Adopted preset · educational comparison

baseline

This construction was adopted after its own inputs, independent 2/1 ms checks and raw waveforms were reviewed through a 1-of-2 gate. It does not establish population representativeness or clinical diagnostic validity.

Overview and circuit

Muscle tension generates pressure; pressure differences drive blood; changing volume feeds back into muscle length and pressure. This 0D model derives waveforms and PV loops from coupled myocardium, geometry, valves and vessels.

Four cavities and eleven main-circuit vascular compartments connect to sixteen coronary compartments. Compartments store blood; links carry hydraulic losses. Names denote lumped regions, not exact catheter positions. The septum is a material wall, not a blood-storage node.

Systemic path

Left ventricle (LV)
Proximal aorta (Ao)
Systemic arteries (SA)
Systemic resistance-vessel side (Art)
Systemic capillary bed (Cap)
Systemic veins (SV)
Vena cava (VC)
Right atrium (RA)

Pulmonary path

Right ventricle (RV)
Proximal pulmonary artery (PA)
Pulmonary resistance-vessel side (PArt)
Pulmonary capillary bed (PCap)
Pulmonary venular side (PVen)
Pulmonary vein / LA inlet (PVein)
Left atrium (LA)

LA —MV→ LV and RA —TV→ RV close the circuit. Arrows define positive flow; permitted reverse flow enters continuity with its sign. Coronaries branch from Ao and return to RA.

Ao/SA/Art separate proximal storage from pressure loss toward the periphery, without representing pulse transit or reflection between compartments. SV is the main venous reservoir; VC is the thoracic-pressure-exposed compartment before RA. PCap receives alveolar pressure, while PVen/PVein receive thoracic pressure, separating pulmonary storage and resistance along the path to LA. This is a functional partition, not a vessel-by-vessel anatomical reconstruction.

All main-circuit compartments: law and pressure reference
Compartment	Pressure relation	External pressure	Displayed counterpart
Left ventricle (LV)	Material / geometry balance	$P_{\mathrm{th}}+P_{\mathrm{peri}}$	LVP
Left atrium (LA)	Material / geometry balance	$P_{\mathrm{th}}+P_{\mathrm{peri}}$	LAP
Right ventricle (RV)	Material / geometry balance	$P_{\mathrm{th}}+P_{\mathrm{peri}}$	RVP
Right atrium (RA)	Material / geometry balance	$P_{\mathrm{th}}+P_{\mathrm{peri}}$	RAP / CVP
Proximal aorta (Ao)	Exponential arterial	0 (reference)	AoP
Systemic arteries (SA)	Exponential arterial	0 (reference)	ABP
Systemic resistance-vessel side (Art)	Exponential arterial	0 (reference)	—
Systemic capillary bed (Cap)	Linear storage	0 (reference)	—
Systemic veins (SV)	Nonlinear venous-type	0 (reference)	—
Vena cava (VC)	Nonlinear venous-type	$P_{\mathrm{th}}$	—
Proximal pulmonary artery (PA)	Exponential arterial	$P_{\mathrm{th}}$	PAP
Pulmonary resistance-vessel side (PArt)	Exponential arterial	$P_{\mathrm{th}}$	—
Pulmonary capillary bed (PCap)	Nonlinear venous-type	$P_{\mathrm{alv}}$	—
Pulmonary venular side (PVen)	Nonlinear venous-type	$P_{\mathrm{th}}$	—
Pulmonary vein / LA inlet (PVein)	Nonlinear venous-type	$P_{\mathrm{th}}$	—

AoP/PAP are Ao/PA intravascular pressures, with no display-only ZcQ. ABP is SA pressure, not a simulated brachial cuff. CVP uses mean RA; PCWP uses mean LA as a proxy, without simulating catheter wedging. PV loops use ventricular transmural pressure.

\dot V_i=\sum_j N_{ij}Q_j,\qquad N_{ij}=\begin{cases}+1&j\text{ enters }i\\-1&j\text{ leaves }i\\0&\text{otherwise}\end{cases},\qquad\sum_{i=1}^{31}V_i=TBV

Construct incidence matrix N from the link tables in preset settings. Each flow leaves one compartment and enters another, conserving total blood. Myocardium and pericardial fluid are excluded. Blood is not added or removed to match pressure or output at fixed TBV.

Mechanisms and equations

These are the shared constitutive definitions. Adopted coefficients, anatomy and initial conditions are in each preset's Settings. Use both documents to reconstruct the selected operating point.

See coefficients, anatomy and initial state

Activation and calcium

Atrial and ventricular activation events drive calcium transients. Calcium generates pressure through myofilaments and chamber mechanics, not through a prescribed pressure waveform.

\dot x_r=-x_r/\tau_r,\quad \dot x_d=-x_d/\tau_d,\qquad [Ca]=\mathrm{Ca}_0+g(x_d-x_r)

Implemented subset for distinct time constants. Events increment both drives equally; their difference gives a rise and subsequent decay.

Equations and assumptions

Regular sinus rhythm includes atrial capture and ventricular conduction. Ventricles use a calibrated biexponential event source; atria retain a separate source. This is not a full action-potential or intracellular calcium-cycling model. Ca₀ is the long event-free limit, not the periodic trough.

$x_r,\;x_d$: Dimensionless calcium-drive states
$\tau_r,\;\tau_d$: Decay time constants (s)
$\mathrm{Ca}_0,\;g$: Event-free calcium limit and amplitude coefficient (µM)

Activation times and state updates

Time is in seconds. In regular sinus rhythm T=60/HR; ventricular activation follows atrial activation by 120 ms (80 ms AV + 40 ms distal conduction). Calcium deposits occur another 12 ms after the corresponding activation. Each wall has two calcium states.

q is deposit strength, equal to 1 for all five walls at baseline. States decay analytically between events and both receive q at an event. The periodic formula below is for regular unit deposits; it does not initialize chamber volumes or Land states.

t_{A,k}=t_{A,0}+kT,\quad t_{V,k}=t_{A,k}+0.120,\quad t_{Ca,w,k}=t_{A\text{ or }V,k}+0.012

x_j(t+\Delta t)=x_j(t)e^{-\Delta t/\tau_j},\quad x_j(t_k^+)=x_j(t_k^-)+q_k\quad(j=r,d)

x_j(t_k^+)=\frac{1}{1-e^{-T/\tau_j}}\quad\text{(periodic, }q_k=1\text{)}

Interval-dependent calcium strength

Ventricles carry a discrete interval-dependent normalized load L, not measured SR calcium concentration. I is the preceding ventricular interval, a recovery and q the next calcium deposit. Coefficients and reference states are tabulated below. L and a at the reference heart rate form a fixed point with q=1.

The construction specified here is regular sinus rhythm without mechanical support. Optional ectopy, pacing and support-device modes are not included in this baseline equation system.

a_k=1-e^{-I_k/\tau_{\mathrm{rec}}},\quad q_k=a_k\beta L_k,\qquad L_{k+1}=L_k-(1-r)q_k+\gamma(1-ha_k)

Myofilaments: calcium to tension

Calcium binding, crossbridge populations, stretch and shortening velocity determine active tension. Force depends on length and loading history even at the same calcium level.

T_a=\frac{h(\lambda)T_{\mathrm{ref}}}{r_s}\left[S(1+\zeta_s)+W\zeta_w\right]

Land-derived active fiber stress output (Pa): W/S are weak/strong populations, ζ their distortions and h a length-dependent factor. This is not cavity pressure.

