Journal of Biomechanics 47 (2014) 729–735
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Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Implementation of intrinsic lumped parameter modeling into computational fluid dynamics studies of cardiopulmonary bypass Tim A.S. Kaufmann n, Michael Neidlin, Martin Büsen, Simon J. Sonntag, Ulrich Steinseifer Department of Cardiovascular Engineering, Institute of Applied Medical Engineering, Helmholtz Institute, RWTH Aachen University, Aachen, Germany
art ic l e i nf o
a b s t r a c t
Article history: Accepted 6 November 2013
Stroke and cerebral hypoxia are among the main complications during cardiopulmonary bypass (CPB). The two main reasons for these complications are the cannula jet, due to altered flow conditions and the sandblast effect, and impaired cerebral autoregulation which often occurs in the elderly. The effect of autoregulation has so far mainly been modeled using lumped parameter modeling, while Computational Fluid Dynamics (CFD) has been applied to analyze flow conditions during CPB. In this study, we combine both modeling techniques to analyze the effect of lumped parameter modeling on blood flow during CPB. Additionally, cerebral autoregulation is implemented using the Baroreflex, which adapts the cerebrovascular resistance and compliance based on the cerebral perfusion pressure. The results show that while a combination of CFD and lumped parameter modeling without autoregulation delivers feasible results for physiological flow conditions, it overestimates the loss of cerebral blood flow during CPB. This is counteracted by the Baroreflex, which restores the cerebral blood flow to native levels. However, the cerebral blood flow during CPB is typically reduced by 10–20% in the clinic. This indicates that either the Baroreflex is not fully functional during CPB, or that the target value for the Baroreflex is not a full native cerebral blood flow, but the plateau phase of cerebral autoregulation, which starts at approximately 80% of native flow. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Computational Fluid Dynamics Lumped parameter Baroreflex Cerebral autoregulation Cannulation Cardiopulmonary bypass
1. Introduction Cardiopulmonary bypass (CPB) remains a standard technique in cardiac surgery. However, cerebral hypoxia and an increased risk for strokes are among the main complications during CPB. The cannula jet has been identified as one of the main reasons due to altered flow conditions in the aortic arch and the sandblast effect (Kapetanakis et al., 2004; Scarborough et al., 2003; Verdonck et al., 1998). It has also been shown that the risk for strokes is significantly increased for patients with impaired cerebral autoregulation (Ono et al., 2012). Since many of the elderly suffer from impaired autoregulation (van Beek et al., 2008), cardiac surgeons face the challenge of an ageing population with the potential to further increase the risk for neurological complications during CPB. Due to the nature of CPB, it is hardly possible to study the underlying mechanisms for these complications in vivo and the analysis is currently limited to in vitro or in silico data (Fukuda et al., 2009; Kaufmann et al., 2009a, 2009b; Laumen et al., 2010; Markl et al., 2007; Tokuda et al., 2008). Numerical simulations provide the possibility of analyzing CPB conditions and have been widely applied to study these phenomena, but they are strongly affected by the boundary conditions assumed during CPB, which
n
Corresponding author. Tel.: +49 241 8080540. E-mail address: [email protected] (T.A.S. Kaufmann).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.005
differ from clinically accessible physiological values (Fukuda et al., 2009; Kaufmann et al., 2009a, 2009b; Tokuda et al., 2008). While the first studies in this field used simple pressure boundary conditions, more recent analyses used boundary conditions based on peripheral resistance (Assmann et al., 2012; Benim et al., 2011; Gallo et al., 2012; Kaufmann et al., 2012a, 2012b). In addition to the usual limitations of in vitro and in silico analyses, hardly any of these studies have included the complex mechanisms of cerebral autoregulation, which is defined as the body's intrinsic ability to provide sufficient cerebral blood flow despite changes in cerebral perfusion pressure (Bellapart et al., 2010; van Beek et al., 2008). The cerebral autoregulation acts by adapting the cerebrovascular resistance (Panerai, 1998) and compliance to changes in perfusion pressure, which is achieved by myogenic, neurogenic or metabolic mechanisms (Chillon and Baumbach, 2002; Paulson et al., 1990). Thereby, the cerebral blood flow is kept constant over a range of cerebral perfusion pressures. In Kaufmann et al. (2012a, 2012b), we presented a mathematical approach to incorporate static cerebral autoregulation into Computational Fluid Dynamics (CFD) studies of CPB by using the relationship between the cerebral perfusion pressure and the cerebral blood flow. While this already delivers feasible results based on the level of autoregulation and cerebral perfusion pressure, this method lacks the timedependent adaptability of cerebrovascular resistance and compliance. A way of including this mechanism is by coupling lumped parameter simulations of the circulatory system into CFD.
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Groundbreaking in this field is the model presented by Kim et al. (2009), who developed and implemented autoregulation mechanisms into CFD simulations of physiological flows and tested them during exercise. This model also includes the Baroreflex (Kim et al., 2010), which acts on the resistance, compliance, heart rate and cardiac contractility. However, Kim et al. only incorporated lumped parameter modeling for physiological conditions. Since the heart is clamped during CPB, the model cannot be applied to CPB without adaptations. In this study, we partly adapt this approach to develop an intrinsic lumped parameter model within our CFD model and apply it to CFD studies of flow conditions during CPB. The model creation and simulation setup was based on Kaufmann et al. (2012a, 2012b). A 3-dimensional CAD model of the human vascular system including aorta and greater vessels was created from imaging data. For simulations of CPB conditions, a 24 FR standard CPB outflow cannula was placed virtually in the CAD model. Prior to the numerical studies, a PIV study was initially performed to validate the effect of vascular resistance during CFD for physiological and CPB conditions. After validation, a simple lumped parameter model was created according to Kim et al. (2010, 2009) and implemented into the CFD studies. It was tested for physiological conditions and then applied to CPB simulations. Furthermore, the baroreflex was implemented to analyze the effect of cerebral autoregulation on cerebral blood flow during CPB.
