Computational evaluation of heart failure and continuous flow left ventricular assist device support in anaemia

Anaemia is common in end‐stage heart failure patients supported with continuous flow left ventricular assist device (CF‐LVAD) and is associated with adverse outcomes such as heart failure readmission. This study evaluates the haemodynamic effects of anaemia on cardiac function and cerebral blood flow in heart failure patients supported with CF‐LVAD using computational simulations. A dynamic model simulating cardiac function, systemic, pulmonary and cerebral circulations, cerebral flow autoregulatory mechanisms and gas contents in blood was used to evaluate the effects of anaemia and iron deficiency in heart failure and during CF‐LVAD support. CF‐LVAD therapy was simulated by a model describing HeartMate 3. Anaemia and iron deficiency were simulated by reducing the haemoglobin level from 15 to 9 g/dL and modifying scaling coefficients in the models simulating heart chamber volumes. Reduced haemoglobin levels decreased the arterial O2 content, which increased cerebral blood flow rate by more than 50% in heart failure and during CF‐LVAD assistance. Reduced haemoglobin levels simulating anaemia had minimal effect on the arterial and atrial blood pressures and ventricular volumes. In contrast, iron deficiency increased end‐diastolic left and right ventricular diameters in heart failure from 6.6 cm to 7 cm and 2.9 cm to 3.1 cm and during CF‐LVAD support from 6.1 to 6.4 cm and 3.1 to 3.3 cm. The developed numerical model simulates the effects of anaemia in failing heart and during CF‐LVAD therapy. It is in good agreement with clinical data and can be utilised to assess CF‐LVAD therapy.

CF-LVAD. 2,36][7] It is defined by reduced haemoglobin in the blood, and the risk of anaemia increases after 65 years of age. 8Anaemia may cause tachycardia, increased cardiac output and hypertrophy in cardiac muscle, formation of collaterals and arteriovenous shunts and hypoxia. 9Anaemia is common in heart failure patients and is associated with diabetes, hypertension, ischemic heart disease, peripheral vascular disease, malignancy and renal insufficiency. 10Ejection fraction in heart failure patients with anaemia varies between 4% and 61%, 11 while anaemia increases mortality risk in heart failure patients. 12Anaemia may cause impaired cerebral oxygen supply and demand, resulting in increased cerebral blood flow. 13Moreover, anaemia may cause cerebral hyperperfusion due to low cerebral oxygen delivery. 14,15naemia is also common in heart failure patients assisted with CF-LVAD and increases mortality risk. 16,17It is associated with suboptimal CF-LVAD support with adverse outcomes such as heart failure readmission. 18Moreover, left and right ventricular unloading may remain suboptimal and cause functional mitral and tricuspid valve regurgitation, and right atrial pressure may decrease during CF-LVAD support because of anaemia. 19Also, suboptimal right ventricular unloading due to anaemia may cause right early ventricular dysfunction. 20Increased reduction of haemoglobin over time also increases the risk of mortality in patients implanted with CF-LVAD. 21,22 numerical model simulating clinical outcomes of anaemia in heart failure patients can help to evaluate possible CF-LVAD support strategies.This study aims assess the haemodynamic effects of anaemia in heart failure patients supported with CF-LVAD using computational simulations.

| MATERIALS AND METHODS
This study does not involve human subjects and/or animals, therefore, does not require ethical approval.
The numerical models in this study simulate blood flow rates in the cardiovascular system and cerebral circulation, blood O 2 and CO 2 contents in arteries and veins, cerebral autoregulation, and CF-LVAD assistance.The cardiovascular system includes ventricular and atrial functions, and blood circulation.Cerebral regulation mechanisms simulate static autoregulation, CO 2 reactivity and blood O 2 content reactivity.CF-LVAD assistance is simulated by a hydrodynamic model describing hydrodynamic characteristics of the HeartMate 3 device.
Simulations were performed in Matlab Simulink R2021b using the ode15s solver.Duration of a cardiac cycle cardiac was 0.8 s in the numerical simulations.All the parameter values used in the simulations are provided in Supplementary Materials 1. Definitions of the numerical models used to simulate a healthy condition, Heart Failure with reduced Ejection Fraction (HFrEF) with and without anaemia and CF-LVAD support with and without anaemia are given in Supplementary Material 2 and provided in the "Numerical Model.rar"file (Data S3).