Equations and assumptions

Based on Land 2017, but not an unchanged reproduction. Ventricular stress scale, calcium affinity and transition rates are calibrated. An extension returns excess strong-bound population to the unbound pool at low calcium. Population conservation does not transfer the original paper's validation to this extension. Passive and viscoelastic components complete wall stress.

$T_a,\;T_{\mathrm{ref}}$: Active fiber stress and its scale (Pa)
$\lambda,\;h(\lambda)$: Fiber stretch and length-dependent factor (dimensionless)
$W,\;S,\;r_s$: Weak/strong populations and reference strong fraction
$\zeta_w,\;\zeta_s$: Dimensionless crossbridge distortions, dependent on shortening and history

J_{\mathrm{exit}}=k_{\mathrm{max}}\left(\frac{\theta^n}{\theta^n+c^n}\right)^p\max(S-rW,0),\quad \dot S|_{\mathrm{exit}}=-J_{\mathrm{exit}},\quad \dot U|_{\mathrm{exit}}=J_{\mathrm{exit}}

Added exit flux: c is calcium-bound troponin fraction, θ its reference, n/p exponents, kmax a rate (s⁻¹), r=kws/ksu the zero-distortion strong/weak ratio, and U the unbound fraction. Only positive excess is transferred.

Six states and conservation

Each wall has c (Ca-bound troponin), b (blocked), W/S (weak/strong binding) and distortions ζw/ζs. U=1−b−W−S is dependent. Valid states require 0<c≤1 and b,U,W,S≥0. b is unrelated to a hydraulic quadratic coefficient.

Ca is free calcium (µM); λ is the Land stretch. gwu/gsu are distortion-dependent detachment rates (s⁻¹). The added exit flux above applies to ventricular walls; Jexit=0 in atria. Detached population returns to U.

\begin{aligned}\dot c&=k_{\mathrm{TRPN}}\{(\mathrm{Ca}/\mathrm{Ca}_{50})^{n_{\mathrm{TRPN}}}(1-c)-c\},\\\dot b&=k_b\min(c^{-n_{\mathrm{Tm}}/2},100)U-k_u c^{n_{\mathrm{Tm}}/2}b,\\\dot W&=k_{\mathrm{uw}}U-(k_{\mathrm{wu}}+k_{\mathrm{ws}}+g_{\mathrm{wu}})W,\\\dot S&=k_{\mathrm{ws}}W-(k_{\mathrm{su}}+g_{\mathrm{su}})S-J_{\mathrm{exit}},\\\dot\zeta_w&=A_w\dot\lambda-c_w\zeta_w,\qquad\dot\zeta_s=A_s\dot\lambda-c_s\zeta_s.\end{aligned}

\begin{aligned}\lambda_c&=\min(\lambda,1.2),\\\mathrm{Ca}_{50}&=\mathrm{Ca}_{50,\mathrm{ref}}+\beta_1(\lambda_c-1),\\h(\lambda)&=\max\{0,1+\beta_0[\lambda_c+\min(\lambda_c,0.87)-1.87]\},\\g_{\mathrm{wu}}&=\gamma_w|\zeta_w|,\quad g_{\mathrm{su}}=\gamma_s\max(-\zeta_s-1,\zeta_s,0).\end{aligned}

Derived rates and wall coupling

The following rates derive from the independent parameters below. rₛ/r𝓌 are reference fractions, θ=TRPN50. k/c rates are in s⁻¹ and A is dimensionless. Ca50 must remain positive. The λ=1.2 saturation and factor limit 100 are part of the adopted law.

With geometric log strain e, λ=s₀ exp(e), where s₀=1 for all baseline walls. The discrete velocity is (λn+1−λn)/Δt. Land Tₐ is multiplied by λ, orientation fo and active fraction fa to obtain Kirchhoff stress. fo=fa=1. Active controls scale Tref; passive controls scale passive stress and viscous modulus.

\begin{aligned}k_b&=\frac{k_u\theta^{n_{\mathrm{Tm}}}}{(1-r_s)(1-r_w)},\quad k_{\mathrm{wu}}=k_{\mathrm{uw}}(1/r_w-1)-k_{\mathrm{ws}},\\k_{\mathrm{su}}&=k_{\mathrm{ws}}r_w(1/r_s-1),\quad A_w=A_s=\frac{A_{\mathrm{eff}}r_s}{(1-r_s)r_w+r_s},\\c_w&=\frac{\phi k_{\mathrm{uw}}(1-r_w)}{r_w},\quad c_s=\frac{\phi k_{\mathrm{ws}}(1-r_s)r_w}{r_s},\\\tau_f&=\lambda f_o f_aT_a+\tau_{\mathrm{pass}}+\tau_{\mathrm{vis}}.\end{aligned}

Land et al. 2017

Passive mechanics and deformation history

Passive stretch resistance and time-dependent viscoelastic relaxation contribute to diastolic pressure alongside active stress.

e=\ln\lambda_g,\quad \tau_{\mathrm{pass}}=\frac{d\Psi}{de},\quad \tau_{\mathrm{vis}}=E_v(e-\alpha),\quad \dot\alpha=\frac{e-\alpha}{\tau_v}

Passive and viscous relations. Land active stress is converted using its stretch and orientation/viability fractions before addition in the same stress convention.

Equations and assumptions

Ventricular passive stress derives from convex strain energy; atria use a separate law. A one-state Maxwell branch represents viscoelastic history. The ventricular prior references organ-level Klotz EDPVR, not direct tissue measurements.

$\lambda_g,\;e,\;\alpha$: Geometric stretch, logarithmic strain, and viscous strain
$\Psi,\;\tau_{\mathrm{pass}},\;\tau_{\mathrm{vis}}$: Energy density (J/m³) and passive/viscous Kirchhoff stresses (Pa)
$E_v,\;\tau_v$: Viscoelastic modulus (Pa) and material time constant (s), distinct from pressure-derived LV τ

Ventricular elastic energy

Let e=ln λg, p=Hδ(e), q=Hδ(−e). K0 is central stiffness; Kt/a govern tensile stiffening; Kc adds compression stiffness. Hδ is a smooth positive part with u=z/δ. The three ventricular walls share this law; energy and derivatives are multiplied by each wall's passive scale sₚ,ｗ, listed in the baseline settings.

H_\delta(z)=\begin{cases}0&z\le0\\\delta(u^3-\tfrac12u^4)&0<z<\delta\\z-\delta/2&z\ge\delta\end{cases}

\Psi(e)=\tfrac12K_0e^2+\frac{K_t}{a^2}(e^{ap}-1-ap)+\tfrac12K_cq^2,\qquad \tau_{\mathrm{pass}}=s_{p,w}\,\frac{d\Psi}{de}

Atrial passive law and viscoelasticity

Atria use a separate equibiaxial reduction; sₚ,ｗ below is each wall's passive multiplier. λg=exp(e); C1/C2/C3 are Pa, C4 dimensionless. The fiber term is tension-only. Energy is Ψ(e)=∫₀ᵉτpass(s)ds. Supported strain is −0.5≤e≤0.5; applying the LA-derived material to RA is an extrapolation.

Each wall has a one-state Maxwell branch: α is viscous strain, Ev modulus, τv relaxation time. The implemented backward-Euler update below is applied once per mechanics step. Listed Ev already includes the passive scale.

