Pressure and flow network

Each connection is represented by a hydraulic resistance. Conductance is G = 1/R. At each node, inflow and outflow must balance:

The scale factor k is calibrated so that the symmetric healthy state produces approximately 50 mL/100 g/min in every bed at MAP 90 mmHg and ICP 10 mmHg. A lesion adds a nonlinear resistance that rises steeply near occlusion; 100% stenosis is represented as near-infinite resistance rather than an exactly disconnected numerical node.

Proximal network resistance scales the native linear resistance of the ICA/basilar inlets and the A1, M1, P1 and terminal arterial segments. It does not replace the separate ACom, PCom or pial-capacity controls. The default value is 100%; increasing it makes changes in source flow produce a larger proximal pressure loss.

Finite source-flow reserve limits additional acute inflow rather than penalizing the collateral-supported resting state. For each ICA and the basilar source, WillisWorks first establishes the resting inflow with the recruited collateral network. The slider then scales the extra flow reserve available above that baseline. This prevents chronic collateral support from being counted twice—once as resting demand and again as capacity exhaustion—while still allowing hypercapnia to exhaust donor reserve and produce steal.

The capacity is intentionally a soft knee, not a hard clamp on flow. This allows high-flow states but makes them progressively more expensive in pressure, so vasodilation in healthy donor beds can lower collateral pressure in an exhausted recipient bed.

ACom and PCom channels carry flow according to their pressure gradient and selected capacity. VB territories share the basilar source with the PCA beds, so basilar restriction and posterior collateral recruitment change all four posterior beds through the same network.

Autoregulation

The grey curve is a familiar pressure-only teaching reference defined in CPP space and displayed against MAP. Coloured territorial curves use the same gently sloped target through the normal range, but the target does not impose a hard upper breakpoint. Their actual high-pressure upturn emerges when the solved resistance required to maintain target flow reaches the territory's available maximum resistance.

Between these reference points, the pressure-only reference is linearly interpolated. The territorial target is then scaled by an oxygen-homeostasis set-point. Higher CMRO₂ demand or lower arterial O₂ content raises the target flow; lower demand or higher O₂ content lowers it. This changes the resistance required at the operating pressure and therefore moves the territorial knees mechanistically rather than translating a pre-drawn curve.

At CMRO₂ 4.0 and CaO₂ 0.20, CBFmetabolic = 50 and the normal territorial curves overlap the grey reference. Raising demand or reducing CaO₂ increases resting vasodilation, consumes lower-pressure reserve and generally moves the operating curve upward and toward higher pressure. The 35–75 mL/100 g/min bounds prevent extreme slider combinations from demanding implausible compensatory flow; residual oxygen stress is then expressed through OEF and supply limitation.

EAR is autoregulatory efficiency. Intact regulation uses 1; pressure-passive flow uses 0; impaired regulation uses an intermediate value. Stenosis, collaterals and bypass alter distal pressure and the resistance already required at baseline, so disease shortens or collapses the usable plateau without assigning arbitrary territorial knees.

Vascular compliance and stiffness

The passive compliance calculation remains distinct from steady-state resistance: in a two-element Windkessel abstraction, resistance determines mean flow whereas compliance stores volume and damps pulsatile pressure. Lower compliance therefore represents stiffer vessels, greater transmission of pulsatile pressure and a shorter passive pressure-buffering lag.

Vascular stiffening can also coexist with endothelial, smooth-muscle and microvascular dysfunction. WillisWorks therefore uses two explicitly empirical stiffness couplings. For the static equilibrium curve, stiffness modestly narrows the available active resistance span and reduces controller efficiency, making the plateau shorter and more sloped without prescribing a new preferred pressure. During a PaCO₂ challenge, stiffness attenuates CO₂ reactivity and adds onset and recovery lag.

At normal compliance, Sstiff = 0, the PaCO₂ response amplitude is unchanged. At the minimum slider value, CO₂-mediated tone change is reduced by 30%, onset is additionally delayed and recovery is prolonged. Compliance above 100% does not create supernormal CO₂ reactivity. Existing stenosis-related reserve exhaustion still acts independently. Hypercapnia raises the flow target, lowers the attainable constrictor ceiling and weakens constrictor-side controller gain in proportion to the normalized CO₂ tone request; hypocapnia lowers the flow target and raises the attainable dilator floor. This lets the pressure limits move or become pressure-passive during challenge rather than remaining fixed.

WillisWorks uses fpulse = 1 Hz, an educational systemic pulse pressure of 40 mmHg and τ₀ = 0.18 s. During an animated MAP simulation, the mean pressure reaching the equilibrium solver is additionally low-pass filtered with a compliance-dependent time constant. Stiffer vessels transmit pressure changes more quickly even though their active challenge response is modeled as smaller and slower. The static Dynamics curves remain equilibrium solutions, but their active resistance range and controller slope now reflect the stiffness-associated factors above.

Background: the passive Windkessel calculation and the stiffness-associated challenge modifier are intentionally distinct. The latter is an educational parameterization of reported associations between arterial stiffness and reduced or slowed cerebrovascular reactivity; it is not a universal patient-specific conversion.

Classic OEF-ceiling model

CMRO₂ demand and CaO₂ first contribute to the global autoregulatory flow set-point described in the previous tab. OEF then handles the remaining delivery mismatch. OEF rises as delivery falls until the selected ceiling is reached; beyond that point, metabolism becomes supply limited.

Flow–diffusion reserve model

This option adds a finite diffusivity-recruitment capacity before the OEF ceiling is reached.

