Simulators & Interactive Hemodynamics

First-Principles Biomechanics

Deriving ESPVR & EDPVR from Spherical Geometry

The left ventricle is approximated as a thick-walled sphere undergoing centrosymmetric radial expansion and contraction with an incompressible myocardium. The relation between intracavity pressure and volume derives directly from the mechanical work principle:

P = dW / dV

where W represents total elastic strain energy stored in the myocardium, calculated by integrating strain energy density functions over the spherical wall domain:

1. Passive Diastolic Mechanics (EDPVR): Characterized by an isotropic exponential strain energy function:

Ψ = (a / b) (e^{b(I₁ − 3)} − 1)

Linear parameter a sets base tissue elasticity; exponent b dictates non-linear stiffening upon stretch (e.g. interstitial fibrosis).

2. Active Systolic Mechanics (ESPVR): Combines passive resistance with active sarcomere contraction via additive stress:

Ψ_a = T_a (λ²/2 − λ₀λ)

T_a represents peak isometric contractility (inotropic state) and λ denotes myocardial fiber strain.

3. Ventriculo-Arterial Coupling: Systemic afterload is coupled as arterial elastance E_a = ESP / SV. The intersection of the E_a line and ESPVR determines stroke volume, ejection fraction, and blood pressure equilibrium.

References

1. Spherical EDPVR & ESPVR Model: Zhang Y, Kalhöfer-Köchling M, Bodenschatz E, Wang Y. Physical model of end-diastolic and end-systolic pressure-volume relationships of a heart. Frontiers in Physiology. 2023;14:1157973. doi:10.3389/fphys.2023.1157973

2. Ventriculo-Arterial Coupling: Sunagawa K, Maughan WL, Burkhoff D, Sagawa K. Left ventricular interaction with arterial load studied in isolated canine ventricle. American Journal of Physiology - Heart and Circulatory Physiology. 1983;245(5):H773–H780. doi:10.1152/ajpheart.1983.245.5.H773

Clinical Interpretation

Parameter Tuning & Clinical Translation

Translate constitutive physics parameters directly into clinical indices measured at the bedside:

Parameter	Physical Meaning	Clinical Translation	Reference & Pathology
Wall Thickness (Δ)	Normalized wall thickness	Relative Wall Thickness (RWT)	0.27 (Normal) · <0.22 (Eccentric dilation) · >0.35 (Concentric hypertrophy)
Contractility (T_a)	Max active stress	Inotropic State	~77 kPa (Normal) · <50 kPa (HFrEF / Shock) · >100 kPa (Hyperdynamic)
Stiffness (a)	Linear scaling factor	Diastolic Compliance	1.15 kPa (Normal) · >1.8 kPa (Global stiffening / aging)
Fibrosis (b)	Exponential stretch	Restrictive Filling	3.8 (Normal) · 6.0–9.0 (Severe fibrosis / restrictive cardiomyopathy)
Preload (EDV)	End-diastolic volume	Venous Filling	130 mL (Normal) · >180 mL (Volume overload / dilated cardiomyopathy)
Afterload (E_a)	Arterial elastance (ESP/SV)	Effective Systemic Load	1.5 mmHg/mL (Normal) · >2.5 mmHg/mL (Vasoconstriction / severe HTN)
Compliance (C_a)	Arterial tree distensibility	Arterial Capacitance	1.47 mL/mmHg (Normal) · <0.80 mL/mmHg (Arterial stiffness)

Back to Advanced Hemodynamics

Non-Invasive Hemodynamic Reconstruction

Bedside Mapping of the LV Pressure-Volume Loop

By combining routinely measured non-invasive clinical parameters—brachial cuff blood pressure and Doppler echocardiographic chamber volumes—the continuous left ventricular pressure-volume loop can be reconstructed without cardiac catheterization:

1. Isovolumic Contraction: Begins at end-diastole upon mitral valve closure (V = EDV, P = LVEDP) and rises isovolumetrically to aortic valve opening (V = EDV, P = DBP).

2. Systolic Ejection: Blood ejects into the aorta along a smooth pressure trajectory peaking at measured SBP, concluding at aortic valve closure (V = ESV, P = ESP), where ESP is anchored to 0.90 × SBP.

3. Isovolumic Relaxation: Pressure drops vertically at constant end-systolic volume to intersect the diastolic filling curve at (V = ESV, P = EDPVR(ESV)).

