Numerical simulation of the blood flow in the human cardiovascular system

Seven Mathematical Models of Hemorrhagic Shock

Although mathematical modelling of pressure-flow dynamics in the cardiocirculatory system has a lengthy history, readily finding the appropriate model for the experimental situation at hand is often a challenge in and of itself. An ideal model would be relatively easy to use and reliable, besides being ethically acceptable. Furthermore, it would address the pathogenic features of the cardiovascular disease that one seeks to investigate. No universally valid model has been identified, even though a host of models have been developed. The object of this review is to describe several of the most relevant mathematical models of the cardiovascular system: the physiological features of circulatory dynamics are explained, and their mathematical formulations are compared. The focus is on the whole-body scale mathematical models that portray the subject's responses to hypovolemic shock. The models contained in this review differ from one another, both in the mathematical methodology adopted and in the physiological or pathological aspects described. Each model, in fact, mimics different aspects of cardiocirculatory physiology and pathophysiology to varying degrees: some of these models are geared to better understand the mechanisms of vascular hemodynamics, whereas others focus more on disease states so as to develop therapeutic standards of care or to test novel approaches. We will elucidate key issues involved in the modeling of cardiovascular system and its control by reviewing seven of these models developed to address these specific purposes.

Porcine and bovine aortic valve comparison for surgical optimization: A fluid-structure interaction modeling study

BACKGROUND

Porcine aortic valve (PAV) and bovine aortic valve (BAV) are commonly used in aortic valve replacement (AVR) surgeries. A detailed comparison for their hemodynamic and structural stress/strain performances would help to better understand valve cardiac function and select valve type and size for AVR outcome optimizations.

METHODS

Eight fluid-structure interaction models were constructed to compare hemodynamic and stress/strain behaviors of PAV and BAV with 4 sizes (19, 21, 23, and 25 mm). Blood flow velocity, systolic cross-valve pressure gradient (SCVPG), geometric orifice area (GOA), flow shear stresses (FSS), and stress/strain were obtained for comparison.

RESULTS

Compared with PAV, BAV has better hemodynamic performance, with lower maximum flow velocity (7.17%) and pressure (9.82%), smaller pressure gradient (mean and peak SCVPG: 8.92% and 9.28%), larger GOA (9.56%) and lower FSS (6.61%). The averages of the mean and peak net pressure gradient values from 4 BAV models were 8.10% and 8.35% lower than that from PAV models. Larger valve sizes for both PAV and BAV had improved hemodynamic performance. Maximum flow velocity, pressure, mean SCVPG and maximum FSS from 25 mm BAV were 36.80%, 15.81%, 39.05% and 38.83% lower than those from 19 mm BAV. The GOA of PAV and BAV 25 mm Valve were 43.75% and 33.07% larger than 19 mm valves, respectively. BAV has lower stress on the leaflets than PAV.

CONCLUSIONS

BAV had better hemodynamic performance and lower leaflets stress than PAV. More patient studies are needed to validate our findings.

Experimental investigation into the effect of compliance of a mock aorta on cardiac performance

Demand for heart transplants far exceeds supply of donated organs. This is attributed to the high percentage of donor hearts that are discarded and to the narrow six-hour time window currently available for transplantation. Ex-vivo heart perfusion (EVHP) provides the opportunity for resuscitation of damaged organs and extended transplantation time window by enabling functional assessment of the hearts in a near-physiologic state. Present work investigates the fluid mechanics of the ex-vivo flow loop and corresponding impact on cardiac performance. A mechanical flow loop is developed that is analogous to the region of the EVHP system that mimics in-vivo systemic circulation, including the body’s largest and most compliant artery, the aorta. This investigation is focused on determining the effect of mock aortic tubing compliance on pump performance. A custom-made silicone mock aorta was developed to simulate a range of in-vivo conditions and a physiological flow was generated using a commercial ventricular assist device (VAD). Monitored parameters, including pressure, tube distension and downstream velocity, acquired using time-resolved particle imaging velocimetry (PIV), were applied to an unsteady Bernoulli analysis of the flow in a novel way to evaluate pump performance as a proxy for cardiac workload. When compared to the rigid case, the compliant mock aorta case demonstrated healthier physiologic pressure waveforms, steadier downstream flow and reduced energetic demands on the pump. These results provide experimental verification of Windkessel theory and support the need for a compliant mock aorta in the EVHP system.

