# Three-Wall Segment (TriSeg) Model Describing Mechanics and Hemodynamics of Ventricular Interaction

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## Abstract

A mathematical model (TriSeg model) of ventricular mechanics incorporating mechanical interaction of the left and right ventricular free walls and the interventricular septum is presented. Global left and right ventricular pump mechanics were related to representative myofiber mechanics in the three ventricular walls, satisfying the principle of conservation of energy. The walls were mechanically coupled satisfying tensile force equilibrium in the junction. Wall sizes and masses were rendered by adaptation to normalize mechanical myofiber load to physiological standard levels. The TriSeg model was implemented in the previously published lumped closed-loop CircAdapt model of heart and circulation. Simulation results of cardiac mechanics and hemodynamics during normal ventricular loading, acute pulmonary hypertension, and chronic pulmonary hypertension (including load adaptation) agreed with clinical data as obtained in healthy volunteers and pulmonary hypertension patients. In chronic pulmonary hypertension, the model predicted right ventricular free wall hypertrophy, increased systolic pulmonary flow acceleration, and increased right ventricular isovolumic contraction and relaxation times. Furthermore, septal curvature decreased linearly with its transmural pressure difference. In conclusion, the TriSeg model enables realistic simulation of ventricular mechanics including interaction between left and right ventricular pump mechanics, dynamics of septal geometry, and myofiber mechanics in the three ventricular walls.

### Keywords

Pulmonary hypertension Septal motion Adaptation Stress Strain Myofiber Cardiac mechanics### Nomenclature

### General

- LV
Left ventricle/ventricular

- RV
Right ventricle/ventricular

- LW
Left ventricular free wall

- SW
Septal wall

- RW
Right ventricular free wall

### Geometry-Related Parameters

*V*_{LV}Left ventricular cavity volume (m

^{3})*V*_{RV}Right ventricular cavity volume (m

^{3})*V*_{w}Wall volume of ventricular wall segment (m

^{3})*V*_{m}Volume of spherical cap, formed by midwall surface of wall segment (m

^{3})*A*_{m}Midwall surface area of curved wall segment (m

^{2})*C*_{m}Curvature of midwall surface (reciprocal of radius) (m

^{−1})*x*_{m}Maximal axial distance from midwall surface to origin (m)

*y*_{m}Radius of midwall junction circle (m)

*z*Ratio of wall thickness to midwall radius of curvature of curved wall segment

*α*Half the opening angle of spherical midwall surface

*ε*_{f}Natural myofiber strain

### Force-Related Parameters

*p*_{LV}Left ventricular cavity pressure (Pa)

*p*_{RV}Right ventricular cavity pressure (Pa)

*p*_{Trans}Transmural pressure difference across curved wall segment (Pa)

*σ*_{f}Cauchy myofiber stress (Pa)

*T*_{m}Representative midwall tension (N m

^{−1})*T*_{x}Axial midwall tension component (N m

^{−1})*T*_{y}Radial midwall tension component (N m

^{−1})

## Introduction

Several mathematical models of ventricular mechanics have been developed to quantify the effect of ventricular interaction on cardiac function. In several lumped models of ventricular hemodynamics,7,13,41,43,48,53,56,60,61 ventricular interaction is described by empirically determined coupling coefficients, quantifying interventricular cross-talk of pressures and volumes. The model designed by Beyar *et al.*7 describes ventricular interaction in a more mechanistic way. In this model, LV and RV cavities are enclosed by three ventricular walls. For given ventricular pressures, the mechanical equilibrium of tensile forces in the junction of the walls is used as constraint to predict ventricular geometry. The model is restricted to description of passive mechanics of walls lacking contractile myofiber properties. More recently, 3D finite element models of the cardiac ventricles were used to simulate ventricular pump function and local tissue mechanics,34,47,66 inherently including ventricular interaction via the septum. In comparison with lumped models of global ventricular mechanics, finite element models allow description of regional wall mechanics and geometry. Consequently, these models are computationally demanding.

For study of beat-to-beat hemodynamics and mechanics of heart and blood vessels, Arts *et al.* previously developed the closed-loop CircAdapt model of heart and circulation.3 In this model, mechanical interaction of the LV and RV has been simulated by a common outer wall, having a transmural pressure equal to RV pressure, encapsulating an inner wall, which represents the left ventricle.1 The inner wall encapsulates the LV cavity only and has a transmural pressure equal to the difference between LV and RV pressure. Under normal ventricular loading conditions, this model enables realistic simulation of global LV and RV pump mechanics.1,3 However, this model setup presumes RV pressure to be substantially lower than LV pressure during the whole cardiac cycle. This condition is not satisfied with pulmonary hypertension or with left-to-right asynchrony of electrical activation. Furthermore, septal geometry cannot be simulated. Thus, this model cannot be used to interpret this measurable signal that contains important information about the difference between LV and RV pressure.35,51 Therefore, we designed the TriSeg model of ventricular mechanics that realistically incorporates ventricular interaction via the interventricular septum.