Chemie). After hardening of the silicone, the printed core was removed to create a transparent model of the cardiovascular system. The process of model building is shown in Fig. 1. In addition to this model, throttles were crafted to set a pressure drop behind each vessel. Since all throttles had the same resistance, a non-physiological flow distribution is expected. The results are therefore discussed only in terms of comparison between experiments and simulations and not in terms of physiological feasibility. Two models were crafted: one with native flow from the heart and one with an implemented cannula for CPB. Using these models, Particle Image Velocimetry (PIV) studies were performed to validate the flow fields obtained in-silico. Stereo 3D PIV was performed for both models in focal planes with a distance of 2.5 mm between. The results were averaged over 300 images in each plane. For simplification, the same pressure drop was assumed for all vessels, resulting in non-physiological flow distributions. The flow in the PIV experiments was also averaged to allow a steady state analysis in the experiments. Additionally, geometric adaptations to the regular model were necessary. To achieve a good agreement in boundary conditions, separate validation simulations were performed in the same geometry and with the same pressure drop as in PIV. Quantitative comparison of experimental and numerical results was performed using significance tests. A water–glycerol mixture of 56.4/43.6 (mGl/mW) tempered at 45 1C was used. Thereby, the refractive index of the silicone (n¼ 1.4095) could be nearly matched and an average Newtonian blood viscosity was achieved in the test fluid. The PIV setup is depicted in Fig. 2. 2.4. Implementation of lumped parameter modeling
2. Materials and methods
The simplest way to describe the flow through a vessel using lumped parameter modeling is an electric circuit of one vascular compliance C and i parallel resistances Ri, for each represented vessel, with i being the individual vessel. This lumped parameter model leads to an ordinary differential equation (ODE) which is given in which is given in (1).
2.1. Basic model creation and mesh generation
_ þ pðtÞ
The process of model building and mesh generation was the same as previously reported (Kaufmann et al., 2012a, 2012b). Based on imaging data, a 3-dimensional model of the human cardiovascular system was generated. The final model consisted of the aortic arch including the descending aorta and the subclavian arteries to represent systemic circulation. The cerebral vessels were represented by the carotid and vertebral arteries. For analysis of CPB conditions, a 24 FR standard CPB outflow cannula was virtually placed in the ascending aorta, representing a standard positioning technique during CPB. A tetrahedral mesh was generated within the geometry (ICEM CFD 14.0, Ansys Germany Inc., Otterfing, Germany). Three layers of prismatic elements were furthermore created around the boundaries representing the vessel walls to resolve the boundary layers of the flow. A mesh independency study with element numbers between 0.6 and 11.1 million was initially performed to minimize discretization errors. The final mesh consisted of approximately three million elements. 2.2. Simulation setup Using these boundary conditions, transient numerical simulations of blood flow in the cardiovascular system were performed using commercial software (ANSYS CFX 14.0, Ansys Germany Inc., Otterfing, Germany). A Specified Blend Factor of 0.75 was set as the advection scheme. With this scheme, at least 75% of all mesh elements are solved second-order-accurate, while up to 25% may be solved firstorder-accurate. The turbulence settings were set according to Kaufmann et al., (2012a, 2012b) using the Shear Stress Transport turbulence model implemented in ANSYS CFX. The time step size was set to 0.01 s. Up to 3 internal iterations were solved per time step until the average changes in the transport equations were smaller than the specified convergence target of 1e-4. The vessel walls were considered rigid with a no slip conditions and blood was modeled as a non-Newtonian fluid. The hematocrit was set to 44%, which resulted in a density of 1056.4 kg/m3. The time for one cardiac cycle was set to 0.8 s with a systolic time of 0.3 s. The inlet representing the heart was set using a simplified inlet profile. For analysis of CPB conditions, a continuous flow of 5 l/min through the CPB outflow cannula was set instead of the pulsatile flow from the native heart. 2.3. Validation Validation of the model is performed with an approach similar to a previous study (Laumen et al., 2010). The main goal was to validate the effect of vascular resistances within CFD simulations. Using a slightly adapted CAD model of the cardiovascular system, a 3dimensional model of this system was printed using rapid prototyping. The model was fixated in a box which was filled with silicone (Elastosils RT601, Wacker
1 1 pðtÞ ¼ qðtÞ RC C
ð1Þ
This ODE can be solved for p in each time step n by using the Eq. (2), with qn being the total flow. Using (2), the pressure at each vessel outlet is given dependent on the vascular resistance and compliance. pn ¼ pn 1 e ðΔt=RCÞ þ qn Rð1 e ðΔt=RCÞ Þ
ð2Þ
The resistance in each vessel is given in (3). Ri ¼
Rves ni
ð3Þ
Rves is the total resistance and ni is the ideal percentage of cardiac output flowing through vessel i. The values for ni and Rves are calculated based on the idealized mass flow to each vessel from Laumen et al. (2010). Using Eqs. (2) and (3) as well as the blood density ρ, the blood flow to each vessel can be calculated as shown in (4). _i¼ m
pðtÞ ρ Ri
ð4Þ
This mass flow calculation can be applied as a boundary condition to each individual outgoing vessel. After the intrinsic lumped parameter model was established for physiological flow conditions, it was applied to the pathological flow conditions during cf-CPB. During CPB, the cerebral vessels were modeled using the aforementioned lumped parameter approach, while the outlets representing the systemic circulation (descending aorta and subclavian arteries) were modeled using the Darcy Weißbach Eq. (5) according to Kaufmann et al. (2012a, 2012b), which calculates the pressure drop based on the normal velocity Un, the fluid density ρ and a dimensionless loss coefficient L, resulting in the model setup shown in Fig. 3. 1 2
ΔP ¼ L ρ U 2n
ð5Þ
2.5. Baroreflex One of the key aspects of autoregulation is the baroreflex, which allows adaptation of the cerebrovascular resistance and compliance based on the cerebral perfusion pressure. If the desired pressure differs from the actual pressure, the sympathetic and parasympathetic nervous system start to change R and C in order to restore sufficient cerebral blood flow. A way of modeling this has been presented by Kim et al. (2010). The sympathetic activity ns and parasympathetic activity np are following equations (6a) and (6b) (Ottesen et al., 2004): ns ¼
1 ð1 þ ðp=pt ÞÞν
ð6aÞ
np ¼
1 ð1þ ðp=pt ÞÞ ν
ð6bÞ
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Fig. 1. Model creation for PIV validation; (a) fixated printed core; (b) silicone block with core; (c) silicone block without core; and (d) final model.
