This paper presents a novel computationally -efficient physical -based solid/fluid constitutive framework for modelling the cardiac cycle using the finite element method. Previous models of the left ventricle simulate the cardiac cycle by applying pressure or nodal displacement s as boundary conditions, which in reality are both ouptuts that are generated due to the interaction between the active contractility of the ventri cular myocardium and the temporally varying fluid domain which it encloses . By modelling the blood cavity using C3D8 solid continuum elements, a Windkessel -type volumetric stress formulation governs the pressure -volume relationship during ejection using a central -difference scheme within th e user -defined material subroutine. We have developed a patient -specific model of the left ventricle directly from CINE Echocardiography scan data and the model can accurately capture the key trends of Inferior Vena Cava Occlusion (IVCO) experiments, demon strating the Frank -Starling mechanism as an output. For the first time, the effect of including nonlinear aortic compliance on ventricular p ressure -volume loops is studied, and it is demonstrated its inclusion is critical to accurately capture key features of IVCO and endovascular aortic repa ir (EVAR) experiments. This mode l acts as an alternati ve to computationally expensive fluid -structure interaction simulations , and overcomes multiple limitations of current finite element models , and can run full cardiac cycles of a patient -specific left ventricle in 5 minutes , rendering it feasible for clinical integration and real time analysis of patient ventricle dynamics. Keywords: left ventr icle, pressure -volume loop, active contractility, patient -specific, solid/fluid constitutive model Introduction Blood pressure in the left ventricle is generated by the interaction of the actively contracting and relaxing myocardial wall with the blood cav ity which it enc loses . The opening and closing of mitral and aortic valves are controlled by a temporally varying competition between left ventricle pressure, and pulmonary and systemic pressures. This competition in turn alters the boundary conditions experienced by the actively contracting left ventricle. The systemic and pulmonary pressures also vary significantly due to external factors, such as the Windkessel effect or inferior vena cava occlusion (IVCO) , rendering the entire heart beat interdependent and complex to model computationally. To simulate the cardiac cycle , the computational biomechanics community rely on either fluid structure interaction ( FSI) or finite element analysis ( FEA) techniques. FSI is a multiphysics , bidirectional coupling betwe en the laws that describe fluid dynamics and structural mechanics [1]. Several commercially available software packages exist including; Comsol, Abaqus, Ansys, and FEbio, each of whi ch, through solving the Navier -Stokes equations, are capable of generating visually impressive particle tracking analyse s, streamlines, and moving fluid visualizations . In the context of cardiac modelling however, FSI simulations require complex boundary c onditions at the inlets and outlets, and are extremely computationally expensive. Take for example a n FSI study by Deng et al., who model the left ventricl e wall, bounded fluid domain and mitral valve [2]. The model neglects active contractility of the ventricular myocardium, and the volume v ersus time plots presented in the results section lack the ability to capture the key featu res of the widely acc epted experimental curves [3]–[8], in particular, the late diastolic plateau evident experimentally before atrial contraction . Another FSI study by Mao et al., modelled the left atrium, ventricle and outflow tract , including the mitral and aortic valves [9]. They too neglect active contractility and prescribe a constant pressure boundary condition at the l eft atrium, which contradicts the unequivocal experimental evidence that the left atrial pressure varies significantly throughout the cardiac cycle [10]–[16]. Notwithstanding the importance of accurately representing the complex physics of the system in the model, FSI models report cardia c simulation times of 36 hours [17], 90 hours [18], and 168 hours [19], limiting their integration into clinical practice at the moment. Although these complicated simulations can be now run locally, typical runtime being in the order of hours/days [20] is still a limiting factor of the technique. FEA, (the secon d approach ), is a numerical technique