Analysis of Cleveland Clinic continuous-flow total artificial heart performance using the Virtual Mock Loop: Comparison with an in vivo study

The Virtual Mock Loop (VML) is a mathematical model designed to simulate mechanism of the human cardiovascular system interacting with mechanical circulatory support devices. Here, we aimed to mimic the hemodynamic performance of Cleveland Clinic's self-regulating continuous-flow total artificial heart (CFTAH) via VML and evaluate the accuracy of the VML compared with an in vivo acute animal study. The VML reproduced 124 hemodynamic conditions from three acute in vivo experiments in calves. Systemic/pulmonary vascular resistances, pump rotational speed, pulsatility, and pulse rate were set for the VML from in vivo data. We compared outputs (pump flow, left and right pump pressure rises, and atrial pressure difference) between the two systems. The pump performance curves all fell in the designed range. There was a strong correlation between the VML and the in vivo study in the left pump flow (r2 = 0.84) and pressure rise (r2 = 0.80), and a moderate correlation in right pressure rise (r2 = 0.52) and atrial pressure difference (r2 = 0.59). Although there is room for improvement in simulating right-sided pump performance of self-regulating CFTAH, the VML acceptably simulated the hemodynamics observed in an in vivo study. These results indicate that pump flow and pressure rise can be estimated from vascular resistances and pump settings.

SIMULATION OF A CLOSED LOOP MODEL EQUIVALENT ELECTRONIC OF NORMAL CARDIOVASCULAR SYSTEM AND VALVULAR AORTIC STENOSIS

This work presents a developed zero-dimensional cardiovascular (CV) system model, based on an electrical analogy, with a detailed compartmental description of the heart and the main vascular circulation which is able to simulate normal and diseased conditions of CV system, especially the stenosis valvular aortic. To know the effect of each parameter on hemodynamics, the number of parameters is increased by adding more segments. The developed model consists of 14 compartments. The results show that the severity of aortic stenosis (AS) effect varies with the effective orifice area and the mean pressure gradient for the case of no AS; the effective orifice area is 4[Formula: see text]cm2 and the mean pressure gradient is 0[Formula: see text]mmHg, while for the case of mild AS, the effective orifice area is 1.5[Formula: see text]cm2 and the mean pressure gradient is 27.24[Formula: see text]mmHg. For the case of moderate AS, the effective orifice area is 1.0[Formula: see text]cm2 and the mean pressure gradient is 44.68[Formula: see text]mmHg. For the case of the severe AS, the effective orifice area is 0.61[Formula: see text]cm2 and the mean pressure gradient is 77.51[Formula: see text]mmHg. It is found that the developed model can estimate an accurate value of the effective orifice area for any value of mean pressure gradient in AS. The results obtained for the CV system under normal and diseased conditions show a good agreement compared to published results.

Cardiovascular Function and Ballistocardiogram: A Relationship Interpreted via Mathematical Modeling

OBJECTIVE

To develop quantitative methods for the clinical interpretation of the ballistocardiogram (BCG).

METHODS

A closed-loop mathematical model of the cardiovascular system is proposed to theoretically simulate the mechanisms generating the BCG signal, which is then compared with the signal acquired via accelerometry on a suspended bed.

RESULTS

Simulated arterial pressure waveforms and ventricular functions are in good qualitative and quantitative agreement with those reported in the clinical literature. Simulated BCG signals exhibit the typical I, J, K, L, M, and N peaks and show good qualitative and quantitative agreement with experimental measurements. Simulated BCG signals associated with reduced contractility and increased stiffness of the left ventricle exhibit different changes that are characteristic of the specific pathological condition.

CONCLUSION

The proposed closed-loop model captures the predominant features of BCG signals and can predict pathological changes on the basis of fundamental mechanisms in cardiovascular physiology.

SIGNIFICANCE

This paper provides a quantitative framework for the clinical interpretation of BCG signals and the optimization of BCG sensing devices. The present paper considers an average human body and can potentially be extended to include variability among individuals.

