title	WillisWorks User and Physiological Model Manual
version	1.0
date	July 2026
authors	Alex A. Bhogal Markus Fahlström
license	GPL-3.0-or-later

WillisWorks

Please let me know if there are issues or suggestions for improvement - via the issues tab or email @ a.bhogal (at) umcutrecht.nl

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User and Physiological Model Manual

Version 1.0 · July 2026
Interactive cerebral hemodynamics teaching simulator
Software conceptualized by Alex A. Bhogal and Markus Fahlström with some help from our intern openAI

WillisWorks is a transparent conceptual laboratory for exploring how arterial anatomy, pressure, resistance, collateral pathways, venous outflow, oxygen exchange, and metabolic demand interact. It is an educational model—not a medical device, patient-specific digital twin, or clinical decision system.

GPL-3.0-or-later · Browser-based · No patient data required

WillisWorks is a work in progress. Parameters may still be optimized and outputs should not be considered as accurate representations of physiology. This is intended as a conceptual tool only

How to use this manual

This document is both a user guide and a model specification. It follows the equations and constants implemented in WillisWorks v1.0. The goal is to let a reader understand what each control changes, why an output moves, and which parts of that response are established physiology versus educational calibration.

Code-fidelity note: Numerical constants identified as calibrations or heuristics are listed in Appendix A rather than presented as universal physiological values.

Model description

The numerical model contains eight tissue beds: bilateral ACA, MCA, PCA and vertebrobasilar (VB) territories. The brain image paints six cortical territories; L VB and R VB are present in the Model, Staging, Dynamics, timelines, comparison tables and exports but are not painted on the brain panel.

Contents

1. Purpose, scope and learning goals
2. Interface and recommended workflow
3. Model architecture
4. Pressure, flow and outflow pressure
5. Autoregulation, PaCO₂ and compliance
6. Finite source reserve, proximal resistance and steal
7. Circle of Willis, leptomeningeal pathways and bypass
8. Rheology, geometry and venous physiology
9. Oxygen exchange, metabolism and neurovascular coupling
10. Territory outputs, staging and Dynamics
11. Hemodynamic simulations
12. Feature and control quick reference
13. Interpretation, performance and limitations
14. Appendix A: Assumption register and rationale
15. Appendix B: Default values and ranges
16. Appendix C: Glossary
17. References

1. Purpose, scope and learning goals

A conceptual laboratory for asking what changes, why it changes, and through which compensatory pathway.

WillisWorks is an interactive cerebral hemodynamics teaching simulator. It links arterial anatomy, pressure loss, distal vascular tone, communicating arteries, pial collateral pathways, venous outflow, oxygen extraction and metabolic demand in one internally consistent zero-dimensional model. Its purpose is not to reproduce a particular patient, but to make otherwise abstract cerebrovascular mechanisms visible, testable and discussable.

Primary learning goals

• Understand why stenosis percentage alone does not determine perfusion: source pressure, lesion geometry, proximal resistance, CoW anatomy, collateral capacity and distal reserve all matter.

• Separate resting CBF from vasodilatory reserve. A territory can maintain near-normal flow while operating near its minimum arteriolar resistance.

• Understand bidirectional PaCO₂ regulation: hypocapnia constricts, hypercapnia dilates, and pressure-limited beds may show steal.

• Distinguish intrinsic territorial reserve from supported perfusion provided by CoW, pial or bypass pathways.

• Relate CBF and arterial oxygen content to OEF, delivered CMRO₂ and extraction limitations.

• Explore how source-flow reserve and proximal network resistance alter pressure redistribution.

• Recognize that staging labels are teaching summaries, not diagnostic classifications.

Intended users and intended use

The simulator is intended for medical students, residents, radiologists, neurologists, neurosurgeons, MRI scientists, physiologists and researchers. It is useful for lectures, case discussions, experimental design and hypothesis generation. It is not intended for diagnosis, treatment selection, perioperative blood-pressure targets, quantitative stroke-risk prediction, or replacement of measured perfusion, CVR, PET, angiography or clinical examination.

Safe interpretation Use the model to compare mechanisms and ask counterfactual questions. Do not infer that an individual patient shares the displayed absolute CBF, reserve or OEF value.

2. Interface and recommended workflow

The interface is organized around interventions on the left, a visual map and simulations in the centre, and numerical interpretation on the right.