The TriSeg model of ventricular mechanics is designed to be incorporated as a module in the existing CircAdapt model simulating mechanics and hemodynamics of the whole circulation.3 According to the principles of CircAdapt, size and mass of the wall segments are determined by adaptation so that mechanical load of the myofibers is normalized to physiological standard levels.3,4 The TriSeg model, as integrated in the CircAdapt model, has been tested by simulation of time-dependent LV and RV mechanics and hemodynamics under normal ventricular loading conditions as well as with acute and chronic pulmonary hypertension (PH). PH has been simulated to test whether the TriSeg model realistically relates septal geometry to transseptal pressure. Simulation results have been compared with previously published experimental data on ventricular hemodynamics9,62 and on the relation between septal geometry and transmural pressure17,51 in healthy volunteers and in patients with chronic PH.

## Methods

### General Design of the TriSeg Model

The design of the new TriSeg model of ventricular mechanics should enable incorporation as a module in the existing CircAdapt model that simulates mechanics and hemodynamics of the whole circulation.3 Therefore, the TriSeg model calculates LV and RV pressures (*p*_{LV} and *p*_{RV}, respectively) as functions of LV and RV cavity volumes (*V*_{LV} and *V*_{RV}, respectively). Mechanics of ventricular interaction is incorporated assuming a simplified ventricular composite geometry (Fig. 2a). The three ventricular walls LW, SW, and RW are modeled to be thick-walled and spherical with a common junction circle with midwall radius *y*_{m} (Fig. 2b). Midwall is defined as the spherical surface that divides wall volume *V*_{w} in an inner and an outer shell of equal volume. Midwall volume *V*_{m} is the volume enclosed by the midwall surface and the plane of the junction circle (Fig. 2c). The center of the junction circle is the origin of the applied cylindrical coordinate system (Fig. 2b). The *x*-direction is perpendicular to the plane of the junction circle and is defined positive toward the RV free wall. The *y*-coordinate represents the radial distance to the *x*-axis. The spherical midwall surface of a wall segment intersects the *x*-axis at value *x*_{m} (Fig. 2c).

*V*

_{m}and boundary radius

*y*

_{m}of a spherical wall segment are used to calculate the axial and radial vector components

*T*

_{x}and

*T*

_{y}, respectively, at the common junction circle from representative midwall tension

*T*

_{m}, applying a model of the mechanics of a curved wall patch. Such patch is defined as a fraction of a spherical wall segment with an arbitrarily shaped boundary and with midwall area

*A*

_{m}and curvature

*C*

_{m}. Curvature is defined as the reciprocal of radius of curvature, having the advantage of being well defined for a flat wall. In the section of curved wall patch mechanics,

*A*

_{m}and

*C*

_{m}are used to calculate representative midwall tension

*T*

_{m}, applying a model of myofiber mechanics. In the myofiber mechanics section, natural myofiber strain

*ε*

_{f}is converted to Cauchy myofiber stress

*σ*

_{f}, using a constitutive law of the myofiber. This model of myofiber mechanics has been heuristically obtained from reported physiological experiments.

### Ventricular Composite Mechanics

In this section, it is shown how cavity volumes *V*_{LV} and *V*_{RV} are used to calculate pressures *p*_{LV} and *p*_{RV}. For that purpose, a model of ventricular wall segment mechanics is applied that renders axial and radial tension components (*T*_{x} and *T*_{y}, respectively) from representative midwall tension as a function of the midwall volume (*V*_{m}) and junction radius (*y*_{m}) of the spherical wall segment. This latter model will be derived in the next section discussing mechanics of the ventricular wall segment.

*y*

_{m}of the wall junction and the enclosed midwall cap volumes

*V*

_{m,LW},

*V*

_{m,SW}, and

*V*

_{m,RW}of the wall segments LW, SW, and RW, respectively. Septal midwall volume

*V*

_{m,SW}and junction radius

*y*

_{m}are initially estimated by the solutions as obtained in the preceding time point, i.e.,

*V*

_{m,SW,est}and

*y*

_{m,est}, respectively:

*V*

_{LV}and

*V*

_{RV}as given directly by the CircAdapt model, render an estimate of ventricular composite geometry. Using the fact that ventricular cavity volume added to half of wall volume

*V*

_{w}of the enclosing wall segments renders the sum of midwall volumes, it is found:

*V*

_{m}is positive if wall curvature is convex to the positive

*x*-direction (Fig. 2b). Thus, for a normal heart, the sign is negative for LW and positive for RW and SW.