\tau_{\mathrm{pass}}=s_{p,w}\left[4C_1(\lambda_g^2-\lambda_g^{-4})+4C_2(\lambda_g^4-\lambda_g^{-2})+\begin{cases}C_3(e^{C_4(\lambda_g-1)}-1)&e>0\\0&e\le0\end{cases}\right]

\alpha_{n+1}=\frac{\alpha_n+(\Delta t/\tau_v)e_{n+1}}{1+\Delta t/\tau_v},\qquad \tau_{vis,n+1}=E_v(e_{n+1}-\alpha_{n+1})

Klotz et al. 2006Organ EDPVR context, not the source of the adopted tissue constitutive law.

Moyer et al. 2015Starting point for atrial passive material, reduced here to equibiaxial deformation.

Chambers: tension to pressure

LV free wall, septum and RV free wall interact through shared geometry, with two atrial walls completing the five-wall model. Pressure follows stress and geometry, rather than a prescribed time waveform.

\delta W=\sum_w V_{\mathrm{m},w}\tau_{\mathrm{f},w}\,\delta\ln\lambda_w,\qquad P_{\mathrm{cavity}}=P_{\mathrm{tm}}+P_{\mathrm{ext}}

Wall virtual work is matched to cavity pressure–volume work. Septal position and junction radius also satisfy force balance.

Equations and assumptions

Ventricles use spherical-cap TriSeg geometry and an energy-conjugate pressure mapping. Pericardial/thoracic external pressure is distinct from transmural pressure. Geometry is not patient-specific 3D imaging and does not resolve local stress or torsion. PV loops use transmural pressure while pressure waveforms use cavity pressure, so they differ when external pressure is present.

$\delta W,\;V_{\mathrm{m},w}$: Virtual work (J) and material volume of wall w (m³)
$\tau_{\mathrm{f},w},\;\lambda_w$: Fiber Kirchhoff stress (Pa) and geometric stretch; this stress τ is distinct from the LVP relaxation time
$P_{\mathrm{cavity}},\;P_{\mathrm{tm}},\;P_{\mathrm{ext}}$: Cavity, transmural, and external pressures, added in the same units

Schematic of spherical caps sharing a junction circle, not measured geometry or wall thickness. h is signed height from the junction plane to each cap apex.

Three spherical caps and fiber length

Mechanics uses SI units: m, m², m³, Pa. VL/VR are cavity blood volumes, Mw wall material volumes, vS the signed septal cap volume, y>0 the junction-circle radius. Wall material is not part of TBV. Cap height h is positive toward RV, usually negative for LV free wall.

For each wall solve h from cap volume, then area Aw, curvature κw and thickness correction zw. Aref,w is reference midwall area. zw is unrelated to Land crossbridge distortion ζ. Geometric ew enters the material law.

v_L=-V_L-\tfrac12(M_L+M_S)+v_S,\qquad v_R=V_R+\tfrac12(M_R+M_S)+v_S

v_w=\frac{\pi h_w(h_w^2+3y^2)}6,\quad A_w=\pi(h_w^2+y^2),\quad\kappa_w=\frac{2h_w}{h_w^2+y^2}

z_w=\frac{3\kappa_wM_w}{2A_w},\qquad e_w=\frac12\ln\frac{A_w}{A_{\mathrm{ref},w}}-\frac{z_w^2}{12}-0.019z_w^4

Internal balance and cavity pressures

F/G are generalized forces for cap volume and junction radius. Multiply each current material stress by geometric strain derivatives; do not differentiate active stress as though it were a conservative potential.

For supplied cavity volumes solve vS/y simultaneously from force balance, then obtain LV/RV transmural pressures. Atria use a spherical one-fiber reduction with reference cavity volume Vref. No prescribed pressure waveform or wall-specific pressure gain is added.

F_w=M_w\tau_{\mathrm{f},w}\left.\frac{\partial e_w}{\partial v_w}\right|_y,\qquad G_w=M_w\tau_{\mathrm{f},w}\left.\frac{\partial e_w}{\partial y}\right|_{v_w}

F_L+F_S+F_R=0,\quad G_L+G_S+G_R=0,\qquad P_{\mathrm{tm},L}=-F_L,\quad P_{\mathrm{tm},R}=F_R

e_A=\frac13\ln\frac{V_A+M_A/2}{V_{\mathrm{ref},A}+M_A/2},\qquad P_{\mathrm{tm},A}=\frac{M_A\tau_{\mathrm{f},A}}{3(V_A+M_A/2)}

Thoracic and pericardial pressure

Occupied heart volume VH sums four cavity blood volumes, five wall volumes and prescribed pericardial fluid. Coronary blood is not added again to this occupancy formula. All cavities share the same external pressure. V0 is reference capacity, P* pressure scale and k stiffness.

H smooths engagement, δ=0.001 and u=(x+δ)/(2δ). H′ differentiates with respect to x. Baseline pressure offset/fluid/respiratory amplitudes are zero. With no engagement and zero thoracic pressure, cavity and transmural pressures coincide.

V_H=\sum_{c=LA,LV,RA,RV}V_c+\sum_w M_w+V_{\mathrm{fluid}},\quad x=(V_H-V_0)/V_0

H(x)=\begin{cases}0&x\le-\delta\\\delta(2u^3-u^4)&|x|<\delta\\x&x\ge\delta\end{cases}

P_{peri}=P_{\mathrm{offset}}+P_*[e^{kH(x)}-1]H'(x),\qquad P_c=P_{\mathrm{tm},c}+P_{\mathrm{th}}+P_{peri}

Lumens et al. 2009 · TriSegBasis for three-wall geometry; the present constitutive laws and pressure mapping are not a reproduction of the entire original model.

Four valves: pressure difference to flow

Mitral, aortic, tricuspid and pulmonary valves share an opening and pressure-loss structure, with valve-specific forward and regurgitant areas.

\Delta P=R Q+B(A)Q|Q|,\qquad B(A)\propto \rho/A^2

Constitutive law on the flowing branch; coefficients include unit conversions. Exact closure supports a pressure difference at Q=0, so this equation alone does not describe every branch.

Equations and assumptions

Flow is solved algebraically without a separate inertial flow state, while a bounded leaflet-opening memory remains. EOA already accounts for contraction/discharge; no extra Cd is applied. This construction has no local pressure-recovery correction. Raw AV/PV node differences are not Doppler maximum-jet or catheter peak-to-peak gradients.

$\Delta P,\;Q$: Simultaneous upstream-minus-downstream pressure (mmHg) and forward-positive flow (mL/s)
$A,\;\rho$: Current effective orifice area and blood density; area depends on maximum area and opening fraction
$R,\;B$: Linear resistance (mmHg·s/mL) and quadratic loss coefficient (mmHg·s²/mL²)

Opening-state dynamics

ξ is opening fraction (0–1), ΔP upstream minus downstream pressure, d deadband, pₒ offset (0 mmHg in all baseline valves) and kₒ opening sensitivity. Fε smooths positive opening drive with ε=0.1 mmHg. Use the opening time constant when target exceeds previous ξ, otherwise the closing constant.

F_\epsilon(z)=\begin{cases}0&z\le0\\z^2/(2\epsilon)&0<z<\epsilon\\z-\epsilon/2&z\ge\epsilon\end{cases},\quad \xi_\infty=1-e^{-k_oF_\epsilon(\Delta P-d-p_o)}

\dot\xi=\frac{\xi_\infty-\xi}{\tau_\xi},\qquad \xi_{n+1}=\frac{\xi_n+(\Delta t/\tau_\xi)\xi_\infty(\Delta P_{n+1})}{1+\Delta t/\tau_\xi}

Area, direction and exact closure

Amax is maximal forward EOA; Ar closed regurgitant EOA. Forward area includes the residual gap plus the opening-dependent part; reverse area is Ar independent of ξ. Changing Ar also affects near-closure forward flow. Baseline Ar=0 in all valves.