| Cardiovascular system
Ventricular functions were simulated using ventricular pressures (p) and volumes (V).The model describing the left ventricular pressure is given below.
Here, p lv,a and p lv,p represent the active and passive pressures.E es,lv is the end-systolic elastance and f act,lv is the activation function.The model describing the left ventricular volume (V lv ) includes radius (r), long axis diameter (l), and a scaling parameter (K) as given below.Derivative of the left ventricular radius (dr/dt) is a function of the inflow and outflow of the left ventricle (Q mv , Q av ).
Right ventricular function was simulated similarly with different parameter values.Detailed information about ventricular functions can be found in Reference 23.
Atrial functions were simulated using elastance (E) and volume (V).Left atrial pressure is given below.
Left atrial volume and change of the atrial radius over time (dr/dt) were modelled similar to the left ventricular volume as given below.
Detailed information about models describing the atria can be found in. 23The circulatory system was modelled with electric analogue elements as described in Reference 24.Systemic arteriolar resistance (R ars ) was controlled by the pressure in aorta (p ao,m ), a sensitivity coefficient (S) and setpoints ( p ao,ars,set , R ars,set ) as described in References 25 and 26.

| Blood gas contents
Arterial and venous O 2 content (C O2 ) was simulated using O 2 binding capacity, saturation (S O2 ) and dissolved O 2 in plasma. 27,28O 2 saturation (S O2 ) in the haemoglobin (Hb) was described using O 2 pressure (p O2 ) in the blood. 29O 2 tension was modelled using the amount of the dissolved in plasma (Sol O2 ) and the solubility coefficient.
In the equations above, x denotes arteries or veins.O 2 pressure corrected for temperature (T), and pH in arteries and veins was modelled using O 2 pressure 30 from Equation (13) and corresponding CO 2 pressure as given below.
Arterial and venous CO 2 pressures (p a,CO2 , p v,CO2 ) were described utilising CO 2 concentration (C x,CO2,p ) in the plasma, reference blood pH and corresponding blood pH in each compartment (pH a , pH v ) and haemoglobin level in the blood (Hb). 31CO 2 concentration (C x,CO2,p ) in the plasma was modelled using the CO 2 solubility coefficient (s), corresponding blood pH (pH x ) and apparent pK of the CO 2 -bicarbonate system ( pK x 0 ) in each compartment. 32pK of the CO 2 -bicarbonate system in the arteries and veins was modelled using reference blood pH, blood temperature (T) and corresponding blood pH in each compartment (pH x ). 32a,CO2 ¼ C a,CO2,p À pH À pH a ð ÞÀ 18:2532

