the simulation, based on Visual Basic platform, of a pneumatic ventricle connected to the basic comprehensive model of the circulatory system (see Chapter 2.6). The pneumatic ventricle is simulated representing the main components of the whole assist device: pressure and vacuum sources, connection tubing, insertion cannulas and the ventricle itself represented as well with its main components: the diaphragm separating air and blood chambers, input and output artificial valves. The second example is based on the LabVIEW platform: in this case a functional model of an IABP is connected to the basic comprehensive model of the circulatory system (see Chapter 2.6). The IABP is in this case represented as flow generator where it is assumed that the areas of downstream and upstream direction flows are equal and that the net average flow is equal to zero. The corresponding equations of both devices must be inserted into the general equation system described in Chapter 2.6. Both devices deserve a short comment. In fact, one of the advantages in using a comprehensive circulatory system model is that it is possible to study the effects of different device control strategies on hemodynamics and ventricular energetics. In the case of pulsatile LVADs (independently on their power sources that can be pneumatic, electric or electromagnetic), it is possible to trace back their behavior to three main control strategies that are in a sense independent of the physical structure of the device: Full-fill---full-empty: a new ejection begins only when the device is completely filled. This control mode implies variable heart rate (HR) or control signals (e.g. driving pressures in the case of pneumatic ventricles). Partial fill---full-empty (sensitivity to preload): this control mode is most physiological as it can realize Starling s law of the heart. The device fills on the base of the venous return and empties completely on the next ejection.

11 28 Full-fill---partial-empty (sensitivity to afterload): this control mode is rarely used as it does not fulfil the Starling s law of the heart and causes the harmful blood stagnation inside the artificial ventricle. Figure 8 shows the effect of different control strategies on the device internal volume. The model could be used in this case to study, beyond hemodynamics and ventricular energetics, the stresses on critical components of the device. The IABP is a device typically used for heart recovery. Its action is aimed at unloading the failing ventricle and improving coronary flow. It consists of a balloon inserted into the descending aorta periodically fully filled and emptied in relation to the ventricular phases. Its immediate effects on hemodynamics are rather poor, what is important is that it is driven to create the conditions for heart recovery that depend widely on coronary flow and ventricular energetics. As one can imagine, in this frame timing is a critical issue: the use of a comprehensive model can help to define criteria for IABP timing optimization [12]. Fully filled Fully Emptied Fully Emptied Partially Filled Partially Emptied Fully Filled Min Max Volume Min Max Volume Min Max Volume Figure 8 : the effect of the device control strategy on its volume volume. The model can be used to study the energy losses and the corresponding stresses on the critical components of the device with the different control strategies PARALLEL PNEUMATIC LVAD SIMULATION VISUAL BASIC PLATFORM Claudio De Lazzari, Libera Fresiello This simulation reproduces a pneumatic VAD connected in parallel to the left ventricle, between the left atrium and the aorta. The simulation includes the VAD driving system (DS) reproduced schematically in Figure 9. The DS is composed of a pressure source and a vacuum source. They are alternatively connected by a 2-position, 3-way electro magnetic valve to the artificial ventricle that contains four basic components (see Figure 10): an air chamber, a blood chamber,

12 29 a flexible diaphragm separating air and blood chambers input and output artificial valves (usually of tilting disk type) The equations representing the pneumatic ventricle are written considering separately the filling and ejection phases. In particular, the pneumatic ventricular filling phase is described as the air outflow from the higher-pressure tank (the pneumatic ventricle) to the lower-pressure tank (connected to the vacuum source). The ventricular ejection phase is described as the air outflow from the higherpressure tank (connected to the pressure source) to the lower-pressure source (the ventricle itself). The basic assumption adopted to reproduce the filling and ejection phases associated with the pneumatic ventricle are [6],[13]: the interaction between air mass flow generator and a compliance (Figure 9,Figure 10,Figure 11) determines the air pressure waveform in the pneumatic ventricle; the gas undergo a first order adiabatic transformation in correspondence of the working point. Under these hypotheses the equation describing the mean air flow (F) is the following: 2 g m Pe Pe F = S Pi R T m 1 Pi Pi 2/m (m+ 1)/m (1)