4. Diastolic Filling: Follows the non-linear passive stiffness curve from ESV back to EDV. Filling pressure is non-invasively derived from the validated tissue Doppler ratio:

PCWP ≈ LVEDP = 1.24 × (E / e') + 1.9 mmHg

Single-Beat Elastance & Energetics

Klotz Single-Beat EDPVR & Suga Energetics

The passive diastolic curve is reconstructed using the single-beat normalised EDPVR equation developed by Klotz and colleagues:

P = α · V^β

where volume intercepts (V₀, V₃₀) and power coefficients (β, α) are calculated from a single measured (EDV, EDP) coordinate, eliminating the historical need for invasive balloon occlusion.

Suga Mechanical Energetics:

Stroke Work (SW): Net mechanical energy delivered to circulating blood per beat, calculated as the integral ∮ P dV (loop area).
Potential Energy (PE): Elastic potential energy stored in myocardium at end-systole between ESPVR and EDPVR.
Pressure-Volume Area (PVA): Total mechanical energy generated per cycle (SW + PE), exhibiting a direct stoichiometric relationship with myocardial oxygen consumption (MVO₂).
Mechanical Efficiency (η): Ratio of external stroke work to total mechanical energy (η = SW / PVA × 100%).

Reference: Klotz S, Hay I, Dickstein ML, Yi GH, Wang J, Maurer MS, Kass DA, Burkhoff D. Single-beat estimation of end-diastolic pressure–volume relationship: a novel method with potential for noninvasive application. Am J Physiol Heart Circ Physiol 2006;291:H403–H412.

Clinical Benchmarks

Preloaded Archetypes & Suggested Settings

Explore five physiologically calibrated phenotypes across hypertensive, ischemic, and cardiomyopathic spectrums:

Clinical Archetype	BP (mmHg)	EF (%)	EDV (mL)	E/e'	Ea / Ees	Hemodynamic & Energetic Profile
1. Normal Reference	120 / 75	60%	110	8.0	0.67	Optimal ventriculo-arterial coupling; high mechanical efficiency (~75%), LVEDP 11.8 mmHg, normal cavity size.
2. Dilated LV (HFrEF)	105 / 65	28%	220	18.0	2.57	Severe systolic uncoupling (Ea/Ees > 2.5); failing pump, depressed Ees (0.60), marked chamber dilation, elevated LVEDP (24.2 mmHg).
3. Concentric (HFpEF)	145 / 85	60%	85	16.0	0.67	Preserved coupling ratio but elevated afterload (Ea 2.56) and steep diastolic EDPVR curve; small cavity, elevated LVEDP (21.7 mmHg).
4. Preserved Cardiac Power	130 / 75	56%	93	14.0	0.79	Compensated state; preserved stroke volume (52.1 mL), effective forward cardiac power output (0.95 W), filling pressure (19.3 mmHg).
5. Reduced Cardiac Power	117 / 69	45%	96	18.0	1.22	Decompensated state; ventriculo-arterial uncoupling, reduced stroke volume (43.2 mL), depressed efficiency and elevated filling pressures (24.2 mmHg).

Clinical Archetypes

Benchmark Cardiac Archetypes in 3D Space

Explore five standardized cardiac phenotypes mapped across the 3D surface topography:

Archetype	E_es (mmHg/mL)	α (1/mL)	Hemodynamic Impact
Normal Reference	2.50	0.030	Normal compliance and inotropic reserve (EF ~55%, SW ~6,350 mmHg·mL).
Compliant Ventricle	2.50	0.024	Enhanced filling capacity (EDV 144 mL); stroke work jumps +59% to 10,120 with preserved EF (~56%).
Stiff Ventricle (HFpEF)	2.00	0.045	Diastolic restriction (EDV 82 mL); stroke work plummets −66% to 2,140 despite "preserved" EF (48%).
High Inotropy	4.50	0.030	Hyperdynamic systolic performance; elevated EF (67%) and elevated BP (136/94 mmHg).
Depressed Inotropy (HFrEF)	1.00	0.030	Systolic failure; reduced EF (33%) and low stroke work (2,260 mmHg·mL) at normal volumes.