Q is mL/s, A cm², pressure mmHg, ρ=1060 kg/m³, cP=133.322387415 Pa/mmHg. Zero area gives exactly Q=0 with supported pressure difference. No area floor or flow smoothing is introduced. R is not rescaled with area.

A=\begin{cases}A_r+\xi(A_{\mathrm{max}}-A_r)&\Delta P\ge0\\A_r&\Delta P<0\end{cases},\quad B(A)=\frac{\rho}{2c_P}\left(\frac{10^{-6}}{10^{-4}A}\right)^2

Q=\begin{cases}0&A=0\text{ or }\Delta P=0\\\operatorname{sgn}(\Delta P)\frac{2|\Delta P|}{R+\sqrt{R^2+4B(A)|\Delta P|}}&\text{otherwise}\end{cases}

Vessels: storage and flow

Compliance stores blood and resistance opposes flow. Systemic and pulmonary vessels form a closed circuit in which volume and pressure are solved together.

\frac{dV_i}{dt}=\sum Q_{\mathrm{in},i}-\sum Q_{\mathrm{out},i},\qquad C_i(P_{\mathrm{tm}})=\frac{dV_i}{dP_{\mathrm{tm},i}}

Volume conservation and differential compliance. Pressure–volume relations, including venous vessels, need not be linear with constant C.

Equations and assumptions

Aortic and pulmonary roots use algebraic flow without local L. AoP and PAP are the Ao and PA node pressures, without a displayed ZcQ addition; SA is a downstream systemic-arterial compartment. Travelling/reflected waves, propagation delay and particular cuff/arterial-line sites are not simulated. Venous tone and TBV are not interchangeable controls: they affect filling and blood distribution.

$V_i,\;Q_{\mathrm{in}},\;Q_{\mathrm{out}}$: Compartment volume (mL) and inflow/outflow (mL/s)
$C_i,\;P_{\mathrm{tm},i}$: Pressure-dependent compliance (mL/mmHg) and transmural vascular pressure (mmHg)

Compartment pressure–volume laws

Pressure is mmHg, volume mL and flow mL/s. p=P−Pext is transmural pressure. Vu is volume at p=0; V−Vu is stressed volume. Listed Vs includes stiffness/compliance scaling: original Vs×0.65/1.42 for Ao/SA/Art, original Vs/1.42 for PA/PArt.

Venous-type compartments join collapsed Cc, open Co and distended Cd smoothly. S(z)=ln(1+exp z), σ(z)=1/(1+exp(−z)); ΔS(p;a,d)=S((p−a)/d)−S(−a/d). Tone u changes Vu=Vu,original−G u. The table includes baseline u=0.15.

The venous inverse saturates at −20/45 mmHg; arterial strain is bounded below by ln(0.05). These are numerical boundaries, not normal ranges; saturated results must not be extrapolated physiologically. Compliance readback is floored at 10⁻⁴ mL/mmHg without replacing V(p).

p=P_0\left(e^{\max[(V-V_u)/V_s,\ln0.05]}-1\right)\quad\text{(arterial)},\qquad p=(V-V_u)/C\quad\text{(linear)}

V(p)=V_u+C_cp+(C_o-C_c)d_o\Delta S(p;p_o,d_o)-(C_o-C_d)d_s\Delta S(p;p_s,d_s)

\frac{dV}{dp}=C_c+(C_o-C_c)\sigma\left(\frac{p-p_o}{d_o}\right)-(C_o-C_d)\sigma\left(\frac{p-p_s}{d_s}\right)

Link flow, collapse and external pressure

Non-valve links solve ΔP=RQ+BQ|Q|. Current baseline links have B=0 and L=0, normally Q=(Pup−Pdown)/R. Systemic/pulmonary multipliers apply only to the listed groups, not all venous resistances.

VC→RA and PCap→PVen use a waterfall and collapse-dependent resistance. M± are smooth max/min with ε=0.25 mmHg; χmin=0.08, w=1 mmHg, pc=0. χ uses raw endpoint pressures, not the already modified downstream pressure.

The node table distinguishes thoracic Pth from alveolar Palv exposure. fresp is Hz and amplitudes mmHg. Workbench PEEP is cmH₂O and is multiplied by 0.7355592401 for these mmHg equations. All respiratory inputs are zero at baseline.

M_\pm(a,b)=\frac{a+b\pm\sqrt{(a-b)^2+\epsilon^2}}2

\begin{aligned}P_{\mathrm{down,eff}}&=M_+(P_{\mathrm{down}},P_{\mathrm{ext}}+p_c),\\\chi&=\chi_{\mathrm{min}}+(1-\chi_{\mathrm{min}})\sigma\left(\frac{M_-(P_{\mathrm{up}}-P_{\mathrm{ext}},P_{\mathrm{down}}-P_{\mathrm{ext}})-p_c}{w}\right),\\Q&=\frac{P_{\mathrm{up}}-P_{\mathrm{down,eff}}}{R/\chi^2}.\end{aligned}

P_{\mathrm{th}}=P_{th,0}+0.20\,PEEP+A_{\mathrm{th}}\sin(2\pi f_{\mathrm{resp}}t),\quad P_{\mathrm{alv}}=PEEP+A_{\mathrm{alv}}\sin(2\pi f_{\mathrm{resp}}t)

Coronary and external-pressure coupling

Coronary inflow and venous return contribute to global volume balance. Myocardial compression modulates coronary flow, while pericardial and thoracic pressures load cavities and vessels externally.

Equations and assumptions

Coronary territories and layers include resistance/storage, intramyocardial pressure, collapse and autoregulation, some with provisional adult priors. Baseline has no mechanical support, valve regurgitation or respiratory oscillation. Fixed-control preload testing holds HR and tone fixed; it is not a bedside fluid challenge with autonomic and whole-body responses.

This branch is repeated for LAD/LCx/RCA, sharing only CV. Art/CV receive common cardiac external pressure; C1/C2 receive territory/layer-specific intramyocardial pressure.

Coronary storage and resistance

Each territory has one large-arterial Art reservoir, splitting into epi/endo paths with proximal C1 and distal C2 storage, then merging into common venous CV and RA. C1/C2 name compartments, not numerical compliances. Sixteen coronary volumes contribute to TBV without duplication.

All coronary storage uses v=V/Vref, m=4, n=2; Cref is tangent compliance at Vref. Collapse resistance uses a different hydraulic reference Vh. With a=0.67, x=min(1,max(0,V/Vh)), compute f below.

Branch flow is pressure difference divided by effective resistance. R1 uses tone θ and C1 collapse; Rm the geometric mean of C1/C2 collapse; R2 C2 collapse. Structural multipliers are 1 and focal stenosis additions 0 at baseline.

p=P_*(v^m-v^{-n}),\quad P_*=\frac{V_{\mathrm{ref}}}{C_{\mathrm{ref}}(m+n)},\quad V>0

f(V)=\{a+(1-a)x^2(3-2x)\}^{-2}

R_{1,eff}=R_1\theta f(V_{C1}),\quad R_{\mathrm{m},eff}=R_m\sqrt{f(V_{C1})f(V_{C2})},\quad R_{2,eff}=R_2f(V_{C2})

Intramyocardial pressure and beat-wise regulation

Art/CV external pressure is Pe=Pth+Pperi; C1/C2 use PIM. Weights wL/wS/wR, depth d and shortening gain K are tabulated. Septal depth s=d for LAD/LCx, 1−d for RCA. eMVC,w updates at the previous accepted mitral closure. F is the valve-section positive part with width 0.005.