Figure 1. Organization of the interface. Output calculations are expressed in the Model, Staging and Dynamics tabs.

A practical workflow

1. Start from Restore defaults. Confirm PaCO₂ 38 mmHg, MAP 90 mmHg and symmetric territorial flow.

2. Introduce one structural perturbation: a lesion, CoW variant, venous obstruction or altered proximal resistance.

3. Inspect the Territory model before applying a challenge. Note distal pressure and intrinsic reserve even if CBF remains preserved.

4. Use the PaCO₂ slider to compare hypocapnia, rest and hypercapnia. The local Reset beside Challenge and anatomy resets PaCO₂ and ACom/PCom capacities only.

5. Add CoW, leptomeningeal or bypass support one route at a time and compare state A versus B.

6. Open Staging to summarize the current state. Open Dynamics to inspect the autoregulatory compensation curve. Open Model to see quantitative outputs.

7. A simulations tab below the brain provides a dynamic representation of flow and oef changes in response to increasing hemodynamic burden.

8. Run a time-resolved simulation only after understanding the corresponding static state.

Teaching controls and Advanced model

Teaching controls expose interventions that are easy to discuss anatomically: lesions, PaCO₂, communicating arteries, CoW variants, leptomeningeal pathways, bypass, metabolic interpretation and autoregulation. Advanced model exposes pressure, source reserve, compliance, venous physiology, rheology, geometry, oxygen exchange and NVC parameters. Both modes modify the same underlying state; switching modes does not reset the model.

Reset behavior

Control	Scope
Restore defaults	Restores the full startup state, including advanced physiology and all simulations. The selected interface view and Dynamics metric are retained.
Challenge and anatomy: Reset	Restores PaCO₂ to 38 mmHg, ACom to 50%, and both PCom capacities to 35%. It does not remove lesions or reset the rest of the model.
Simulation Reset	Returns the selected simulation to its start state while retaining the configured simulation type and settings.
Simulation Off	Removes simulation-specific modifiers and returns to the saved static base state.

3. Model architecture

A steady lumped-parameter network with active distal resistance and separate oxygen/metabolic calculations.

Figure 2. Simplified topology. Arterial sources and communicating pathways are represented by resistances; each distal bed has a terminal connection and controllable tissue resistance.

Numerical compartments

The model contains three arterial boundary sources (left ICA, right ICA and basilar), seven proximal arterial nodes and eight distal beds. The bilateral VB beds connect independently to the basilar node and compete with both PCA beds for posterior inflow. They do not receive cortical leptomeningeal or EC–IC bypass support.

Why a zero-dimensional model?

A resistance network sacrifices spatial detail but makes pressure redistribution, reserve and collateral dependence transparent. Electrical-circuit and Windkessel analogies are widely used in cardiovascular modeling [13,17,18]. The present solver is appropriate for mechanism comparison, but it cannot reproduce patient-specific geometry, local wall shear, three-dimensional jets, pulse-wave propagation, collateral transit-time dispersion or infarct topology.

Order of calculation

1. Create arterial, communicating, pial, bypass and outflow resistances from the current controls.

2. Solve nodal pressure and source inflow, including the finite source-reserve feedback when active.

3. Iteratively update territorial arteriolar resistance toward the pressure-, metabolism- and PaCO₂-dependent flow target.

4. Calculate collateral flows, source fractions, reserve and pressure readouts.

5. Calculate oxygen delivery, extraction, delivered metabolism, CTH/shunt effects and the mitochondrial PO₂ proxy.

6. Classify the state and update Model, Staging, Dynamics, map and export objects.

Equilibrium versus time Most views show an equilibrium solution. Simulations add prescribed trajectories and simple response lags; the ordering of calculations should not be interpreted as the exact temporal order of physiological events in vivo.

4. Pressure, flow and outflow pressure

Flow follows pressure gradients through a conductance network; territory-specific downstream pressure is the greater of local ICP and venous pressure.

Conservation and nodal pressure

Each segment has hydraulic resistance R and conductance G = 1/R. At every free node, inflow and outflow sum to zero. The resulting linear system is solved for nodal pressures.

Core network equations

Σj Gij(Pi − Pj) + Gi,source(Pi − Psource) + Gi,sink(Pi − Pout)= 0
Qterritory = k · max(Pdistal − Pout, 0) / Rtissue

The scale factor k is calibrated so that the symmetric default state produces approximately 50 mL/100 g/min in each bed at MAP 90 mmHg and ICP 10 mmHg. Internal resistance values are therefore normalized model units rather than directly measured mmHg·min·100 g/mL.