Right Atrial Cardiac Cycle

The Venous Pressure Waveform

In the absence of venous obstruction, pressure measured in the large central veins (the CVP) faithfully reflects pressure and flow changes within the right atrium. The waveform has five distinct phasic events: three positive deflections (the a, c, and v waves) and two descents (x and y).

Component	Timing	Biophysical Mechanism
a wave	End-diastole	Active right atrial contraction. Dominant positive deflection in normal right-heart physiology.
c wave	Early systole	Isovolumic right ventricular contraction; tricuspid valve closure and systolic dome bulging.
x descent	Mid-systole	Atrial relaxation and systolic annular descent pulling the atrial floor toward the apex (caval S wave).
v wave	Late systole	Passive venous inflow against a closed tricuspid valve. Height determined by compliance and inflow volume.
y descent	Early diastole	Tricuspid valve opening and rapid ventricular passive filling down the pressure gradient (caval D wave).

Mechanics & Compliance

The Atrial Pressure-Volume Loop

Like the ventricle, right atrial function can be depicted as a pressure-volume loop, forming a characteristic figure-of-eight on its side:

The A Loop (Left): Initiated by the P wave, running counter-clockwise. Atrial contraction raises pressure while emptying the chamber—the booster pump function, contributing 15–30% of ventricular stroke volume.

The V Loop (Right): Runs clockwise, representing passive filling and emptying. It comprises the reservoir phase (venous return stored while the tricuspid valve is shut) and the conduit phase (blood draining into the ventricle once the tricuspid valve opens).

Static Atrial Compliance: The steepness of the pressure-volume slope from the minimum of the A loop to the maximum of the V loop defines chamber compliance. When stiff or distended, pressure spikes rapidly, producing a tall v wave and exaggerated y descent.

Clinical Scenarios

Three Validated Physiological Archetypes

Use the interactive simulation console below to toggle between these three preloaded states and explore how waveform morphology, tricuspid inflow velocity, and atrial PV loops change in lockstep:

State 01

1. Normal Hemodynamics

The a wave exceeds the v wave; the systolic x descent is the dominant trough. The atrial PV loop shows a balanced figure-of-eight with high compliance and low filling pressures.

State 02

2. Right Ventricular Failure

Blunted systolic annular descent abolishes the x descent. Elevated filling pressures cause a dilated, non-compliant atrium with a towering v wave (> a wave) and rapid, steep y descent into the stiff ventricle.

State 03

3. Constrictive Pericarditis

Both x and y descents are exaggerated; filling into the constrained ventricle is rapid in early diastole but abruptly halted. Markedly elevated pressures throughout, shifting the PV loop upward with steep elastance.

Oxygen Transport Physiology

The Direct & Indirect Fick Principle

First described by Adolf Eugen Fick in 1870, the Fick principle states that the uptake or release of a substance by an organ is the product of blood flow to that organ and the arteriovenous concentration difference of the substance. For the heart:

Cardiac Output (CO) = VO₂ / (C_aO₂ − C_vO₂)

where VO₂ is total whole-body oxygen uptake (mL/min), and C_aO₂ − C_vO₂ represents the arteriovenous oxygen content difference across the systemic capillary bed:

C(a−v)O₂ = 1.34 × Hb × (S_aO₂ − S_vO₂) + 0.0031 × (P_aO₂ − P_vO₂)

The direct Fick method remains the gold standard in diagnostic right heart catheterization, especially in severe tricuspid regurgitation, low cardiac output states, or intracardiac shunting where thermodilution fails.

Intracardiac Shunts & Resistance

Pulmonary vs. Systemic Flow (Qp / Qs)

In the presence of intracardiac shunts (e.g. ASD, VSD) or extracardiac communications (PDA), the Fick principle enables independent quantification of pulmonary flow (Q_p) and systemic flow (Q_s):

Q_p / Q_s = (S_aO₂ − S_mvO₂) / (S_pvO₂ − S_paO₂)

where S_mvO₂ is the mixed venous oxygen saturation sampled from the cavae or RA, and S_paO₂ is the pulmonary arterial saturation.

The interactive Fick suite integrates congenital shunt quantification (ASD, VSD, PDA, step-up oximetry runs), catheterization laboratory cardiac output with multi-method VO₂ estimation, extracorporeal life support native cardiac output modeling, and Severinghaus oxyhemoglobin dissociation.

Bedside Physiology: Cardiovascular Mechanics & Visualisations