The shortening reference updates when accepted MV flow changes from >1 to ≤1 mL/s, using that endpoint's strains for subsequent steps. This is not an imaging leaflet-contact time or an interpolated metric closure time.

Tone is held within a cycle and updated at completion. Q̄m is signed cycle-mean flow through Rm; Qtarget is the resting target. ℓ=ln θ, T cycle length, τa=25 s, θmin=4/45 and θmax=2. Baseline demand=1 and hyperemia=0 give the update below. It does not autonomously alter contractility from oxygen demand.

P_{\mathrm{IM}}=P_e+(w_Ld+w_Ss)P_{\mathrm{tm},L}+[w_Rd+w_S(1-s)]P_{\mathrm{tm},R}+K\sum_w w_wF_{0.005}(1-e^{e_w-e_{\mathrm{MVC},w}})

\bar Q_{\mathrm{m},k}=\frac{1}{T}\sum_{n\text{ in cycle }k}\Delta t_n Q_{m,n+1}

\ell_{k+1}=\operatorname{clip}_{\ln\theta_{\mathrm{min}},\ln\theta_{\mathrm{max}}}\left[\ell_k+\frac{T}{\tau_a}\ln\max\left(\frac{\bar Q_{\mathrm{m},k}}{Q_{\mathrm{target}}},0.05\right)\right]

Oxygen supply and consumption

Beat-mean flow, hemoglobin, inspired oxygen and prescribed consumption determine systemic oxygen balance separately from myocardial contraction.

\dot D_{\mathrm{O_2}}=10\,\mathrm{CO}\,C_{\mathrm{aO}_2},\qquad \dot V_{\mathrm{O_2}}=10\,\mathrm{CO}(C_{\mathrm{aO}_2}-C_{\mathrm{vO}_2})

Equations and assumptions

Alveolar gas, oxygen dissociation and shunt mixing determine arterial content. Fick balance gives required mixed-venous content; negative required content is infeasible. Local diffusion and full metabolic adaptation are not resolved.

$\mathrm{\mathrm{CO}}$: Beat-mean systemic flow (L/min)
$C_{\mathrm{aO}_2},\;C_{\mathrm{vO}_2}$: Arterial and mixed-venous oxygen content (mL O₂/dL)
$\dot D_{\mathrm{O}_2},\;\dot V_{\mathrm{O}_2}$: Oxygen delivery and prescribed consumption (mL O₂/min); 10 converts L to dL

Beat-mean oxygen balance

Oxygen transport is an algebraic beat-mean readout without feedback to blood volume or tension. P is O₂ pressure (mmHg), S saturation, Hb g/dL, C mL O₂/dL. Alveolar and end-capillary PO₂ are equated. PB is barometric pressure and R respiratory exchange ratio.

CO is L/min, VO₂ mL/min and s an oxygen-mixing shunt fraction, not an added hydraulic connection. End-capillary Cc, venous Cv and arterial Ca satisfy Fick balance and content mixing. Nonpositive flow or negative required content makes the evaluation unavailable.

P_{\mathrm{AO}_2}=F_{\mathrm{IO}_2}(P_B-47)-P_{\mathrm{aCO}_2}/R>0,\quad S(P)=\frac{P^{2.7}}{26.8^{2.7}+P^{2.7}}

C(P)=1.34\,Hb\,S(P)+0.0031P,\quad D=\frac{VO_2}{10\mathrm{CO}},\quad C_a=C_c-\frac{s}{1-s}D,\quad C_v=C_a-D

Analysis: relationships beyond one loop

A PV loop is the simulated beat trajectory. ESPVR, EDPVR, Starling/Guyton curves and PVA/PE are derived through separate loading protocols.

Equations and assumptions

Analyses branch without mutating the live simulation. ESPVR uses the reduced-preload limb through baseline; EDPVR and Starling retain high-volume conditions too. Estimates depend on the finite loading range, settlement and event definitions; linearity and load independence are not guaranteed. Unavailable analysis stays unassessed, not an exact-output placeholder.

Coupling and time integration

Blood occupies 31 compartments, constrained by fixed TBV (30 independent volume degrees of freedom). Each wall has six Land states, one viscous strain and two calcium states; four valves have opening states. Six coronary tones and event/load/previous-MVC memories are retained. Pressures, flows and internal geometry follow simultaneous algebraic constraints.

\begin{aligned}V_{n+1}-V_n&=\Delta t\,NQ_{n+1},\\z_{n+1}-z_n&=\Delta t\,f(z_{n+1},Ca_{n+1},\lambda_{n+1},\dot\lambda_{n+1}),\\0&=g(V_{n+1},z_{n+1},P_{n+1},Q_{n+1},v_{S,n+1},y_{n+1}).\end{aligned}

z collects Land/viscous/opening states; f is the documented evolution and g the constitutive, hydraulic and force-balance constraints. The coupled system uses backward Euler, with exact inter-event calcium propagation. Candidate volume, geometry, stress, pressure and flow must satisfy continuity together; vascular updates do not keep stale cavity pressures.

Nominal step is 2 ms, split at activation, calcium and control-window boundaries. Coronary tone updates from completed-cycle flow integrals. Invalid populations, nonfinite states, volume imbalance or failed nonlinear solves are not accepted. Another integrator may approximate the same continuous equations, but finite-step peaks/events and saved baseline parity require separate checks.

Scope and limitations

Compartments provide spatially lumped values, not local jets, travelling/reflected waves, intraventricular pressure gradients or regional ischemia and tissue heterogeneity. Matching labels do not ensure equivalence to measurements at different sites or by different methods.

Defined equations, numerical stability and physiological validity are separate claims. Each preset records admission conditions, evidence comparisons and tested control scope. This is an educational/research model, not a tool for patient diagnosis or treatment.

Assessment of this setting · 2 ms

Operating targets, construction checks and source comparisons have different roles. A reference warning is not a failed required condition.

CIOperating target2.95 L/min/m²Pass

Use signed native AoV NET output/BSA under a distinct metric ID. Interpretation as whole-circulation CO requires a settled unassisted nonshunting state; no positive-only flow substitution or distal-CMR-plane equivalence.

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 2.5 ≤ x ≤ 4 L/min/m²

resting adult RHC reference · published-reference-interval: 2.5 ≤ x ≤ 4 L/min/m²

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, CI

CVP / mean RAPOperating target3.08 mmHgPass

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 2 ≤ x ≤ 6 mmHg

resting adult RHC reference · published-reference-interval: 2 ≤ x ≤ 6 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, mean RAP

mean PAPOperating target17.89 mmHgPass

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 8 ≤ x ≤ 20 mmHg

resting adult RHC reference · published-reference-interval: 8 ≤ x ≤ 20 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, mean PAP

AoP maxConstruction / load guard111.3 mmHgPass

Model Ao root pressure is invasive-like. Source cSBP is cuff-calibrated noninvasive estimation; authors explicitly distinguish it from higher invasive intra-aortic SBP. Published P10/P90 are context, not 95% normal cutoffs or a model calibration target.

Keep a baseline away from low/high systemic load and high native end-filling pressure. Ao bounds are retained design choices, not derived from Herbert. LV native flow cessation may precede the pressure upstroke; <=16 is an approximate end-filling design ceiling, not validated catheter LVEDP equivalence or a lower normal limit.

