Model & case documentation

HFrEF · chronic LV-dilated case

One resting construction with a larger left ventricle and reduced ejected fraction.

A case for research and education, not for patient diagnosis or treatment.

Case overview

HFrEF identifies heart failure with reduced EF; it does not itself specify chronicity, dilation or etiology. This case selects a chronic LV-dilated phenotype, not acute infarction, regional ischemia or the time course of remodeling.

2022 AHA/ACC/HFSA Guideline for the Management of Heart FailureSection 2.2, Table 4
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

The intended lesson is greater residual LV blood, retained resting forward output and elevated left atrial pressure. RV behavior and filling need not match a universal HFrEF pattern. Intentions and observed results are separated below.

Circuit and pressure interpretation

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

The link tables define incidence matrix N. 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.

Model mechanisms

Myocardium generates tension; geometry and equilibrium determine pressure and blood flow. Waveforms are not prescribed. Shared equations and this case's effective coefficients follow.

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.

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.

\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.

$x_r,\;x_d$: 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)

Adopted calcium source
Wall	$\tau_r$ (s)	$\tau_d$ (s)	$\mathrm{Ca}_0$ (µM)	g (µM)
Left atrium (LA)	0.0125	0.3	0.0631545706813	0.604696429446
LV free wall (LVFW)	0.11853637422	0.131707082467	0.120706159272	12.0780349398
Ventricular septum (SEP)	0.11853637422	0.131707082467	0.120706159272	12.0780349398
RV free wall (RVFW)	0.11853637422	0.131707082467	0.120706159272	12.0780349398
Right atrium (RA)	0.0125	0.3	0.0631545706813	0.604696429446

Interval-strength coefficients and reference state
Symbol / meaning	Adopted value
τrec (s)	0.5
β	0.8
r	0.5
h	0.2
γ	0.598072916413
$T_{\mathrm{ref}}$ (s)	0.857142857143
aref	0.819907687852
Lref	1.52456187266

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.

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=\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.

$T_a,\;T_{\mathrm{ref}}$: 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}

These are model reference coefficients. Active multipliers apply to each wall's Tref; effective values are also tabulated in the passive/viscoelastic section.

Independent Land parameters: ventricles and atria
Symbol / meaning	Ventricular walls	Atria	Unit
$k_{\mathrm{TRPN}}$ Troponin rate	100	100	s⁻¹
$n_{\mathrm{TRPN}}$ Ca binding exponent	2	2	1
$\mathrm{Ca}_{50,\mathrm{ref}}$ Ca50,ref	0.6	0.86	µM
$k_u$ Thin-filament activation rate	1000	1000	s⁻¹
$n_{\mathrm{Tm}}$ Thin-filament exponent	5	5	1
$\theta$ reference bound fraction	0.35	0.35	1
$k_{\mathrm{uw}}$ Unbound→weak	104	182	s⁻¹
$k_{\mathrm{ws}}$ Weak→strong	4.8	36	s⁻¹
$r_w$ reference weak fraction	0.5	0.5	1
$r_s$ reference strong fraction	0.25	0.25	1
$\gamma_s$ strong distortion detachment	8.5	8.5	s⁻¹
$\gamma_w$ weak distortion detachment	615	615	s⁻¹
$\phi$ distortion relaxation scale	2.23	2.23	1
$A_{\mathrm{eff}}$ velocity sensitivity	26.5	25	1
$\beta_0$ force-length dependence	2.3	2.3	1
$\beta_1$ affinity-length dependence	-1.2	-2.4	µM
$T_{\mathrm{ref}}$ tension scale	238816.546281	11661.1510106	Pa
$T$ Source temperature (not a variable kinetic factor)	310.15	310.15	K

Derived Land coefficients
	Ventricle	Atrium
$k_b$	14.0058333333	14.0058333333
$A_w$	10.6	10
$A_s$	10.6	10
$k_{\mathrm{wu}}$	99.2	146
$k_{\mathrm{su}}$	7.2	54
$c_w$	231.92	405.86
$c_s$	16.056	120.42

Ventricular added exit: kmax=60 s⁻¹, p=16; atria have no added exit. Baseline settings compare adopted values with source values and calibration provenance. These are not independently measured constants from one subject.

Land et al. 2017

Passive mechanics and deformation history

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

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.

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.

$\lambda_g,\;e,\;\alpha$: 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})

Reference passive coefficients (wall multipliers applied separately)
Symbol / meaning	Adopted value
$K_0$ (Pa)	1200
$K_t$ (Pa)	22345.9396953
a	18
$K_c$ (Pa)	0
δ	0.001
$C_1$ (Pa)	1650
$C_2$ (Pa)	0
$C_3$ (Pa)	15
$C_4$	13.37

Wall multipliers and effective material coefficients
Wall	Active scale	Passive scale	$T_{\mathrm{ref}}$ (Pa)	$E_v$ (Pa)	$\tau_v$ (s)
Left atrium (LA)	1	1	11661.1510106	5947	0.3
LV free wall (LVFW)	0.35	1.04	83585.7911985	19307.6	0.3
Ventricular septum (SEP)	0.35	1.04	83585.7911985	19307.6	0.3
RV free wall (RVFW)	1	1.04	238816.546281	19307.6	0.3
Right atrium (RA)	1	1	11661.1510106	5947	0.3

The 0.3 s branch is a reduction informed by healthy ovine RV Prony data, not identified in every human wall. Acta Biomaterialia 2022

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.

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=\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.

$\delta W,\;V_{\mathrm{m},w}$: 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}

Ventricular wall geometry constants
Wall	M (m³)	$A_{\mathrm{ref}}$ (m²)
LV free wall (LVFW)	0.0000838442958008	0.0107575058279
Ventricular septum (SEP)	0.0000447169577604	0.00455984429148
RV free wall (RVFW)	0.0000360873694207	0.012911294586

Atrial geometry constants
Wall	M (mL)	$V_{\mathrm{ref}}$ (mL)
Left atrium (LA)	25.9829059829	22.8900425619
Right atrium (RA)	23.3998100665	32.5968711181

Pericardial constants
Symbol / meaning	Adopted value
$V_0$ (m³)	0.000600126542735
$P_*$ (Pa)	500
k	8
$P_{\mathrm{offset}}$ (Pa)	0
$V_{\mathrm{fluid}}$ (m³)	0

BSA 1.9 m²; density 1053 kg/m³. LV free-wall/septal reference area and tissue volume are fixed at 1.15 and 1.25 times reference. RV free wall and atria are unchanged. This is neither evolving remodeling nor measured patient anatomy. Actual tissue occupies the unchanged pericardial bag; capacity, stiffness and fluid are not retuned.