Territory-specific outflow pressure

A territory first receives a local ICP that may include lateral and posterior gradients. Venous obstruction increases local effective venous pressure. The downstream pressure used for flow is the larger of the two, consistent with a simplified vascular-waterfall or Starling-resistor concept [17,18].

Outflow and perfusion pressure

PIClocal,t = ICP + lateral offsett + posterior offsett
Pvenous,t = CVP + obstruction incrementt
Pout,t = max(PIClocal,t, Pvenous,t)
CPPt = Pdistal,t − Pout,t

Stenosis and near-occlusion

Stenosis severity is translated into a viscous resistance term plus a bounded inertial/jet-loss term. The relation becomes steep near 95–100% so that near-occlusion is possible without numerically disconnecting a node.

Implemented lesion mapping

Rviscous = 3.2s², s < 0.95
Rviscous = 3.2(0.95)² + 220[(s−0.95)/0.05]³, s ≥ 0.95
Rjet = min{85, 0.055s² / max(1−s,0.025)²}
Rlesion = Rviscous + Rjet

Important assumption This is not a clinical conversion from angiographic percent stenosis to pressure loss. Real pressure loss depends on lumen geometry, length, eccentricity, flow rate, separation, collateral demand and rheology.

5. Autoregulation, PaCO₂ and compliance

The healthy reference is a gently sloped CPP-based standard; territorial curves emerge from actual pressure loss, target flow and available resistance range.

Figure 3. Pressure-only autoregulatory reference profiles for each territory.

Healthy reference and pressure controller

The familiar 60–150 mmHg MAP range is retained as a fixed healthy reference at ICP 10 mmHg. The grey reference is calculated from the standard reference state and does not move with the current lesion, PaCO₂, ICP, metabolic or simulation settings. It is expressed internally in CPP space and has a gentle slope rather than a perfectly horizontal plateau [1–3].

Reference point	CPP	MAP at ICP 10	CBF
Lower operational knee	50 mmHg	60 mmHg	47.5
Preferred operating point	80 mmHg	90 mmHg	50.0
Upper operational knee	140 mmHg	150 mmHg	52.5

For each territory, the controller calculates the resistance required to reach its current target flow and moves from the reference resistance toward that value. The result is constrained by calibrated minimum and maximum resistance. The territorial pressure target continues with the same gentle slope above the upper reference point rather than switching to a pressure-proportional branch at CPP 140 mmHg. The actual lower and upper pressure-passive limbs therefore emerge when the required resistance reaches Rmin or Rmax.

Territorial autoregulation

Rrequired = (Pdistal − Pout) / (CBFtarget / k)
Rcontroller = Rreference + EAR(Rrequired − Rreference)
Rarteriole = clamp(Rcontroller, Rmin, Rmax)
Pressure-passive limb begins when Rrequired lies outside [Rmin, Rmax]

At the symmetric normal baseline, the calibrated territorial curves overlap the fixed grey reference, including the low- and high-pressure tails. Stenosis, collateral support, PaCO₂, outflow pressure, compliance and metabolic demand can then alter the territorial curves without moving the reference itself.

Metabolic and NVC flow target

The pressure-only reference is scaled by the flow required to support metabolic demand at a preferred OEF of 0.40. Neural activation increases both represented demand and a feed-forward flow request; NVC efficiency and endothelial function determine how strongly the flow component is expressed [9–12,21]. Parameters can be modified under the Advanced Model controls (Oxygen, capillaries & neurovascular coupling).

Metabolic targeting

Demand scale = 1 + 0.25 · neural activation
NVC flow scale = 1 + 0.30 · activation · NVC efficiency · endothelial
function
CBFmetabolic = clamp[CMRO₂demand/(CaO₂ · 0.40) · NVC flow scale, 35,
95]
CBFtarget(MAP) = CBFreference(MAP) · CBFmetabolic / 50

Unified PaCO₂ regulation

PaCO₂ is the only global vasoactive challenge. A sigmoid is used because the human CBF–CO₂ response flattens at hypocapnic and hypercapnic extremes rather than remaining indefinitely linear [6–8]. Rest is defined at 38 mmHg, between common resting values and close to the reported midpoint of the human response.