Multiscale systems biology of trauma-induced coagulopathy

Trauma with hypovolemic shock is an extreme pathological state that challenges the body to maintain blood pressure and oxygenation in the face of hemorrhagic blood loss. In conjunction with surgical actions and transfusion therapy, survival requires the patient's blood to maintain hemostasis to stop bleeding. The physics of the problem are multiscale: (a) the systemic circulation sets the global blood pressure in response to blood loss and resuscitation therapy, (b) local tissue perfusion is altered by localized vasoregulatory mechanisms and bleeding, and (c) altered blood and vessel biology resulting from the trauma as well as local hemodynamics control the assembly of clotting components at the site of injury. Building upon ongoing modeling efforts to simulate arterial or venous thrombosis in a diseased vasculature, computer simulation of trauma-induced coagulopathy is an emerging approach to understand patient risk and predict response. Despite uncertainties in quantifying the patient's dynamic injury burden, multiscale systems biology may help link blood biochemistry at the molecular level to multiorgan responses in the bleeding patient. As an important goal of systems modeling, establishing early metrics of a patient's high-dimensional trajectory may help guide transfusion therapy or warn of subsequent later stage bleeding or thrombotic risks. This article is categorized under: Analytical and Computational Methods > Computational Methods Biological Mechanisms > Regulatory Biology Models of Systems Properties and Processes > Mechanistic Models.

Global sensitivity analysis of a model for venous valve dynamics

Chronic venous disease is defined as dysfunction of the venous system caused by incompetent venous valves with or without a proximal venous obstruction. Assessing the severity of the disease is challenging, since venous function is determined by various interacting hemodynamic factors. Mathematical models can relate these factors using physical laws and can thereby aid understanding of venous (patho-)physiology. To eventually use a mathematical model to support clinical decision making, first the model sensitivity needs to be determined. Therefore, the aim of this study is to assess the sensitivity of the venous valve model outputs to the relevant input parameters. Using a 1D pulse wave propagation model of the tibial vein including a venous valve, valve dynamics under head up tilt are simulated. A variance-based sensitivity analysis is performed based on generalized polynomial chaos expansion. Taking a global approach, individual parameter importance on the valve dynamics as well as importance of their interactions is determined. For the output related to opening state of the valve, the opening/closing pressure drop (dp_valve,0) is found to be the most important parameter. The venous radius (r_vein,0) is related to venous filling volume and is consequently most important for the output describing venous filling time. Finally, it is concluded that improved assessment of r_vein,0 and dp_valve,0 is most rewarding when simulating valve dynamics, as this results in the largest reduction in output uncertainty. In practice, this could be achieved using ultrasound imaging of the veins and fluid structure interaction simulations to characterize detailed valve dynamics, respectively.

Study of cardiovascular function using a coupled left ventricle and systemic circulation model

To gain insight into cardio-arterial interactions, a coupled left ventricle-systemic artery (LV–SA) model is developed that incorporates a three-dimensional finite-strain left ventricle (LV), and a physiologically-based one-dimensional model for the systemic arteries (SA). The coupling of the LV model and the SA model is achieved by matching the pressure and the flow rate at the aortic root, i.e. the SA model feeds back the pressure as a boundary condition to the LV model, and the aortic flow rate from the LV model is used as the input for the SA model. The governing equations of the coupled system are solved using a combined immersed-boundary finite-element (IB/FE) method and a Lax–Wendroff scheme. A baseline case using physiological measurements of healthy subjects, and four exemplar cases based on different physiological and pathological scenarios are studied using the LV–SA model. The results of the baseline case agree well with published experimental data. The four exemplar cases predict varied pathological responses of the cardiovascular system, which are also supported by clinical observations. The new model can be used to gain insight into cardio-arterial interactions across a range of clinical applications.