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.

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=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.

$\Delta P,\;Q$: 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}

Four-valve coefficients (L=0)
Valve	R (mmHg·s/mL)	$A_{\mathrm{max}}$ (cm²)	$k_o$ (mmHg⁻¹)	d (mmHg)	$\tau_{\mathrm{open}}$ (s)	$\tau_{\mathrm{close}}$ (s)
MV	0.0027	5.5	2	0.6	0.024	0.016
AV	0.0015	3.5	3	0	0.006	0.008
TV	0.0035	8	2	0	0.018	0.01
PV	0.005	4	2	0	0.01	0.006

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.

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.

\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.

$V_i,\;Q_{\mathrm{in}},\;Q_{\mathrm{out}}$: 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)

Effective arterial and linear coefficients
Compartment	$P_0$ (mmHg)	$V_s$ (mL)	C (mL/mmHg)
Proximal aorta (Ao)	50	68.661971831	—
Systemic arteries (SA)	50	183.098591549	—
Systemic resistance-vessel side (Art)	45	54.9295774648	—
Systemic capillary bed (Cap)	—	—	15
Proximal pulmonary artery (PA)	20	42.2535211268	—
Pulmonary resistance-vessel side (PArt)	20	63.3802816901	—

Effective venous-type coefficients
Compartment	$V_u$ (mL)	$C_c / C_o / C_d$ (mL/mmHg)	$p_o / p_s$ (mmHg)	$d_o / d_s$ (mmHg)	Vu,元 / G (mL)
Systemic veins (SV)	1538.409	15 / 130 / 35	-2 / 16	1.5 / 4	1653.909 / 770
Vena cava (VC)	150.091	5 / 45 / 12	-1 / 12	1 / 3	169.591 / 130
Pulmonary capillary bed (PCap)	105	1 / 2 / 1	0 / 14	1 / 3	105 / 0
Pulmonary venular side (PVen)	160	1.2 / 3 / 1.2	-1 / 14	1 / 3	160 / 0
Pulmonary vein / LA inlet (PVein)	215	1.5 / 4 / 1.5	-1 / 14	1 / 3	215 / 0

Main-circuit links and effective non-valve resistance
Link: positive direction	R (mmHg·s/mL)	Multiplier group	Additional law
Left atrium (LA) → Left ventricle (LV)	See valve table	—	Valve opening / direction
Left ventricle (LV) → Proximal aorta (Ao)	See valve table	—	Valve opening / direction
Right atrium (RA) → Right ventricle (RV)	See valve table	—	Valve opening / direction
Right ventricle (RV) → Proximal pulmonary artery (PA)	See valve table	—	Valve opening / direction
Proximal aorta (Ao) → Systemic arteries (SA)	0.05581056	Systemic ×1.2	—
Systemic arteries (SA) → Systemic resistance-vessel side (Art)	0.089296896	Systemic ×1.2	—
Systemic resistance-vessel side (Art) → Systemic capillary bed (Cap)	0.72553728	Systemic ×1.2	—
Systemic capillary bed (Cap) → Systemic veins (SV)	0.15	—	—
Systemic veins (SV) → Vena cava (VC)	0.05	—	—
Vena cava (VC) → Right atrium (RA)	0.04	—	waterfall + χ (Pth)
Proximal pulmonary artery (PA) → Pulmonary resistance-vessel side (PArt)	0.00625	Pulmonary ×0.625	—
Pulmonary resistance-vessel side (PArt) → Pulmonary capillary bed (PCap)	0.025	Pulmonary ×0.625	—
Pulmonary capillary bed (PCap) → Pulmonary venular side (PVen)	0.03	—	waterfall + χ (Palv)
Pulmonary venular side (PVen) → Pulmonary vein / LA inlet (PVein)	0.01	—	—
Pulmonary vein / LA inlet (PVein) → Left atrium (LA)	0.03025	—	—

Vascular coefficients and resistance partition are lumped construction/calibration values, not direct measurements at specified human vascular sites. Do not apply baseline multipliers a second time to the listed R.

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]

Coronary compartment PV constants
Compartment	$V_{\mathrm{ref}}$ (mL)	$C_{\mathrm{ref}}$ (mL/mmHg)	$P_*$ (mmHg)	$V_h$ (mL)
LAD large arterial storage (Art)	1.43391058398	0.00153221919053	155.973178516	1.53221919053
LAD epi C1	0.48107452286	0.0030959673913	25.8979107366	0.496293624668
LAD epi C2	0.276429025657	0.0928790217391	0.496037785643	0.496293624668
LAD endo C1	0.48107452286	0.0030959673913	25.8979107366	0.496293624668
LAD endo C2	0.276429025657	0.0928790217391	0.496037785643	0.496293624668
LCx large arterial storage (Art)	0.943218749158	0.00100788562723	155.973178516	1.00788562723
LCx epi C1	0.316448260285	0.00203651086957	25.8979107366	0.32645930444
LCx epi C2	0.181833541592	0.061095326087	0.496037785643	0.32645930444
LCx endo C1	0.316448260285	0.00203651086957	25.8979107366	0.32645930444
LCx endo C2	0.181833541592	0.061095326087	0.496037785643	0.32645930444
RCA large arterial storage (Art)	1.01084431002	0.00108014758225	155.973178516	1.08014758225
RCA epi C1	0.339136519084	0.00218252173913	25.8979107366	0.349865320892
RCA epi C2	0.194870385107	0.0654756521739	0.496037785643	0.349865320892
RCA endo C1	0.339136519084	0.00218252173913	25.8979107366	0.349865320892
RCA endo C2	0.194870385107	0.0654756521739	0.496037785643	0.349865320892
Common coronary venous storage (CV)	4.12994828733	0.098037492	7.0210355295	4.9018746