that subdivides the structure of interest into multiple sub -domains, known as elements that are connected to each other via nodes. T he FE method formulation of a boundary value problem results in a system of algebraic equations . The simple equations that model the finite elements are then assembled into a larger system of equations that models the entire problem. Currently, there are two main approaches to modelling a cardiac cycle using FEA in the literature; (i) via the application of pressure boundary conditions [21]–[24], and (ii) via the application of displacement boundary conditions [25]–[28] to the myocardial wall to impose a v olume change. However, there is a fundamental flaw in both of these approaches , as both pressure and volume should be outputs of the model (and not inputs) , as they come from the interaction of the actively contracting ventricular myocardium with the blood volume which it encloses . The pressure -volume (P - V) relationship also depends on the stage of the cardiac cycle, and specifically whether the mitral or aortic valves are open or closed. In FEA models without an actual ventricular blood domain, a constrain t equation must be solved during the isovolum etric phase to determine the necessary ventricular pressure to retain the volume of the ventricles . It should be noted however, that the isovolumetric contraction and relaxation sections of a P -V loop are almost never entirely vertical, indi cating some volume change [29]–[32]. Once the ventricular pressure exceeds the aortic pressure, the aortic valve will open and the relationship between pressure and volume i s often related via a wi ndkessel -type circulatory model. The problem with current FEA models that attempt to model this windkessel -type circulatory behaviour is that pressure or volume are treated as inputs to these simulations . For example, t he endocardium is loaded with a spatially uniform pressure determined by the arterial windkessel pressure, atrial pressure, or volumetric constraint pressure depending on the phase of the cardiac cycle in [33]. Moreover, a spatially uniform pressure is not obtained in the left ventricle accordin g to experimental evidence [34]. Kerkhoffs et al., describes the four major modules required to couple the FE and circulatory model (pressure update, circulatory model, perturbation of circulatory model, and FE model). This iterative approach requires that pressures be updated such that cavity volumes from the FE and circulation models are within a given tolerance, a highly computationally expensive process [35]. Recent studies have used a time -varying elastance model for the ventricle to simulate the full cardiac c ycle [36], [37] , however this model as sumes an exponential end -diastolic P-V relation ship and a linear end -systolic P-V relationship, and the P-V relationship for any points between these two is described by a smooth time -varying function [38]. Experimental evidence of P -V loops however, shows a highly non -linear end-systolic P -V relationship [30], [ 39], [40] . Others have reco gnised the relationship between pressure, volume, and active stress ; Cervantes et al., report an approach whereby for increasing pressures , the active fibre stretch is tuned such that the resulting volume yields a point on the PV loop, however this requires t he PV loop to be predefined [41]. Hassaballah et al., modelled the blood cavity using a 9 -noded hydrostatic fluid element, with 8 nodes on the internal surface of the left ventricle cavity and a 9th (pressure) node at the centroid of the ventricle base, however a predefined value of pressure was used which require s the definition of fluid or fluid link elements [42]. The living heart model (LHM) (Dassault Syste mes), is a n anatomically accurate electro -mechanical model of the entire heart and great vessels, comprised of ~7.5M nodes, and uses surface -based fluid cavity representation s of the four heart chambers [43]. The Abaqus surface -based -fluid -cavity feature however, is only available is Abaqus/Explicit, which does not enforce a Newton -Raphson convergence algorithm and requires additional definitions of fluid -exchange and inflator features [44]. Moreover, a single LHM simulation takes 40 hours using 160 CPU cores to solve [45] In this study we describe a novel approach that addresses all of the abovementioned drawbacks with conventional techniques for simulating a cardiac cycle computationally . We mesh the entire fluid and wall of the left ventricle but don’t perform an FSI analysis, we model the