| Cerebral autoregulation mechanisms
The static autoregulation function used in the cerebral circulation controls the pial arterial resistance (R pc,pao ) utilising a sensitivity coefficient (S Rpc,ao ), change in the pial circulation resistance (ΔR pc,ao ) and set points (R pc,set, p ao,cbf,set ). 25 ΔR pc,pao ¼ S Rpc,pao p ao,cbf ,set À p ao,m R pc,set , The effect of the arterial O 2 content (C a,O2 ) on the cerebrovascular resistance was modelled by fitting a linear function in Curve Fitting Toolbox 3.6 in Matlab 2021b.
Here, R pc,O2 represent pial arterial resistance controlled by arterial O 2 content (C a,O2 ) and c 1 and c 2 are the fit coefficients in the model.Cerebral blood flow rate and O 2 content data were taken from. 34Simulations were performed using F I G U R E 1 Electrical circuit representation of the cardiovascular system model with the CF-LVAD, and block diagram of the cerebral blood flow rate regulatory mechanisms.R, L and C represent resistance, inertance and compliance, MV, AV, TV, and PV are the mitral, aortic, tricuspid and pulmonary valves, p and Ca represent pressure and arterial content, and MAP is the mean arterial pressure.Subscripts la, lv, ra and rv are the left and right atria and ventricles, ao, aa, ars, cs, and vs represent aorta, aortic arch, systemic arterioles, capillaries and veins, ap, arp, and vp represent pulmonary arteries, arterioles and veins.In the cerebral circulation, l and r represent left and right, ica, va, ba, are the internal carotid, vertebral and basilar arteries, pca, sca, mca, aca, acha are the posterior, superior cerebellar, middle, anterior and the anterior choroidal arteries, pcoa, acoa are the posterior and anterior communicating arteries, 1 and 2 represent proximal and distal components of the cerebral arteries, pc, cc, vc are the pial, cerebral capillary, and cerebral vein circulations.CO 2 and O 2 carbon dioxide and oxygen.
the cardiovascular system model to simulate cerebral blood flow rate data by adjusting the pial arterial resistance (R pc ).Corresponding pial arterial resistance (R pc ) values and arterial O 2 content (C a,O2 ) were used to find the coefficients (c 1 , c 2 ) in Equation (21).The data points used in the curve fitting Table 1.
Interaction of the static cerebral autoregulation, cerebrovascular CO 2 reactivity and effect of arterial O 2 content on the pial arterial resistance was modelled as given below.
Minimum and maximum values for the samples for haemoglobin level (Hb) and K coefficients (K la , K lv , K ra , and K rv ) in the sensitivity analyses.
Lower value 9 0.5 0.5 0.5 0.5 Scatter plots and histograms for haemoglobin level (Hb) and K coefficients (K la , K lv , K ra , and K rv ) in the sensitivity analyses.

| CF-LVAD support
A numerical model describing the pressure (H CF-LVAD ) and flow rate (Q CF-LVAD ) relation across HeartMate 3 device 35 was incorporated into the cardiovascular system model to simulate heart pump support.CF-LVAD support was simulated as shown below. 36 CF-LVAD was operated at 5600 rpm speed, and artificial pulse was simulated every 2 s by decreasing the pump speed 2000 rpm for 0.15 s and then increasing the pump speed 4000 rpm for 0.20 s in the simulations.Haemodynamic variables were averaged for 4 s considering 0.8 s heartbeat period and CF-LVAD speed variation over 2 s in the simulations.The electrical circuit representation of the cardiovascular system model with the CF-LVAD, and block diagram of the regulatory mechanisms are given in Figure 1.

| Simulation of iron deficiency anaemia and heart failure with reduced ejection fraction
Heart failure with reduced ejection fraction (HFrEF) was simulated by reducing the left ventricular endsystolic elastance, parameter A in the left ventricle model whilst the left ventricular zero-pressure volume was increased and the aortic pressure set point in the cerebral flow autoregulatory function was decreased as described in Reference 33.Also, the K coefficient in the left ventricular volume (K lv , Equation 4) was as described in Reference 23 in the model simulating HFrEF.The altered static cerebral autoregulatory curve is restored with CF-LVAD support. 37Therefore, the aortic pressure set point in the cerebral flow autoregulatory function was increased to its original value during CF-LVAD support with and without iron deficiency anaemia.The effect of anaemia was simulated by reducing the haemoglobin level from 15 g/dL to 9 g/dL, as reported in. 19ron deficiency decreases Adenosine Triphosphate (ATP) production in the heart and causes mitochondrial dysfunction. 38Moreover, iron deficiency may alter oxygen utilisation and energy production. 39As a result, cardiomyocytes generate less force and cardiomyocyte contractility decreases. 38Although iron deficiency may not significantly affect the left ventricular ejection fraction, 34 it may increase the left ventricular diastolic diameter in HFrEF patients. 40This effect was modelled by decreasing K coefficients in the left ventricle from 0.95 to 0.855, in the right ventricle from 1.75 to T A B L E 3 Mean and systolic aortic pressures, mean pulmonary arterial and atrial pressures, end-diastolic and end-systolic ventricular volumes and diameters, cardiac output, and mean cerebral blood flow rate in the cardiovascular system models simulating HFrEF and CF-LVAD support with and without anaemia.