All coronary links and reference resistance (before tone / collapse)
Link: positive direction	R (mmHg·s/mL)
Proximal aorta (Ao) → LAD large arterial storage (Art)	20.749926765
LAD large arterial storage (Art) → LAD epi C1	58.0014068184
LAD epi C1 → LAD epi C2	35.7263939068
LAD epi C2 → Common coronary venous storage (CV)	11.9087979689
LAD large arterial storage (Art) → LAD endo C1	42.2821588661
LAD endo C1 → LAD endo C2	32.1859404566
LAD endo C2 → Common coronary venous storage (CV)	10.7286468189
Proximal aorta (Ao) → LCx large arterial storage (Art)	31.1248901474
LCx large arterial storage (Art) → LCx epi C1	94.8761406353
LCx epi C1 → LCx epi C2	53.5895908603
LCx epi C2 → Common coronary venous storage (CV)	17.8631969534
LCx large arterial storage (Art) → LCx endo C1	64.2296097205
LCx endo C1 → LCx endo C2	48.2789106849
LCx endo C2 → Common coronary venous storage (CV)	16.0929702283
Proximal aorta (Ao) → RCA large arterial storage (Art)	29.049897471
RCA large arterial storage (Art) → RCA epi C1	89.1455943902
RCA epi C1 → RCA epi C2	50.0169514696
RCA epi C2 → Common coronary venous storage (CV)	16.6723171565
RCA large arterial storage (Art) → RCA endo C1	74.0663831102
RCA endo C1 → RCA endo C2	45.0603166393
RCA endo C2 → Common coronary venous storage (CV)	15.0201055464
Common coronary venous storage (CV) → Right atrium (RA)	2.44019138756

Coronary coupling weights and shortening gain
Territory	$w_L / w_S / w_R$	K (mmHg)	Rest Qtarget: epi / endo (mL/s)
LAD	0.75 / 0.25 / 0	81.1546499712	0.485355450237 / 0.538744549763
LCx	1 / 0 / 0	79.6541200248	0.323570300158 / 0.359163033175
RCA	0 / 0.2 / 0.8	84.2078441934	0.346682464455 / 0.384817535545

Layer depths are 0.25 (epi) and 0.75 (endo). Volume allocation starts from porcine morphometry and C1/C2 from canine effective compliance, with selected adjustments from beating-boundary checks. This is not a uniquely identified human circuit.

Kassab et al. 1994 · Spaan et al. 2000

Oxygen supply and consumption

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

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.

\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})

$\mathrm{\mathrm{CO}}$: 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.

Coupled equations and saved initial state

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.

Saved setting's initial conditions

The settled launch state below is at t₀=133.716 s。If shifting time to zero, shift event and control times equally. Keeping volumes but zeroing material/Ca states is not equivalent. Display uses 12 significant digits; CSV retains full stored precision.

Initial blood volume (mL)
Compartment	$V(t_0)$
Proximal aorta (Ao)	60.7974182576
Systemic resistance-vessel side (Art)	49.5877122232
Systemic capillary bed (Cap)	313.053642764
Left atrium (LA)	57.3167860333
Left ventricle (LV)	213.427061319
Proximal pulmonary artery (PA)	27.1194113675
Pulmonary resistance-vessel side (PArt)	40.731619674
Pulmonary capillary bed (PCap)	136.094800018
Pulmonary vein / LA inlet (PVein)	278.462837908
Pulmonary venular side (PVen)	207.053540772
Right atrium (RA)	30.7552157612
Right ventricle (RV)	125.003400925
Systemic arteries (SA)	161.089364255
Systemic veins (SV)	2782.872251
Vena cava (VC)	439.462971682
Common coronary venous storage (CV)	4.55465316263
LAD large arterial storage (Art)	1.50693981091
LAD endo C1	0.539500512265
LAD epi C1	0.539906216416
LAD endo C2	0.1749080481
LAD epi C2	0.347495239095
LCx large arterial storage (Art)	0.991648299263
LCx endo C1	0.355230192399
LCx epi C1	0.353524530672
LCx endo C2	0.118505343692
LCx epi C2	0.274266149514
RCA large arterial storage (Art)	1.06060659051
RCA endo C1	0.381837588606
RCA epi C1	0.379841461835
RCA endo C2	0.29507311099
RCA epi C2	0.298029779845

Initial Land states (dimensionless)
Wall	c	b	W	S	$\zeta_w$	$\zeta_s$
Left atrium (LA)	0.630118752156	0.0523595807967	0.339386559246	0.226111867685	-0.0376362413446	-0.132825098168
LV free wall (LVFW)	0.0957846106489	0.995452731342	0.00172103618242	0.000873637958577	0.00868286045424	0.198264518171
Ventricular septum (SEP)	0.08570938425	0.996535668194	0.0013034013353	0.000680225512146	0.0090911069066	0.19738001841
RV free wall (RVFW)	0.0607390609521	0.998434912027	0.000592784245004	0.000342902432599	0.00189767943141	0.126710084624
Right atrium (RA)	0.236055565802	0.855892199458	0.0498238625624	0.0458255926901	0.0128329189468	0.0397999847835

Initial strain / calcium states (dimensionless)
Wall	$e(t_0)$	$\alpha(t_0)$	$x_r(t_0)$	$x_d(t_0)$
Left atrium (LA)	0.224222423872	0.328669449654	0.0000590490054998	0.7071118549
LV free wall (LVFW)	0.148723187106	0.109740931974	0.986355850024	0.988542892627
Ventricular septum (SEP)	0.130136120695	0.0847944683576	0.986355850024	0.988542892627
RV free wall (RVFW)	0.0616866713113	0.011184412826	0.986355850024	0.988542892627
Right atrium (RA)	-0.0141547882043	0.0427606904407	0.0000590490054998	0.7071118549

Initial opening / geometry
Quantity	Value
AV ξ	8.3941454043e-26
MV ξ	0.588170415645
PV ξ	5.4145877253e-35
TV ξ	0.0286221508103
$v_S$ (m³)	0.0000564792328315
y (m)	0.0369639529204

Coronary tone and partial-window integrals
Territory / layer	$\theta(t_0)$	$\int Q_m\,dt$ (mL)
LAD epi	0.792442538489	0.000873481607701
LAD endo	0.693930902783	0.000876155822331
LCx epi	0.827693238772	0.000539011765354
LCx endo	0.691037509045	0.000591086208632
RCA epi	0.787718542087	0.000591683288816
RCA endo	0.792238438707	0.0006995759526

Retained timing and discrete memory
Quantity	Value
Next atrial activation (s)	134.439428571
Last ventricular activation (s)	133.702285714
$L(t_0)$	1.52456187266
Coronary window start (s)	133.714285714
Elapsed coronary window (s)	0.00171428571431
$e_{\mathrm{MVC},\mathrm{LVFW}}$	0.151058531376
$e_{\mathrm{MVC},\mathrm{RVFW}}$	0.062000493637
$e_{\mathrm{MVC},\mathrm{SEP}}$	0.132741561282

There are no pending calcium/conduction events at this instant. Next calcium deposits follow the next atrial activation by 12 ms (atria) or 132 ms (ventricles). Coronary windows are T=60/70 s from original time zero. Recompute several cycles to verify consistency; this table does not establish uniqueness or stability in every scenario.