Fig. 2. Schematic of PIV setup; (a) physiological flow and (b) CPB support.
Using ns and np, one can calculate the changes in resistance and compliance as first order ODEs (7a) and (7b).
τR
τC
3. Results 3.1. PIV validation
dR þ R ¼ α R ns p β R np p þ γ R dt
ð7aÞ
dC þ C ¼ α C ns p β C np p þ γ C dt
ð7bÞ
The variables τ, α, β and γ can be estimated by assuming that 0.6 o R6o 1.4 mmHg s/ml and 1.26 o C6o 1.6 ml/mmHg (Kim et al., 2010, 2009; Ottesen et al., 2004). The values are given in Table 1. The simulations for CPB were performed with and without Baroreflex to study the effect of cerebral autoregulation on cerebral blood flow during CPB.
The mass flow distribution from the PIV validation is shown in Fig. 4. The error bars indicate the standard deviation based on the overall deviation of the mass flow to the outlets and the inlet flow. For CPB support, the flow distributions in the experiments and simulations are not significantly different (p4 0.999). For physiological flow, a deviation occurs only in the right carotid artery, which is significantly reduced in the PIV measurements (po0.001). Subsequently, the flow to the other vessels is evenly increased. This
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Fig. 4. Flow distribution during physiological and CPB support for experiments (PIV) and simulation (SIM). LC: left carotid artery; LS: left subclavian artery; LV: left vertebral artery; RC: right carotid artery; RS: right subclavian artery; RV: right vertebral artery.
The averaged flow distribution during physiological flow and CPB support is shown in Table 2. The overall flow to the right side is higher as for the left side due to slightly larger vessels on the right side. The total cerebral blood flow for simulated physiological flow is 850 ml/min (17%). For CPB without baroreflex, this is reduced to 507 ml/min (10%). In general, the flow to the brachiocephalic trunk is reduced due to the venturi effect of the cannula jet (Kaufmann et al., 2009a, 2009b), which also affects the right subclavian artery. An exemplary flow field caused by this cannula jet is given in Fig. 8, showing highly increased velocities and non-physiological flow condition during CPB. 3.3. CPB with baroreflex Fig. 3. Model setup for CPB simulations.
Table 1 Variables for cerebral autoregulation. Parameters
Values [Unit]
γR αR βR τR γC αC βC τC
1 [mmHg s/ml] 4 [mmHg s/ml] 4 [mmHg s/ml] 60 [s] 1.4 [ml/mmHg] 2 [ml/mmHg] 2 [ml/mmHg] 60 [s]
results in an overall still non-significant difference between experiments and simulations (p¼0.56) for physiological conditions. The Figs. 5 and 6 show exemplary flow fields during physiological (5) and CPB (6) conditions in a specific plane in the cardiovascular system which is indicated by the red line on the left (5) and middle (6) picture, respectively. For physiological conditions, the flow fields and velocities correspond very well between PIV and CFD. For CPB conditions, the flow fields correspond equally well, however the jet of the cannula is not as clearly visible in the PIV results.
3.2. Physiological flow and CPB without autoregulation The effect of the intrinsic lumped parameter model on the physiological systemic and cerebral outflow is shown in Fig. 7. The flow to the arterial vessels is delayed due to the Windkessel effect of the arterial compliance.
The effect of the baroreflex can be seen from Fig. 9. Due to the decreased CBF, the cerebrovascular resistance is reduced and the cerebrovascular compliance is increased. This adaptation happens in the first 12 s and results in an adaptation of CBF over time as well, as shown in Fig. 10. After approximately 15 s, the CBF is restored to the native level.
4. Discussion Stroke and cerebral hypoperfusion are among the main complications during CPB. They are mainly related to the cannula jet and impaired autoregulation (Ono et al., 2012). One of the effects is a decreased cerebral blood flow and thereby decreased cerebral oxygen supply. This has been studied in the past using CFD. However, numerical studies with simplified boundary conditions overestimate the loss in cerebral blood flow during CPB (Kaufmann et al., 2009a, 2009b). In Kaufmann et al. (2012a, 2012b), we found that a fully established cerebral autoregulation provides approximately 80–90% of native cerebral blood flow during CPB. In this study, we incorporated intrinsic lumped parameter modeling based on the work from (Kim et al., 2010, 2009) into CFD studies of cardiopulmonary bypass. Additionally, a PIV validation was performed to analyze the effect of vascular resistance on the validity of our CFD results. The results of the in vitro and in silico studies were in good agreement for the overall flow distribution and the local flow fields. The only deviation found in the validation was the decreased mass flow to the right carotid artery during physiological flow in the PIV studies. Since all other vessels had roughly the same flow going through them due to the throttles, this is most likely an impurity in the model. The comparatively higher flow in the other vessels is a consequence of this condition.