| Sensitivity analysis
Sensitivity analysis was done to evaluate the effects of the haemoglobin level and K coefficients in all the heart chambers on the cerebral and total blood flow rates, mean system and pulmonary arterial pressures, ventricular volumes, and diameters in the cardiovascular system models simulating HFrEF and CF-LVAD support.Fifty samples with uniform distributions were generated for each parameter in Simulink Sensitivity Analyzer.Minimum and maximum values for the samples in the sensitivity analyses are given in Table 2. Scatter plots showing the sample values for the variables against each other and histograms for the samples are given in Figure 2.
Scatter plots with fit lines were used to assess the correlation between variables and output parameters.Bar charts were used to evaluate the influence of the variables.

| RESULTS
Firstly, the model was developed to simulate the relation between arterial O 2 content and pial arterial resistance (Figure 3).Coefficients c 1 and c 2 in Equation ( 22) were 0.09991 and À 0.9821, whereas the coefficient of determination was 0.99.Mean and systolic aortic pressures, mean pulmonary arterial and atrial pressures, end-diastolic and end-systolic ventricular volumes and diameters, cardiac output, and mean cerebral blood flow rate in the cardiovascular system models simulating HFrEF and CF-LVAD support with and without anaemia are given in Table 3.
F I G U R E 9 Scatter plots with linear lines in the sample plots for each parameter and haemodynamic variable assessed in the sensitivity analysis for the cardiovascular system models simulating HFrEF and anaemia.Hb, K la , Kl v, K ra , and K rv represent haemoglobin and the K coefficients in the equations describing the left and right ventricular and atrial volumes, MAP, SPAo, MPAP, MLAP, and MRAP represent mean aortic pressure, systolic aortic pressure, mean pulmonary arterial pressure, mean left and right atrial pressures, respectively, LVEDV, LVESV, RVEDV, and RVESV represent left, and right ventricular end-diastolic and end-systolic volumes, LVEDD, LVESD, RVEDD, and RVESD represent left and right ventricular end-diastolic and end-systolic diameters, CO and CBF represent cardiac output and cerebral blood flow rate.
Reduced haemoglobin levels in anaemia decreased the mean and systolic aortic pressures in heart failure and CF-LVAD support.End-diastolic left ventricular volume decreased during CF-LVAD support due to decreased haemoglobin level.Anaemia reduced the end-systolic left ventricular volume in heart failure and CF-LVAD support.End-diastolic right ventricular volume increased because of anaemia.Iron deficiency increased end-diastolic and end-systolic volumes in the left and right ventricles.Cardiac output increased slightly in anaemia because autoregulation mechanisms controlling blood flow rate changed arteriolar and pial circulatory resistances.Also, the cerebral blood flow rate increased profoundly because of anaemia.Ventricular, atrial and arterial pressures, ventricular and atrial volumes and diameters in the cardiovascular system model simulating a healthy condition are given in Figure 4.
The systolic left ventricular and aortic pressures were around 120 mmHg.The systolic right ventricular and pulmonary arterial pressures were around 35 mmHg.The left ventricular volume changed between 56 and 124 mL and the left ventricular diameter changed between 3.4 and 5.1 cm.Internal carotid, vertebral and middle cerebral arterial flow rates are given in Figure 5.
The internal carotid arterial blood flow rate changed between 155 and 500 mL/min, the vertebral arterial blood flow rate changed between 47 and 150 mL/min, and the middle cerebral arterial blood flow rate changed between 80 and 250 mL/min in the cardiovascular system simulating a healthy condition.Ventricular, atrial and arterial pressures in the cardiovascular system models simulating HFrEF and CF-LVAD support with and without anaemia are given in Figure 6.
Systolic left ventricular and aortic pressures decreased, whilst left atrial pressure increased in HFrEF with and without anaemia.CF-LVAD support further decreased systolic left ventricular pressure and increased the overall aortic pressure in HFrEF with and without anaemia.There was a slight increase in the systolic right ventricular and pulmonary arterial pressures in HFrEF with and without anaemia, whilst CF-LVAD support reduced these pressures.Ventricular and atrial volumes and diameters in the cardiovascular system models simulating HFrEF and CF-LVAD support with and without anaemia are given in Figure 7.  K coefficients resulted in higher heart chamber diameters under CF-LVAD support.Internal carotid, vertebral and middle cerebral arterial flow rates in the cardiovascular system models simulating HFrEF and CF-LVAD support with and without anaemia are given in Figure 8.