Save parameter and initial-state tables (CSV)

Settings

HR: 70 bpm
TBV: 4935 mL
BSA: 1.9 m²
LV tissue mass: 135.4 g

LV free-wall/septal active tension is 0.35/0.35 and systemic resistance 1.2 times reference. One is a model reference, not an absolute unit of normal contractility. This tissue mass does not establish pathological hypertrophy. Calcium, crossbridge kinetics and passive coefficients are unchanged.

Complete inputs and material provenance

Saved primitive inputs
Parameter	Value	Unit
Heart rate (HR)	70	bpm
Total blood volume (TBV)	4,935	mL
Systemic vascular resistance (SVR)	1.2	1
Common ventricular active tension	See individual walls	1
Venous tone	0.15	1
PEEP	0	cmH2O
Pulmonary vascular resistance (PVR)	0.625	1
Arterial stiffness	1.42	1
LA active tension	1	1
LV free wall active tension	0.35	1
Ventricular septum active tension	0.35	1
RV free wall active tension	1	1
RA active tension	1	1
LA passive stiffness	1	1
LV free wall passive stiffness	1.04	1
Ventricular septum passive stiffness	1.04	1
RV free wall passive stiffness	1.04	1
RA passive stiffness	1	1
Mitral valve (MV) maximum EOA	5.5	cm2
Mitral valve (MV) reverse EROA	0	cm2
Aortic valve (AoV) maximum EOA	3.5	cm2
Aortic valve (AoV) reverse EROA	0	cm2
Tricuspid valve (TV) maximum EOA	8	cm2
Tricuspid valve (TV) reverse EROA	0	cm2
Pulmonary valve (PV) maximum EOA	4	cm2
Pulmonary valve (PV) reverse EROA	0	cm2
Hemoglobin (Hb)	15	g/dL
Inspired oxygen fraction (FiO₂)	0.21	1
Arterial PCO₂ (PaCO₂)	40	mmHg
Respiratory exchange ratio (RER)	0.8	1
Barometric pressure	760	mmHg
True shunt fraction	0.02	1
Target oxygen consumption (VO₂)	250	mL O2/min
pericardium.reference-capacity-scale	1	1
pericardium.pressure-scale	1	1
pericardium.exponential-stiffness-scale	1	1
pericardium.prescribed-fluid-volume-ml	0	mL
coronary.focal-diameter-loss-fraction.LAD	0	1
coronary.focal-diameter-loss-fraction.LCx	0	1
coronary.focal-diameter-loss-fraction.RCA	0	1
coronary.structural-r1-resistance-scale.LAD.subepicardial	1	1
coronary.structural-r1-resistance-scale.LAD.subendocardial	1	1
coronary.structural-r1-resistance-scale.LCx.subepicardial	1	1
coronary.structural-r1-resistance-scale.LCx.subendocardial	1	1
coronary.structural-r1-resistance-scale.RCA.subepicardial	1	1
coronary.structural-r1-resistance-scale.RCA.subendocardial	1	1
coronary.structural-rm-resistance-scale.LAD.subepicardial	1	1
coronary.structural-rm-resistance-scale.LAD.subendocardial	1	1
coronary.structural-rm-resistance-scale.LCx.subepicardial	1	1
coronary.structural-rm-resistance-scale.LCx.subendocardial	1	1
coronary.structural-rm-resistance-scale.RCA.subepicardial	1	1
coronary.structural-rm-resistance-scale.RCA.subendocardial	1	1
LV contractility	0.35	1

Grouped contractility sets the individual wall inputs; it does not multiply them again. Cases retain primitive inputs, fixed anatomy and state, not only knob positions. Switching geometry loads the target case's own state.

Land et al. 2017

Material calibration before case tension scaling
Parameter	Source	Model reference
kTRPN	0.1 1/ms	100 1/s
nTRPN	2 dimensionless	2 dimensionless
CaT50Ref	0.805 uM	0.6 uM
ku	1 1/ms	1000 1/s
nTm	5 dimensionless	5 dimensionless
TRPN50	0.35 dimensionless	0.35 dimensionless
kuw	0.182 1/ms	104 1/s
kws	0.012 1/ms	4.8 1/s
rw	0.5 dimensionless	0.5 dimensionless
rs	0.25 dimensionless	0.25 dimensionless
gammaS	0.0085 1/ms	8.5 1/s
gammaW	0.615 1/ms	615 1/s
phi	2.23 dimensionless	2.23 dimensionless
Aeff	25 dimensionless	26.5 dimensionless
beta0	2.3 dimensionless	2.3 dimensionless
beta1	-2.4 uM	-1.2 uM
Tref	120 kPa	238816.54628141236 Pa
temperatureK	37 degC	310.15 K

Fixed pericardial and coronary construction

Added tissue occupies the same pericardial bag, without automatic enlargement. Coronary reference storage/resistance and absolute oxygen demand are not rescaled to current mass. Adequate perfusion or oxygen supply in the enlarged heart is not established.

Case assessment

Rest comparisons and launch states were calculated in this exact model. PVA, passive curves, control diagnostics and slow-tail observations refer to prior experiments with the same construction, retaining their original model identity and provenance.

2 ms · launch grid

Both grids meet required conditions and case preferences. This qualifies a selected operating point, not a diagnosis or clinical validation. Population statistics and model selection intervals are distinct.

Saved baseline and case comparison
Metric	Comparator baseline2 ms, fixed	This case2 ms	Unit
LVEF	55.8	28.5	%
LV EDVI	75.6	113.4	mL/m²
LV ESVI	33.5	81.1	mL/m²
CI	2.95	2.26	L/min/m²
SVI	42.2	32.3	mL/m²
Ao node mean	93.6	81.0	mmHg
mean LAP	8.4	16.6	mmHg
mean RAP	3.1	4.3	mmHg
PA node mean	17.9	23.9	mmHg
RVEF	56.9	49.0	%
RV EDVI	74.1	65.8	mL/m²
RV ESVI	31.9	33.5	mL/m²
LV end-filling P	10.9	25.5	mmHg
LV end-filling Ptm	10.9	22.9	mmHg
LV +dP/dt	2,564	1,113	mmHg/s
LV −dP/dt	-1,539	-824	mmHg/s
ICT	89.1	48.0	ms
ET	258.0	280.0	ms
IRT	92.0	128.0	ms
Tei	0.70	0.63
E/A (flow)	0.94	0.86
τ Weiss	32.0	51.3	ms
τ Glantz	52.4	—	ms

The comparator baseline remains its saved 2 ms record; selecting the 1 ms case does not recompute it. Unrounded values are included in the JSON export.