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Fig. 5. Flow field during physiological flow for PIV (upper) and CFD (lower) in a specific plane in the cardiovascular system (left).
Fig. 6. Flow field during CPB conditions for PIV (left) and CFD (right) showing the cannula jet in a specific plane in the cardiovascular system (middle).
Table 2 Blood flow distribution during physiological flow and CPB.
Fig. 7. Cardiac output, systemic and cerebral flow with lumped parameter model.
The PIV validation included vascular resistances behind each vessel. However, the effect of systemic compliance was neglected. A study with a similar setup, but including a fully functional mock circulation loop around the PIV model, is currently ongoing.
Vessel
Physiological blood flow [l/min]
CPB support without baroreflex [l/min]
Descending aorta Right subclavian Left subclavian Right carotid Left carotid Right vertebral Left vertebral
3.52 0.39 0.27 0.42 0.28 0.08 0.07
3.84 0.20 0.45 0.25 0.17 0.05 0.04
The simulated results for physiological flow proved the feasibility of the incorporated lumped parameter model. For physiological flow, the flow results are in good agreement with previous in silico and clinical data. During CPB however, the loss in cerebral perfusion is approximately 40%. The lumped parameter model alone is therefore already able to avoid the unrealistic negative backflow seen in earlier studies (Kaufmann et al., 2009a, 2009b),
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Fig. 8. Simulated flow field during CPB; left/middle: max. velocities; right: velocities in distal vessels.
Fig. 9. Effect of cerebral autoregulation on resistance and compliance during CPB.
Fig. 10. Cerebral blood flow restoration due to the Baroreflex. LC: left carotid artery; LV: left vertebral artery; RC: right carotid artery; RV: right vertebral artery.
but cannot explain the still relatively high cerebral blood flow during CPB. Once the Baroreflex is incorporated into the model, the cerebral blood flow during CPB is restored, even up to native flow levels. However, one can argue that the Baroreflex may not be fully functional during CPB due to medication and altered flow conditions. Therefore, the simple incorporation of the Baroreflex is not sufficient to analyze patient-specific CPB conditions. It needs to be combined with the results from static autoregulation testing (van Beek et al., 2008) and an approach similar to Kaufmann et al. (2012a, 2012b).
In this study, the target function of the Baroreflex is the native blood flow, and it changes C and R within their determined limits to restore native cerebral flow. A more realistic target would be to reach the plateau phase instead, which starts approximately at 80% of the native flow (Kaufmann et al., 2012a, 2012b). This would be a combination of static and dynamic autoregulation (van Beek et al., 2008) and will be performed in the future. Another limitation of this study is the rigid wall aorta of the CFD domain. While the lumped parameter model incorporates a vascular compliance, the CFD part of the model neglects this effect. Even though the results for physiological flow conditions, which are probably more affected by arterial compliance due to the pulsatile native flow, are in good agreement with clinical data, it is unknown how strong the effect of arterial compliance is on pathological conditions like CPB. There are several obstacles to a fully coupled fluid–structure-interaction model of CPB conditions which have not yet been overcome. Firstly, the numerical costs for such a model are very high compared to standard CFD. Secondly, the patient-specific material properties of human tissue are not always known, and calcification and atherosclerosis cannot be taken into account, which are common in CPB patients, especially the elderly. Besides these limitations, the methodology presented in this study can be used to further understand the effect of cerebral autoregulation on cerebral blood flow during CPB. It can be used to study the effect of dynamic autoregulation during several pathological conditions, for instance continuous flow and pulsatile flow CPB. It will furthermore be transferred to flow conditions during Ventricular Assist Device support and thereby support the development process of these devices at early stages.
T.A.S. Kaufmann et al. / Journal of Biomechanics 47 (2014) 729–735
Conflict of interest statement No conflicts of interests.