Development of a cardiopulmonary mathematical model incorporating a baro-chemoreceptor reflex control system

This article describes the development of a comprehensive mathematical model of the human cardiopulmonary system that combines the respiratory and cardiovascular systems and their associated autonomous nervous control actions. The model is structured to allow the complex interactions between the two systems and the responses of the combined system to be predicted under different physiological conditions. The cardiovascular system model contains 13 compartments, including the heart chambers operating as a pump and the blood vessels represented as distensible tubes configured in a serial and parallel arrangement. The accurate representation of the hemodynamics in the system and the good fit to published pressure and flow waveforms gave confidence in the modelling approach adopted for the cardiovascular system prior to the incorporation of the baroreflex control and the respiratory models. An improved baroreceptor reflex model is developed in this research, incorporating afferent, central and efferent compartments. A sigmoid function is included in the efferent compartment to produce sympathetic and parasympathetic nerve outflow to the effector sites. The baroreflex action is modelled using physiological data, its interaction with the chemoreflex control is explained and the simulation results presented show the ability of the model to predict the static and dynamic hemodynamic responses to environmental disturbances. A previously published respiratory model that includes the mechanics of breathing, gas exchange process and the regulation of the system is then combined with the cardiovascular model to form the cardiopulmonary model. Through comparison with published data, the cardiopulmonary model with the baro-chemoreflex control is validated during hypoxia and hypercapnia. The percentage difference between the predicted and measured changes in the heart rates and the mean arterial pressures are within 3% in both cases. The total peripheral resistance correlates well for hypoxia but is less good for hypercapnia, where the predicted change from normal condition is around 7% compared with a measured change of 23%. An example showing the application of the proposed model in sport science is also included.

A simple, versatile valve model for use in lumped parameter and one-dimensional cardiovascular models

Lumped parameter and one-dimensional models of the cardiovascular system generally employ ideal cardiac and/or venous valves that open and close instantaneously. However, under normal or pathological conditions, valves can exhibit complex motions that are mainly determined by the instantaneous difference between upstream and downstream pressures. We present a simple valve model that predicts valve motion on the basis of this pressure difference, and can be used to investigate not only valve pathology, but a wide range of cardiac and vascular factors that are likely to influence valve motion.

Mathematical multi-scale model of the cardiovascular system including mitral valve dynamics. Application to ischemic mitral insufficiency

Background

Valve dysfunction is a common cardiovascular pathology. Despite significant clinical research, there is little formal study of how valve dysfunction affects overall circulatory dynamics. Validated models would offer the ability to better understand these dynamics and thus optimize diagnosis, as well as surgical and other interventions.

Methods

A cardiovascular and circulatory system (CVS) model has already been validated in silico, and in several animal model studies. It accounts for valve dynamics using Heaviside functions to simulate a physiologically accurate "open on pressure, close on flow" law. However, it does not consider real-time valve opening dynamics and therefore does not fully capture valve dysfunction, particularly where the dysfunction involves partial closure. This research describes an updated version of this previous closed-loop CVS model that includes the progressive opening of the mitral valve, and is defined over the full cardiac cycle.

Results

Simulations of the cardiovascular system with healthy mitral valve are performed, and, the global hemodynamic behaviour is studied compared with previously validated results. The error between resulting pressure-volume (PV) loops of already validated CVS model and the new CVS model that includes the progressive opening of the mitral valve is assessed and remains within typical measurement error and variability. Simulations of ischemic mitral insufficiency are also performed. Pressure-Volume loops, transmitral flow evolution and mitral valve aperture area evolution follow reported measurements in shape, amplitude and trends.

Conclusions

The resulting cardiovascular system model including mitral valve dynamics provides a foundation for clinical validation and the study of valvular dysfunction in vivo. The overall models and results could readily be generalised to other cardiac valves.

Review of zero-D and 1-D models of blood flow in the cardiovascular system

Background

Zero-dimensional (lumped parameter) and one dimensional models, based on simplified representations of the components of the cardiovascular system, can contribute strongly to our understanding of circulatory physiology. Zero-D models provide a concise way to evaluate the haemodynamic interactions among the cardiovascular organs, whilst one-D (distributed parameter) models add the facility to represent efficiently the effects of pulse wave transmission in the arterial network at greatly reduced computational expense compared to higher dimensional computational fluid dynamics studies. There is extensive literature on both types of models.