This is not a load-matched comparison. Reduced EF can coexist with forward output; a smaller Tei than baseline does not demonstrate better contractility. Mean LA, end-filling cavity and transmural pressures differ. E/A uses volumetric flow, not Doppler velocity.

Intervals and rationale

LVEF · Required28.46 / 20–40 % · Met

Reduced EF defines the chosen phenotype. The 20% lower bound selects severity; it is not a diagnostic lower limit.

Method: native-valve-closure-volumes

2022 AHA/ACC/HFSA Guideline for the Management of Heart FailureSection 2.2, Table 4
Association between systolic ejection time and outcomes in heart failure by ejection fractionMethods: Echocardiographic measurements; Table 2, HFrEF

LV EDVI · Required113.4 / 90–180 mL/m² · Met

Selects a chronic LV-dilated volume range, not a requirement for all HFrEF or an imaging diagnostic boundary.

Method: native-valve-closure-volumes

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

CI · Required2.258 / 1.8–3.5 L/min/m² · Met

Allows lower or retained output; reduced EF is not equated with extreme low output.

Method: exact-beat-aortic-valve-net-flow

Cardiac output and cardiac index measured with cardiovascular magnetic resonance in healthy subjects, elite athletes and patients with congestive heart failureMethods: patient population / CMR; Results: CHF patients
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean LAP · Required16.61 / 3–25 mmHg · Met

Retains the broad screen against extreme filling. Elevated pressure is a case preference, not a required condition.

Method: exact-beat-mean-left-atrial-pressure

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean RAP · Required4.261 / 0–12 mmHg · Met

Selects loading for an LV-dominant case; it proves neither normal RVEF nor absence of RV involvement.

Method: exact-beat-mean-right-atrial-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

Ao node mean · Required81.04 / 60–110 mmHg · Met

Avoids extreme arterial loading. Site and method differences prevent treating it as clinical MAP diagnosis.

Method: exact-beat-mean-aortic-node-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest
Prognostic value of myocardial strain and late gadolinium enhancement on cardiovascular magnetic resonance imaging in patients with idiopathic dilated cardiomyopathy with moderate to severely reduced ejection fractionMethods; Tables 1–2

LVEF · Case preference28.46 / 25–35 % · Met

Meet EF jointly with the other primary preferences, without ranking EF above all other objectives.

Method: native-valve-closure-volumes

Association between systolic ejection time and outcomes in heart failure by ejection fractionMethods: Echocardiographic measurements; Table 2, HFrEF
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

LV EDVI · Case preference113.4 / 110–150 mL/m² · Met

Selects dilation and a large residual volume. ESVI is not added as a duplicate objective.

Method: native-valve-closure-volumes

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

CI · Case preference2.258 / 2.2–2.8 L/min/m² · Met

Selects one case with retained resting net output, not a disease-specific normal CI interval.

Method: exact-beat-aortic-valve-net-flow

Cardiac output and cardiac index measured with cardiovascular magnetic resonance in healthy subjects, elite athletes and patients with congestive heart failureMethods: patient population / CMR; Results: CHF patients
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

Ao node mean · Case preference81.04 / 80–100 mmHg · Met

A resting arterial-pressure preference, strongly dependent on output and vascular load. Missing it does not invalidate HFrEF; the lower bound was not lowered to admit a known 77-mmHg candidate.

Method: exact-beat-mean-aortic-node-pressure

Prognostic value of myocardial strain and late gadolinium enhancement on cardiovascular magnetic resonance imaging in patients with idiopathic dilated cardiomyopathy with moderate to severely reduced ejection fractionMethods; Tables 1–2
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean LAP · Case preference16.61 / 12–20 mmHg · Met

Intentionally selects higher filling than the mean PAWP of 10–12 mmHg in treated cohorts. Lavine supports the existence of such cases, not a frequency weight. Pressure alone does not diagnose edema or clinical congestion; native LV end-filling cavity and transmural pressures are also reported. Higher is not rewarded beyond the interval.

Method: exact-beat-mean-left-atrial-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion
Importance of the left ventricular filling pressure on diastolic filling in idiopathic dilated cardiomyopathyAbstract

τ Weiss · Case preference51.31 / 40–75 ms · Met

A broad design interval informed by 54±14 ms, not a normal/abnormal boundary. Use only quality-qualified Weiss fits and retain the window and baseline comparison. With calcium/crossbridge rates fixed, tau changes can arise from geometry, tension, load and regression window; intrinsic relaxation change is not identified. No substitution by Glantz or extra reward for longer tau.

Method: main-wire-lv-relaxation-tau-v1:Weiss-zero-asymptote-time-weighted-log-linear

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

Relaxation, pressure rates and filling

P(t)=P_{\infty}+A e^{-(t-t_0)/\tau},\qquad \mathrm{Tei}=\frac{ICT+IRT}{ET}

Tau is fitted to cavity-pressure decay. Weiss fixes the asymptote at zero; Glantz estimates it. Glantz does not pass fit quality here and is shown as unavailable. Weiss is prolonged despite unchanged calcium/crossbridge kinetics; effects of loading, geometry and the observation window are not identified intrinsic relaxation disease.

Weiss: 51.31 ms · R² 0.9985 · window 25.98 ms · 15 samples.

The window runs from the midpoint of the minimum-dP/dt interval to the next end-filling LV pressure plus 5 mmHg, before mitral opening. Time-weighted linear regressions use log-pressure versus time for Weiss and dP/dt versus pressure for Glantz. Required support is 6 samples, 15 ms and 10 mmHg decay; minimum R² is 0.97/0.95, and reconstructed-pressure RMSE must be ≤5% of the decay. These are measurement-usability design rules, not disease-normality limits.

Glantz R² describes the dP/dt-versus-pressure regression. A small reconstructed-pressure error does not override that fit-quality requirement.

dP/dt uses accepted-step cavity-pressure differences divided by elapsed time, not an analytic derivative or a filtered sensor. ICT/ET/IRT follow valve events/forward flow, not interchangeable clinical acquisition methods.