CO₂ sigmoid and normalized tone

S(PaCO₂) = 1 / {1 + exp[−(PaCO₂−38)/6]}
Fraw = 0.55 + 0.90S
FCO₂ = Fraw / Fraw(38)
Tone request = (FCO₂ − 1) · endothelial function · stiffness
efficiency
CBFtarget,acute = CBFtarget · (1 + Tone request)

Hypercapnia raises the acute flow target, shifts the attainable maximum resistance toward Rmin and weakens constrictor-side controller gain. This allows the upper pressure limit to move or become more pressure-passive instead of remaining fixed near MAP 150 mmHg. Hypocapnia lowers the acute flow target and shifts the attainable minimum resistance toward Rmax. In a pressure-limited bed that is already near its dilatory limit, stronger donor-bed dilation and finite source reserve may produce steal [8].

Profiles and compliance

Setting	Implementation	Interpretation
Normal autoregulation	Normal CPP target, calibrated Rmin/Rmax and full controller efficiency; the grey reference remains fixed.	Symmetric default curves overlap the grey healthy reference.
Chronic hypertension	CPP reference points 70/100/165 mmHg.	Literature-informed right shift; not patient-specific [4].
Impaired autoregulation	Controller efficiency 0.28.	More pressure-passive, sloped response.
Autoregulation off	Controller efficiency 0 and fixed active resistance.	Demonstrates pressure-passive flow.
Low compliance	Narrows active range, reduces controller and challenge efficiency, alters lag.	Represents stiffness-associated damping loss and modest vascular dysfunction [13–15].

6. Finite source reserve, proximal resistance and steal

Steal becomes prominent when additional vasodilatory demand approaches the flow reserve of the supplying ICA or basilar source.

Figure 4. A vasoactive stimulus leads to vascular steal in the left territories due to competition for limited inflow supply.

Why finite source reserve was added

With fixed-pressure sources and purely linear proximal resistance, healthy territories can often obtain more flow during hypercapnia without substantially lowering pressure in a compromised bed. This biases the model toward increased total inflow and limits steal. Human and conceptual MRI work emphasizes that redistribution becomes important when major-vessel flow capacity is constrained [8].

Source-flow-reserve model

Each source—left ICA, right ICA and basilar—has a healthy reference capacity calibrated at 1.75 times its healthy resting inflow. During a challenge, the model treats the current collateral-supported resting inflow as the baseline and applies the capacity limit to additional acute demand.

Source reserve

Qcap,s = Qrest,s + Fcapacity · (Qcap,healthy,s −
Qhealthy,rest,s)
us = max(Qs − Qrest,s, 0) / max(Qcap,s − Qrest,s, ε)
ΔPsource = soft-knee(us), beginning at u = 0.65
Psource,effective = Psource − ΔPsource

The pressure-drop request is 8·knee³ below the reference capacity and gains linear and quadratic terms after utilization exceeds 100%; drop is capped at 78% of the available source-to-outflow pressure. This is a soft knee, not a hard flow clamp.

Proximal network resistance

The proximal-resistance slider scales native ICA/basilar inlet, A1, M1, P1 and terminal arterial resistances. It does not override ACom, PCom, pial or lesion-specific resistance. Increasing it magnifies pressure loss for a given flow change and makes competition among beds more visible.

Interpreting steal

True modeled steal is a fall in territorial CBF relative to its own static resting state during vasodilation elsewhere. It is most likely when the recipient bed has little remaining dilation, depends on a limited collateral route, and shares a source whose acute reserve is being used. A complete occlusion is not necessarily the strongest steal state: some resting collateral flow must remain available to be redistributed.

MRI caution Negative BOLD CVR is not identical to true CBF steal, and single-delay ASL can underestimate flow when arterial transit time changes. Compare the model primarily with quantitative, delay-aware flow measurements when possible.

7. Circle of Willis, leptomeningeal pathways and bypass

Primary, secondary and external support pathways are represented separately so their pressure and reserve consequences can be compared.

ACom and PCom pathways

Communicating-artery flow follows the solved pressure gradient. Capacity sliders alter pathway resistance; zero capacity makes a path functionally absent. Named CoW variants modify selected A1, P1 or source resistances. These controls represent functional conductance rather than a measured diameter [22–24].