fluid using C3D8 solid continuum elements. We develop patient -specific solid/ fluid model of the left ventricle directly from CINE Echocardiography scan data using Matlab (Mathworks Inc.) , with a user defined constitutive la w that governs the pressure and volume of the left ventricle during each of the key phases of the cardiac cycle; (i) isovolumetric contraction, (ii) ejection, (iii) isovolumetric relaxation, and (iv) filling. Our framework allows for the simulation of Inferior Vena Cava Occlusion ( IVCO ) expe riments, the integration of non linear aortic com pliance, and the simulation of endovascular aortic repair ( EVAR ) or aortic stenting , on the P -V relationship of the left ventricle. Excellent agreement is observed with exper imental studies throughout , and o ur full patient -specific left ventricle finite element model runs a full cardiac cycle in less than 5 minutes, rendering the approach far more computationally efficient and clinically feasible than previous models to date . Methodology In this paper, a sol id-fluid left ventricle finite element model is created that can accurately capture the key physics of a cardiac cycle through a PV loop. First, a framework is designed to generate patient specific FE models directly from TriPlanar Echo cardiography scan da ta. A physically based constitutive law for the myocardium is presented, that accounts for the individual contributions of the wall constituents using a constrained mixture approach (ie collagen, cardiomyocytes, groundmatrix ). Then a continuum based fluid const itutive model is presented which describes the P-V relationship throughout each phase of the cardiac cycle , as a function of bulk fluid in compressibility during isovolumetric and filling phases and a function of aortic compliance and resistance during ejec tion. Multiplanar CINE Echo cardiography derived Finite Element Mesh As part of this study , a MATLAB framework was created for generating patient specific finite element mesh es of the left ventricle using multi -planar CINE echocardiography dataset s. Captur ing images of the left ventricle using 2 -Chamber, 3 -Chamber and 4 -Chamber planes, results in three image datasets each separated by an angular orientation separation of 60⁰ (See Figure 1(a) ). The frames corresponding to end diastole in each of the datasets is used for mesh generation. The boundary of the inner ventricular wall is defined in each image ( Figure 1(b) ), and rotation of the 2-Chamber and 4 -Chamber boundary coordinates by 60⁰ and 120⁰ respectively results in three splines , which meet at the apex and define the three -dimensional internal morphology of the ventricle ( Figure 1(c) ). Regularized resampling of each spline is performed such that each spline contains the same number of points. At each level along the long axis of the ventricle, another sp line is then fit to pass through each of these six points (2 -red,2 - green, 2 -blue). The point spacing of these splines defines the mesh density, as each point a node in the initial surface mesh of the finite element model. This surface mesh is then offset o utwards to form the wall of the ventricular myocardium using C3D8 solid continuum elements. Figure 1: Flowchart of Finite Element Mesh generation from multi -planar echocardiography. Reference Configuration If we assume that the reference configuration for the finite element simulation is that of end diastole (MVc in Figure 2(a) below), following isovolumetric contraction, ejection, and isovolumetric relaxation, the final filling step would be driven only by the relaxation of the actively contracted cardiomyocyt es. However, if we assume that the reference configuration is somewhere below the end diastolic configuration and incorporate the phase of late filling of the ventricle due to atrial contraction as a precursor step to isovolumetric contraction, ejection, a nd isovolumetric relaxation, the final filling step is now driven by both relaxation of the cardiomyocytes and elastic recoil of the compressed groundmatrix of the ventricular myocardium . This elastic recoil or restoring forces have been widely reported i n the literature . The concept of an equilibrium volume implies that when the end systolic volume is less than the equilibrium volume of the ventricle, the restoring forces will aid in filling [46], or if mitral flow is prevented, will produce a negative pressure [47], [48] . It should be also noted that as the end systolic volume decrea ses the magnitude of the negative pressure increases. Nikolic et al., found that in 