Flow E/A is 0.8579, A-dominant as in baseline (0.9379). Lavine's high-filling-pressure DCM group had a greater early filling fraction and a lower atrial fraction, a different direction. The flow-versus-Doppler distinction alone does not resolve this mismatch. A-dominance is a model inflow trait shared with baseline, not an established HFrEF finding reproduced here.

End-filling LV cavity pressure exceeds mean LA pressure by 8.847 mmHg. Kato's difference of cohort means was 4 mmHg, not a distribution of individual differences. Mean LA is not substituted for PCWP or end-filling LV pressure; this discrepancy remains an inflow-mechanism limitation.

Importance of the left ventricular filling pressure on diastolic filling in idiopathic dilated cardiomyopathyAbstract
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

Waveform and settlement

LVP/RVP each have one significant peak. Peak position and PV-loop roundness are not forced into a universal normal interval. This does not rule out small fluctuations or ringing under every control combination.

LVP: 1 peaks · variation 1.217 · peak 71.53% ET.
RVP: 1 peaks · variation 1.237.

Three consecutive periodic-boundary passes were followed by a separate audit beat. 156 cycles · 0.0009783.

An additional 60 beats from the earlier saved state changed coronary tone 0.693→0.6484. CI changed 0.0004584 L/min/m² and mean Ao-node pressure -0.01326 mmHg. Small hemodynamic changes do not establish complete stationarity.

This extension was run on lab-003 and checked against the then-current lab-004 numerical state, solver predictor, construction and initial observations. It is not this model's launch record. It is retained as historical supporting evidence for the same construction, not a fresh execution.

PV analysis and passive accommodation

These loading analyses use a historical 2 ms launch with the same construction, not a rerun from this model's launch. Selecting the 1 ms record does not change the grid of these analyses. Pressure is transmural.

LV · J/beat	baseline	HFrEF
SW	1.080	0.627
PE	0.382	0.907
PVA	1.462	1.534

SW falls while PE contributes more. Model load-family ESPVR/EDPVR are not catheter Ees or an isolated passive material law. PE includes a low-volume tangent extension. 2.56 mL.

A separate fully passive, viscously relaxed comparison at fixed RV volume 140 mL found 150.8→184.3 mL LV volume at 10 mmHg LV transmural pressure. It is neither RV-pressure-matched nor a dynamic EDPVR.

The PVA-derived oxygen estimate uses current LV mass, but remains illustrative: a canine relation and an uncalibrated contractility-dependent intercept do not validate disease metabolism, ischemia or mechanical efficiency.

Controls and qualification scope

The 53 controls are inherited, not a certification of all combinations. Single edits to LV tension 0.25/0.75/1.33, TBV4200/7000 mL and HR60 passed interaction, continuation and checkpoint tests. The altered states need not retain HFrEF targets.

Known limit: TBV7000 mL cold initialization failed at 2 ms. Changing from a saved state and independent 1 ms cold initialization succeeded. No automatic time-step fallback is implemented. Standard case launch uses its qualified 2 ms saved state.

Case assessment

1 ms · independent fine grid

Saved baseline and case comparison
Metric	Comparator baseline2 ms, fixed	This case1 ms	Unit
LVEF	55.8	28.5	%
LV EDVI	75.6	113.4	mL/m²
LV ESVI	33.5	81.1	mL/m²
CI	2.95	2.26	L/min/m²
SVI	42.2	32.3	mL/m²
Ao node mean	93.6	81.1	mmHg
mean LAP	8.4	16.6	mmHg
mean RAP	3.1	4.3	mmHg
PA node mean	17.9	23.9	mmHg
RVEF	56.9	49.1	%
RV EDVI	74.1	65.8	mL/m²
RV ESVI	31.9	33.5	mL/m²
LV end-filling P	10.9	25.2	mmHg
LV end-filling Ptm	10.9	22.7	mmHg
LV +dP/dt	2,564	1,126	mmHg/s
LV −dP/dt	-1,539	-828	mmHg/s
ICT	89.1	48.0	ms
ET	258.0	279.0	ms
IRT	92.0	130.0	ms
Tei	0.70	0.64
E/A (flow)	0.94	0.86
τ Weiss	32.0	50.8	ms
τ Glantz	52.4	—	ms

The comparator baseline remains its saved 2 ms record; selecting the 1 ms case does not recompute it. Unrounded values are included in the JSON export.

Intervals and rationale

LVEF · Required28.48 / 20–40 % · Met

Reduced EF defines the chosen phenotype. The 20% lower bound selects severity; it is not a diagnostic lower limit.

Method: native-valve-closure-volumes

2022 AHA/ACC/HFSA Guideline for the Management of Heart FailureSection 2.2, Table 4
Association between systolic ejection time and outcomes in heart failure by ejection fractionMethods: Echocardiographic measurements; Table 2, HFrEF

LV EDVI · Required113.4 / 90–180 mL/m² · Met

Selects a chronic LV-dilated volume range, not a requirement for all HFrEF or an imaging diagnostic boundary.

Method: native-valve-closure-volumes

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

CI · Required2.261 / 1.8–3.5 L/min/m² · Met

Allows lower or retained output; reduced EF is not equated with extreme low output.

Method: exact-beat-aortic-valve-net-flow

Cardiac output and cardiac index measured with cardiovascular magnetic resonance in healthy subjects, elite athletes and patients with congestive heart failureMethods: patient population / CMR; Results: CHF patients
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean LAP · Required16.6 / 3–25 mmHg · Met

Retains the broad screen against extreme filling. Elevated pressure is a case preference, not a required condition.

Method: exact-beat-mean-left-atrial-pressure

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean RAP · Required4.253 / 0–12 mmHg · Met

Selects loading for an LV-dominant case; it proves neither normal RVEF nor absence of RV involvement.

Method: exact-beat-mean-right-atrial-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

Ao node mean · Required81.13 / 60–110 mmHg · Met

Avoids extreme arterial loading. Site and method differences prevent treating it as clinical MAP diagnosis.

Method: exact-beat-mean-aortic-node-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest
Prognostic value of myocardial strain and late gadolinium enhancement on cardiovascular magnetic resonance imaging in patients with idiopathic dilated cardiomyopathy with moderate to severely reduced ejection fractionMethods; Tables 1–2

LVEF · Case preference28.48 / 25–35 % · Met

Meet EF jointly with the other primary preferences, without ranking EF above all other objectives.

