Ann Thorac Surg 1999;67:676-682
© 1999 The Society of Thoracic Surgeons
Original Articles
Simulation of arterial hemodynamics after partial prosthetic replacement of the aorta
Robert Bauernschmitt, MDb,
Stephan Schulz, PhDb,
Andreas Schwarzhaupt, PhDa,
Uwe Kiencke, PhDa,
Christian-Friedrich Vahl, MDa,
Rüdiger Lange, MDa,
Siegfried Hagl, MDa
a Department of Cardiac Surgery, University of Heidelberg, Heidelberg, Germany
b Institute for Industrial Information Technique, University of Karlsruhe, Karlsruhe, Germany
Accepted for publication August 11, 1998.
Address reprint requests to Dr Bauernschmitt, German Heart Center Munich, Lazarett Str 36 80636 Munich, Germany
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Abstract
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Background. Replacing parts of the aorta with a noncompliant vascular prosthesis results in marked alterations of the aortic input impedance and influences arterial hemodynamics. We propose a mathematical model of circulation that can predict hemodynamic changes after simulation of vascular grafting.
Methods. A new mathematical model of the human arterial system was developed on a 75-MHz Pentium personal computer using Matlab software. The human arterial tree was delineated according to a 128-branch design encompassing bifurcations and physical properties of the arterial wall. A digitized aortic flow wave was chosen as the input signal to the system. After determination of the modules of elasticity of native vascular tissue and standard prostheses in technical experiments, replacement of any part of the aorta with a prosthesis was simulated by increasing the elasticity in the parts desired.
Results. During control conditions, the model displayed a physiologic distribution of flow and pressure waves throughout the arterial system. Simulated replacement of the aorta resulted in an increase in pressure amplitude and a partial loss of the aortic "Windkessel" function. Calculation of the aortic input impedance showed an increase in the characteristic impedance, whereas the peripheral resistance remained unaltered.
Conclusions. This mathematical model of the arterial circulation is useful for simulating hemodynamic changes after implantation of vascular grafts. The results of the model analysis are consistent with those in previous experimental work.
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Introduction
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The aorta is a compliant vessel that transforms the pulsatile hydraulic energy provided by the ventricle into a more steady kind of flow by elastic distention and contraction: this property is known as the "Windkessel" function of the aorta and major conduit arteries. Bypassing or replacing the aorta with a noncompliant synthetic prosthesis to aneurysms, complex coarctation, or hypoplastic aortic arch must result in marked alterations in arterial hemodynamics and left ventricular load [1, 2]. Several studies [3, 4] reported increased characteristic impedance and increased afterload caused by the changes in aortic vascular properties, whereas cardiac output and mean aortic pressure remained unaltered. Clinical observations documented left ventricular hypertrophy after bypass procedures, most likely a result of increasing aortic input impedance [5].
Measurement and prediction of these changes in a patient may be important to establish perioperative and postoperative strategies of treatment. However, this requires invasive recording of flow and pressure waves for the calculation of spectral components of impedance, which restricts this method, if used at all, to the operating theater or the catheterization laboratory. Measurements made during operation, at catheterization, or in an experiment, however, are influenced by trauma, anesthesia, volume depletion, and similar conditions and may not be representative of the real pathophysiologic situation. Noninvasive methods to determine exact spectral components of flow and pressure may not yet provide reliable results [6].
To increase the understanding of hemodynamic changes after replacement of the aorta with a synthetic graft, an analytical mathematical model would be useful. In the model, the arterial tree is distinctly constructed so as to encompass bifurcations, the major physical properties of the arterial wall, and the blood flowing through the vessels. As the majority of changes in flow and pressure waves are due to reflections and reflections are generated wherever there is a change in impedance or elasticity, multibranch models provide the closest approximation to the physiologic situation [7, 8].
Flow and pressure waves are supposed to establish themselves depending only on the input "flow" signal to the system. The concept of the model should allow the module of elasticity in each part of the arterial system to increase, so that replacement of specific parts of the aorta or the great arteries can be mathematically simulated.