13 30 Pressure Source Pressure Regulator Pressure Air Tank dp dt Regulator 2 Position, 3 Way Valve To the pneumatic ventricle TS Vacuum Source Vacuum Regulator Vacuum Air Tank Generator TD Figure 9: general layout of the driving unit system of the pneumatic ventricle. T S (T D ) represents the systole (diastole) time of the drive unit In equation (1) F represents the mean air flow, S is the tube cross-section, Pi is the absolute pressure of gas in higher pressure tank, Pe is the absolute pressure of gas in the lower pressure tank, m is the adiabatic exponent, R is the gas constant, T is the absolute gas temperature and finally g is the gravity acceleration. Equation (1) can be used if the condition Pe Pi is met, as it usually happens in pneumatic ventricles. Figure 10 shows a schematic representation of the pneumatic ventricle. The device is connected to the atrium through the atrial cuff and is connected to the aorta through the dacron tube. In the pneumatic ventricle the diaphragm divides the two chambers. One chamber contains the blood received from the left atrium and successively pumped into the aorta. This chamber is connected to the atrium and to the aorta by two valves. Ped (Ved) is the pressure (volume) in the chamber. Another chamber contains the air pumped from the pressure (vacuum) air tank. Pair (Vair) is the pressure (volume) inside this second chamber.

14 Diaphragm position corresponding the minimum systolic volume 31 atrial cuff Vvad Pvad diaphragm air air tube Vair Pair epoxy housing Diaphragm position corresponding the maximum systolic volume Figure 10: schematic representation of the pneumatic ventricle. Vair (Pair) represents the volume (pressure) of the air inside the air chamber connected to the driving system. Vvad (Pvad) represents the volume (pressure) into the blood chamber. Pressure tan -1 Ced Vmin Vmax Volume tan -1 Ces Figure 11: pressure-volume relationship for the pneumatic ventricle diaphragm in its end-systolic and end-diastolic configurations. Vmax (Vmin) is the maximum (minimum) pneumatic volume and Ced (Ces) is the diaphragm compliance in end-diastolic (end-systolic) positions Figure 11 shows the effect of the diaphragm compliance on the interaction between air and liquid (blood) in the pneumatic ventricle. The pressure drop across the diaphragm depends on its compliance which can be different in end-systolic and end-diastolic positions. In particular, the pressure drop is assumed equal to zero far from the minimum and the maximum ventricular

15 32 volumes. Outside of these two limits, it increases with the diaphragm compliance. Differentiating equation (1) and combining it with the diaphragm characteristic (Figure 11), it is possible to calculate pressures inside air and blood chambers [6]. Pressure inside blood chamber can be used to calculate LVAD input and output flows that are combined with the equation system described in Chapter 2.6. The numerical model of the pneumatic ventricle is a software module that can be connected to the circulatory system model in different ways: LVAD in parallel connection, the object of this chapter: the assistance aspirates blood from the left atrium and ejects it into the aorta. Left ventricular assist device (LVAD) in serial connection: the assistance aspirates blood from the left ventricle and ejects it into the aorta. Right ventricular assist device (RVAD) in serial connection: the assistance aspirates blood from the right ventricle and ejects it into the pulmonary artery. RVAD in serial connection: the assistance aspirates blood from the right atrium and ejects it into the pulmonary artery. By a simultaneous LVAD-RVAD parallel connection it is possible to realize a biventricular assist device (BVAD) (Figure 12).

16 33 Rvi Pvad Pair Rvo LVAD Cair Pd Pv Cair Pulmonary Venous Section Atrium Ventricle Systemic Arterial Section Left Heart Coronary Circulation Pulmonary Arterial Section Right Heart Ventricle Atrium Systemic Venous Section Rvo Pvad Pair Rvi RVAD Cair Pd Pv Cair Figure 12: connection of the pneumatic ventricle model to the circulatory system model in parallel LVAD, RVAD or BVAD modes. After the connection of the VAD numerical model to the cardiocirculatory system model, it is possible to select different control algorithms (see Figure 8) can be selected. It is further possible to control the VAD parameters (driving pressure and vacuum, S/D and its delays from the onset of natural ventricle contraction) synchronized or delayed with the ECG. It is possible to optimize the device performance by selecting. Tubing connections and insertion cannulas are simulated as well. The number of the VAD parameters makes the simulation realistic and permits for example to find the optimal set of parameters for the best device performance or the best effect on natural ventricle variables.