PaCO₂ challenge and reserve

PaCO₂ is now the single vasoactive challenge variable. The response is sigmoidal, centred near resting values, and normalized to no additional tone request at 38 mmHg. Raising PaCO₂ consumes vasodilatory reserve; lowering it recruits vasoconstriction. Stenosis, baseline metabolic vasodilation, stiffness and endothelial dysfunction constrain the achievable response through the same resistance bounds.

Leptomeningeal, bypass and moyamoya support

Ipsilateral leptomeningeal pathways connect ACA–MCA, MCA–PCA and a weaker ACA–PCA border zone. The transhemispheric option adds a medial ACA/pericallosal bridge; it does not create a direct convexity MCA-to-MCA vessel. EC–IC bypass is an external pressure source with strongest support to MCA and weaker ACA/PCA support. VB beds do not receive cortical pial or EC–IC bypass support.

Pial pathways are recruited from the resting distal pressure gradient. The resting collateral throughput is treated as an established protective component. During an acute PaCO₂ challenge, 35% of the pathway remains pressure-responsive while 65% of the resting transfer is retained as committed support. This prevents donor vasodilation or a finite source reserve from abruptly withdrawing an already recruited collateral supply, but still permits additional redistribution and eventual reversal when pressure changes are sufficiently large. Recruitment is recalculated after anatomy, lesion, pressure or collateral-control changes.

Staging

Derdeyn position combines resting reserve, OEF and preserved metabolism. The “supported perfusion” index additionally considers distal pressure and modeled pial/bypass support. These are conceptual teaching classifications rather than clinical diagnostic thresholds.

Hemodynamic simulations

Suzuki progression: interpolates terminal ICA/ACA/MCA narrowing, a mid-stage rise and late regression of basal moyamoya vessels, increasing posterior dependence and later ECA/transdural support.

Blood pressure & autoregulation: samples equilibrium territory curves across MAP using CPP internally. Chronic hypertension uses a right-shifted resistance calibration; impaired regulation reduces controller efficiency. CMRO₂ and CaO₂ scale the territorial flow target. Stiffness modestly narrows and slopes the static curve and also affects animated pressure transmission, challenge kinetics and estimated pulsatility.

PaCO₂ challenge: the selected target is approached from a resting value of 38 mmHg and then returned to baseline. A normalized sigmoid maps PaCO₂ to constrictor or dilator tone, allowing both hypocapnic and hypercapnic challenges. Lower compliance and endothelial dysfunction attenuate the response, while each territory remains limited by its own resistance bounds and collateral-dependent pressure.

Steno-occlusive progression: progressively increases selected large-artery resistance while allowing user-selected communicating and pial collateral adaptation.

Oxygen delivery: changes haemoglobin, oxygen saturation or metabolic demand and recomputes CaO₂, OEF and CMRO₂ through the selected metabolic model.

Important limitations

The extended model includes a continuous PaCO₂ response, regional venous/ICP outflow pressure, haematocrit- and calibre-dependent resistance, reduced-order stenotic jet loss, finite source-flow capacity, fixed acute collateral conductance, CTH, shunting and neurovascular coupling. These remain lumped educational abstractions. The model still lacks patient-specific segmented geometry, true one-dimensional wave propagation, full CFD, a venous collateral network, cellular biochemical signalling, spatial diffusion PDEs, reperfusion injury and validated clinical infarction prediction. Suzuki grade is not a unique measurement of CBF, CVR, OEF, symptoms or elapsed calendar time.

Performance: slider updates are synchronized to browser animation frames, the autoregulatory solver stops after numerical convergence, and inactive ECA collateral iterations are skipped. Full PaCO₂-response and Dynamics curves are still recomputed from the active network, but redundant solves are avoided.

Extended physiology

This branch adds bounded teaching layers to the existing lumped network. Parameters are normalized so the v20g healthy state remains the default. These additions improve conceptual completeness but do not convert WillisWorks into a patient-specific 1-D wave solver, CFD model, tissue digital twin or clinical predictor.

CO₂ regulation and neurovascular coupling

The PaCO₂ response is sigmoidal and normalized to no additional tone request at 38 mmHg. Hypercapnia consumes vasodilatory reserve; hypocapnia constricts. The midpoint and range are literature-informed teaching values rather than individualized fitted parameters. Acetazolamide can be represented only as an effective positive CO₂-equivalent vasoactive shift; the displayed value should not be interpreted as measured arterial PaCO₂ after acetazolamide. Neural activation increases both energy demand and a separately adjustable feed-forward flow request.

Venous pressure and spatial ICP

Each territory has a local Starling-resistor-style outflow pressure. Side and posterior ICP gradients and selected venous obstruction can therefore produce regional CPP differences.

Rheology, geometry and stenotic losses

Large-vessel and microvascular resistance are scaled around the calibrated baseline using haematocrit, representative calibre and path length. The apparent microvascular viscosity uses a normalized Pries-type Fåhræus–Lindqvist relation. The stenosis term adds a bounded inertial/jet-loss component inspired by reduced-order stenosis models; it is not CFD and does not resolve turbulence spatially.

Capillary transit heterogeneity, shunting and mitochondrial oxygen

Increasing capillary transit-time heterogeneity lowers effective extraction efficiency. A functional shunt fraction bypasses exchange. The mitochondrial PO₂ result is a dimensioned teaching proxy, not a direct biophysical tissue PO₂ calculation.

Derived tissue-state indices