72 of 75 runs in 11 dogs, the equilibrium volume was greater than the end systolic volume, and that diastolic suction was a key feature of the filling phase [49]. Brecher found that when ventricular pressure became negative during mitral valve occlusion, that fluid was aspirated into the heart through a cannula from a reservoir below heart level [50]. Finally, Par k et al., point out that if the expansion of the ventricle was a direct result of transmitral flow, these should be in phase with each other. However, they report that for the greater part of the rapid filling phase, left ventricular pressure falls while cavity size increases [51]. This is also observable in the recordings of Fukuta & Little [52]. The abovementioned studies inform our decision in this paper to define the reference volume as ~60 % of the end diastolic volume by multiplying the nodes in our finite elem ent mesh by a scalar multiplier, α (Figure 2(b) ). Figure 2: (a) Schematic of PV loop showing phases of the cardiac cycle; (1.) isovolumetric contraction, (2.) ejection, (3.) isovolumetric relaxation, and (4.) filling. AV and MV represent the aortic and mitral valves respectively where subscripts ‘o’ and ‘c’ indicate an open or closed position. Each of these phases of the cardiac cycle are described in detail in Section 2.4 The Reference Configuration is depicted as 𝑽𝟎. (b) Scaling of ventricle mesh from end diastolic to reference c onfiguration, where α is the scale factor multiplied by each set of nodal coordinates. Myocardium Constitutive Law The constitutive model for the ventricular myocardium employs a rule of mixtures approach where each component of the wall undergoes the sam e strain, but the stresses are additive, and the sum of constituent densities equals unity [53]. The constituents of the wall explicitly modelled include collagen, elastin, cardiomyocytes, and ground -matrix. A bilinear strain energy function describes the strain stiffening behaviour of fibrillar collagen under tension 𝝈col=𝑉𝑐𝑜𝑙𝑓∗{ 0 𝜀<0 (𝑘1𝜀)∗(𝒂𝑐𝑜𝑙⊗𝒂𝑐𝑜𝑙) 0≤𝜀<𝜀𝑡 (𝑘1𝜀+𝑘2(𝜀−𝜀𝑡))∗(𝒂𝑐𝑜𝑙⊗𝒂𝑐𝑜𝑙) 𝜀𝑡≤𝜀 (1) where 𝑉𝑐𝑜𝑙𝑓 is the volume fraction of collagen in the wall, 𝑘1 is the initial stiffness of collagen, 𝑘2 is the secondary stiffness of collagen, 𝜀𝑡 is the transition strain , 𝜀 is the strain, and 𝒂𝑐𝑜𝑙 denotes the current direction of deformed collagen fibres with respect to the circumferential axis of the artery, where 𝒂𝑐𝑜𝑙= 𝐅𝒂𝑐𝑜𝑙0 (𝐅=deformation gradient, and 𝒂𝑐𝑜𝑙0 is the direction of undeformed collagen fibres in the r eference configuration) [54]. Elastin fibres may also contribute to the anisotropy of the myocardium with a mean directionality defined by 𝒂𝑒𝑙𝑎 such that the stress is 𝝈𝑒𝑙𝑎=𝑉𝑒𝑙𝑎𝑓∗((√𝐼4𝑒−1)∗𝑘𝑒)∗(𝒂𝑒𝑙𝑎⊗𝒂𝑒𝑙𝑎). (2) 𝐼4𝑒 is the elastin fibre anisotropic invariant, i.e. the square of stretc h of an elastin fibre, ( 𝐼4𝑒=(𝒂𝑒𝑙𝑎∙ 𝐂𝒂𝑒𝑙𝑎), where 𝐂 is the right Cauchy -Green Tensor ). As described above for collagen, 𝒂𝑒𝑙𝑎=𝐅𝒂𝑒𝑙𝑎0. For anisotropic invariant of the type 𝐼4𝑘, we use full invariant rather than the is ochoric invariant. This is necessary for reasons outlined in our previous paper [55]. 𝑉𝑒𝑙𝑎𝑓 is the volume fraction of elastin in the wall, and 𝑘𝑒 is the stiffness of elastin. We also incorporate the active component of the wall as represented by 𝜎𝑐𝑚𝑦. The contractile stress generated by a single cardiomyocyte (𝜎𝑎𝑐𝑡) is 25 kPa [56]. Cardiomyocytes also contribute to the anisotropy of the vessel through 𝒂𝑐𝑚𝑦 𝝈𝑐𝑚𝑦 =𝑉𝑐𝑚𝑦𝑓∗(𝜎𝑎𝑐𝑡∗(𝒂𝑐𝑚𝑦 ⊗𝒂𝑐𝑚𝑦)) (3) where 𝑉𝑐𝑚𝑦𝑓 is the volume fraction of SMCs in the wall, 𝜎𝑎𝑐𝑡 is the active stress of an individual cardiomyocyte and 𝒂𝑐𝑚𝑦 =𝐅𝒂𝑐𝑚𝑦0. The active stress versus time waveform is shown in Figure 3(b) below. Finally, the stress in the isotropic ground -matrix can be defined as 𝝈mat =𝑉𝑚𝑎𝑡𝑓∗{ 𝐸1(𝜆̅𝑖−1) (𝜆̅𝑖−1)≤𝜀𝑡1 𝑝𝑚(𝜆̅𝑖−1)2+𝑞𝑚(𝜆̅𝑖−1)+𝑟𝑚 𝜀𝑡1<(𝜆̅𝑖−1)<𝜀𝑡2 𝐸2(𝜆̅𝑖−1−𝜀𝑡2)+(𝑝𝑚(𝜀𝑡2)2+𝑞𝑚(𝜀𝑡2)2+𝑟𝑚) (𝜆̅𝑖−1)≥𝜀𝑡2 (4) where 𝑝𝑚=(𝐸1−𝐸2)/2(𝜀𝑡1−𝜀𝑡2) (5) 𝑞𝑚=𝐸1−2𝜀𝑡1𝑝𝑚 (6) 𝑟𝑚=(𝐸1−𝑞𝑚)𝜀𝑡1−𝑝𝑚(𝜀𝑡1)2 (7) 𝑉𝑚𝑎𝑡𝑓 is the volume fraction of matrix in the wall, 𝐸1 and 𝐸2 are the initial and secondary stiffness of the matrix, 𝜀𝑡1 and 𝜀𝑡2 are the initial and secondary transition strains . Equations 5 -7 must be enforced for continuity. Figure 3(a) shows the passive myocardium model simulation for uniaxial tension comp ared to experimental data, while Figure 3(b) details the cardiomyocyte active contractility versus time curve for each phase of the cardiac cycle. The physical parameter values for the passive model are shown in Figure 3(c) . Figure 3: (a) Simulated (SIM) vs experimental (EXP) passive myocardium stress -strain data for uniaxial