*i*,

*V*

_{m,i}and

*y*

_{m,i}are used to calculate axial and radial tension components

*T*

_{x,i}and

*T*

_{y,i}, as well as transmural pressure

*p*

_{Trans,i}. This is done by applying a model of ventricular wall segment mechanics as discussed in the next section. The axial tension components of the three wall segments

*T*

_{x,LW},

*T*

_{x,SW}, and

*T*

_{x,RW}are summed to calculate the net axial tension

*T*

_{x,Tot}in the junction. Similarly, the net radial tension

*T*

_{y,Tot}in the junction is obtained. The initial estimates of

*y*

_{m}and

*V*

_{m,SW}(Eqs. 1 and 2) are adjusted numerically so that for the summed tension components it holds:

*p*

_{LV}is caused by the definition of

*p*

_{Trans}, which is positive for a negative

*x*-gradient of pressure in the wall segment.

### Mechanics of the Ventricular Wall Segment

In this section, it is shown for a wall segment (Fig. 2c) how *V*_{m} and *y*_{m} are used to calculate axial and radial midwall tension components *T*_{x} and *T*_{y}, and transmural pressure *p*_{Trans}. For that purpose, a model of the mechanics of a curved wall patch is applied. Such curved wall patch is defined as a fraction of a spherical wall segment with midwall surface area *A*_{m} and curvature *C*_{m}. This latter model renders representative midwall tension from midwall surface area *A*_{m} and curvature *C*_{m} and will be derived in the next section discussing mechanics of the curved wall patch.

*y*

_{m}of the junction circle and distance

*x*

_{m}between plane of junction circle and center of midwall surface (Fig. 2c).

*V*

_{m},

*A*

_{m}, and

*C*

_{m}depend on

*x*

_{m}and

*y*

_{m}by

*x*

_{m}is calculated from

*y*

_{m}and

*V*

_{m}by solving Eq. (9). Second,

*A*

_{m}and

*C*

_{m}are determined with Eqs. (10) and (11). Next,

*A*

_{m}and

*C*

_{m}are used to calculate representative midwall tension

*T*

_{m}, applying a model of the mechanics of a curved wall patch as discussed in the next section.

*x*

_{m}and

*y*

_{m}, and midwall tension

*T*

_{m}are used to calculate axial (

*T*

_{x}) and radial (

*T*

_{y}) tension components:

*α*represents half opening angle of the circular wall segment (Fig. 2c).

*p*

_{Trans}equals total axial force. Consequently,

*p*

_{Trans}is calculated by multiplying

*T*

_{x}by the ratio of length of the junction contour (2

*πy*

_{m}) to area of the junction circle (

*πy*

_{m}

^{2}):

### Mechanics of the Curved Wall Patch

In this section, it is shown how midwall surface area *A*_{m} and curvature *C*_{m} are used to calculate representative midwall tension *T*_{m}. For that purpose, a constitutive model of myofiber mechanics is applied. This latter model renders Cauchy myofiber stress from natural myofiber strain and is presented in Appendix B.

In the ventricular module, originally implemented in the CircAdapt model,3 the one-fiber model developed by Arts *et al.*2 was used to relate ventricular pump mechanics, as described by cavity pressure and volume, to myofiber mechanics, as described by myofiber stress and strain. When assembling three wall segments to a ventricular composite with two cavities, contractile function of a wall segment, as described by representative midwall tension and area change, should be put between mechanics of cavity and myofiber. The following conditions should be satisfied. First, when folding a wall segment to a completely closed spherical surface, the relation between pump mechanics and myofiber mechanics should be equivalent to the equations of the one-fiber model. Second, like in the one-fiber model, where contractile myofiber work equals ventricular pump work, summed pump work of both cavities should be equal to summed work as generated by the three walls. Furthermore, within each wall segment, work as delivered by the wall through wall tension and changes of geometry, should be equal to the work generated by the myofibers.

*ε*

_{f}depends on midwall surface area

*A*

_{m}and curvature

*C*

_{m}by the following approximation (relative error <1%), as derived in Appendix A:

*A*

_{m,ref}represents reference midwall surface area. Note that the dimensionless curvature parameter

*z*is closely related to the dimensionless ratio of wall thickness to radius of curvature. With a constitutive model of the myofiber (Appendix B), natural myofiber strain

*ε*

_{f}is used to calculate Cauchy myofiber stress

*σ*

_{f}. In Appendix A, it is derived how myofiber stress and wall segment geometry are used to calculate representative midwall tension

*T*

_{m}. It holds by approximation (relative error <2%):

### Implementation of TriSeg Model in CircAdapt Model

The TriSeg model of ventricular mechanics is incorporated in the existing CircAdapt model of the whole circulation.3 This latter model supplies the required hemodynamic boundary conditions, i.e., LV and RV cavity volumes. The CircAdapt model is designed as a network of modules representing cardiac chambers, valves, large blood vessels, and peripheral resistances. Extended with the TriSeg model, the CircAdapt model allows beat-to-beat simulation of time-dependent ventricular mechanics and hemodynamics, e.g., ventricular cavity volumes and pressures, geometries and representative myofiber mechanics of ventricular walls, and flows through valves. An important feature of the existing CircAdapt model is that the number of independent parameters is reduced by incorporating adaptation of cavity size and wall mass of cardiac chambers and blood vessels to mechanical load so that stresses and strains in the walls of heart and blood vessels are normalized to tissue-specific physiological standard levels.