Method: native-valve-closure-volumes

Association between systolic ejection time and outcomes in heart failure by ejection fractionMethods: Echocardiographic measurements; Table 2, HFrEF
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

LV EDVI · Case preference113.4 / 110–150 mL/m² · Met

Selects dilation and a large residual volume. ESVI is not added as a duplicate objective.

Method: native-valve-closure-volumes

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

CI · Case preference2.261 / 2.2–2.8 L/min/m² · Met

Selects one case with retained resting net output, not a disease-specific normal CI interval.

Method: exact-beat-aortic-valve-net-flow

Cardiac output and cardiac index measured with cardiovascular magnetic resonance in healthy subjects, elite athletes and patients with congestive heart failureMethods: patient population / CMR; Results: CHF patients
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

Ao node mean · Case preference81.13 / 80–100 mmHg · Met

A resting arterial-pressure preference, strongly dependent on output and vascular load. Missing it does not invalidate HFrEF; the lower bound was not lowered to admit a known 77-mmHg candidate.

Method: exact-beat-mean-aortic-node-pressure

Prognostic value of myocardial strain and late gadolinium enhancement on cardiovascular magnetic resonance imaging in patients with idiopathic dilated cardiomyopathy with moderate to severely reduced ejection fractionMethods; Tables 1–2
Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest

mean LAP · Case preference16.6 / 12–20 mmHg · Met

Method: exact-beat-mean-left-atrial-pressure

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fractionMethods; Tables 1–2, HFrEF supine rest
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion
Importance of the left ventricular filling pressure on diastolic filling in idiopathic dilated cardiomyopathyAbstract

τ Weiss · Case preference50.81 / 40–75 ms · Met

Method: main-wire-lv-relaxation-tau-v1:Weiss-zero-asymptote-time-weighted-log-linear

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

Relaxation, pressure rates and filling

P(t)=P_{\infty}+A e^{-(t-t_0)/\tau},\qquad \mathrm{Tei}=\frac{ICT+IRT}{ET}

Weiss: 50.81 ms · R² 0.9986 · window 25.78 ms · 28 samples.

Glantz R² describes the dP/dt-versus-pressure regression. A small reconstructed-pressure error does not override that fit-quality requirement.

Flow E/A is 0.8582, A-dominant as in baseline (0.9379). Lavine's high-filling-pressure DCM group had a greater early filling fraction and a lower atrial fraction, a different direction. The flow-versus-Doppler distinction alone does not resolve this mismatch. A-dominance is a model inflow trait shared with baseline, not an established HFrEF finding reproduced here.

End-filling LV cavity pressure exceeds mean LA pressure by 8.65 mmHg. Kato's difference of cohort means was 4 mmHg, not a distribution of individual differences. Mean LA is not substituted for PCWP or end-filling LV pressure; this discrepancy remains an inflow-mechanism limitation.

Importance of the left ventricular filling pressure on diastolic filling in idiopathic dilated cardiomyopathyAbstract
Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathyMethods: Data Analysis; Table I; Results and Discussion

Waveform and settlement

LVP: 1 peaks · variation 1.214 · peak 71.38% ET.
RVP: 1 peaks · variation 1.231.

Three consecutive periodic-boundary passes were followed by a separate audit beat. 155 cycles · 0.0009788.

An additional 60 beats from the earlier saved state changed coronary tone 0.6971→0.6525. CI changed 0.0004628 L/min/m² and mean Ao-node pressure -0.01312 mmHg. Small hemodynamic changes do not establish complete stationarity.

PV analysis and passive accommodation

LV · J/beat	baseline	HFrEF
SW	1.080	0.627
PE	0.382	0.907
PVA	1.462	1.534

SW falls while PE contributes more. Model load-family ESPVR/EDPVR are not catheter Ees or an isolated passive material law. PE includes a low-volume tangent extension. 2.56 mL.

Controls and qualification scope

Evidence and reproducibility

Source populations, methods and limitations

2022 AHA/ACC/HFSA Guideline for the Management of Heart Failure

EF classification in the 2022 heart-failure guideline.

The guideline's EF-based classification is used as a construction convention.

Neither a claim about the latest definition nor a clinical diagnosis from low EF alone.

Cardiac output and cardiac index measured with cardiovascular magnetic resonance in healthy subjects, elite athletes and patients with congestive heart failure

157 patients with CHF and EF ≤40%, age 60±13 years. Moderate or greater MR, AR and aortic velocity >3 m/s were among the exclusions.

Supine CMR; whole-cycle phase-contrast flow in the ascending aorta at the pulmonary-artery bifurcation.

CI and EF were weakly related. Flow measured beyond the coronary origins is not identical to model AV net output. Low CI is not a necessary HFrEF finding.

Beneficial Effects of Heart Rate Reduction on Cardiac Mechanics and Energetics in Patients With Left Ventricular Dysfunction

14 patients with LV dysfunction: 10 with previous infarction and four with DCM. EF 34±12%, range 23–53%; not all had EF ≤40%.

Control HR 83±10; conductance volume and Millar cavity pressure. Linear ESPVR during IVC occlusion; loop area for work; Raff–Glantz relaxation.

Diuretics and vasodilators were withheld for at least 24 hours. EDP differs between Tables 1 and 2; Ees indexing units also differ between text and table. These values cannot directly define model thresholds or validate transmural work and a nonlinear common-isochrone ESPVR.

Comparison of haemodynamic response to muscle reflex in heart failure with reduced vs. preserved ejection fraction

10 treated HFrEF patients with improved symptoms; EF 23±6%, HR 72±17. PAWP ≥25 mmHg, resting SBP ≥160 mmHg, AF, severe valve disease and recent AMI were excluded.

Right-heart catheterization and 333-Hz microchip/conductance PV, calibrated to MRI volume. Senzaki single-beat Ees; MAP estimated from LV end-systolic and estimated diastolic pressures.

Small, selected, treated cohort. Do not index individual volumes using the cohort's mean BSA. PAWP is not measured mean LA pressure. Nonzero-asymptote tau is not Weiss tau and requires method checking before comparison with Glantz.

Association between systolic ejection time and outcomes in heart failure by ejection fraction

171 ambulatory HFrEF patients; median EF 30%, interquartile range 25–35%.

Three echocardiographic measurements. SET from LVOT Doppler; PEP includes ECG timing; relaxation calculated as R-to-E minus PEP and SET.