tension, using the myocardium constitutive law described above. (b) The active stress ( 𝝈𝒂𝒄𝒕) versus time profile for each phase of the cardiac cycle ; (1.) isovolumetric contaction, (2.) ejection, (3.) isovolumetric relaxation, (4a.) early filling, and (4b.) late filling. (c) The parameters and values of the constitutive law correspondi ng to the stress -strain response shown in (a). Fluid Constitutive Law Consideration of the interaction between the ventricular myocardium and the internal blood is a key feature of this model. Fundamentally, by neglecting the fluid and allowing the ventri cle to contract against an empty cavity will result in an underestimation of the level of active contractility required to reach the end systolic configuration. In this study the fluid volume is meshed with C3D8 solid continuum elements where the outermost fluid element layer share nodes with the innermost layer of the myocardium. We tie these nodes in order to simulate contact between the wall and the blood. The physics of the blood changes in each st age of the cardiac cycle depending on the opening and closing of the aortic and mitral valve s. The detail of such is described below according to each stage . In each stage the shear modulus of the blood is negligible compared to the volumetric. Step 4b (Late Fillin g): In late diast ole, the left atrium contracts while the mitral valve is opened, resulting in an increase in system pressure and ventricular volume. This pressure increase in the ventricle due to atrial contraction is modelled by applying a outwards pressure on the inner myocardium wall of 15mmHg, which ex pands the fluid domain outwards also (the interface nodes are shared between the blood and the wall). We allow the fluid to deform freely with a low bulk modulus ( 𝜅=1.0e -08 MPa ) according to 𝜎ℎ=𝜅(𝐽−1)𝐽 (8) Where 𝐽 is the Jacobian (determinant of the deformation gradient tensor). We then employ a pressure transfer equilibrium step, whereby we remove the pressure boundary condition from the wall, and increase the bulk modulus to 𝜅=1 MPa, which results in the wall generating an internal fluid stress of 15mmHg as desired. This step ends at the onset of active contractility. Throughout the rest of th is study we denote (𝜅⇧=1.0 𝑀𝑃𝑎 ) and (𝜅⇩=1.0𝑒−08 𝑀𝑃𝑎 ). Step 1 (Isovolumetric Contraction): During isovolumetric contraction both the mitral valve and the aortic valve are closed. The active contractility of the myocardium increases and acts against the incompressible fluid volume which it encloses. We model this by using the simple volumetric formulation 𝜎ℎ=𝜅⇧(𝐽−1)𝐽 (9) As the ventricle contracts against the high bulk modulus fluid, pressure is generated in the fluid with negligible volumetric deformation of the elements, and we observe the characteristic right vertical line of the PV loop. This stage ends when the ventricular pressure exceeds the aortic pressure. Step 2 (Ejection): Once the ventricular fluid pressure exceeds that of the aortic pressure, the aortic valve opens and blood is ejected into the systemic circulation. The physics of the blood in this step is captured using a Windkessel formulation that relates the pressure to the flow 𝑃=𝑃𝑡+(−𝑅𝐹𝑑 𝑉𝑒−𝑃𝑡𝑑𝑡 𝑅𝐶+0.5𝑑𝑡) (10) 𝜎ℎ=−𝑃 (11) where 𝑃𝑡 is the previous pressure, 𝑅 is the aortic resistance (), 𝐶 is the aortic compliance ( 1.0 cm3/mmHg ), 𝐹 is a scale factor mapping element to ventricle volume (𝐹=𝑉𝑣𝑜/𝑉𝑒𝑜), 𝑑𝑉𝑒 is the incremental change in volume of the element (𝑑𝑉𝑒=𝐽(𝑉𝑒𝑛−𝑉𝑒𝑝)), and 𝑑𝑡 is the time increment . This stage ends when the ejection time has passed (0.25s) . Step 3 (Isovolumetric Relaxation): During isovolumetric r elaxation, again, both the mitral valve and the aortic valve are closed. The active contractility of the myocardium this time decreases rapidly and attempts to expand the incompressible fluid volume which it encloses. We model this stage also by using the simple volumetric formulation 𝜎ℎ=𝜅⇧(𝐽−1)𝐽 (12) Step 4a (Early Filling ): Once the pressure in the ventricle drops below the pulmonary pressure, the mitral valve opens and blood can enter the ventricle. We model this stage also by using the simple volumetric formulation 𝜎ℎ=𝜅⇩(𝐽−1)𝐽 (13) If we assume that by the end of the isovolumetric relaxation phase, the active contractility is fully removed ( 𝜎𝑎𝑐𝑡=0), then the expansion of the ventricle in this step is driven primarily by the restoring forces of the ventricle (the compr ession springback of the groun dmatrix which has been compressed below its reference configuration by the active contractility in Step 3). With a low bulk