*V*

_{w}, reference midwall surface area

*A*

_{m,ref}, and reference sarcomere length

*L*

_{s,ref}(Table 1:

*Parameter values for model initialization*). After model initialization, these parameters will be obtained for each wall segment by load adaptation as described in the “Simulations” section. The actual parameter values chosen for initialization (Table 1) are noncritical due to load adaptation. However, for fast convergence to steady-state adaptation, these estimates were chosen in physiological range.

Input parameter values for NORM and PH simulations

Parameter | Unit | Value |
---|---|---|

| ||

Mean systemic arterial blood pressure | kPa | 12.2 |

Mean systemic blood flow | mL s | 85 |

Cardiac cycle time | s | 0.850 |

Mean pulmonary arteriovenous pressure drop | kPa | 1.5 (NORM) |

| ||

Mean systemic blood flow | mL s | 255 |

Cardiac cycle time | s | 0.425 |

| ||

Mean flow velocity in large blood vessels (rest) | m s | 0.17 |

Maximum vascular wall stress (exercise) | kPa | 500 |

Maximum sarcomere length (exercise) | μm | 2.2 |

Minimum sarcomere length (exercise) | μm | 1.75 |

Maximum passive myofiber stress (exercise) | kPa | 10 (LW), 8 (SW), 20 (RW) |

| ||

Mean pulmonary arteriovenous pressure drop | kPa | 3.0 (PHAc1 and PHCh1), 4.5 (PHAc2 and PHCh2), 6.0 (PHAc3 and PHCh3) |

| ||

Wall volume ( | mL | 75 (LW), 40 (SW), 30 (RW) |

Reference midwall surface area ( | cm | 80 (LW), 45 (SW), 100 (RW) |

Reference sarcomere length ( | μm | 2.0 (LW, SW, and RW) |

Septal midwall volume ( | mL | 42 |

Radius of midwall junction circle ( | cm | 3.3 |

The set of differential equations that describe the CircAdapt model, including the TriSeg module, was solved by numerical integration with time steps of 2 ms using the ODE113 function in MATLAB 7.1.0 (MathWorks, Natick, MA). Simulation time of a single cardiac cycle was less than 7 s on a Windows XP™ platform (version 2002) with a 2.00 GHz Intel^{®} Core™2 Duo T7250 processor and 1 GB of RAM.

### Simulations

The TriSeg model, as integrated in the CircAdapt model, was tested by simulation of human ventricular mechanics and hemodynamics with (1) normal ventricular loading conditions (NORM), (2) acute pulmonary hypertension (PHAc), and (3) chronic pulmonary hypertension including adaptation (PHCh). All simulations are presented with similar values of mean systemic blood flow (cardiac output), cardiac cycle time (heart rate), and mean systemic arterial blood pressure, simulating hemodynamics at rest (Table 1: *Hemodynamics at rest*).

#### NORM Simulation

The diameter of large blood vessels was assumed to be determined by adaptation to mean chronic circumstances, which state is most closely described by the condition of rest (Table 1: *Hemodynamics at rest*). However, geometry of the heart and wall thickness of large blood vessels were considered to be a result of adaptation to a state of moderate exercise, thus simulating the effect of relatively short periods of training by exercise. The applied simulation protocol of adaptation has been described earlier.3 Briefly recapitulating, with resting hemodynamics (Table 1: *Hemodynamics at rest*), diameters of the large blood vessels were adapted until mean blood flow velocity reached the set point value (Table 1: *Set point values for adaptation*). Next, a moderate state of exercise was simulated by tripling cardiac output and doubling heart rate (Table 1: *Interventions with exercise*). Under these circumstances, wall volume (*V*_{w}) and reference midwall surface area (*A*_{m,ref}) of all cardiac wall segments were adapted until maximum and minimum sarcomere length as well as maximum passive myofiber stress reached the set point values of adaptation with exercise (Table 1). Also, wall thickness of the blood vessels was adapted until maximum wall stress was equal to the set point value (Table 1). Next, hemodynamics were returned to rest conditions and the above-mentioned adaptation protocol was repeated two or three times until steady-state geometry was reached (<1% deviation from set point values). In total, finding the steady-state adapted NORM simulation required simulation of about 100–200 cardiac cycles.

#### PHAc and PHCh Simulations

The NORM simulation was used as point of departure for simulation of ventricular mechanics and hemodynamics with increased pulmonary resistance. Steady-state simulations (PHAc1, PHAc2, and PHAc3), representing increasing degrees of acute pulmonary hypertension, were obtained by acute increase of mean pulmonary arteriovenous pressure drop in three steps without adaptation (Table 1: *Interventions with PH simulations*). Finally, adaptation was applied to the three steady-state PHAc simulations following the same protocol used for the NORM simulation. This resulted in three steady-state simulations (PHCh1, PHCh2, and PHCh3, respectively), representing gradually increasing degrees of chronic pulmonary hypertension at rest.