In the cardiovascular system model simulating heart failure, the peak internal carotid, vertebral, and middle cerebral arterial flow rates were around 376 mL/min, 113 mL/min and 190 mL/min.Decreased haemoglobin levels increased the peak internal carotid, vertebral, and middle cerebral arterial flow rates.CF-LVAD support reduced the amplitude of the cerebral blood flow rate signals, and the maximal internal carotid, vertebral, and middle cerebral arterial flow rates were around 320, 97, and 162 mL/min.Again, decreased haemoglobin levels during CF-LVAD support increased the maximal internal carotid, vertebral, and middle cerebral arterial flow rates.Scatter plots with linear fit lines in the sensitivity analysis for the evaluated parameters in the cardiovascular system models simulating HFrEF with and without anaemia are given in Figure 9.
Aortic and pulmonary arterial pressures, mean left and right atrial pressures, end-systolic and end-diastolic ventricular volumes, cardiac output and cerebral blood flow rate were sensitive to the changes in the haemoglobin level.Although haemoglobin level affects these variables in the cardiovascular system model, only cerebral blood flow rate changed with the haemoglobin level.Ventricular diameters were sensitive to the K coefficient modified to simulate the effect of iron deficiency in the heart chambers.Bar charts showing the parameter influence on the evaluated haemodynamic variables in the cardiovascular system models simulating HFrEF with and without anaemia are given in Figure 10.
Haemoglobin level was the most influential parameter on aortic and pulmonary arterial pressures, mean left and right atrial pressures, end-systolic and end-diastolic ventricular volumes, cardiac output and cerebral flow rate in the cardiovascular system models simulating heart failure.K lv and K rv coefficients were the most influential parameters on the end-systolic and end-diastolic left and right ventricular diameters, respectively.K la was the second most influential parameter on the end-systolic and end-diastolic left ventricular diameters, whilst haemoglobin level was the second most influential parameter on the end-systolic and end-diastolic right F I G U R E 1 2 Bar charts showing the parameter influence on the evaluated haemodynamic variables in the cardiovascular system models simulating HFrEF, CF-LVAD support and anaemia.Hb, K la , Kl v, K ra , and K rv represent haemoglobin and the K coefficients in the equations describing the left and right ventricular and atrial volumes, MAP, SPAo, MPAP, MLAP, and MRAP represent mean aortic pressure, systolic aortic pressure, mean pulmonary arterial pressure, mean left, and right atrial pressures, respectively, LVEDV, LVESV, RVEDV, and RVESV represent left, and right ventricular end-diastolic and end-systolic volumes, LVEDD, LVESD, RVEDD, and RVESD represent left and right ventricular end-diastolic and end-systolic diameters, MPO and CBF represent mean output and cerebral blood flow rate.C, RC, PC, and RPC denote correlation, rank correlation, partial correlation, and rank partial correlation.ventricular diameters.Linear fit curves in the sample scatter plots in the sensitivity analysis for the evaluated parameters in the cardiovascular system models simulating CF-LVAD support with and without anaemia are given in Figure 11.
Aortic and pulmonary arterial pressures, mean left and right atrial pressures, end-systolic and end-diastolic ventricular volumes, cardiac output and cerebral flow rate, were sensitive to the changes in the haemoglobin level in the cardiovascular system model simulating CF-LVAD support as in the cardiovascular system model simulating heart failure.Again only the cerebral blood flow rate changed with the haemoglobin level.Ventricular diameters were sensitive to the K coefficient modified to simulate the effect of iron deficiency in the heart chambers in the cardiovascular system model simulating CF-LVAD support.Bar charts showing the parameter influence on the evaluated haemodynamic variables in the cardiovascular system models simulating CF-LVAD support with and without anaemia are given in Figure 12.
Haemoglobin level was the most influential parameter on aortic and pulmonary arterial pressures, mean left and right atrial pressures, end-systolic and end-diastolic ventricular volumes, cardiac output and cerebral flow rate in cardiovascular system models simulating CF-LVAD support as in the cardiovascular system models simulating heart failure.K lv and K rv coefficients were the most influential parameters on the end-systolic and end-diastolic left and right ventricular diameters, respectively.K la was the second most influential parameter on the end-systolic and end-diastolic left ventricular diameters, whilst haemoglobin level was the second most influential parameter on the end-systolic and end-diastolic right ventricular diameters.