modulus of the fluid, the relaxation of the myocardium to the reference configuration results in a flu id expansion with negligible pressure increase. Alternatively, if we assume that by the end of the isovolumetric relaxation phase, the active contractility is not fully removed ( 𝜎𝑎𝑐𝑡>0), then the expansion of the ventricle in this step is driven by a combination of the restoring forces and the relaxing residual active contractility. Results Paramteric Analysis As described in the previous section, our model is physically bas ed, allowing for a physiologically based calibration of the model rather than a black -box fitting of parameters. As a consequence the only unknown in the model is the 𝜎𝑎 versus time profile. In order for the PV loops to behave according to experiments , we know that 𝜎𝑎 must increase during isovolumetric contraction, continue to increase at a reduced slope in ejection, and then decrease rapidly during isovolumetric relacation until the mitral valve opens at such a point we enter the filling stage. Figure 4 shows the influence of active stress magnitude and rate of contractility build -up on the PV loops, in addition to the influence of aortic compliance ( 𝐶) and peripheral resistance ( 𝑅) in a parametric analysis. It is observed that decreasing 𝐶, or equivalently increasing aortic stiffness results in a reduced stroke volume and increased peak left ventricular pressure ( Figure 4: Row 1 ). Increasing 𝑅 results in a significant reduction in stroke volume with little effect on peak pressure (Figure 4: Ro w 2). Increasing the magnitude of 𝜎𝑎 increases stroke volume, and peak left ventricular pressure (Figure 4: Row 3). Finally the rate of contractility build -up alters the shape of the PV loop only in the ejection phase (Figure 4: Row 4). Figure 4: (a) Parametric analysis on Compliance, Resistance, ac tive stress magnitude and rate. Unless otherwise stated parameters follow those in Figure 3(c). Inferior Vena Cava Occlusion (IVCO) IVCO is commonly performed to measure baseline contractility clinically [57]–[59]. Occluding the IVC reduces venous return to the right atrium which results in a decreased left atrium pressure in subsequent cardiac cycles. 4 key features are evide nt from the IVCO experiments ; (i) a reduction in aortic valve opening pressure (ii) a reduction in a ortic valve closing pressure, (iii) a reduction in stroke volume, and (iv) a change in ejection slope from bilinear at the beginning to linear at the end of the IVCO experiment . Figure 5 shows the experimental PV loop adapted from Pinsky et al., 2018 (Exp) , with our model simulation (Sim) superimposed. Our computat ional model prediction for IVCO captures the first 3 of the 4 features desc ribed above with a linear Compliance (C). Figure 5: Simulation (Sim) compared to Experimental study (Exp) of Garcia & Pinsky (2018) [30]. Simulated: Our model captures the first 3 of the 4 key experimental features; (i) reduction in aortic valve opening pressure (*); (ii) reduction in aortic valve closing pressure (**), and (iii) reducting stroke volume (***) with every additional cardiac cycle using linear compliance . Nonlinear Aortic Comp liance Evidence exists that the aorta exhibits a nonlinear compliance both ex-vivo and in-vivo within each cardiac cycle, with a high compliance at low pressures and a low compliance at high pressures [60]– [63]. Figure 6 below shows the effect of integrating nonlinear aortic compliance on the ventricular PV loop when compared to the linear complianc e model shown in Figure 5 . In order to capture the fourth feature of the IVCO experiments ( (iv) a change in ejection slope from bilinear at the beginning to linear at the end of the IVCO experiment), finite element models of the left ventricle , and specifi cally the constitutive law must include a non linear compliance term for the aorta . Figure 6: Incorporation of the nonlinear aortic compliance term allows the model to capture the fourth feature of the experimental curves ( See Figure 5 (Exp) [30]) ; a change in ejection slop e from bilinear at the beginning to linear at the end of the IVCO experiment . Endovascular Aortic Repair (EVAR) A recent study by Concannon et al., revealed that aortic stenting expands the aorta beyond the transition strain that separates low from high compl iance regimes [64]. The outward force of the stent pushes the aorta into the low compliance regime and equilibrates with the compressive forces of the wall at a new equilibrium configuration such that in all subsequent cycles , the pressure area curve