### Simulation Data Analysis

#### Ventricular Pump Mechanics and Hemodynamics

Simulated time courses of normal (NORM) LV and RV pressures, volumes, and flows were compared with physiological data obtained in normal subjects.26,29,54,62,63 Area of ventricular pressure–volume relation was calculated to quantify ventricular pump stroke work (*W*_{stroke}) for the LV and RV. Time courses of blood flow velocities through the mitral, aortic, tricuspid, and pulmonary valves were used to study the effects of chronic pulmonary hypertension on ventricular hemodynamics. For comparison with clinical data obtained in healthy subjects and in patients with chronic pulmonary hypertension,9,62 several timing parameters of RV hemodynamics were quantified for the NORM and PHCh simulations. RV ejection time (ET) was quantified as the time from pulmonary valve opening to closure, acceleration time of pulmonary flow (AT) as the time from pulmonary valve opening to moment of maximal pulmonary flow velocity, RV isovolumic contraction time (ICT) as the time from tricuspid valve closure to pulmonary valve opening, and RV isovolumic relaxation time (IRT) as the time from pulmonary valve closure to tricuspid valve opening.

#### Tissue Mechanics

Area of myofiber stress–strain relation was calculated to quantify stroke work density (*w*_{stroke}) for the LW, SW, and RW. Stroke work density was defined as contractile myofiber stroke work per unit of tissue volume.

#### Ventricular Wall Geometry

End-diastolic wall thicknesses (*H*_{ed}) were calculated as wall volume divided by end-diastolic midwall surface area. End-diastole was defined as the moment of mitral valve closure. Furthermore, time courses of LW, SW, and RW midwall curvatures were used to assess acute and chronic effects of increase of pulmonary resistance on wall curvatures. The relation between septal geometry and the transseptal pressure gradient in the TriSeg model was compared with the relation found in patients with and without chronic pulmonary hypertension.17 For that purpose, curvature ratio (CR) was calculated as the SW/LW midwall curvature ratio at the moment of aortic valve closure, whereas transmural pressure ratio (PR) was defined as the difference between maximum LV and RV pressures divided by maximum LV pressure.

## Results

### Simulation of Normal Physiology (NORM)

*H*

_{ed,LW}and

*H*

_{ed,SW}were almost equal and three times as large as

*H*

_{ed,RW}(Table 2).

Simulation results: ventricular wall thickness

Parameter | Unit | NORM | PHAc | PHCh | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | |||

| ||||||||

| mm | 7.9 | 7.8 | 7.6 | 7.3 | 8.3 | 8.2 | 8.3 |

| mm | 6.8 | 6.9 | 7.0 | 6.9 | 6.7 | 6.9 | 6.5 |

| mm | 2.3 | 2.1 | 2.0 | 1.7 | 3.3 | 4.2 | 5.1 |

### Simulations of Acute Pulmonary Hypertension (PHAc)

*H*

_{ed,RW}decreased maximally by 26% in PHAc3 relative to NORM,

*H*

_{ed,LW}decreased maximally by 8%, whereas

*H*

_{ed,SW}increased only 1%.

Simulation results: pulmonary artery pressures and ventricular work

Parameter | Unit | NORM | PHAc | PHCh | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | |||

| ||||||||

| Pa m | 1.04 | 1.04 | 1.03 | 1.02 | 1.05 | 1.04 | 1.04 |

| Pa m | 0.22 | 0.32 | 0.42 | 0.51 | 0.35 | 0.48 | 0.61 |

| ||||||||

| kPa | 9.41 | 9.54 | 9.32 | 8.66 | 8.88 | 8.90 | 8.78 |

| kPa | 8.94 | 8.28 | 7.54 | 6.15 | 9.19 | 8.66 | 8.49 |

| kPa | 7.91 | 11.66 | 15.94 | 21.59 | 8.59 | 9.13 | 9.48 |

| ||||||||

| kPa | 2.04 | 3.56 | 5.11 | 6.77 | 3.53 | 5.00 | 6.47 |

### Simulations of Chronic Pulmonary Hypertension (PHCh)

*H*

_{ed,RW}increased up to 120% in PHCh3, whereas changes of

*H*

_{ed,LW}and

*H*

_{ed,SW}were small (<5%) (Table 2).