Method and Results

The purpose of this review article is to summarise published 0D and 1D models of the cardiovascular system, to explore their limitations and range of application, and to provide an indication of the physiological phenomena that can be included in these representations. The review on 0D models collects together in one place a description of the range of models that have been used to describe the various characteristics of cardiovascular response, together with the factors that influence it. Such models generally feature the major components of the system, such as the heart, the heart valves and the vasculature. The models are categorised in terms of the features of the system that they are able to represent, their complexity and range of application: representations of effects including pressure-dependent vessel properties, interaction between the heart chambers, neuro-regulation and auto-regulation are explored. The examination on 1D models covers various methods for the assembly, discretisation and solution of the governing equations, in conjunction with a report of the definition and treatment of boundary conditions. Increasingly, 0D and 1D models are used in multi-scale models, in which their primary role is to provide boundary conditions for sophisticate, and often patient-specific, 2D and 3D models, and this application is also addressed. As an example of 0D cardiovascular modelling, a small selection of simple models have been represented in the CellML mark-up language and uploaded to the CellML model repository http://models.cellml.org/. They are freely available to the research and education communities.

Conclusion

Each published cardiovascular model has merit for particular applications. This review categorises 0D and 1D models, highlights their advantages and disadvantages, and thus provides guidance on the selection of models to assist various cardiovascular modelling studies. It also identifies directions for further development, as well as current challenges in the wider use of these models including service to represent boundary conditions for local 3D models and translation to clinical application.

Morphometric and simulation analyses of right hepatic vein reconstruction in adult living donor liver transplantation using right lobe grafts

The incidence of clinically significant right hepatic vein (RHV) stenosis after adult living donor liver transplantation has been higher than expected. In this study, an assessment of the risk factors for the development of RHV stenosis in this context was undertaken. Hepatic anatomy, surgical techniques, and the incidence of RHV stenosis 1 year after transplantation were evaluated retrospectively in 225 recipients of right lobe grafts. These patients underwent independent RHV reconstruction, which was facilitated by the application of computed tomography morphometry and computational simulation analyses. Three types of preparation of the orifice of the graft RHV and 7 types of preparation for venoplasty of the recipient RHV were used. The frequency of high, middle, and low sites of RHV insertion into the inferior vena cava (IVC) was 56.0%, 36.4%, and 7.6%, respectively, for donors, and 26.7%, 58.7%, and 14.7%, respectively, for recipients. Nine patients (4%) developed RHV stenosis of early onset that required stent insertion during the first 2 postoperative weeks; in 12 patients (5.3%), RHV stenosis of delayed onset occurred. Inappropriate matching of RHV sites of insertion correlated with the incidence of stenosis of early onset (P = 0.039). Technical refinements to avoid adverse consequences of inappropriate ventrodorsal matching of RHV sites of insertion include making the recipient RHV orifice wide and enlarging the recipient IVC by a customized incision and patch venoplasty after anatomical assessment of the RHV and IVC of the graft and recipient.

Modeling intravasation of liquid distension media in surgical simulators

During therapeutic hysteroscopy and transurethral resection of the prostate, intravasation of the liquid distension media into the vascular system of the patient occurs. We present a model which allows the integration of the intravasation process into surgical simulator systems. A linear network flow model is extended with a correction for non-Newtonian blood behavior in small vessels and an appropriate handling of vessel compliance. We employ a fast lookup scheme in order to allow for real-time simulation. Cutting of tissue is accounted for by adjusting pressure boundary conditions for all cut vessels. We investigate the influence of changing distention fluid pressure settings and of the position of tissue cuts. In addition, we quantify the intravasation occurring with different approaches of fluid control, and we compare the performance of direct and iterative solvers applied to the non-linear system of the compliant model. Our simulation predicts significant intravasation only on the venous side, and just in cases when larger veins are cut. The implemented methods allow the realistic control of bleeding for short-term and of the total resulting intravasation volume for long-term complication scenarios. While the simulation is fast enough to support real-time training, it is also adequate for explaining intravasation effects which were previously observed on a phenomenological level only.