follows that of the low compliance slope. In this study, we model this EVAR comput ationally by removing the high compliance regime from the nonlinear compliance model (as occurs following stenting). We observe the following trends on the ventricular PV loop: (i) the stroke volume decreases, (ii) peak systolic pressure increas es, and (iii) aortic valve opening pressure reduces (diastolic decay reduces) . These features can be observed in Figure 7 where control represents the healthy aorta preoperatively with a nonlinear shaped PV relationship during ejecton, while EVAR represent s the stented aorta postoperatively. Figure 7: Simulation of the effect of aortic stenting (EVAR) on left ventricular PV loop. In the control case we observe healthy nonlinear PV relationship during ejection, however this is lost postoperstively as the slope follows the low -compliance regime slope resulting in increased systolic pressure, reduced stroke volume and reduced aortic valve opening pressure or increased diastolic decay. Discussion In this study we describe a novel approach that surpasses conven tional techniques for simulating a cardiac cycle. We mesh the entire fluid and wall of the left ventricle and model the fluid using C3D8 solid continuum elements. We develop a patient -specific solid /fluid model of the left ventricle directly from CINE Echo cardiography scan data using Matlab (Mathworks Inc.) with a physically based user defined material subroutine or constitutive law that governs the pressure and volume of the left ventricle during each of the key phases of the cardiac cycle; (i) isovolumetric contract ion, (ii) ejection, (iii) isovolumetric relaxation, and (iv) filling . Crucially, this physically -based solid/fluid umat approach predicts the key features of experimental PV loops with pressure and volume as outputs of the model and not inputs as studies t o date have incorrectly assumed . Our framework allows for the simulation of IVCO experiments, the integration of nonlinear aortic com pliance, and the simulation of EVAR (aortic stenting) on the P -V relationship of the left ventricle. Excellent agreement is seen with exper iments throughout , with each cardiac cycle exhibiting a run time of 5 minutes, rendering the approach computationally far more efficient and clinically feasible than current methods. Physical based prediction of P -V loop As shown in Figure 4, the parametric study highlights the phy sical nature of the interaction of the solid/fluid umats in generating P -V loops that capture the key trents observed experimentally. Increasing aortic compliance results in a larger stroke volume, and reduced peak ventricular pressure, as would be expected when the ventricle is pumping blood into a more compliant outflow tract. Decreasing compliance, on the other hand, results in a reduction in stroke volume and an increase in peak systolic ventricular pressure as the active contractility in the myocardial wall is acting against a more rigid conduit. This has also been observed experimentally by Ioannou and colleagues who subjected porci ne aortae to banding to decrease vessel compliance which lead to a 41% increase in peak systolic pressure [65]. Parametric analysis on the peripheral resistance term ( 𝑅) showed little change on peak systolic pressure but significant increase in stroke volume with decreasing 𝑅. This has also been observed clinically where a reduction in systemic vascular resistance as patients moved from rest to exercise resulted in an increase in cardiac output [66]. An increase in the magnitude of the active contractility term ( 𝜎𝑎) results in an increased stroke volume, which has been observed in drug infusion studies where dobutamine (positive inotropic agent) resulted in a lower end -systolic volume and verapamil (calcium channel blocker) resulted in a higher end -systolic vo lume [67]. Modelling IVCO Experiments The key physics of the cardiopulmonary system emebedded in this model allows for accurate simulation of IVCO experiments, and proves the robustness of the framework to deal with transient reductions in inflow boundary conditions which result from restricting the inferior vena cava. This results in a reduction in left atrial filling pressure which results in drop in aortic valve opening pressure, closing pressure and stroke volume experimentally . Our linear compliance model can ca pture each of these key features, in contrast to previous models which for example, prescribe d the volume in an airbag to model the left ventricle and the end systolic volume does not shift