Acknowledgment No acknowledgments. References Assmann, A., Benim, A., Gül, F., Lux, P., Akhyari, P., Boeken, U., Joos, F., Feindt, P., Lichtenberg, A., 2012. Pulsatile extracorporeal circulation during on-pump cardiac surgery enhances aortic wall shear stress. J. Biomech. 45 (1), 156–163. Bellapart, J., Geng, S., Dunster, K., Timms, D., Barnett, A., Boots, R., Fraser, J., 2010. Intraaortic balloon pump counterpulsation and cerebralautoregulation: an observational study. BMC Anesthesiol. 10 (1), 325–329. Benim, A., Nahavandi, A., Assmann, A., Schubert, D., Feindt, P., Suh, S., 2011. Simulation of blood flow in human aorta with emphasis on outlet boundary conditions. Appl. Math. Model. 35 (7), 3175–3188. Chillon, J-M., Baumbach, G., 2002. Autoregulation: arterial and intracranial pressure. In: Edvinsson, L., Krause, D. (Eds.), Cerebral Blood Flow and Metabolism. Lippincott Williams & Wilkins, Philadelphia, pp. 395–412. Fukuda, I., Osanai, S., Shirota, M., Inamura, T., Yanaoka, H., Minakawa, M., Fukui, K., 2009. Computer-simulated fluid dynamics of arterial perfusion in extracorporeal circulation: From reality to virtual simulation. Int. J. Arts Org. 32 (6), 362–370. Gallo, D., Santis, G., Negri, F., Tresoldi, D., Ponzini, R., Massai, D., Deriu, M., Segers, P., Verhegghe, B., Rizzo, G., Morbiducci, U., 2012. On the use of in vivo measured flow rates as boundary conditions for image-based hemodynamic models of the human aorta: implications for indicators of abnormal flow. Annu. Biomed. Eng. 40, 729–741. Kaufmann, T., Hormes, M., Laumen, M., Timms, D., Linde, T., Schmitz-Rode, T., Moritz, A., Dzemali, O., Steinseifer, U., 2009a. The impact of aortic/subclavian outflow cannulation for cardiopulmonary bypass and cardiac support: a computational fluid dynamics study. Artif. Organs 33 (9), 727–732. Kaufmann, T., Hormes, M., Laumen, M., Timms, D., Linde, T., Schmitz-Rode, T., Moritz, A., Dzemali, O., Steinseifer, U., 2009b. Flow distribution during cardiopulmonary bypass in dependency on the outflow cannula positioning. Artif. Org. 33 (11), 988–992.
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Contents lists available at ScienceDirect
Journal of Biomechanics journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com
Implementation of intrinsic lumped parameter modeling into computational fluid dynamics studies of cardiopulmonary bypass Tim A.S. Kaufmann n, Michael Neidlin, Martin Büsen, Simon J. Sonntag, Ulrich Steinseifer Department of Cardiovascular Engineering, Institute of Applied Medical Engineering, Helmholtz Institute, RWTH Aachen University, Aachen, Germany
art ic l e i nf o
a b s t r a c t
Article history: Accepted 6 November 2013
Stroke and cerebral hypoxia are among the main complications during cardiopulmonary bypass (CPB). The two main reasons for these complications are the cannula jet, due to altered flow conditions and the sandblast effect, and impaired cerebral autoregulation which often occurs in the elderly. The effect of autoregulation has so far mainly been modeled using lumped parameter modeling, while Computational Fluid Dynamics (CFD) has been applied to analyze flow conditions during CPB. In this study, we combine both modeling techniques to analyze the effect of lumped parameter modeling on blood flow during CPB. Additionally, cerebral autoregulation is implemented using the Baroreflex, which adapts the cerebrovascular resistance and compliance based on the cerebral perfusion pressure. The results show that while a combination of CFD and lumped parameter modeling without autoregulation delivers feasible results for physiological flow conditions, it overestimates the loss of cerebral blood flow during CPB. This is counteracted by the Baroreflex, which restores the cerebral blood flow to native levels. However, the cerebral blood flow during CPB is typically reduced by 10–20% in the clinic. This indicates that either the Baroreflex is not fully functional during CPB, or that the target value for the Baroreflex is not a full native cerebral blood flow, but the plateau phase of cerebral autoregulation, which starts at approximately 80% of native flow. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Computational Fluid Dynamics Lumped parameter Baroreflex Cerebral autoregulation Cannulation Cardiopulmonary bypass
1. Introduction Cardiopulmonary bypass (CPB) remains a standard technique in cardiac surgery. However, cerebral hypoxia and an increased risk for strokes are among the main complications during CPB. The cannula jet has been identified as one of the main reasons due to altered flow conditions in the aortic arch and the sandblast effect (Kapetanakis et al., 2004; Scarborough et al., 2003; Verdonck et al., 1998). It has also been shown that the risk for strokes is significantly increased for patients with impaired cerebral autoregulation (Ono et al., 2012). Since many of the elderly suffer from impaired autoregulation (van Beek et al., 2008), cardiac surgeons face the challenge of an ageing population with the potential to further increase the risk for neurological complications during CPB. Due to the nature of CPB, it is hardly possible to study the underlying mechanisms for these complications in vivo and the analysis is currently limited to in vitro or in silico data (Fukuda et al., 2009; Kaufmann et al., 2009a, 2009b; Laumen et al., 2010; Markl et al., 2007; Tokuda et al., 2008). Numerical simulations provide the possibility of analyzing CPB conditions and have been widely applied to study these phenomena, but they are strongly affected by the boundary conditions assumed during CPB, which
n
Corresponding author. Tel.: +49 241 8080540. E-mail address: [email protected] (T.A.S. Kaufmann).
0021-9290/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jbiomech.2013.11.005
differ from clinically accessible physiological values (Fukuda et al., 2009; Kaufmann et al., 2009a, 2009b; Tokuda et al., 2008). While the first studies in this field used simple pressure boundary conditions, more recent analyses used boundary conditions based on peripheral resistance (Assmann et al., 2012; Benim et al., 2011; Gallo et al., 2012; Kaufmann et al., 2012a, 2012b). In addition to the usual limitations of in vitro and in silico analyses, hardly any of these studies have included the complex mechanisms of cerebral autoregulation, which is defined as the body's intrinsic ability to provide sufficient cerebral blood flow despite changes in cerebral perfusion pressure (Bellapart et al., 2010; van Beek et al., 2008). The cerebral autoregulation acts by adapting the cerebrovascular resistance (Panerai, 1998) and compliance to changes in perfusion pressure, which is achieved by myogenic, neurogenic or metabolic mechanisms (Chillon and Baumbach, 2002; Paulson et al., 1990). Thereby, the cerebral blood flow is kept constant over a range of cerebral perfusion pressures. In Kaufmann et al. (2012a, 2012b), we presented a mathematical approach to incorporate static cerebral autoregulation into Computational Fluid Dynamics (CFD) studies of CPB by using the relationship between the cerebral perfusion pressure and the cerebral blood flow. While this already delivers feasible results based on the level of autoregulation and cerebral perfusion pressure, this method lacks the timedependent adaptability of cerebrovascular resistance and compliance. A way of including this mechanism is by coupling lumped parameter simulations of the circulatory system into CFD.