A functional cardiovascular model with disorders

This paper introduces a functional model of the cardiovascular system that is capable of describing its behavior in normal as well as pathologic cases. The developed model includes all the main compartments of the circulatory system and also the baroreflex-feedback regulatory mechanism. The model response to the incorporation of two critical cardiovascular disorders namely hypertension and acute congestive heart failure is realistic and within the expected range of the results of the literature experimental data.

Parameter estimation by descent and genetic algorithm methods of an in-vitro stenosis bypass model

The development of improved hemodynamic impedance models can greatly aid the understanding of arterial disease progression and its remediation. This paper leverages the recent progress in advanced manufacturing techniques to engage in in-vitro experimentation with physiologically relevant geometries and flows that correspond to arterial stenosis. The measurements of pressures and flow obtained by these experiments were then used to estimate flow-to-pressure transfer functions, aimed at determining a lumped-parameter impedance model of the physical system, by applying conventional descent as well as recently developed Genetic Algorithm methods. The resulting transfer functions can now be utilized for further studies with regard to the hemodynamics relevant to arterial stenosis.

Affine versus non-affine fibril kinematics in collagen networks: theoretical studies of network behavior

The microstructure of tissues and tissue equivalents (TEs) plays a critical role in determining the mechanical properties thereof. One of the key challenges in constitutive modeling of TEs is incorporating the kinematics at both the macroscopic and the microscopic scale. Models of fibrous microstructure commonly assume fibrils to move homogeneously, that is affine with the macroscopic deformation. While intuitive for situations of fibril-matrix load transfer, the relevance of the affine assumption is less clear when primary load transfer is from fibril to fibril. The microstructure of TEs is a hydrated network of collagen fibrils, making its microstructural kinematics an open question. Numerical simulation of uniaxial extensile behavior in planar TE networks was performed with fibril kinematics dictated by the network model and by the affine model. The average fibril orientation evolved similarly with strain for both models. The individual fibril kinematics, however, were markedly different. There was no correlation between fibril strain and orientation in the network model, and fibril strains were contained by extensive reorientation. As a result, the macroscopic stress given by the network model was roughly threefold lower than the affine model. Also, the network model showed a toe region, where fibril reorientation precluded the development of significant fibril strain. We conclude that network fibril kinematics are not governed by affine principles, an important consideration in the understanding of tissue and TE mechanics, especially when load bearing is primarily by an interconnected fibril network.

Effects of atrial contraction, atrioventricular interaction and heart valve dynamics on human cardiovascular system response

Various simulation models of different complexity have been proposed to model the dynamic response of the human cardiovascular system. In a related paper we proposed an improved numerical model to study the dynamic response of the cardiovascular system, and the pressures, volumes and flow-rates in the four chambers of the heart, which included the effects of atrial contraction, atrioventricular interaction, and heart valve dynamics. This paper investigates the effects of each one of these aspects of the model on the overall dynamic system response. The dynamic response is studied under different situations, with and without including the effect of various features of the model, and these situations are studied and compared among themselves and to detailed aspects of expected healthy-system response. As an important contribution with potential clinical applications, this paper examines the corresponding effects of atrioventricular interaction, and heart valve opening and closing dynamics to the general system dynamic response. This isolation of physical cause-effect relationships is difficult to study with purely experimental methods. The simulation results agree well with results in the open literature. Comparison shows that introduction of these new features greatly improves the simulation accuracy of the effects of a, v and c waves, and in predicting regurgitant valve flow, the dichrotic notch, and E/A velocity ratio.

A concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction

Numerical modeling of the human cardiovascular system has always been an active research direction since the 19th century. In the past, various simulation models of different complexities were proposed for different research purposes. In this paper, an improved numerical model to study the dynamic function of the human circulation system is proposed. In the development of the mathematical model, the heart chambers are described with a variable elastance model. The systemic and pulmonary loops are described based on the resistance-compliance-inertia concept by considering local effects of flow friction, elasticity of blood vessels and inertia of blood in different segments of the blood vessels. As an advancement from previous models, heart valve dynamics and atrioventricular interaction, including atrial contraction and motion of the annulus fibrosus, are specifically modeled. With these improvements the developed model can predict several important features that were missing in previous numerical models, including regurgitant flow on heart valve closure, the value of E/A velocity ratio in mitral flow, the motion of the annulus fibrosus (called the KG diaphragm pumping action), etc. These features have important clinical meaning and their changes are often related to cardiovascular diseases. Successful simulation of these features enhances the accuracy of simulations of cardiovascular dynamics, and helps in clinical studies of cardiac function.