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Material and methods
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Components of model
All simulations were developed on a 75-MHz personal computer with 8MB RAM. Matlab software along with the toolbox simulink were used; all programs were written in Matlab language.
The model was based on the idea of segmentation of the arterial tree, each segment described by its inductive, capacitive, and resistive properties. The arterial system was delineated according to a 128-branch model of the arterial tree as proposed by Avolio [9] (Fig 1). The basic principle of the model is to translate the properties of arterial hemodynamics into an electrotechnical analogue. Flow is interpreted as the intensity of current, and pressure is interpreted as voltage (Fig 2). The simplified and linearized Navier-Stokes equations link flow and pressure in terms of capacitance and inductance (Appendix 1).
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Fig 1. Model of human arterial tree and estimated distribution of perfusion (modified from Avolio [9]). The inflow into each terminal branch is calculated from physiologic data measured in humans at rest. Flow distribution to the terminal branches is given as the percentage of the total cardiac output. The thoracic aorta is represented by segments 1, 2, and 5 (ascending aorta and aortic arch) and segments 11, 21, and 34 (descending aorta).
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Fig 2. (A) Arterial segment. Flow (q) results in pressure (p) depending on resistance of vessel and blood. (B) Electrotechnical analogue. Resistance (R) and inductance (L) are combined serially, and capacitance (C) is a parallel connection. (p = voltage; q intensity of current.)
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For simplification of mathematical handling, the arterial tree was implemented into a matrix on the basis of the graph theory. Calculation of hemodynamic variables in a specific segment of the arterial tree starts at the top of the tree by subtracting the segment resistances and multiplying the percent values of the assumed blood distribution at the bifurcations according to the current- divisor-rule. When the calculation reaches an end segment, a terminal resistance has to be added to this segment. These resistance values reflect the Ohm-like resistance of arterioles and capillaries in different organs.
With 128 arterial segments, a system of 256 difference equations and differential equations results. To solve the system, the equations were transformed into the frequency domain using Laplace’s transformation.
To avoid stability problems, only transfer functions with integration terms have been used. Pressure in one segment of the arterial system is calculated from the particular inflow to that segment and the elastic properties of the virtual vessel. Flow, pressure, and resistance in each part of the system are determined on the basis of the assumed inflow signal; venous pressure as the end point of pressure measurement is assumed to be zero. In the arterial branches, pressure at the beginning of a new segment is considered equal to the pressure at the end of the supplying segment. The sum of flows in all branches equals the flow of the supplying segment.
Total arterial compliance, defined as the ratio of total arterial volume variation to corresponding arterial pressure variation, was considered to be constant throughout the simulation.
The friction resistance of the blood is calculated according to the law of Hagen Poiseuille (Appendix 2). The values for resistance added together represent the total arterial load or impedance. The influence of the venous system on the arterial load is overlooked in this model. The arterial system ends with a terminal resistance factor calculated according to the physiologic inflow into the respective branch. Inflow into each particular branch and the corresponding pressure curves are determined according to the physical properties of the simulated vessel and the amount of blood passing through it.
For all calculations, the physical properties of a male patient with a height of 175 cm and a weight of 75 kg were introduced. Stroke volume was assumed to be 69 mL, and heart rate was 85 beats per minute. No severe level of artherosclerosis was introduced. For calculation of the friction resistance of blood, a hematocrit level of 45% was assumed at normal body temperature. All these variables can be changed easily, if necessary.
Calculation of elasticity of vascular prostheses
To determine the module of elasticity in synthetic prostheses, a fluid-filled tube model was used. An integrated roller-pump provided a constant flow within the model; static pressure was adjustable. Commercially available prostheses were mounted into the tube system. A microtip-catheter (Miller Instruments Inc, Houston, TX) was introduced at each end of the prosthesis. A pressure pulse wave with a sharp rise was produced within the tube system, and pulse wave velocity was calculated using the transit time from the first to the second pressure catheter. As the pulse wave velocity in a fluid-filled tube has a linear correlation with the module of elasticity of the system, calculation of the elasticity of synthetic prostheses and of biologic tissue was possible from these data. Because the module of elasticity of stretched knitted and woven grafts did not differ significantly, a mean value was used for further calculations.