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Groundbreaking in this field is the model presented by Kim et al. (2009), who developed and implemented autoregulation mechanisms into CFD simulations of physiological flows and tested them during exercise. This model also includes the Baroreflex (Kim et al., 2010), which acts on the resistance, compliance, heart rate and cardiac contractility. However, Kim et al. only incorporated lumped parameter modeling for physiological conditions. Since the heart is clamped during CPB, the model cannot be applied to CPB without adaptations. In this study, we partly adapt this approach to develop an intrinsic lumped parameter model within our CFD model and apply it to CFD studies of flow conditions during CPB. The model creation and simulation setup was based on Kaufmann et al. (2012a, 2012b). A 3-dimensional CAD model of the human vascular system including aorta and greater vessels was created from imaging data. For simulations of CPB conditions, a 24 FR standard CPB outflow cannula was placed virtually in the CAD model. Prior to the numerical studies, a PIV study was initially performed to validate the effect of vascular resistance during CFD for physiological and CPB conditions. After validation, a simple lumped parameter model was created according to Kim et al. (2010, 2009) and implemented into the CFD studies. It was tested for physiological conditions and then applied to CPB simulations. Furthermore, the baroreflex was implemented to analyze the effect of cerebral autoregulation on cerebral blood flow during CPB.
Chemie). After hardening of the silicone, the printed core was removed to create a transparent model of the cardiovascular system. The process of model building is shown in Fig. 1. In addition to this model, throttles were crafted to set a pressure drop behind each vessel. Since all throttles had the same resistance, a non-physiological flow distribution is expected. The results are therefore discussed only in terms of comparison between experiments and simulations and not in terms of physiological feasibility. Two models were crafted: one with native flow from the heart and one with an implemented cannula for CPB. Using these models, Particle Image Velocimetry (PIV) studies were performed to validate the flow fields obtained in-silico. Stereo 3D PIV was performed for both models in focal planes with a distance of 2.5 mm between. The results were averaged over 300 images in each plane. For simplification, the same pressure drop was assumed for all vessels, resulting in non-physiological flow distributions. The flow in the PIV experiments was also averaged to allow a steady state analysis in the experiments. Additionally, geometric adaptations to the regular model were necessary. To achieve a good agreement in boundary conditions, separate validation simulations were performed in the same geometry and with the same pressure drop as in PIV. Quantitative comparison of experimental and numerical results was performed using significance tests. A water–glycerol mixture of 56.4/43.6 (mGl/mW) tempered at 45 1C was used. Thereby, the refractive index of the silicone (n¼ 1.4095) could be nearly matched and an average Newtonian blood viscosity was achieved in the test fluid. The PIV setup is depicted in Fig. 2. 2.4. Implementation of lumped parameter modeling
2. Materials and methods
The simplest way to describe the flow through a vessel using lumped parameter modeling is an electric circuit of one vascular compliance C and i parallel resistances Ri, for each represented vessel, with i being the individual vessel. This lumped parameter model leads to an ordinary differential equation (ODE) which is given in which is given in (1).
2.1. Basic model creation and mesh generation
_ þ pðtÞ
The process of model building and mesh generation was the same as previously reported (Kaufmann et al., 2012a, 2012b). Based on imaging data, a 3-dimensional model of the human cardiovascular system was generated. The final model consisted of the aortic arch including the descending aorta and the subclavian arteries to represent systemic circulation. The cerebral vessels were represented by the carotid and vertebral arteries. For analysis of CPB conditions, a 24 FR standard CPB outflow cannula was virtually placed in the ascending aorta, representing a standard positioning technique during CPB. A tetrahedral mesh was generated within the geometry (ICEM CFD 14.0, Ansys Germany Inc., Otterfing, Germany). Three layers of prismatic elements were furthermore created around the boundaries representing the vessel walls to resolve the boundary layers of the flow. A mesh independency study with element numbers between 0.6 and 11.1 million was initially performed to minimize discretization errors. The final mesh consisted of approximately three million elements. 2.2. Simulation setup Using these boundary conditions, transient numerical simulations of blood flow in the cardiovascular system were performed using commercial software (ANSYS CFX 14.0, Ansys Germany Inc., Otterfing, Germany). A Specified Blend Factor of 0.75 was set as the advection scheme. With this scheme, at least 75% of all mesh elements are solved second-order-accurate, while up to 25% may be solved firstorder-accurate. The turbulence settings were set according to Kaufmann et al., (2012a, 2012b) using the Shear Stress Transport turbulence model implemented in ANSYS CFX. The time step size was set to 0.01 s. Up to 3 internal iterations were solved per time step until the average changes in the transport equations were smaller than the specified convergence target of 1e-4. The vessel walls were considered rigid with a no slip conditions and blood was modeled as a non-Newtonian fluid. The hematocrit was set to 44%, which resulted in a density of 1056.4 kg/m3. The time for one cardiac cycle was set to 0.8 s with a systolic time of 0.3 s. The inlet representing the heart was set using a simplified inlet profile. For analysis of CPB conditions, a continuous flow of 5 l/min through the CPB outflow cannula was set instead of the pulsatile flow from the native heart. 2.3. Validation Validation of the model is performed with an approach similar to a previous study (Laumen et al., 2010). The main goal was to validate the effect of vascular resistances within CFD simulations. Using a slightly adapted CAD model of the cardiovascular system, a 3dimensional model of this system was printed using rapid prototyping. The model was fixated in a box which was filled with silicone (Elastosils RT601, Wacker