CoW capacity mapping

Rcommunicating(c) = 0.16 + 20(1−c)²
Qcommunicating = k(Pdonor − Precipient) / Rcommunicating

Leptomeningeal recruitment

Pial templates link ACA–MCA, MCA–PCA and a weaker ACA–PCA route in each hemisphere. The transhemispheric option adds an ACA bridge. Conductance is recruited from the resting distal pressure difference, consistent with the directional importance of pressure gradients but not with a universal measured threshold [22,23].

Resting pial conductance

Recruitment = smoothstep[(|ΔPrest| − 3.5 mmHg) / 12 mmHg]
Rpial = edge factor · [0.42 + 10(1−capacity)²]
Edge factors: ACA–MCA 1.00; MCA–PCA 1.10; ACA–PCA 2.50; trans-ACA
1.35

Fixed acute conductance and the 65/35 split

In an acute PaCO₂ challenge, a vessel recruited at rest should not disappear instantly because the pressure gradient changes. The model therefore retains 65% of established resting pial flow and lets 35% remain pressure responsive. This avoids counting donor competition twice while still allowing collateral flow to weaken or reverse.

Acute pial flow

Qpial,acute = 0.65Qpial,rest + 0.35 · kΔPacute/Rpial

Teaching calibration The 3.5-mmHg onset, 12-mmHg recruitment span, edge weights and 65/35 split were selected for stable, interpretable rescue and redistribution. They are not universal human thresholds.

EC–IC bypass

Bypass adds an external distal pressure source, strongest to the MCA bed and weaker to ipsilateral ACA and PCA beds. At default hyperemia, the source pressure is approximately MAP−2 mmHg. Resistance weights are 0.34/potency for MCA, 0.94/potency for ACA and 1.02/potency for PCA. VB beds are not supplied. This is a teaching model of functional pressure support, not a graft-flow or surgical-outcome calculation [30].

8. Rheology, geometry and venous physiology

Resistance is allowed to vary with haematocrit, apparent viscosity, representative diameter, path length and outflow pressure (experimental modeling).

Haematocrit and apparent viscosity

Large-vessel viscosity is represented by a normalized haematocrit-dependent factor. Microvascular apparent viscosity uses a Pries-type relationship at a representative diameter of 38 μm, normalized to Hct 0.42 [16].

Rheology and Poiseuille-like geometry

Rgeometry ∝ μapparent(Hct,D) · L / D⁴
Large-vessel viscosity proxy = (1 + 2.5Hct) / (1 + 2.5·0.42)
Representative microvascular diameter = 38 μm · diameter scale

The D⁴ dependence makes the diameter slider intentionally powerful. It is a representative global calibre control, not a literal uniform change in every vessel.

Compliance-dependent calibre

A small pressure strain changes representative diameter as systemic CPP moves away from the healthy calibration point of 80 mmHg. When autoregulation is active, the CPP used for this passive geometry term is bounded by the lower and upper reference points of the selected profile. This prevents pressure-dependent calibre from creating an additional second tail after the active resistance limits have already been reached. With autoregulation off, the unbounded current CPP is used.

Pressure-dependent calibre

CPPeffective = clamp(CPP, CPPlower, CPPupper), when autoregulation is on
Strain = 0.035 · compliance · (CPPeffective−80)/50 ·
(1−0.65·stiffness)
Diameter multiplier is clamped to 0.60–1.45; path length to
0.60–1.80

The strain is attenuated by stiffness and remains a modest static wall-mechanics effect rather than a full pressure–area or viscoelastic model.

Venous obstruction and spatial ICP

CVP is normally below ICP and therefore may not set the effective outflow pressure. Venous obstruction can add up to 15 mmHg to selected territories. Lateral and posterior ICP gradients are applied before taking the maximum of venous and local ICP pressure. This permits asymmetric or posterior pressure limitation, but it is not a venous sinus network and does not model venous collateral channels [17,18].

9. Oxygen exchange, metabolism and neurovascular coupling

Both metabolic modes share Fick balance and the same arterial network, but differ in how extraction capacity is represented.

Figure 5. Shared oxygen cascade (experimental modeling). CTH, shunt and endothelial/NVC settings alter exchange or flow demand before derived metrics are calculated.

Arterial oxygen content and Fick balance

The oxygen simulation derives arterial oxygen content from haemoglobin and saturation; the Advanced CaO₂ control sets content directly. The dissolved term is fixed because PaO₂ is not independently modeled [9].