Simulation of replacement of aorta and replacement sites
The modules of elasticity in specific parts of the arterial model were replaced with the elasticity values of the prosthesis obtained from the tube-model studies (Table 1). The concept of the mathematical model allow increases in the elasticity (which means increasing stiffness) in each part of the arterial tree and therefore, simulation of the hemodynamic consequences of vascular grafting of any arterial branch. In this investigation, virtual replacement of the ascending aorta and arch (segments 1, 2, and 5; see Fig 1) and of the descending thoracic aorta (segments 11, 21, and 34; see Fig 1) was performed. Usually, an "ideal" prosthesis exactly matching the diameter of the replaced segment was simulated; additional calculations were done for a large prosthesis (30 mm) replacing the ascending aorta and aortic arch and a small prosthesis (16 mm) replacing the descending thoracic aorta.
Calculation of aortic input impedance
From the spectra of flow and pressure, the arterial impedance spectrum representing the total arterial load of the segment analyzed is calculated using the ratio of the respective pressure modulus to flow modulus at a given frequency. The impedance modulus at zero frequency represents total peripheral resistance (mean pressure divided by mean flow), and the modulus of the first harmonic of impedance can be considered an index of ventriculoarterial impedance matching according to Kelly and Fitchett [6]. Characteristic impedance, which is supposed to be influenced by vessel wall distensibility, is expressed by the impedance moduli in the frequency range between 2 and 12 Hz [10].
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Results
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General performance of model
After serial and parallel combination of all values for resistance, impedance, and capacitance of the arterial system and after application of a flow curve as the input signal, the system automatically displayed the distribution of flow and pressure curves throughout the entire arterial system according to the physical properties of the vessels. Starting from a real, digitized flow curve as the input signal, the model created almost physiologic pressure curves including reflections from near and remote sites. Flow and pressure curves were observed in each part of the arterial circulation and changed their shapes and amplitudes according to the particular branch in which the observation was made. To simplify the comparison between different conditions, all flow and pressure curves were recorded in the aortic arch (segment 5).
The distinct modeling in the frequency domain enabled performance of fast-Fourier analyses of pressure and flow. Reflection phenomena and the resulting interferences under the conditions applied were clearly demonstrated.
Simulation of prostheses
Altering the modules of elasticity in various parts of the aorta did not influence the stability of simulation. The decrease in distensibility caused changes in the morphology of flow and pressure curves that resulted in an increased amplitude of pulse pressure and a partial loss of the aortic "Windkessel" function (as expressed by a decrease in the negative flow component at the end of diastole) (Figs 3–5). The amplitude of pulse pressure increased from 138/89 mm Hg during control conditions to 144/89 mm Hg after simulated replacement of the descending thoracic aorta. The difference between systolic and diastolic pressure was even higher when the ascending aorta and aortic arch was replaced (Table 2).
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Fig 3. (A) Simulated flow and pressure waves in distal ascending aorta during control conditions. (B) Impedance spectrum as obtained by Fourier transformation of flow and pressure waves.
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Fig 4. (A) Simulated flow and pressure waves after elasticity of segments 1, 2, and 5 of arterial tree was increased (simulation of prosthetic replacement of ascending aorta and aortic arch). Note the increase in pulse pressure amplitude and the decrease in negative aortic flow compared with control conditions (see Fig 3). (B) Impedance spectrum. Peripheral resistance remains unaltered, but there is an increase in impedance values between 2 and 12 Hz (estimated characteristic impedance).
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Fig 5. (A) Simulated flow and pressure waves after elasticity of segments 11, 21, and 34 was increased (simulation of descending thoracic aorta replacement). Increase in pulse pressure amplitude and loss of negative flow are less marked than in Figure 4. (B) Impedance spectrum shows an increase in characteristic impedance.