A ventricular-vascular coupling model in presence of aortic stenosis

In patients with aortic stenosis, the left ventricular afterload is determined by the degree of valvular obstruction and the systemic arterial system. We developed an explicit mathematical model formulated with a limited number of independent parameters that describes the interaction among the left ventricle, an aortic stenosis, and the arterial system. This ventricular-valvular-vascular (V3) model consists of the combination of the time-varying elastance model for the left ventricle, the instantaneous transvalvular pressure-flow relationship for the aortic valve, and the three-element windkessel representation of the vascular system. The objective of this study was to validate the V3 model by using pressure-volume loop data obtained in six patients with severe aortic stenosis before and after aortic valve replacement. There was very good agreement between the estimated and the measured left ventricular and aortic pressure waveforms. The total relative error between estimated and measured pressures was on average (standard deviation) 7.5% (SD 2.3) and the equation of the corresponding regression line was y = 0.99 x − 2.36 with a coefficient of determination r2 = 0.98. There was also very good agreement between estimated and measured stroke volumes ( y = 1.03 x + 2.2, r2 = 0.96, SEE = 2.8 ml). Hence, this mathematical V3 model can be used to describe the hemodynamic interaction among the left ventricle, the aortic valve, and the systemic arterial system.

Adaptation to mechanical load determines shape and properties of heart and circulation: the CircAdapt model

With circulatory pathology, patient-specific simulation of hemodynamics is required to minimize invasiveness for diagnosis, treatment planning, and followup. We investigated the advantages of a smart combination of often already known hemodynamic principles. The CircAdapt model was designed to simulate beat-to-beat dynamics of the four-chamber heart with systemic and pulmonary circulation while incorporating a realistic relation between pressure-volume load and tissue mechanics and adaptation of tissues to mechanical load. Adaptation was modeled by rules, where a locally sensed signal results in a local action of the tissue. The applied rules were as follows: For blood vessel walls, 1) flow shear stress dilates the wall and 2) tensile stress thickens the wall; for myocardial tissue, 3) strain dilates the wall material, 4) larger maximum sarcomere length increases contractility, and 5) contractility increases wall mass. The circulation was composed of active and passive compliances and inertias. A realistic circulation developed by self-structuring through adaptation provided mean levels of systemic pressure and flow. Ability to simulate a wide variety of patient-specific circumstances was demonstrated by application of the same adaptation rules to the conditions of fetal circulation followed by a switch to the newborn circulation around birth. It was concluded that a few adaptation rules, directed to normalize mechanical load of the tissue, were sufficient to develop and maintain a realistic circulation automatically. Adaptation rules appear to be the key to reduce dramatically the number of input parameters for simulating circulation dynamics. The model may be used to simulate circulation pathology and to predict effects of treatment.

Computational fluid dynamic study of flow optimization in realistic models of the total cavopulmonary connections

OBJECTIVES AND BACKGROUND

In the Fontan circulation, pulmonary and systemic vascular resistances are in series. The influence of various inferior vena cava to pulmonary artery connections in this unique circulatory arrangement was evaluated using computation fluid dynamics methods.

METHODS

Realistic three-dimensional models of total cavopulmonary connections were created from angiographic measurements to include the hepatic vein, superior vena cava, and branches of the pulmonary arteries. Steady-state finite volume analyses were performed using identical in vivo boundary conditions. Computational solutions calculated the percent hydraulic power dissipation and left-to-right pulmonary arterial flow distribution.

RESULTS

Simulations of the lateral tunnel, intra-atrial tube, extracardiac conduit with left and right pulmonary artery anastomosis demonstrated extracardiac conduit with left pulmonary artery anastomosis having the lowest energy loss. Varying the extracardiac conduit from 10 to 30 mm resulted in the least energy dissipation at 20 mm. Serial dilation of the lateral tunnel pathway showed a small incremental worsening of energy loss.