Oxygen content and Fick balance

CaO₂ [mL O₂/mL] = [1.34·Hb(g/dL)·SaO₂ + 0.30] / 100
DO₂ = CBF · CaO₂
OEFrequired = CMRO₂demand / DO₂
CMRO₂delivered = min(CMRO₂demand, DO₂ · OEF)

Capillary transit-time heterogeneity and shunt

CTH reduces the attainable extraction fraction when capillary transit times become more heterogeneous; a functional shunt removes a fraction of flow from exchange. These effects are motivated by capillary-flow models showing that heterogeneous transit can reduce oxygen extraction efficacy [19,20].

Implemented exchange penalties

CTH penalty = 1 / [1 + 0.48(CTH−1)^1.25], for CTH ≥ 1
Effective OEFmax = OEFmax · CTH penalty · (1−0.55·shunt)
Exchange flow = CBF · (1−shunt)

Two metabolic interpretations

Model	Core rule	Best use / caution
Classic OEF ceiling	OEF rises to the effective selected ceiling; delivered CMRO₂ is delivery × OEF.	Clear demonstration of compensated hypoperfusion and metabolic failure.
Flow–diffusion reserve	OEF = 1−exp(−D/flow); D is capped by a normalized diffusivity reserve.	Shows that extraction depends on transport capacity [10–12].

Mitochondrial PO₂ proxy

The displayed mitochondrial PO₂ is a bounded teaching proxy derived from oxygen-delivery margin, extraction stress and CTH penalty. It is not calculated from tissue diffusion geometry, capillary PO₂ distributions or mitochondrial respiration kinetics.

Derived proxy

Capacity = DO₂ · effective OEFmax
Mito-PO₂ proxy = clamp[20 + 12·oxygen margin + 12(1−extraction stress) −
8(1−CTH penalty), 0, 45]

Neurovascular coupling and endothelial function

Neural activation increases demand by up to 25% and requests up to 30% additional flow before NVC and endothelial scaling. Endothelial function also attenuates CO₂ reactivity. This makes metabolic demand, feed-forward coupling and vascular responsiveness separable, while remaining a simplified abstraction of the neurovascular unit [21].

10. Territory outputs, staging and Dynamics

Begin with continuous pressure, flow and reserve; use categorical labels as summaries rather than as endpoints.

Territory model

Each of the eight territory rows reports CBF, OEF, delivered CMRO₂, a state badge, intrinsic and available reserve, source composition and small CBF/OEF/CMRO₂ histories. The wide CBF bar provides a rapid visual comparison. Intrinsic reserve uses a violet palette; available reserve uses green. Hyperemia is shown in green, whereas metabolic failure remains red.

Intrinsic versus available reserve

Intrinsic reserve measures unused vasodilatory range in the territorial bed. Available reserve additionally reflects current support from communicating, pial, bypass and ECA pathways. A collateralized bed can therefore maintain adequate supported perfusion while remaining intrinsically exhausted and donor dependent.

Staging rules

State	Implemented rule	Meaning
Normal	Reserve ≥0.55; CBF 45–57; no metabolic deficit.	Default teaching range.
Reduced reserve	Reserve <0.55.	Compensation is being used.
Stage I / exhausted	Reserve <0.18.	Bed is close to minimum resistance.
Compensated hypoperfusion	CBF <45 without Stage-II OEF elevation.	Low flow with incomplete or model-dependent extraction response.
Stage II	CBF <45 and OEF > normal OEF +0.08.	Elevated extraction supports metabolism [25,26].
Metabolic failure	Delivered CMRO₂ <95% of represented demand.	Selected extraction capacity cannot meet demand.
Hyperemic response	CBF >57.	Flow above the reference range; displayed as a positive green state.

All categorical boundaries are WillisWorks teaching rules and are not PET diagnostic cutoffs.

Stages are representative and are meant for teaching/visualization purposes only

Dynamics

Dynamics is normally independent of the active simulation. The View dropdown selects Autoregulation curve, Suzuki progression, Steno-occlusive progression or Oxygen delivery. Selecting the Blood pressure & autoregulation simulation automatically opens the Dynamics panel with the Autoregulation curve as the initial view; the user can subsequently select another view. The Metric dropdown retains CBF, OEF, CMRO₂, intrinsic/available reserve, distal pressure, mitochondrial PO₂ proxy. Full curves use 43 samples and include the exact current operating point; a 17-point preview is used while dragging.