*r*

^{2}= 1.00, SEE = 0.008) between CR and PR with similar slope as the relation found in patients (

*r*

^{2}= 0.73, SEE = 0.044). In the simulations, however, an offset was found, shifting the relation to higher CR values (Fig. 10). It is noted (not shown) that although acute change (±20%) of mean systemic flow or mean arterial blood pressure affected CR and PR, their interdependence followed the linear relation as derived from the chronic simulations within 2% deviation. The same holds for the data points derived from the PHAc1 and PHAc2 simulations, while for the PHAc3 simulation the deviation was more than 10% (Table 4, not shown in Fig. 10).

Simulation results: curvature ratio and transmural pressure ratio

Parameter | Unit | NORM | PHAc | PHCh | ||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 1 | 2 | 3 | |||

CR | – | 0.94 | 0.85 | 0.72 | 0.54 | 0.81 | 0.67 | 0.55 |

PR | – | 0.78 | 0.68 | 0.57 | 0.46 | 0.65 | 0.52 | 0.40 |

## Discussion

The newly designed TriSeg model of ventricular mechanics incorporates mechanical interaction of the LV free wall, RV free wall, and septal wall resulting in a strong coupling of LV and RV pump mechanics and hemodynamics. The TriSeg model was successfully implemented as a module in the available CircAdapt model, simulating cardiac mechanics and hemodynamics of the closed-loop circulation. Effects of ventricular interaction on cardiac mechanics and hemodynamics were assessed by simulation of pulmonary hypertension in the acute phase as well as in the chronic phase, the latter implying adaptation of ventricular geometry to mechanical load. For chronic pulmonary hypertension, simulated ventricular geometry, hemodynamics, and septal mechanics agreed surprisingly well with corresponding measurements in patients.

### Model Assumptions

In the TriSeg model, ventricular geometry was approximated by three thick-walled spherical segments encapsulating the LV and RV cavities (Fig. 2a). In reality, the ventricular cavities are enclosed by truncated ellipsoidal muscular walls and the noncontractile basal sheet with valves.58 The simplification to spherical segments without a noncontractile sheet resulted in a general underestimation of ventricular dimensions. Although the fact that there are indications that bending stiffness of the myocardium may be important for simulation of septal geometry,7,21 we neglected this effect for simplicity. Despite these inaccuracies, relative changes of dimensions during the cardiac cycle as a result of adaptation were simulated realistically.

In the TriSeg model, myofiber strain was estimated from midwall curvature, area, and wall volume by application of the one-fiber model2 to a spherical wall segment (Eq. 15 and Appendix A). The one-fiber model has been shown to be insensitive to actual wall geometry by assuming conservation of energy and homogeneity of fiber stress in the wall.2 So, we expected that the present relation for transmural pressure as a function of midwall surface area and curvature was also applicable to the real, more irregular cardiac geometry, although this fact has not been proven.

The analytically derived dependency of myofiber strain on wall segment geometry (Eq. A7) appeared continuous and differentiable around *z* = 0. To avoid numerical inaccuracy near zero curvature, because of zero-division, a fourth-order Taylor series approximation (Eq. 15) was used instead. For similar reasons, Eq. (A4) for midwall tension was also approximated by a fourth-order Taylor series (Eq. 16). Within the physiological range of ventricular geometry, the errors of the approximations as compared to the analytically derived relations were smaller than 1% and 2% for strain and tension, respectively. With respect to total ventricular pump work, total myofiber stroke work was overestimated less than 2.6%, whereas total tensile stroke work at the midwall surface was underestimated less than 0.5%.

In the NORM and PHCh simulations, a load-controlling adaptation mechanism was applied to render size and mass of each wall segment. The applied adaptation rules required prescribed values for maximum and minimum sarcomere lengths and for maximum passive myofiber stress in each wall segment (Table 1). Maximum and minimum sarcomere lengths were derived from experiments on isolated cardiac muscle of the rat.15,31,64 Maximum passive myofiber stress was chosen as adaptation stimulus because experimental data obtained in dogs with chronic volume overload suggested that end-diastolic myofiber stress and ejection strain were important mechanical stimuli for hypertrophy, while peak systolic myofiber stress appeared irrelevant.22 However, when assuming similar values of maximum passive myofiber stress in the three ventricular walls of the TriSeg model, ventricular geometry did not develop anatomically accurate. Therefore, in our model, we adjusted levels of maximum diastolic stress per wall segment (Table 1) so that after adaptation the weight ratios for the walls agreed with findings in healthy volunteers.19,28

In this study, the external pressure surrounding the LV and RV free walls was assumed to be zero. The real heart is surrounded by the pericardium, which constrains increase of total heart volume during volume overload. Under resting conditions, the pericardium is believed to play a minor role, setting pericardial pressure close to zero.42,46,67 However, with acute increase of total heart volume, the pericardium affects cardiac hemodynamics and interaction of the cardiac chambers significantly.6,11,25 In our simulations of acute pulmonary hypertension (PHAc), the effect of the pericardium should be considered because of severe RV dilatation (Fig. 5). In the PHAc3 simulation, RV cavity volume even increased by 120% relative to the NORM simulation. Together with the changes of right atrial and LV cavity volumes (45% increase and 10% decrease, respectively), total heart volume increased by 25%. In the chronic PH simulations, the effect of the pericardium on ventricular mechanics is likely to be of minor importance, because measurements demonstrated adaptive dilatation of the pericardium in patients with chronic pulmonary hypertension.8,24