CONCLUSIONS

Maximizing energy conservation in a low-energy flow domain, such as the Fontan circulation, can be significant to its fluid dynamic performance. Although computational modeling cannot predict postoperative failure or functional outcome, this study confirms the importance of local geometry of the surgically created pathway in the total cavopulmonary connection.

Theoretical study of inspiratory flow waveforms during mechanical ventilation on pulmonary blood flow and gas exchange

A lumped two-compartment mathematical model of respiratory mechanics incorporating gas exchange and pulmonary circulation is utilized to analyze the effects of square, descending and ascending inspiratory flow waveforms during mechanical ventilation. The effects on alveolar volume variation, alveolar pressure, airway pressure, gas exchange rate, and expired gas species concentration are evaluated. Advantages in ventilation employing a certain inspiratory flow profile are offset by corresponding reduction in perfusion rates, leading to marginal effects on net gas exchange rates. The descending profile provides better CO2 exchange, whereas the ascending profile is more advantageous for O2 exchange. Regional disparities in airway/lung properties create maldistribution of ventilation and a concomitant inequality in regional alveolar gas composition and gas exchange rates. When minute ventilation is maintained constant, for identical time constant disparities, inequalities in compliance yield pronounced effects on net gas exchange rates at low frequencies, whereas the adverse effects of inequalities in resistance are more pronounced at higher frequencies. Reduction in expiratory air flow (via increased airway resistance) reduces the magnitude of upstroke slope of capnogram and oxigram time courses without significantly affecting end-tidal expired gas compositions, whereas alterations in mechanical factors that result in increased gas exchanges rates yield increases in CO2 and decreases in O2 end-tidal composition values. The model provides a template for assessing the dynamics of cardiopulmonary interactions during mechanical ventilation by combining concurrent descriptions of ventilation, capillary perfusion, and gas exchange.

Mathematical modelling of the human foetal cardiovascular system based on Doppler ultrasound data

A lumped parameter model of the human foetal circulation primarily based on blood velocity data derived from the Doppler analysis was developed in this study. It consists of two major parts, the heart and the foetal vascular circulation. The heart model accounts for both ventricular and atrial contractility. The circulation was divided into 19 compliant vascular compartments in order to describe all of the clinically monitored sites. The model parameters refer to the final gestation period and were derived either from literature on foetal sheep circulation or from anatomical dimension monitoring of the human foetus. No control mechanism is incorporated into the model. The model was validated by comparing several index values of simulated velocity curves to those of the experimental Doppler waveforms. The mean and maximum percentual errors in the estimation of the experimental results by the model are 7.7% and 20.1%, respectively. Velocity and pressure tracings of the foetal circulation were investigated, as well as regional blood flow rate distribution.

A mathematical model of circulation in the presence of the bidirectional cavopulmonary anastomosis in children with a univentricular heart

The bidirectional cavopulmonary anastomosis is used as a staged procedure or a definitive palliation of univentricular hearts. It is often performed in the presence of an additional blood flow arising from the native pulmonary outflow tract. In this paper, the effects of the severity of the pulmonary outflow obstruction and the pulmonary arteriolar resistance are analysed with regard to the haemodynamics in the superior vena cava and the blood distribution into the lungs. A computer model has been developed, which can represent both the preoperative and the postoperative (systemic and pulmonary) circulations in a patient with a double-outlet univentricular heart. It is particularly detailed in the region of the large vessels and includes components that account for local three-dimensional effects due to the actual shape of the anastomosis. Results have indicated that the mean pressure in the superior vena cava increases from 8.2 to 19.2 mmHg with pulmonary arteriolar resistance ranging from 0.8 to 7.9 Woods units and pulmonary outflow obstruction ranging from 50 to 100%. The percentage flow distribution to the right lung has turned out to be heavily affected by the flow competition and has ranged from 43 to 50% of the total flow to the lungs in the systolic phase, and from 51 to 62% in the diastolic phase. The model allows routinely used clinical indices to be computed, as well as the evaluation of new indices, which is potentially helpful in the clinical assessment of postoperative haemodynamics (e.g. the right-to-left lung flow ratio and the superior vena cava-to-pulmonary flow ratio).