PEP is not ICT; Doppler endpoints differ from flow-zero crossings. HR dependence prevents adopting the interquartile interval as an HR70 gate. Table 2 gives HFrEF relaxation 93.3 [67.3, 122] ms, while the abstract reverses the group medians. That inconsistent statistic is contextual only, not a gate or objective.

Diastolic Determinants of Excess Mortality in Heart Failure With Reduced Ejection Fraction

12,421 patients with Stage B/C LV dysfunction, including EF <50%; not an exclusively EF ≤40% cohort.

Doppler diastolic measures; filling patterns differ between E/e′ groups.

Supports not requiring restrictive filling in every HFrEF case. The 0D model does not measure e′, so E/e′ must not be invented.

Correlation between left ventricular contractility and relaxation in patients with idiopathic dilated cardiomyopathy

38 resting idiopathic DCM patients in sinus rhythm, NYHA II–III, and nine controls. Stabilized by treatment, then medications withheld for at least three days. Observed EF 17–64%; not all meet modern HFrEF classification.

High-fidelity LV pressure at 3-ms intervals. Zero-asymptote Weiss and Raff–Glantz tau from minimum dP/dt to EDP+5 mmHg. Conductance volume and four to eight IVC-occlusion beats for Ees.

Small selected cohort. Acquisition differs from time-weighted accepted-step regression and valve-event measurements. Ees and tau were not correlated; some patients retained relaxation. Inconsistent assignment of preserved TL/TD proportions in the text prevents using a finding frequency.

Prognostic value of myocardial strain and late gadolinium enhancement on cardiovascular magnetic resonance imaging in patients with idiopathic dilated cardiomyopathy with moderate to severely reduced ejection fraction

172 idiopathic DCM patients with EF <40%; HR 83±20 and MAP 84±13 mmHg.

CMR and contemporaneous clinical measurements. LV-volume-derived and RV-volume-derived output differ.

Clinical MAP and exact Ao-node cycle mean differ in site and method. Do not transfer LV-volume-derived CI to net AV flow or derive net CI from ratios of cohort means.

Functional and structural abnormalities in patients with dilated cardiomyopathy

12 DCM patients and 10 controls. Seven DCM patients had normal material stiffness and five had increased stiffness; group mean EF 37/36% and LVEDP 18/22 mmHg.

Simultaneous ventriculography and high-fidelity pressure, viscoelastic pressure–volume/stress–strain analysis, and right-heart biopsy.

Chamber pressure–volume behavior differs from tissue stiffness. Small-group abstract statistics do not directly specify model material coefficients or required intervals.

Relation between ventriculoarterial coupling and myocardial energetics in patients with idiopathic dilated cardiomyopathy

Invasive PV in 23 idiopathic DCM patients; simultaneous coronary-sinus oxygen consumption in 16.

Conductance PV, IVC-occlusion Ees and a double-thermistor coronary-sinus catheter.

Statistics are mean±SE. Inferring Ees/Ea from EF does not provide independent evidence. SW/PVA and SW/MVO2 differ; transmural model work alone establishes neither metabolic efficiency nor increased PVA. Institution and authors overlap with Kato.

Importance of the left ventricular filling pressure on diastolic filling in idiopathic dilated cardiomyopathy

33 controls; 14 DCM patients with normal and 26 with elevated filling pressure, classified by mean pulmonary capillary pressure.

Filling pressure and pulsed transmitral Doppler; mitral-regurgitation effects also examined.

Filling pressure and MR affect inflow differently. The 15-mmHg group boundary is not an identical threshold for mean LA or LVEDP. Doppler velocity and total volumetric valve flow must remain distinct.

Comparison of Doppler indexes of left ventricular diastolic function with simultaneous high fidelity left atrial and ventricular pressures in idiopathic dilated cardiomyopathy

Idiopathic DCM at baseline and during amrinone. Sample size was not stated in the available abstract.

Simultaneous LA/LV micromanometer pressures, transmitral Doppler and M-mode/2D echocardiography.

E/A can fall despite improved relaxation. Intervention mechanisms were not individually isolated. This is not validation of a numerical target or drug model.

Characteristics and prognostic value of right ventricular (dys)function in patients with non-ischaemic dilated cardiomyopathy assessed with cardiac magnetic resonance imaging

216 nonischemic DCM patients with CMR EF <50%; median EF 37% [25–44]. RV dysfunction in 38% and MR in 63%, mostly mild.

Single-center retrospective CMR cohort; preexisting RV disease and pulmonary hypertension excluded.

Includes EF >40%, treatment, AF and MR. The 38% finding is not a universal HFrEF frequency or occurrence probability. LV/RV association does not determine causal direction.

Transcription correction: Ishihara 1994 mean Ea/Ees was corrected from 3.14 to 3.24 against the primary abstract. The original record is preserved. This contextual correction changes neither selection intervals, assessment results nor model values.

Identity, export and adoption status

This saved snapshot contains the assessed values, criteria and provenance and survives retirement of authoring code. Generating it or meeting intervals is not an adoption approval.

Model: circleheart.main-wire-integrated-transaction-v3.static-anatomy.standard-73
Surface: circleheart.main-wire.surface.static-anatomy.standard-73.workbench-v1
Case: standard73-hfref-chronic-dilated-v1
sourceSha256: c42db7ce5ea71fa6fc3ea301700af1b25e0ff4c673aba120bd2b9598ee2d79cf
evidenceSha256: 38bd2a289ae86f2d13bcfed72a70827e4ce55ab52056c473ec458acaaaf078df
referenceSha256: 45f553dffa52e0cdc25f51f065762474d41fd692692852f079bd8e1e648bd00a
artifactSha256: f40388e5918e968ed9db093d7c9cd72467da332212c575d08164958e206f5124
artifactRevisionId: 38e31e94e7b25e71d0bb99c1a7e9a27dc094985c3920d12c2fdf22afa34abce4
checkpointSha256: c97b528ac709354762a2c19f5cf4f52b32b22aa746542146cc8c99e39f102cf3

Formal registration still requires at least one unconditional approval from the two reviewers and a new exact release's own artifact, captures and Surface binding. Existing Standard72 checkpoints will not be relabeled.

analysis/main-wire-integrated-v3-formal-fixed-tbv-pressure-volume-relations-v1
analysis/main-wire-integrated-v3-guyton-starling-structural-orientation-v1
derivation/suga-pva-measured-load-display-exact-anatomy-mvo2-v14

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