1 1 pðtÞ ¼ qðtÞ RC C
ð1Þ
This ODE can be solved for p in each time step n by using the Eq. (2), with qn being the total flow. Using (2), the pressure at each vessel outlet is given dependent on the vascular resistance and compliance. pn ¼ pn 1 e ðΔt=RCÞ þ qn Rð1 e ðΔt=RCÞ Þ
ð2Þ
The resistance in each vessel is given in (3). Ri ¼
Rves ni
ð3Þ
Rves is the total resistance and ni is the ideal percentage of cardiac output flowing through vessel i. The values for ni and Rves are calculated based on the idealized mass flow to each vessel from Laumen et al. (2010). Using Eqs. (2) and (3) as well as the blood density ρ, the blood flow to each vessel can be calculated as shown in (4). _i¼ m
pðtÞ ρ Ri
ð4Þ
This mass flow calculation can be applied as a boundary condition to each individual outgoing vessel. After the intrinsic lumped parameter model was established for physiological flow conditions, it was applied to the pathological flow conditions during cf-CPB. During CPB, the cerebral vessels were modeled using the aforementioned lumped parameter approach, while the outlets representing the systemic circulation (descending aorta and subclavian arteries) were modeled using the Darcy Weißbach Eq. (5) according to Kaufmann et al. (2012a, 2012b), which calculates the pressure drop based on the normal velocity Un, the fluid density ρ and a dimensionless loss coefficient L, resulting in the model setup shown in Fig. 3. 1 2
ΔP ¼ L ρ U 2n
ð5Þ
2.5. Baroreflex One of the key aspects of autoregulation is the baroreflex, which allows adaptation of the cerebrovascular resistance and compliance based on the cerebral perfusion pressure. If the desired pressure differs from the actual pressure, the sympathetic and parasympathetic nervous system start to change R and C in order to restore sufficient cerebral blood flow. A way of modeling this has been presented by Kim et al. (2010). The sympathetic activity ns and parasympathetic activity np are following equations (6a) and (6b) (Ottesen et al., 2004): ns ¼
1 ð1 þ ðp=pt ÞÞν
ð6aÞ
np ¼
1 ð1þ ðp=pt ÞÞ ν
ð6bÞ
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Fig. 1. Model creation for PIV validation; (a) fixated printed core; (b) silicone block with core; (c) silicone block without core; and (d) final model.
Fig. 2. Schematic of PIV setup; (a) physiological flow and (b) CPB support.
Using ns and np, one can calculate the changes in resistance and compliance as first order ODEs (7a) and (7b).
τR
τC
3. Results 3.1. PIV validation
dR þ R ¼ α R ns p β R np p þ γ R dt
ð7aÞ
dC þ C ¼ α C ns p β C np p þ γ C dt
ð7bÞ
The variables τ, α, β and γ can be estimated by assuming that 0.6 o R6o 1.4 mmHg s/ml and 1.26 o C6o 1.6 ml/mmHg (Kim et al., 2010, 2009; Ottesen et al., 2004). The values are given in Table 1. The simulations for CPB were performed with and without Baroreflex to study the effect of cerebral autoregulation on cerebral blood flow during CPB.
The mass flow distribution from the PIV validation is shown in Fig. 4. The error bars indicate the standard deviation based on the overall deviation of the mass flow to the outlets and the inlet flow. For CPB support, the flow distributions in the experiments and simulations are not significantly different (p4 0.999). For physiological flow, a deviation occurs only in the right carotid artery, which is significantly reduced in the PIV measurements (po0.001). Subsequently, the flow to the other vessels is evenly increased. This
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Fig. 4. Flow distribution during physiological and CPB support for experiments (PIV) and simulation (SIM). LC: left carotid artery; LS: left subclavian artery; LV: left vertebral artery; RC: right carotid artery; RS: right subclavian artery; RV: right vertebral artery.
The averaged flow distribution during physiological flow and CPB support is shown in Table 2. The overall flow to the right side is higher as for the left side due to slightly larger vessels on the right side. The total cerebral blood flow for simulated physiological flow is 850 ml/min (17%). For CPB without baroreflex, this is reduced to 507 ml/min (10%). In general, the flow to the brachiocephalic trunk is reduced due to the venturi effect of the cannula jet (Kaufmann et al., 2009a, 2009b), which also affects the right subclavian artery. An exemplary flow field caused by this cannula jet is given in Fig. 8, showing highly increased velocities and non-physiological flow condition during CPB. 3.3. CPB with baroreflex Fig. 3. Model setup for CPB simulations.
Table 1 Variables for cerebral autoregulation. Parameters
Values [Unit]
γR αR βR τR γC αC βC τC
1 [mmHg s/ml] 4 [mmHg s/ml] 4 [mmHg s/ml] 60 [s] 1.4 [ml/mmHg] 2 [ml/mmHg] 2 [ml/mmHg] 60 [s]
results in an overall still non-significant difference between experiments and simulations (p¼0.56) for physiological conditions. The Figs. 5 and 6 show exemplary flow fields during physiological (5) and CPB (6) conditions in a specific plane in the cardiovascular system which is indicated by the red line on the left (5) and middle (6) picture, respectively. For physiological conditions, the flow fields and velocities correspond very well between PIV and CFD. For CPB conditions, the flow fields correspond equally well, however the jet of the cannula is not as clearly visible in the PIV results.