Design interval, not a normal range: 90 ≤ x ≤ 140 mmHg

women 20-29 · published-10th-90th-percentiles: 80 ≤ x ≤ 110 mmHg

men 20-29 · published-10th-90th-percentiles: 92 ≤ x ≤ 115 mmHg

women 30-39 · published-10th-90th-percentiles: 84 ≤ x ≤ 119 mmHg

men 30-39 · published-10th-90th-percentiles: 88 ≤ x ≤ 120 mmHg

women 40-49 · published-10th-90th-percentiles: 87 ≤ x ≤ 123 mmHg

men 40-49 · published-10th-90th-percentiles: 90 ≤ x ≤ 123 mmHg

women 50-59 · published-10th-90th-percentiles: 93 ≤ x ≤ 127 mmHg

men 50-59 · published-10th-90th-percentiles: 96 ≤ x ≤ 126 mmHg

women 60-69 · published-10th-90th-percentiles: 97 ≤ x ≤ 129 mmHg

men 60-69 · published-10th-90th-percentiles: 97 ≤ x ≤ 128 mmHg

women 70+ · published-10th-90th-percentiles: 100 ≤ x ≤ 131 mmHg

men 70+ · published-10th-90th-percentiles: 99 ≤ x ≤ 130 mmHg

Establishing reference values for central blood pressure and its amplification in a general healthy population and according to cardiovascular risk factors · Methods, invasive versus cuff-calibrated pressure distinction

AoP minConstruction / load guard77.52 mmHgPass

Source assumes DBP consistency for calibration, but does not publish a central-DBP normal interval. Do not turn the brachial mean +/- SD into a verified Ao-node cutoff.

Design interval, not a normal range: 60 ≤ x ≤ 90 mmHg

Establishing reference values for central blood pressure and its amplification in a general healthy population and according to cardiovascular risk factors · Methods, DBP calibration; no central-DBP reference interval

LV end-filling PConstruction / load guard10.92 mmHgPass

Native inlet-closure LV pressure; not validated catheter LVEDP equivalence.

Design interval, not a normal range: x ≤ 16 mmHg

Recommendations for the Evaluation of Left Ventricular Diastolic Function by Echocardiography and for Heart Failure With Preserved Ejection Fraction Diagnosis: An Update From the American Society of Echocardiography · Section2, Table1 p539 (>16), Figure1 p540

AV ETReference258 msReference

Native accumulated positive AoV-flow duration, not mitral-leaflet color-TDI timing. Opening-to-closure interpretation requires the separate morphology/timing observer to establish one forward episode; this comparison does not enforce that condition. No HR correction or method equivalence inferred.

Source comparison, not an automatic rejection threshold; account for method differences.

Copenhagen pooled healthy adults; HR 63 +/- 10 · published-reference-interval: 248 ≤ x ≤ 336 ms

Age- and sex-based normal reference ranges of the cardiac time intervals: the Copenhagen City Heart Study · Methods, Cardiac time intervals; Table 2, pooled LVET 95% prediction interval

PV ETReference258 msReference

Native accumulated positive PV-flow duration; the separate morphology/timing observer must establish one forward episode. Doppler PA versus RVOT station, respiration and HR differ. No adult ET bounds inferred from acceleration time, tissue S-wave duration or LVET.

Source comparison, not an automatic rejection threshold; account for method differences.

Reference values for pulsed Doppler signals from the blood flow velocity on both sides of the pulmonary valve · Abstract: population and recording stations; no numeric adult ET interval verified

PAP maxReference26.17 mmHgReference

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Source comparison, not an automatic rejection threshold; account for method differences.

resting adult RHC reference · published-reference-interval: 15 ≤ x ≤ 30 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, systolic PAP; section 5.1.12.1

PAP minReference12.06 mmHgOutside reference

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Source comparison, not an automatic rejection threshold; account for method differences.

resting adult RHC reference · published-reference-interval: 4 ≤ x ≤ 12 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, diastolic PAP; section 5.1.12.1

mean LAP (PCWP surrogate)Reference8.4 mmHgReference

Observed quantity remains LA mean, not a wedge measurement or LVEDP. PAWP <=15 is the guideline clinical reference upper limit, not a healthy-cohort distribution or a verified model LA-to-PAWP transfer. No lower bound is invented.

Source comparison, not an automatic rejection threshold; account for method differences.

adult RHC PAWP clinical reference · clinical-upper-limit: x ≤ 15 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, PAWP; section 5.1.12.1

LV EDVIOperating target75.64 mL/m²Pass

Native valve-closure cavity blood volumes, BSA indexed; compared with anatomical bSSFP CMR excluding papillary/trabecular myocardium from the blood pool. No validated image segmentation or age/sex assignment; separate marginal ranges are not a joint distribution. EF is derived from EDV and ESV.

For this sex-unspecified generic baseline only, automatic eligibility requires all six valid anatomical CMR comparisons inside BOTH declared sex strata. Otherwise require demographic/method review, not automatic disease rejection. This conservative design intersection does not assign sex or claim joint population normality; preset/patient fitting must use its own profile.

Adopted intersection of both sex strata (design choice): 46 ≤ x ≤ 91 mL/m²

men, pooled adult ages · published-reference-interval: 46 ≤ x ≤ 104 mL/m²

women, pooled adult ages · published-reference-interval: 46 ≤ x ≤ 91 mL/m²

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 2, anatomical segmentation, sex-specific indexed volume/EF rows

LV ESVIOperating target33.47 mL/m²Pass

Adopted intersection of both sex strata (design choice): 11 ≤ x ≤ 34 mL/m²

men, pooled adult ages · published-reference-interval: 11 ≤ x ≤ 41 mL/m²

women, pooled adult ages · published-reference-interval: 11 ≤ x ≤ 34 mL/m²

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 2, anatomical segmentation, sex-specific indexed volume/EF rows

LVEFOperating target55.75 %Pass

Adopted intersection of both sex strata (design choice): 55 ≤ x ≤ 79 %

men, pooled adult ages · published-reference-interval: 53 ≤ x ≤ 79 %

women, pooled adult ages · published-reference-interval: 55 ≤ x ≤ 80 %

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 2, anatomical segmentation, sex-specific indexed volume/EF rows

RV EDVIOperating target74.09 mL/m²Pass

Adopted intersection of both sex strata (design choice): 49 ≤ x ≤ 99 mL/m²

men, pooled adult ages · published-reference-interval: 49 ≤ x ≤ 117 mL/m²

women, pooled adult ages · published-reference-interval: 47 ≤ x ≤ 99 mL/m²

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 8, anatomical segmentation, sex-specific indexed volume/EF rows

RV ESVIOperating target31.92 mL/m²Pass

Adopted intersection of both sex strata (design choice): 12 ≤ x ≤ 43 mL/m²

men, pooled adult ages · published-reference-interval: 12 ≤ x ≤ 56 mL/m²

women, pooled adult ages · published-reference-interval: 11 ≤ x ≤ 43 mL/m²

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 8, anatomical segmentation, sex-specific indexed volume/EF rows

RVEFOperating target56.92 %Pass

Adopted intersection of both sex strata (design choice): 49 ≤ x ≤ 77 %

men, pooled adult ages · published-reference-interval: 44 ≤ x ≤ 77 %

women, pooled adult ages · published-reference-interval: 49 ≤ x ≤ 77 %

Society for Cardiovascular Magnetic Resonance reference values (normal values) in cardiovascular magnetic resonance: 2025 update · Table 8, anatomical segmentation, sex-specific indexed volume/EF rows

SVIReference42.17 mL/m²Reference

Signed native AoV NET volume/BSA. CI = HR * SVI / 1000, so this is a coupled comparison, not a second independent fitting objective. The CI-conditional interval is reported separately; it is not this published SVI interval.