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Fourier transformations of flow and pressure curves and calculation of their impedance spectra verified these observations. Mean flow and mean pressure were unchanged after simulated implantation of prostheses, but there was a marked increase in the amplitude of the impedance spectrum between 2 and 12 Hz, which was caused more by a reduction in the moduli of flow wave amplitudes than by an increase in the pressure spectral components (see Figs 3–5).
The modulus of the first harmonic of the impedance spectrum, which is considered an indicator of ventriculoarterial impedance matching, was increased from 0.2 mm Hg · s-1 · mL-1 during control simulation (see Fig 3) to 0.29 mm Hg · s-1 · mL-1 after simulated implantation of a prosthesis, irrespective of the location (see Figs 4, 5).
Simulation of total replacement of the thoracic aorta led to more pronounced changes. The pressure amplitude was even higher, and the modulus of the first harmonic showed a further increase to 0.5 mm Hg · s-1 · mL-1 (Fig 6).
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Fig 6. (A) Simulated flow and pressure waves after elasticity of segments 1, 2, 5, 11, 21, and 34 was increased (simulation of total replacement of thoracic aorta). Increase in pulse pressure amplitude and loss of negative flow are more pronounced than in Figures 4 and 5. (B) Impedance spectrum shows an increase in characteristic impedance; the modulus of the first harmonic reached 0.5 mm Hg · s-1 · mL-1.
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The simulation of a 30-mm prosthesis for replacement of the ascending aorta and aortic arch had only minor effects on the hemodynamic situation (Fig 7). Likewise, reduction of the prosthesis diameter to 16 mm for replacement of the descending aorta did not produce major changes.
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Fig 7. Hemodynamics after simulated replacement of ascending aorta and arch with (A) 26-mm prosthesis and (B) 30-mm prosthesis. Only minor differences in diastolic pressure were noted.
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The calculation of pressure in peripheral arteries revealed changes basically matching those in aortic hemodynamics. Because of the increased influence of peripheral reflections, the overall pressure level was higher than in the central recordings (Table 3).
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Comment
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Replacing parts of the aorta with a noncompliant vascular prosthesis changes the elastic properties of the arterial system and thereby interferes with ventriculoarterial coupling [11]. The arterial system, acting as a "hydraulic filter" to convert the pulsatile volume output of the heart into a more steady flow, loses a certain amount of distensibility, and the pulsatile load on the left ventricle increases despite a constant peripheral resistance [12].
Experimental and clinical studies of alterations occurring after implantation of vascular prostheses are difficult to perform. In the experimental scenario, the arterial input impedance can be altered by factors such as an extended surgical procedure per se, blood loss, and anesthesia, and reproducible results of conditions before and after the intervention can be difficult to obtain [13]. Determining the aortic input impedance in humans requires invasive procedures and is therefore restricted to cardiac catheterization or the operating theater. Noninvasive methods to determine exact flow and pressure spectra in patients may not yield completely reliable results and must be restricted to the few very experienced researchers.
Mathematical modeling can become an important tool to study and to predict the interactions of the cardiovascular system with certain surgical interventions [14]. The present study is an attempt to develop a mathematical model of arterial circulation based on the physical properties of the arterial wall and the rheologic properties of blood to achieve some intuitive understanding of arterial hemodynamics after operations on the great arterial vessels.
In this simulation model, flow and pressure waveforms establish themselves depending on only the input signal (the "physiologic," digitized flow curve), the physical properties of vessels and blood, and the terminal resistances. Estimation of a pulse wave velocity is possible, and these data are comparable to values measured experimentally in adult humans [14, 15], which indicates a sufficient approximation of the simulated arterial properties to the real physiologic situation. Differential terms and nonlinearities are avoided, which results in high stability and fast performance of the model.
In an analysis of vascular dynamics, the aortic input impedance spectrum provides the most comprehensive description of the total hydraulic load imposed on the left ventricle. Modulation in the frequency domain allows Fourier transformation of flow and pressure and calculation of the impedance spectrum, which is essential for any analysis of the pulsatile character of cardiovascular action.