3.2. Physiological flow and CPB without autoregulation The effect of the intrinsic lumped parameter model on the physiological systemic and cerebral outflow is shown in Fig. 7. The flow to the arterial vessels is delayed due to the Windkessel effect of the arterial compliance.
The effect of the baroreflex can be seen from Fig. 9. Due to the decreased CBF, the cerebrovascular resistance is reduced and the cerebrovascular compliance is increased. This adaptation happens in the first 12 s and results in an adaptation of CBF over time as well, as shown in Fig. 10. After approximately 15 s, the CBF is restored to the native level.
4. Discussion Stroke and cerebral hypoperfusion are among the main complications during CPB. They are mainly related to the cannula jet and impaired autoregulation (Ono et al., 2012). One of the effects is a decreased cerebral blood flow and thereby decreased cerebral oxygen supply. This has been studied in the past using CFD. However, numerical studies with simplified boundary conditions overestimate the loss in cerebral blood flow during CPB (Kaufmann et al., 2009a, 2009b). In Kaufmann et al. (2012a, 2012b), we found that a fully established cerebral autoregulation provides approximately 80–90% of native cerebral blood flow during CPB. In this study, we incorporated intrinsic lumped parameter modeling based on the work from (Kim et al., 2010, 2009) into CFD studies of cardiopulmonary bypass. Additionally, a PIV validation was performed to analyze the effect of vascular resistance on the validity of our CFD results. The results of the in vitro and in silico studies were in good agreement for the overall flow distribution and the local flow fields. The only deviation found in the validation was the decreased mass flow to the right carotid artery during physiological flow in the PIV studies. Since all other vessels had roughly the same flow going through them due to the throttles, this is most likely an impurity in the model. The comparatively higher flow in the other vessels is a consequence of this condition.
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Fig. 5. Flow field during physiological flow for PIV (upper) and CFD (lower) in a specific plane in the cardiovascular system (left).
Fig. 6. Flow field during CPB conditions for PIV (left) and CFD (right) showing the cannula jet in a specific plane in the cardiovascular system (middle).
Table 2 Blood flow distribution during physiological flow and CPB.
Fig. 7. Cardiac output, systemic and cerebral flow with lumped parameter model.
The PIV validation included vascular resistances behind each vessel. However, the effect of systemic compliance was neglected. A study with a similar setup, but including a fully functional mock circulation loop around the PIV model, is currently ongoing.
Vessel
Physiological blood flow [l/min]
CPB support without baroreflex [l/min]
Descending aorta Right subclavian Left subclavian Right carotid Left carotid Right vertebral Left vertebral
3.52 0.39 0.27 0.42 0.28 0.08 0.07
3.84 0.20 0.45 0.25 0.17 0.05 0.04
The simulated results for physiological flow proved the feasibility of the incorporated lumped parameter model. For physiological flow, the flow results are in good agreement with previous in silico and clinical data. During CPB however, the loss in cerebral perfusion is approximately 40%. The lumped parameter model alone is therefore already able to avoid the unrealistic negative backflow seen in earlier studies (Kaufmann et al., 2009a, 2009b),
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Fig. 8. Simulated flow field during CPB; left/middle: max. velocities; right: velocities in distal vessels.
Fig. 9. Effect of cerebral autoregulation on resistance and compliance during CPB.
Fig. 10. Cerebral blood flow restoration due to the Baroreflex. LC: left carotid artery; LV: left vertebral artery; RC: right carotid artery; RV: right vertebral artery.
but cannot explain the still relatively high cerebral blood flow during CPB. Once the Baroreflex is incorporated into the model, the cerebral blood flow during CPB is restored, even up to native flow levels. However, one can argue that the Baroreflex may not be fully functional during CPB due to medication and altered flow conditions. Therefore, the simple incorporation of the Baroreflex is not sufficient to analyze patient-specific CPB conditions. It needs to be combined with the results from static autoregulation testing (van Beek et al., 2008) and an approach similar to Kaufmann et al. (2012a, 2012b).
In this study, the target function of the Baroreflex is the native blood flow, and it changes C and R within their determined limits to restore native cerebral flow. A more realistic target would be to reach the plateau phase instead, which starts approximately at 80% of the native flow (Kaufmann et al., 2012a, 2012b). This would be a combination of static and dynamic autoregulation (van Beek et al., 2008) and will be performed in the future. Another limitation of this study is the rigid wall aorta of the CFD domain. While the lumped parameter model incorporates a vascular compliance, the CFD part of the model neglects this effect. Even though the results for physiological flow conditions, which are probably more affected by arterial compliance due to the pulsatile native flow, are in good agreement with clinical data, it is unknown how strong the effect of arterial compliance is on pathological conditions like CPB. There are several obstacles to a fully coupled fluid–structure-interaction model of CPB conditions which have not yet been overcome. Firstly, the numerical costs for such a model are very high compared to standard CFD. Secondly, the patient-specific material properties of human tissue are not always known, and calcification and atherosclerosis cannot be taken into account, which are common in CPB patients, especially the elderly. Besides these limitations, the methodology presented in this study can be used to further understand the effect of cerebral autoregulation on cerebral blood flow during CPB. It can be used to study the effect of dynamic autoregulation during several pathological conditions, for instance continuous flow and pulsatile flow CPB. It will furthermore be transferred to flow conditions during Ventricular Assist Device support and thereby support the development process of these devices at early stages.
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Conflict of interest statement No conflicts of interests.
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