to the left [68] as in experimental studies of IVCO [30], [69] . The reduction in stroke volume that occurs as a consequence of a reduced left ventricle filling pressure during IVCO is an important output of t he model and highlights its ability to capture the Frank -Starling mechanism [70] in-silico . Effect of Nonlinear Aortic Compliance on PV loop Each of the key features of the IVCO experimen ts bar one can be captured by the linear compliance model described above. Evidence exists however, that the aorta exhibits nonlinear compliance in-vivo within each cardiac cycle [63]. Integration of this n onlinear compl iance of the aorta in the user defined material subroutine results in a slope change during ejection commensura te with the transition pressure or strain whereby the collagen fibres begi n to affect the overall modulus of the vessel wall [53], [71], [72]. Once the aortic valve opens, the left ventricle and aorta become a unified conduit and exhibit the same syste m pressure, while the overall aortic compliance affecting the ventricle outflow is dependent on the presssure in the aorta, where compliance is now a nonlinear function of pressure. Integration of this key feature in the model allows for prediction of the experimental shape change seen clinically [30], [69] . This phenomenon can be explained as the transient reduction in left atrial filling pressure as a consequence of occlusion of the inferior vena cava results in the left ventricle opening the a ortic valve at a lower pressure each sysle, and the pressure during ejection no longer passes the transition pres sure leading to a slope change, or essentially, the aorta no longer enters the low compliance regime. Simulation of Endovascular Aortic Repair (EVAR) The current framework readily allows for simulation of the effects of endovacular aortic repair or stenting on left ventricular performance. This is particulary relevant clinically, with recent studies uncovering a link between proximal aortic stent ing and left ventricular hypertophy [73], [74] and cardiac deaths [75]. Moreover animal studies have shown that placing a graft in the native porcine aorta diminishes the windkessel ef fect [76], as evident in this model when the high -compliance regime is removed . In a previous study the authors reported the mechanism by which stenting (EVAR) alters vessel constitutive beha viour [64]. The o utward radial force of the stent stretches artery be yond transition strain to a new equilibrium configuration and as such the vessel no longer exhibits a high compliance regime at low pressures, and operates only in the low compliance regime, given the dacron enforced limit is not reached. Simulation of th is required removal of the high compliance regime from the constitutive law such that the windkessel formulation only sees the high stiffness throughout the entire cardiac cycle. This results in a reduction in stroke volume and an increase in systolic pres sure and reduction in diastolic pressure. This h as been shown in both experimental [77], [78] and clinical [79]–[81] studies to date. Takaseya et al., reported that EVAR in 15 aortic aneurysm patients resulted in an increase in systolic blood pressure from 90.4 to 100.4mmHg (p<0.05) [82], while Kelly et al., observed an increase in systolc pressure from 110 -162mmHg and decrease in diastolic pressure from 76-63mmHg following dacron graft insertion in dogs [83]. Sarraf et al., also observed a reduction in end diastolic pressure following transcatheter aortic valve implantation [84], which acts the same as a stent -graft in altering the mechanical behaviour of the vessel wall through expansion , suggesting this left ventricular response to device -artery compliance mismatch may not be unique to EVAR . Conclusion In conclusion, this study has de veloped a novel computationally -efficient physical based solid/fluid constitutive framework for modelling the cardiac cycle using the finite element method. We have developed a patient -specific model of the left ventricle directly from CINE Echocardiography scan data using Matlab with a user defined constitutive law that g overns the pressure and volume of the left ventricle during each of the key phases of the cardiac cycle predicting the PV loop as a result of the interaction between the active contractility of the ventricular myocardium and the fluid domain. The active an isotropic hyperelastic model includes nonlinear aortic compliance and is capable of simulating complex experim ents such as IVCO and EVAR in clinically feasible timeframes.