17 34 Assisted Pathological Assisted Pathological Assisted Pathological Figure 13: graphical and numerical output produced using the CARDIOSIM software. In this figure a pathological condition and an assisted condition realized by the activation of a LVAD are reproduced. The assistance is realized by connecting in parallel to the left heart the LVAD. Pla (Pra) represents the mean (during the cardiac cycle) left (right) atrial pressure. Pvs (Pvp) represents the mean systemic (pulmonary) venous pressure The results of a simulation where a LVAD is connected in parallel to the cardiocirculatory system model are shown in Figure 13. The simulation is based on the basic circulatory system model (Chapter 2.6), schematically represented in Figure 12. Before starting the LVAD, a pathological condition was reproduced. Successively the LVAD was activated and synchronized with the natural heart using a delay of 200 msec (in respect to the ventricular systole beginning). The left (right) ventricular P-V loop is shown in the upper left (right) panel in Figure 13. In each panel two ventricular loops, before (pathological) and after the onset of the assistance, are presented. The wide loop in the left panel represents the left ventricular P-V loop before the onset of the assistance. The narrow loop (smaller SV) represents the left ventricular P-V loop after the

18 35 assistance onset. The third panel in Figure 13 shows the systemic arterial pressure (Pas) and the left ventricular pressure (Plv) time courses. Also in this case two sets of waveforms are presented: the first one corresponding to the pathological condition, the second one to the assisted condition. The numerical values reported in the right panel refers to the assisted condition (with LVAD). During the assistance among the others, HR was set to 70 beats/min, the mean systemic arterial pressure (Pas) was equal to 93 mmhg, the systolic aortic pressure was mmhg and the diastolic aortic pressure was 86.5 mmhg. The effect of the assistance can be evaluated considering the ventricular end systolic pressure rise that can be observed in the left ventricular P-V loop. During the assistance, the total flow was 5.11 l/min corresponding to the mean right atrial input flow (Qri) and to the mean right ventricular output flow (Qro). The total flow was produced in part by the left natural ventricle (Qlo=Qli=1.8 l/min) and in part by the LVAD (3.31 l/min). The lower panel in Figure 13 shows the working parameters of the device. Without going into further details, this simulation permits to understand the importance of comprehensive modeling applied, in this case, to parallel LVAD assistance. Beyond being in good agreement with experimental data [14],[15],[16], the simulation permits to estimate parameters that are not easily measurable in clinical or experimental conditions both in the natural ventricle (volumes, flows, energetics) and in the assist device (volumes, flows and pressures) IABP SIMULATION LABVIEW PLATFORM Gianfranco Ferrari As it was said above, the graphical programming typical for LabVIEW permits to build complex Figure 14 : complete screen output of the main user interface applications that are easily changeable. Another important feature of LabVIEW is that it makes it possible to build applications in a modular form which is especially useful for hybrid modeling.

19 36 This library of computer models includes open and closed loop (see Figure 14) circuits used for different applications [4],[9],[10] including functional models of assist devices (see paragraph 4.4). The IABP assistance is among them. IABP is a widely used, well assessed and easy to implant mechanical heart assist device [17]: for these reasons it is still object of several studies aimed at improving its performance and widening its scope of application [18],[19]. One of the issues of interest is the effect of IABP on cardiac performance in relation to its timing. This is a study where the use of a comprehensive circulatory model could give valuable information. The application shortly sketched here uses the basic circulatory model of chapter 2.6 connected to an IABP simulated (see chapter 4.4) as a zero average flow generator. The flow is related to the balloon volume that can be set in the model. Other Figure 15 : IABP simulation. Main user interface with IABP simulation data parameters that can be set are balloon inflation and deflation times. The flow generator representing the IABP is connected to the systemic arterial compliance of the basic model (Chapter 2.6). Of course this is a rather simple connection as in the basic model the systemic arterial tree is represented by a single compliance. The results, as it will be shown, are anyway realistic even if a more complex model could permit a more accurate representation of the balloon insertion. Figure 15 shows a screen output of the model during the IABP simulation experiment. This experiment adopts a procedure similar to the procedure adopted for the previous experiment in paragraph 4.5. Starting from a control condition, the system was forced into a pathological condition [20]. The pathological condition was then assisted changing IABP filling and emptying times. The next Figure 16 shows the ventricular and circulatory parameter values and two samples of ventricular P-V loops and ventricular and arterial pressure time courses before and after the onset