Source comparison, not an automatic rejection threshold; account for method differences.

resting adult RHC reference · published-reference-interval: 33 ≤ x ≤ 47 mL/m²

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, SVI; section 5.1.12.1

Period-1Numerical quality1 Pass

Whether the exact periodic classifier established the required period-one terminal state. Period-one settlement is numerical admissibility, not physiological normality or independent model validation.

Fail closed before interpreting any derived baseline measurement. No clinical population interval applies to an exact periodic-classifier contract.

Design interval, not a normal range: 1 ≤ x ≤ 1

LVP ringingConstruction / load guard1 Pass

Algorithmic peak count, total variation and episode count of pressure during the associated semilunar-valve forward-flow episode. Prominence and variation depend on accepted sampling and the thresholded forward-flow episode. Real arterial reflection can produce a systolic shoulder or secondary rise; the current lumped model has no explicit propagation/reflection mechanism, so that observation does not explain its numerical or coupled-mode ringing.

Retain the artifact guards motivated by Standard65-to-68 ringing, separately from unvalidated contour reference corridors. This is a current-model construction requirement, not a claim that every healthy human trace has one peak. No primary healthy-cohort distribution supports the exact peak-count or variation cutoffs. They remain transparent construction guards; no clinical normality follows from passing.

Design interval, not a normal range: 1 ≤ x ≤ 1

LVP contourReference20.46 %Reference

Central accepted-sample-index pressure range divided by full ejection pressure range, jointly with the first maximum's normalized sample index. The check reports peak index when that alone fails; it is not PV-loop curvature or a time-weighted shape measurement. Sample-index phase is not elapsed-time phase on nonuniform accepted steps. Pressure versus time and pressure versus volume have different curvature because ejection flow varies. Resolution, pressure loading and the selected episode affect these summaries; RV and LV cannot be assigned the same physiological contour solely by analogy.

Demote the frozen central-range and peak-index corridors to descriptive warnings: no matched normal distribution supports the limits, and a single compound failure previously conflated late pressure peak with flatness. Numerical thresholds are not widened to admit a candidate. No matched healthy LV or RV reference interval was identified for centralRangeFraction 0.08-0.35 or peakPhase01 0.2-0.8. These values remain visible warnings, not independent scientific validation or permission to accept unexplained ringing.

Historical corridor, context only: 8 ≤ x ≤ 35 %

Aortic input impedance in normal man: relationship to pressure wave forms · Author-institution abstract; ascending aortic pressure waveform groups
The quest for load-independent left ventricular chamber properties: exploring the normalized pressure-volume loop · Methods; Tables 1 and 4; Figures 4-5, PDF pp. 6-7

RVP ringingConstruction / load guard1 Pass

Design interval, not a normal range: 1 ≤ x ≤ 1

RVP contourReference18.98 %Reference

Historical corridor, context only: 8 ≤ x ≤ 35 %

Aortic input impedance in normal man: relationship to pressure wave forms · Author-institution abstract; ascending aortic pressure waveform groups
The quest for load-independent left ventricular chamber properties: exploring the normalized pressure-volume loop · Methods; Tables 1 and 4; Figures 4-5, PDF pp. 6-7

AV mean ΔPConstruction / load guard4.19 mmHgPass

Time-weighted mean and maximum raw LV-minus-Ao node pressure difference during native aortic forward flow. This hydraulic node gradient is neither a Doppler Bernoulli gradient nor a simultaneous catheter LV-to-recovered-aortic pressure difference; pressure recovery and spatial acceleration are not observed.

Constrain the non-stenotic baseline while retaining the model's explicit pressure-station limitation. No matching healthy-population source establishes the exact mean or peak cutoffs for these model pressure stations. Stenosis diagnostic thresholds would not validate them.

Design interval, not a normal range: 0 ≤ x ≤ 5 mmHg

AV peak ΔPConstruction / load guard7.82 mmHgPass

Design interval, not a normal range: 0 ≤ x ≤ 10 mmHg

LV +dP/dtReference2,564.3 mmHg/sOutside reference

Maximum and minimum accepted-step finite-difference derivative of absolute intracavitary LV pressure over the completed beat. Accepted-step bandwidth, pressure loading, HR, preload, medication and catheter filtering affect extrema. Published negative dP/dt is often a positive magnitude, whereas this model stores a signed minimum; this is not a transmural derivative.

Retain plausible contraction and relaxation rates while adjusting aortic ejection morphology. The positive corridor overlaps small normal-LV patient series, but the frozen negative corridor excludes their resting means. Neither series supplies a population-normal acceptance interval; keep both corridors as warnings rather than validated pass/fail physiology.

Historical corridor, context only: 1,200 ≤ x ≤ 2,500 mmHg/s

Contractile and relaxation reserve of the left ventricle. I. Normal left ventricle · Abstract; resting values before ergometer exercise
A clinical study of left ventricular relaxation · Methods p. 757; normal-control table p. 759; abstract

LV −dP/dtReference-1,538.9 mmHg/sOutside reference

Historical corridor, context only: -1,400 ≤ x ≤ -700 mmHg/s

Contractile and relaxation reserve of the left ventricle. I. Normal left ventricle · Abstract; resting values before ergometer exercise
A clinical study of left ventricular relaxation · Methods p. 757; normal-control table p. 759; abstract

MV E/AReference0.94 Reference

Ratio of peak native mitral forward volume flow in early and atrial filling windows anchored to the observed atrial-capture event. Volume-flow peaks are not leaflet-tip Doppler velocities; varying effective valve area can change their ratio. A complete post-capture inlet closure and identifiable E/A windows are required, and age, rhythm, HR and loading affect interpretation.

Retain the recorded corridor as context only: native volumetric E/A is not Doppler velocity E/A. Positive resolved waves remain required; no numeric widening or candidate-specific target change. The fixed corridor is provisional, not an age-specific Doppler normal interval. Better event anchoring does not remove the flow-versus-velocity observation mismatch.

Historical corridor, context only: 0.8 ≤ x ≤ 2

Recommendations for the Evaluation of Left Ventricular Diastolic Function by Echocardiography and for Heart Failure With Preserved Ejection Fraction Diagnosis: An Update From the American Society of Echocardiography · Tables 3 and 5 (p. 551)

LV ICTReference89.14 msOutside reference

ICT from exact MV closure to AoV zero-flow opening, IRT from exact AoV closure to MV zero-flow opening, and (ICT + IRT) divided by the matched aortic ejection duration. Closure landmarks and trace-interpolated openings must describe one complete beat. Tei is algebraically linked to ICT, IRT and the separately checked ejection time, so these are not independent constraints.

Demote method-mismatched ICT/IRT/Tei corridors to context; retain complete ordered hydraulic events, positivity and exact Tei arithmetic. ET remains an explicitly chosen baseline operating target, not proof of a population-normal timing pattern. The intervals remain provisional hydraulic timing targets. Matching some published endpoints does not establish a measurement-equivalence study or historical source derivation.

Historical corridor, context only: 20 ≤ x ≤ 70 ms

Age- and sex-based normal reference ranges of the cardiac time intervals: the Copenhagen City Heart Study · Table 2; Methods, Cardiac time intervals

LV IRTReference92 msReference

Historical corridor, context only: 59 ≤ x ≤ 134 ms

Age- and sex-based normal reference ranges of the cardiac time intervals: the Copenhagen City Heart Study · Table 2; Methods, Cardiac time intervals

LV TeiReference0.7 Outside reference

Historical corridor, context only: 0.29 ≤ x ≤ 0.65

Age- and sex-based normal reference ranges of the cardiac time intervals: the Copenhagen City Heart Study · Table 2; Methods, Cardiac time intervals

PV mean ΔPConstruction / load guard4.45 mmHgPass

Time-weighted mean and maximum raw RV-minus-PA node pressure difference while native PV flow is positive over the completed beat. The model's hydraulic gradient is not a Doppler Bernoulli or recovered catheter gradient; an explicit outlet node does not provide spatial velocity, pressure recovery or an invasive sensor model.