| DISCUSSION
In this study, the effects of decreased haemoglobin level and iron deficiency on cardiac variables and cerebral blood flow rate in heart failure and during CF-LVAD support were investigated utilising numerical simulations.Ejection fraction in HFrEF decreases below 40%, 41 whereas mean arterial pressure becomes around 80 mmHg. 42The cardiovascular system model simulating HFrEF yielded similar results.As simulated in this study, anaemia and iron deficiency can occur together in heart failure 38 ; however, the pathophysiological consequences of anaemia and iron deficiency are different. 7Reduced haemoglobin level in blood decreases systemic vascular resistance and causes vasodilation. 7As a result, arterial blood pressure decreases and ejection fraction may increase. 7Clinical studies show that mean arterial blood pressure is slightly lower in anaemic heart failure patients than in non-anaemic patients 43 as in the simulations (Table 3).If the mean right atrial pressure remains below 10 mmHg in anaemic heart failure patients, mean pulmonary arterial pressure may not be affected by the decreased haemoglobin level. 43The mean right atrial pressure in the numerical model simulating HFrEF in the simulations was 6 mmHg, and the mean pulmonary arterial pressures were the same for anaemic and non-anaemic heart failure conditions (Table 3).Slight differences have been reported in ejection fraction in anaemic and non-anaemic heart failure patients, where these differences are non-significant. 12,44As in the simulations, the ejection fraction in anaemic heart failure patients was higher than in non-anaemic heart failure patients 38,39 (Table 3).
Iron deficiency causes structural and functional myocardial remodelling, contributing to ventricular dysfunction. 7Iron deficiency may not affect ejection fraction; however, it may increase left ventricular end-diastolic diameter. 40educed K coefficients simulating the effects of iron deficiency increased diameters of the heart chambers.
Oxygen content profoundly affects the cerebral blood flow rate, which increases during chronic anaemia to supply the brain with sufficient oxygen. 45On the other hand, cerebral blood flow rate decreases in patients with HFrEF. 46lthough there was a decrease in the cerebral blood flow rate in the numerical model simulating non-anaemic heart failure, reduced haemoglobin levels in heart failure resulted in a higher cerebral blood flow rate than in the cardiovascular system model simulating a healthy condition (Table 3).
Sensitivity analysis showed that reduced haemoglobin level in the blood is correlated with changes in aortic, pulmonary arterial and atrial pressures, ventricular volumes, cardiac output and cerebral blood flow rate, whereas iron deficiency is correlated with increased ventricular diameters in the numerical model simulating heart failure.However, haemoglobin levels only profoundly affected the cerebral blood flow rate (Figure 9).Iron deficiency profoundly affects the ventricular diameters (Figures 9 and 10).Results in the sensitivity analyses confirm reported clinical data 7,[43][44][45] and validate the numerical simulations.
Mean and systolic aortic pressure during CF-LVAD support was higher in the numerical model simulating only heart failure than in the cardiovascular system model simulating CF-LVAD support and anaemia.Therefore, CF-LVAD support did not alter the effect of anaemia in systemic circulation.End-diastolic left ventricular volume was slightly higher in the numerical model simulating CF-LVAD support and heart failure than in the cardiovascular system model simulating CF-LVAD support and anaemia (Table 3).An increase in end-diastolic left ventricular volume may be due to CF-LVAD pressure and flow characteristics.Under CF-LVAD support, end-diastolic right ventricular volumes were slightly higher in the cardiovascular system model simulating anaemia.Again, CF-LVAD support did not influence the effect of anaemia on the right ventricular volume.Imamura et al. 19 reported that the mean right atrial pressure was 1 mmHg lower in anaemia patients during CF-LVAD support.Simulations yielded the same mean right atrial pressure in the cardiovascular system model simulating CF-LVAD support with and without anaemia (Table 3).Imamura et al. 19 showed that end-diastolic left ventricular diameter and right ventricular enlargement were different under CF-LVAD support in anaemia patients.Simulations also yielded higher left and right ventricular diameters at the end of systole and diastole under CF-LVAD support in anaemia (Table 3).Enlarged left and right ventricles may result in functional heart valve insufficiency and can increase further morbidity in these patients.
The cerebral blood flow rate in the cardiovascular system model simulating only CF-LVAD support was similar to the healthy condition (Table 3); however, it increased by more than 50% in the cardiovascular system model simulating CF-LVAD support and anaemia, whereas there was a slight increase in cardiac output.The increased cerebral blood flow rate in heart failure and during CF-LVAD support in anaemic patients suggest that the blood flow in the lower body is compromised to ensure cerebral perfusion and oxygen delivery.
Iron deficiency is correlated with increased ventricular diameters in the numerical model simulating CF-LVAD support (Figure 11) similar with the cardiovascular system model simulating heart failure.Again, haemoglobin levels only profoundly affected the cerebral blood flow rate (Figure 11), and iron deficiency profoundly affects the ventricular diameters (Figures 11 and 12).However, the range of evaluated variables changes due to CF-LVAD support.Therefore, CF-LVAD support does not alter the effect of anaemia in the numerical model used in this study.
There are number of limitations in the study.Iron deficiency affects cardiomyocyte mitochondrial respiration and adenosine triphosphate production. 38A recent study 47 shows that iron-deficiency anaemia impairs cardiac contraction by downregulating ryanodine receptor 2, which controls calcium ion flow out of the sarcoplasmic reticulum.These mechanisms have not been modelled, but the effect of altered myocardial contraction was simulated by modifying the K coefficients (K lv , K rv , K la , and K ra ) in the equations which describe heart chamber volumes.Modifying the K coefficients yielded physiological results for iron deficiency in the numerical models. 40Anaemia is associated with a change in haematocrit and may alter blood viscosity. 48Altered blood viscosity affects the mechanical behaviour of CF-LVAD where increased viscosity increases the motor current for high rotor speeds. 49he CF-LVAD model used in this study simulated only the hydraulic part of the pump and the mechanical part of the CF-LVAD was not modelled.