20 37 of IABP assistance. The tables show the parameters setting before (CNT) and after (HF) the onset of a severe heart failure. Other significant data are reproduced in the two boxes in right part of the figure. The first shows the time courses of systemic arterial pressure before and after the onset of the assistance. The second one presents the corresponding left ventricular P-V loops. The effect of Left v. CNT HF Right v. CNT/HF E maxl mmhg cm E maxr V 0r V 0l cm R as HR bts min -1 CI ml min -1 m Main circulatory parameters g cm -4 s /2700 C as mmhg cm R vs g cm -4 s -1 Aut.Contr. R ap g cm-4 s C ap mmhg cm Pressure [mmhg] AoP [mmhg] HF Coronary Flow HF IABP Time [s] HF HF IABP Volume [cm 3 ] Figure 16 : circulatory and ventricular parameters settings to reproduce the pathological condition [20] 1.4 T1 T2 T3 T4 T5 1.3 f % e % CI HF HF - T1 HF - T2 HF - T3 HF - T4 HF - T5 Figure 17 : the effect of IABP timing on Cardiac Index (CI) the assistance strongly depends on timing. The data presented in Figure 16 correspond to the best IABP timing shown in the next figure. Figure 17 presents the effect of different timing (from HF

21 38 T1 to HF T5) on Cardiac Index CI. The corresponding IABP control parameter setting is reported in the table inserted in the picture: f and e are the beginning of the filling and emptying phases of the balloon expressed as percentage of the cardiac cycle duration. The effect of the IABP assistance on the arterial pressure is evidenced by the reduction of systolic arterial pressure, the diastolic augmentation and the reduction of the end-diastolic pressure, typical of IABP assistance. The analysis of ventricular P-V loops shows other important effects of the IABP assistance. The most evident are the reduction of the end-systolic ventricular volume together with the counterclockwise rotation of the arterial elastance line. A smaller effect can be observed on the ventricular end-diastolic volume, slightly reduced by the assistance. Stroke volume (that can be easily detected from the P-V loop) is increased by about 15%. It is to be pointed out here that the increase of cardiac output obtained during the assistance is limited: it is only the onset of feedback mechanisms to improve coronary perfusion and to permit a general recovery of the left ventricle and the further improvement of hemodynamic conditions. Also in this case, the use of a comprehensive circulatory model permits to estimate the assistance effects over a wide number of circulatory and ventricular variables offering a more complete view on the effects of this important heart assist device CONCLUSION This chapter has illustrated the main design requirements of digital computer circulatory models. The two examples reported here show the possibilities of modeling in giving altogether information that can be usually retrieved from separate experiments. It is possible in this way to analyze the effects of a single parameter or of a set of parameters on specific circulatory and ventricular variables. In the case of mechanical circulatory assistance discussed in the two examples reported in paragraphs 4.5 and 4.6, it is possible to evaluate the mutual influence between assistance control parameters and circulatory conditions. These possibilities open new scenarios for the application of circulatory modeling permitting a comprehensive view on many hemodynamic phenomena including the mutual interaction between the circulatory system and mechanical circulatory assistance. The introduction of the hybrid modeling platform concept (outlined in Chapter 1 and discussed in Chapter 8) including to the respiratory system model reinforces these possibilities creating a sort of standardized modeling procedure. References [1] Heldt T, Shim EB, Kamm RD, Mark RG. Computational modeling of cardiovascular response to orthostatic stress. J Appl Physiol 2002;92(3): [2] Fitz-Clarke JR. Computer simulation of human breath-hold diving: cardiovascular adjustments. Eur J Appl Physiol 2007;100(2):

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