Record the existing Standard70 right-heart gradient sentinels without promoting them to left-objective groups or changing their numerical limits. No source establishes the exact mean and peak healthy cutoffs for these pressure stations. These remain non-stenotic construction guards, not clinical normal-gradient intervals.

Design interval, not a normal range: 0 ≤ x ≤ 5 mmHg

PV peak ΔPConstruction / load guard7.37 mmHgPass

Design interval, not a normal range: 0 ≤ x ≤ 10 mmHg

RV +dP/dtReference585.6 mmHg/sReference

Maximum and minimum accepted-step finite-difference derivative of absolute intracavitary RV pressure over the completed beat. These load- and bandwidth-dependent extrema are neither transmural pressure derivatives nor the mean RV-to-RA pressure-gradient rise inferred over a selected TR velocity interval. Signed negative extrema must not be confused with published pressure-fall magnitudes.

Reclassify the existing Standard70 RV derivative corridors as reference warnings without altering the recorded numerical bounds. No verified source establishes either frozen normal corridor. Sparse normal-PAP invasive data and a method-specific abnormal Doppler threshold justify contextual warnings, not hard healthy-population acceptance limits.

Historical corridor, context only: 300 ≤ x ≤ 1,000 mmHg/s

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RV dP/dt section, pp. 158-159
Effect of chronic pressure overload on the maximal rate of pressure fall of the right ventricle · Abstract; normal-PAP and pulmonary-hypertension groups

RV −dP/dtReference-371.9 mmHg/sReference

Historical corridor, context only: -700 ≤ x ≤ -150 mmHg/s

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RV dP/dt section, pp. 158-159
Effect of chronic pressure overload on the maximal rate of pressure fall of the right ventricle · Abstract; normal-PAP and pulmonary-hypertension groups

TV E/AReference1.1 Reference

Ratio of peak native tricuspid forward volume flow in early and atrial filling windows anchored to the observed atrial-capture event. Requires identifiable windows and an observed post-capture inlet closure. Volume-flow peaks do not equal Doppler velocities if effective valve area changes; respiration, HR, rhythm, age and loading remain relevant.

Demote the native tricuspid flow-ratio corridor to context because Doppler velocity and respiratory averaging are different observations; resolved positive filling waves remain required. The numerical corridor resembles a guideline Doppler interval but remains provisional for the native-flow observer; the source upper endpoint is exclusive whereas the frozen gate is inclusive.

Historical corridor, context only: 0.8 ≤ x ≤ 2

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · Table 1; RV diastolic function, pp. 168-169

RV ICTReference28 msReference

ICT from exact TV closure to PV zero-flow opening, IRT from exact PV closure to TV zero-flow opening, and (ICT + IRT) divided by the matched pulmonary ejection duration. One complete valve-event sequence is required. Tei is algebraically linked to ICT, IRT and the separately checked pulmonary ET; PW-Doppler, TDI and hydraulic timings are different observations.

Demote the unsupported hydraulic right ICT/IRT/Tei numeric corridors to context. Ordered complete valve events and algebraic consistency remain mandatory; a short RV isovolumic phase is not automatically pathological. No primary source was identified for the chosen joint ICT/IRT/Tei corridors. The source below supplies method-specific context, not equivalent normal intervals; all three remain provisional physiological targets.

Historical corridor, context only: 20 ≤ x ≤ 90 ms

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RV MPI, pp. 158-159; IVRT, pp. 168-169

RV IRTReference64 msReference

Historical corridor, context only: 30 ≤ x ≤ 120 ms

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RV MPI, pp. 158-159; IVRT, pp. 168-169

RV TeiReference0.36 Reference

Historical corridor, context only: 0.25 ≤ x ≤ 0.65

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RV MPI, pp. 158-159; IVRT, pp. 168-169

PAP peaksConstruction / load guard1 Pass

Algorithmic full-cycle PA pressure peak count, thresholded PV forward-episode count, primary-episode PV flow peak count and maximal post-episode PA rebound. The episode threshold and peak-prominence algorithm define these observations; the post-episode rebound is not an invasive dicrotic-notch metric. One peak does not characterize the entire pressure or flow contour.

Record Standard70 pulmonary ringing and re-ejection guards as construction-only waveform criteria. No primary normal distribution supports the exact one-peak, one-episode or rebound cutoffs. A qualitative Doppler flow shape must not be cited as proof of these pressure-waveform thresholds.

Design interval, not a normal range: 1 ≤ x ≤ 1

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RVOT Doppler flow, p. 158

PV flow episodesConstruction / load guard1 Pass

Design interval, not a normal range: 1 ≤ x ≤ 1

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RVOT Doppler flow, p. 158

PV flow peaksConstruction / load guard1 Pass

Design interval, not a normal range: 1 ≤ x ≤ 1

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RVOT Doppler flow, p. 158

PAP closure reboundConstruction / load guard0 mmHgPass

Design interval, not a normal range: 0 ≤ x ≤ 0.5 mmHg

Guidelines for the Echocardiographic Assessment of the Right Heart in Adults and Special Considerations in Pulmonary Hypertension: Recommendations from the American Society of Echocardiography · RVOT Doppler flow, p. 158

LV τ (Weiss)Reference32.04 msReference

Time-weighted exponential fit of intracavitary LVP from the minimum-dP/dt midpoint to next EDP +5 mmHg, before MVO; zero asymptote. Not interchangeable with free-asymptote Glantz τ.

τ and fit usability, residuals and window sensitivity are separate checks. Source pressure is not smoothed.

Prolongation context, not a normal distribution: x ≤ 48 ms

Recommendations for the Evaluation of Left Ventricular Diastolic Function by Echocardiography and for Heart Failure With Preserved Ejection Fraction Diagnosis: An Update From the American Society of Echocardiography · Invasive relaxation reference; Weiss asymptote/method must be matched

Assessment of this setting · 1 ms

Operating targets, construction checks and source comparisons have different roles. A reference warning is not a failed required condition.

CIOperating target2.95 L/min/m²Pass

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 2.5 ≤ x ≤ 4 L/min/m²

resting adult RHC reference · published-reference-interval: 2.5 ≤ x ≤ 4 L/min/m²

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, CI

CVP / mean RAPOperating target3.08 mmHgPass

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 2 ≤ x ≤ 6 mmHg

resting adult RHC reference · published-reference-interval: 2 ≤ x ≤ 6 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, mean RAP

mean PAPOperating target17.87 mmHgPass

Absolute lumped pressure, no respiratory cycle; compare with supine end-expiratory catheter values at the specified zero. No catheter transfer function or peripheral waveform is simulated.

Resting unassisted nonshunting sinus research construction, BSA1.9, HR60 or70, zero intrathoracic reference. Scientific eligibility for exact-model promotion, not public mint or clinical normality.

Adopted, source-informed: 8 ≤ x ≤ 20 mmHg

resting adult RHC reference · published-reference-interval: 8 ≤ x ≤ 20 mmHg

2022 ESC/ERS Guidelines for the diagnosis and treatment of pulmonary hypertension · Table 11, mean PAP

AoP maxConstruction / load guard111.4 mmHgPass