### Comparison of Model Simulations with Measurements

The relation between curvature ratio CR and transmural pressure ratio PR as extracted from the NORM and PHCh simulations (Fig. 10) agreed quite well with the relation as found in a patient group consisting of patients with and without chronic pulmonary hypertension.17 The relations were about linear with equal slope. In the simulations, however, an offset was found, shifting the relation to higher CR values, implying overestimation of septal curvature for a given right-to-left pressure ratio.

The overestimation of septal curvature may have many causes. In the model, wall geometry was considered spherical having a clear unique radius of curvature by definition. The real ventricular walls are not spherical,58 implying that radii of curvature along the circumferential and base-to-apex direction are different. Also, the junction of the real ventricular walls is smoothed over the wall boundaries, thus further hindering a clear definition of curvature. Besides an error in the curvature ratio as derived from the simulations, the offset might originate from a systematic error in the pressure or curvature measurements obtained in patients as discussed by Dellegrottaglie *et al.*17 Nonetheless, the linear relationship between CR and PR might be a useful tool for noninvasive estimation of systolic RV cavity pressure in patients with RV pressure overload.17,35,51 In the simulations, the linear relation appeared also valid (<2% deviation) for the PHAc1 and PHAc2 simulations (Table 4) as well as after acute changes (±20%) of mean arterial pressure and cardiac output in the chronic pulmonary hypertension simulations (not shown in Fig. 10). The latter findings suggested that the linear relation between CR and PR is insensitive to acute changes of hemodynamic status.

Experimental studies with acute manipulation of transseptal pressure difference as well as clinical studies among patients with chronic pulmonary hypertension showed that septal curvature and position of the septum between the LV and RV free walls depend instantaneously on transseptal pressure difference.10,20,21,35,36,52 This dependency was found during systole as well as diastole. Similar dependencies were found in our simulations of pulmonary hypertension. Septal curvature decreased with increase of pulmonary resistance (Figs. 5 and 6). Furthermore, leftward shift of the septum, increase of RV volume, and decrease of LV volume with acute increase of pulmonary resistance (Fig. 5), as predicted by the TriSeg model, were also predicted by Kerckhoffs *et al.*34 using a finite element model of the ventricles coupled to a lumped circulation model.

Simulations and clinical observations were in agreement concerning changes in timing of cardiac flow events due to pulmonary hypertension (Figs. 7 and 8). For example, acceleration time of pulmonary flow velocity was significantly decreased and varied linearly with mean pulmonary artery pressure.9,14 Furthermore, RV isovolumic contraction and relaxation times increased while pulmonary ejection time decreased.62,70 Moreover, tricuspid E/A-ratio was decreased in patients with chronic pulmonary hypertension indicating deterioration of RV diastolic function.71

Ventricular wall volumes and end-diastolic wall thicknesses (Table 2) in the NORM simulation were all about 25% smaller than values measured in healthy volunteers.19,28,57 This difference most likely resulted from an overestimation of contractility used in the sarcomere mechanics model (Appendix B, Eq. B7), causing walls to be thinner after adaptation to mechanical load. Furthermore, pulmonary acceleration and ejection times as derived from the NORM simulation (Fig. 8) were underestimated as compared to data obtained in healthy individuals.9,27 A probable cause is inaccuracy in the model description of sarcomere mechanics, which is primarily derived from experiments on isolated cardiac muscle of the rat.15,31,64 Human myocardium under *in vivo* conditions is likely to behave differently.

In the PHCh simulations, LV pump function and hemodynamics were relatively unaffected by increase of pulmonary resistance (Figs. 6 and 7). Also in rats with chronic pulmonary hypertension, resting LV pump function remains unaffected as long as the myocardium is able to compensate for increased tissue load by structural adaptation.23 Furthermore, in the PHCh simulations, the RV free wall hypertrophied and RV cavity volume decreased (Table 2 and Fig. 6). These geometric changes were in agreement with experimental observations in rats with mild chronic pulmonary hypertension.49 In the latter study, chronic pulmonary artery banding resulted in an increase of RV systolic pressure from 33 to 71 mmHg. This increase of RV afterload resulted in 76% increase of thickness of the RV free wall and 14% decrease of RV free wall area, which suggests a small reduction in RV cavity volume. In our simulations, RV systolic pressure increased from 27 mmHg in the NORM simulation to 72 mmHg in the PHCh3 simulation (Fig. 6), end-diastolic wall thickness increased 120% (Table 2), and RV cavity volume decreased 8%. In another animal study in rats,44 it was shown that structural adaptation to chronic increase of RV afterload was associated with concentric hypertrophy up to a certain level. Beyond this level, however, the myocardium could not fully compensate for further load increase. Consequently, the RV cavity dilated, mainly due to dilatation of the RV free wall. In patients with severe chronic pulmonary hypertension, RV free wall hypertrophy is found to occur together with RV dilatation.12,30 Secondary to RV failure, systolic and diastolic LV function deteriorate by ventricular interaction.38,39 As a result, cardiac performance deteriorates and exercise capacity reduces.12,45 These phenomena indicating RV failure were also found in our PHAc simulations.