| CONCLUSION
The effects of anaemia on cardiac variables during heart failure and CF-LVAD support were evaluated utilising numerical simulations.Cerebral blood flow rate increased profoundly due to reduced arterial O 2 content, whereas cardiac output increased slightly.Therefore, blood flow in the lower body is compromised.Also, iron deficiency results in increased left and right ventricular volume and can cause functional heart valve insufficiency.The developed cardiovascular system model replicates clinical observations such as reduced ejection fraction and cardiac output in heart failure and increased cerebral blood flow rate and ventricular diameters in anaemia.Therefore, it can be used to evaluate mechanical circulatory support and test different scenarios in patients implanted with CF-LVAD.ORCID Selim Bozkurt https://orcid.org/0000-0002-4151-8653

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I G U R E 3 (A) Data points for arterial O 2 content and cerebral blood flow rate, (B) Data points for arterial O 2 content and pial arterial resistance and fit model.
Note: LVEDV, LVESV, RVEDV and RVESV represent left, and right ventricular end-diastolic and end-systolic volumes, LVEDD, LVESD, RVEDD and RVESD represent left, and right ventricular end-diastolic and end-systolic diameters.Abbreviations: CO, cardiac, CBF, cerebral blood flow rate; EF, ejection fraction; HFrEF, heart failure with reduced ejection fraction; MAP, mean aortic pressure; MLAP and MRAP, mean left and right atrial pressures; MPAP, mean pulmonary arterial pressure; MPO, mean pump outputs; SPAo, systolic aortic pressure.F I G U R E 4 (A) Left ventricular, left atrial and aortic pressures ( p lv , p la , and p ao ), (B) right ventricular, right atrial and pulmonary arterial pressures (p rv , p ra , and p ap ), (C) left and right ventricular and atrial volumes (V lv , V rv , V la , and V ra ), (D) left and right ventricular and atrial diameters (D lv , D rv , D la , and D ra ) in the cardiovascular system model simulating a healthy condition.1.575 and in the left and right atria from 1.2 to 1.08.Parameters in the numerical model are given in Supplementary Materials 1 and 2.