After simulated implantation of a synthetic prosthesis, the model did not display any changes in peripheral resistance, but there was an increase in characteristic impedance and pulse pressure amplitude and a loss of the aortic "Windkessel" function. The results of this mathematical analysis are in agreement with those in previous clinical and experimental work. Morita and coworkers [3] described similar changes after extraanatomic bypass of the aorta with a Dacron graft in open-chest dogs: unaltered peripheral resistance, increased blood pressure amplitude, and increased characteristic impedance. Kim and associates [4] noted the same changes in patients who had undergone previous aortic replacement. In addition, the clinical finding of left ventricular hypertrophy after aortic replacement described by Maeta and Hori [5] can be explained by the model; the increase in the modulus of impedance amplitudes and the loss of the aortic "Windkessel" function must result in an increased left ventricular load.
Even the variations in the increases in pulse pressure amplitude between simulated replacement of the ascending aorta and aortic arch (segments 1, 2, and 5) and simulated replacement of the descending thoracic aorta (segments 11, 21, and 34) reflect pathophysiologic conditions. As the ascending aorta is the most "compliant" part of the vessel, replacement of this segment should result in a higher increase in pressure amplitude than replacement of any other part of the vessel. The changes described were even more pronounced when simulation of total replacement of the thoracic aorta was performed.
Increasing or reducing the aortic diameter by simulation of larger or smaller prostheses did not substantially change the results. An exact simulation of possible problems related to different diameters of the prostheses, however, would require the means to simulate flow turbulences occurring at anastomotic mismatches; these calculations are not yet possible.
In summary, the model appears useful for predicting changes in aortic input impedance without the disturbances caused by various circumstances unrelated to the implantation of the prosthesis that inevitably occur during experimental procedures. The concept of this model allows adaption to specific conditions in each patient without any basic modification. It is possible to simulate any site of replacement as well as varying patient properties such as height, weight, hematocrit, amount of atherosclerosis, heart rates, stroke volumes, and levels of arterial pressure. In the future, it will be possible to extend the model in such a way as to allow specific predictions of hemodynamic changes in an individual. The clinician may be able to run a program that offers rapid preoperative estimates of the hemodynamics that will be associated with the intervention planned.
There are limitations to the model. It is focused on the arterial system only; the heart and the venous system are not included. This has the disadvantage that it cannot offer predictions on the circulatory system as a whole. On the other hand, reducing the model to arterial hemodynamics alone results in a lower level of mathematical complexity without losing data concerning the particular site of interest.
The concept of the model is purely linear. A future goal is the integration of nonlinear influences, such as the baroreflex; this will increase the degree of complexity but most likely will not change the basic hemodynamic statements. In the current model, pressure in peripheral arteries is markedly higher than in central arteries (see Table 3). This is in part due to the increasing influence of peripheral reflected waves and in part to the lack of baroreflex control, which would act as a down-regulator of peripheral pressures.
In most experimental studies, however, nonlinear responses of the organism such as the baroreflex have to be suppressed by drug application or surgical resection to allow meaningful spectral analyses [3]. Hence, the model is very comparable to these studies. Nonlinear responses of the arterial wall, which can occur during variations in pulse frequency and pressure amplitude and lead to scatters in the aortic input impedance spectrum, cannot yet be represented by the model [16]. Total arterial compliance was considered to be constant throughout our simulation, although there is evidence that arterial wall stiffness increases with increasing pressure [17]. Capello and colleagues [18] have proposed an algorithm for the estimation of pressure-dependent compliance, which soon will be incorporated into our model.
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Acknowledgments
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This study was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 414, "Computer- and Sensor-aided Surgery."
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Appendix 1. Simplified and linearized Navier-Stokes equations
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Difference-differential equations that link flow and pressure in terms of inductance (L), capacitance, (C), and resistance (R), where d = thickness of vessel wall, e = Young’s nodules for the wall, k = segment number, l = vessel length, p = pressure, q = flow, r = vessel radius, t = time, 
= fluid mass density, and µ = fluid viscosity.
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Appendix 2. Law of Hagen Poiseuille
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R = 8 µl/r4, where R = resistance, µ = fluid viscosity, l = vessel length and r = vessel radius.
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References
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Abstract
Introduction
