Standard 73: mechanisms

Circuit connections, myocardial laws, pressure/flow computation and analysis methods.

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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.