F I G U R E 5
Flow rates in the (A) internal carotid (Q ica ), (B) vertebral (Q va ) and (C) middle cerebral arteries (Q mca ) in the cardiovascular system model simulate a healthy condition.F I G U R E 6 Left ventricular, left atrial and aortic pressures ( p lv , p la , and p ao ) in the cardiovascular system models simulating (A) HFrEF, (B) HFrEF and anaemia, (C) HFrEF during CF-LVAD support, (D) HFrEF and anaemia during CF-LVAD support, right ventricular, right atrial and pulmonary arterial pressures ( p rv , p ra , and p ap ) in the cardiovascular system models simulating (E) HFrEF, (F) HFrEF and anaemia, (G) HFrEF during CF-LVAD support, and (H) HFrEF and anaemia during CF-LVAD support.

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I G U R E 7 Left and right ventricular and left and right atrial volumes (V lv , V rv , V la , and V ra ) in the cardiovascular system models simulating (A) HFrEF, (B) HFrEF and anaemia, (C) HFrEF during CF-LVAD support, (D) HFrEF and anaemia during CF-LVAD support, right ventricular, left and right ventricular and left and right atrial diameter (D lv , D rv , D la , and D ra ) in the cardiovascular system models simulating (E) HFrEF, (F) HFrEF and anaemia, (G) HFrEF during CF-LVAD support, and (H) HFrEF and anaemia during CF-LVAD support.F I G U R E 8 Internal carotid, vertebral and middle cerebral arterial flow rates (Q ica , Q va , and Q mca ) in the cardiovascular system models simulating (A) HFrEF, (B) HFrEF and anaemia, (C) HFrEF during CF-LVAD support, and (D) HFrEF and anaemia during CF-LVAD support.

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I G U R E 1 0 Bar charts showing the parameter influence on the evaluated haemodynamic variables in the cardiovascular system models simulating HFrEF and anaemia.Hb, K la , Kl v, K ra , and K rv represent haemoglobin and the K coefficients in the equations describing the left and right ventricular and atrial volumes, MAP, SPAo, MPAP, MLAP, and MRAP represent mean aortic pressure, systolic aortic pressure, mean pulmonary arterial pressure, mean left and right atrial pressures respectively, LVEDV, LVESV, RVEDV, and RVESV represent left, and right ventricular end-diastolic and end-systolic volumes, LVEDD, LVESD, RVEDD, and RVESD represent left and right ventricular end-diastolic and end-systolic diameters, CO and CBF represent cardiac output and cerebral blood flow rate.C, RC, PC, and RPC denote correlation, rank correlation, partial correlation and rank partial correlation.Heart failure resulted in increased left ventricular volume and diameter.CF-LVAD support reduced left ventricular volume and diameter whilst reducing the end-systolic right ventricular volume.Increased left atrial volumes in the cardiovascular system models simulating heart failure with and without anaemia reduced with CF-LVAD support.Reduced K coefficients simulating the effect of iron efficiency increased heart chamber diameters.CF-LVAD support reduced left ventricular and atrial diameters and increased right ventricular and atrial diameters.Again, relatively low

F I G U R E 1 1
Scatter plots with linear lines in the sample scatter plots for each parameter and haemodynamic variable assessed in the sensitivity analysis for the cardiovascular system models simulating HFrEF, CF-LVAD support and anaemia.Hb, K la , Kl v, K ra , and K rv represent haemoglobin and the K coefficients in the equations describing the left and right ventricular and atrial volumes, MAP, SPAo, MPAP, MLAP, and MRAP represent mean aortic pressure, systolic aortic pressure, mean pulmonary arterial pressure, mean left and right atrial pressures respectively, LVEDV, LVESV, RVEDV, and RVESV represent left, and right ventricular end-diastolic and end-systolic volumes, LVEDD, LVESD, RVEDD, and RVESD represent left and right ventricular end-diastolic and end-systolic diameters, MPO and CBF represent mean pump output and cerebral blood flow rate.