11. Hemodynamic simulations

Time-resolved demonstrations built from the same static network. Playback time is pedagogical, not biological disease time.

General behavior

Run advances the selected trajectory; Reset returns it to the start; Off removes its modifiers. The progress slider allows manual inspection. Timeline metrics are stored as the trajectory advances. Dynamics normally remains user selected; choosing Blood pressure & autoregulation opens the Autoregulation curve once as the most relevant starting view but does not lock the user to it.

Suzuki progression

Figure 6. Normalized Suzuki anchor functions. Continuous trajectories interpolate between these grade anchors.

Moyamoya disease involves progressive terminal ICA steno-occlusion and evolving basal, posterior and external collateral pathways. Suzuki and Takaku described six angiographic stages [27–29]. WillisWorks uses independent left and right start/end grades. Interpolation modifies ICA and branch severity, basal collateral support, posterior recruitment, PCA burden and ECA support. EC–IC conversion scales the external component.

Interpretation Angiographic grade is not equivalent to perfusion, symptoms or progression rate. The anchor values reproduce a recognizable sequence; they are not longitudinal measurements or probabilities.

Blood pressure and autoregulation

Select Normal, Chronic hypertension or Impaired autoregulation and specify start/end MAP. Selecting this simulation automatically opens Dynamics with the Autoregulation curve displayed. Commanded MAP is transmitted through a compliance-dependent first-order lag: τ = 0.55 + 1.65·relative compliance seconds. At each step the complete pressure network and resistance controller are solved. The same equilibrium law is used in both directions; the lag provides history but not a full dynamic autoregulation model [1,5].

Steno-occlusive progression

Progress raises the selected M1, ICA or tandem lesion smoothly toward 98%. The optional collateral-adaptation slider begins after early disease and increases CoW and pial capacity according to a normalized trajectory. This is a simulation-specific educational modifier, not the removed chronic remodeling module. Watch reserve fall before resting CBF and compare ICA versus M1 location.

Oxygen delivery

Anaemia changes haemoglobin, hypoxaemia changes SaO₂, and increased demand changes CMRO₂. A smooth progression recalculates CaO₂, metabolic target flow, extraction and delivered metabolism. The model does not include systemic cardiac-output, ventilatory, 2,3-DPG or transfusion effects; it isolates cerebral oxygen-content mechanisms [9].

Simulation	Primary controls	Best metric to watch
Suzuki progression	Left/right grade trajectory, duration, EC–IC conversion.	Intrinsic vs available reserve; source fractions; distal pressure.
Pressure & autoregulation	Profile, start/end MAP, compliance.	CBF and distal pressure; compare falling vs rising sweeps.
Steno-occlusive progression	Side, M1/ICA/tandem, adaptation, duration.	Reserve before CBF; donor-recipient asymmetry.
Oxygen delivery	Anaemia, hypoxaemia or demand; metabolic model.	CaO₂, OEF, delivered CMRO₂ and mito-PO₂ proxy.

12. Feature and control quick reference

Controls are grouped by the mechanism they modify. Defaults and exact ranges are tabulated in Appendix B.

Teaching controls

Control	What it changes	Use
Primary / second lesion	Adds independent resistance to ICA, M1, A1, P1 or basilar segments.	Build unilateral, bilateral or tandem disease; increase severity gradually.
PaCO₂ challenge	Sets arterial PaCO₂ from hypocapnia through hypercapnia.	Test constriction, dilation, reserve and steal. Rest is 38 mmHg.
ACom / PCom capacity	Changes functional communicating-artery conductance.	Test anterior cross-filling and anterior–posterior redistribution.
CoW configuration	Applies named A1/P1/PCom/source-resistance variants.	Use one anatomical pattern at a time.
Activate leptos	Adds recruited cortical pial links.	Compare recipient rescue, donor burden and acute challenge response.
Lepto pattern / strength	Selects ipsilateral or transhemispheric links and functional capacity.	Treat as conductance, not angiographic collateral grade.
L/R EC–IC bypass	Adds an external distal pressure source.	Compare pre/post support; not a surgical-outcome predictor.
Metabolic model	Classic or Flow–diffusion.	Choose the oxygen-extraction interpretation.
Autoregulation	Enables or disables active distal resistance control.	Disable to demonstrate pressure-passive flow.