In the PHAc simulations, the myocardial tissue was unable to compensate for increased RV afterload by load adaptation. As a result, mechanical myofiber load was inhomogeneously distributed over the ventricular walls (Table 3 and Fig. 5). RW stroke work density increased with increase of RV afterload, whereas LW and SW stroke work densities decreased. In the PHAc3 simulation, RW stroke work density was increased by almost 200% with respect to the NORM simulation. When assuming stroke work density to be correlated to oxygen consumption,59 this implies an increase of oxygen demand by the RV free wall. Since oxygen supply is not included in our model, the potential effect of perfusion imposed limitations on myocardial performance is unknown in our simulations of pulmonary hypertension.

In patients with pulmonary hypertension, tricuspid regurgitation often occurs due to RV and tricuspid annular dilatation.18,65 In our simulations, tricuspid valve regurgitation was not included. In the PHAc3 simulation, RV end-diastolic volume was increased by almost 100% with respect to the NORM simulation. It is likely that the absence of tricuspid regurgitation resulted in underestimation of RV volume overload in the PHAc simulations.

The TriSeg model was successfully integrated as a module in the CircAdapt model of the closed-loop cardiovascular system. As shown previously, the CircAdapt environment is flexible by its modular setup and enables realistic simulation of cardiovascular mechanics and hemodynamics under normal as well as various pathological conditions.3,33,40 A set of physiological adaptation rules, expressing structural adaptation of the system to mechanical load, makes the model self-structuring and reduces the number of independent model parameters. For each ventricular wall segment in the TriSeg model, wall volume, midwall area, and reference sarcomere length were varied so that mechanical myofiber load was normalized to a known physiological level, which was assumed to be the same for all ventricular walls. The field of application of the CircAdapt model was substantially enlarged by implementation of the TriSeg model. In the future, the combined model may be used to study fundamental research questions concerning ventricular interaction and its role in cardiac pathologies. For example, specific material properties may be changed per wall segment in order to study effects of heterogeneity of wall properties on cardiac mechanics and hemodynamics, e.g., asynchronous mechanical activation (and pacing) or a localized myocardial infarct.33,40

Although the TriSeg model can be easily modified to include a description of inhomogeneous myocardial wall properties, a finite element model of cardiac mechanics is more accurate and better suited to describe local inhomogeneities in mechanical load. For patient-specific modeling of the circulation, the model should simulate many cycles in order to find a best match with the available set of measurements. For that purpose, the TriSeg model is to be preferred because calculation effort was about 1,000 times less than that of a finite element approach. The finite element model is absolutely needed to estimate and evaluate possible errors introduced by the applied simplifications of the TriSeg model. Currently, the CircAdapt model with the TriSeg module will be evaluated in its possibility to simulate hemodynamics of pulmonary hypertension patient specifically.

## Conclusions

We presented the TriSeg model of ventricular mechanics and hemodynamics incorporating mechanical interaction of the LV free wall, RV free wall, and septal wall, resulting in a strong coupling of LV and RV hemodynamics. The model enables calculation of LV and RV pressures given the respective cavity volumes. LV and RV hemodynamics are related to myofiber mechanics in the three ventricular walls, satisfying the principle of conservation of energy. The three ventricular walls are mechanically coupled satisfying equilibrium of tensile forces in their junction. After implementation as a module in the lumped closed-loop CircAdapt model of heart and circulation, the TriSeg model enables simulation of ventricular hemodynamics and wall mechanics as functions of time.

Simulations of cardiac mechanics and hemodynamics during normal ventricular loading, acute pulmonary hypertension, and chronic pulmonary hypertension after load adaptation agreed with clinical data as obtained in normal subjects and in chronic pulmonary hypertension patients. With increasing levels of chronic pulmonary hypertension, the TriSeg model predicted increase of systolic pulmonary flow acceleration, increase of isovolumic contraction and relaxation times, and linear decrease of septal-to-LV free wall curvature ratio. Summarizing, the TriSeg model realistically describes ventricular mechanics including the interaction between left and right ventricular pump mechanics, dynamics of septal geometry, and contractile myofiber function in the three ventricular walls.

## Notes

### Acknowledgments

We gratefully acknowledge the financial support of Actelion Pharmaceuticals Nederland B.V. (Woerden, The Netherlands) and the Netherlands Heart Foundation Grant 2007B203.

### Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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