Advanced controls: pressure and circulation

Control	Effect	Caution
MAP	Changes arterial boundary pressure.	Does not include systemic baroreflex or cardiac output.
ICP, CVP and gradients	Change territory-specific effective outflow pressure.	Reduced venous/Starling abstraction, not a sinus network.
Proximal network resistance	Scales native inlet and A1/M1/P1/terminal resistance.	Independent of CoW and pial controls.
Source flow reserve	Scales acute capacity above the current resting inflow.	Soft pressure-drop calibration, not measured Qmax.
Vascular compliance	Changes pulse damping, MAP lag and stiffness-associated active function.	Relative index, not physical compliance units.
Vasodilatory capacity	Scales distance from reference resistance to Rmin.	100% reproduces calibrated healthy capacity.
Venous obstruction	Raises local venous outflow pressure.	Location categories are simplified.

Advanced controls: rheology, oxygen and coupling

Control	Effect	Caution
Haematocrit	Changes large- and microvascular apparent viscosity.	Does not include all systemic responses to anaemia/polycythaemia.
Diameter / path length	Scales representative geometry and resistance.	Global proxy; D⁴ makes diameter highly influential.
CMRO₂ / CaO₂	Change metabolic demand, delivery and target flow.	High target flow is clamped at 95.
OEF ceiling / diffusivity reserve	Limit extraction in Classic or Flow–diffusion models.	Normalized capacities, not patient measurements.
CTH / shunt	Reduce effective extraction and exchange flow.	Reduced exchange penalties, not capillary-network simulation.
Neural activation / NVC efficiency	Separate demand and feed-forward flow components.	Simplified whole-territory NVC.
Endothelial function	Attenuates CO₂ reactivity, NVC and challenge efficiency.	Aggregate functional index.

Display, comparison and export

Control	Function
Map metric and palette	Colours six cortical territories; always read the colourbar.
Preset state	Loads a complete teaching state; manual edits change it to Custom.
Model / Staging / Dynamics	Switches fixed-height right-panel representations.
Dynamics View	Selects autoregulation, Suzuki, stenosis or oxygen series independently of the simulation.
Dynamics Metric	Selects CBF, OEF, CMRO₂, reserve, pressure or mitochondrial PO₂ proxy.
Save state A / B	Stores two current states in browser memory for comparison.
PNG / CSV / JSON	Exports the map, territorial results or full parameter/result object.

13. Interpretation, performance and limitations

A plausible output is a model prediction under its assumptions, not evidence that the same quantitative response occurs in vivo.

How to interpret a surprising result

1. Check effective outflow pressure and distal arterial pressure before looking only at CBF.

2. Check intrinsic reserve: preserved flow may already require near-maximal dilation.

3. Check source-reserve utilization and proximal resistance when challenge responses appear too small or steal appears strong.

4. Check whether collateral support raises absolute flow even if CVR relative to the rescued baseline remains negative.

5. Check CaO₂, demand, CTH, shunt and OEF ceiling before calling a low CMRO₂ value pressure failure.

6. Use A/B comparison and change one mechanism at a time.

Major limitations

• No patient-specific vessel geometry, measured flow boundary conditions or parameter fitting.

• No one-dimensional wave propagation, inertance, full pulse-wave reflections or cardiac-output model.

• No spatial capillary network, venous sinus network, true tissue PO₂ diffusion or mitochondrial kinetics.

• No validated conversion from angiographic stenosis percentage, collateral grade or bypass potency to hydraulic resistance.

• No grey/white-matter or border-zone subcompartments; focal severe steal can therefore be averaged within a territory.

• No explicit transit-time effect on ASL signal and no BOLD biophysical signal model.

• No embolic mechanism, infarct-core evolution or patient-specific neuronal injury model.

• Simulation durations and progression anchors are pedagogical, not biological time scales.

Responsible use WillisWorks is most valuable when its assumptions are made explicit. The appendix below is intentionally detailed so users can identify which conclusions are robust to model structure and which depend on a calibration choice.

Appendix A: Assumption register and rationale

A systematic audit of defaults, formulas, thresholds and structural choices implemented in v21j.

The tables below distinguish physiological principles from reduced abstractions and WillisWorks-specific calibrations. “Why chosen” describes the implementation rationale, not a claim that the value is universally normal.

A1. Baseline state and normalization

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
MAP 90 mmHg	Literature-informed default	A recognizable normotensive operating point and the preferred point of the healthy reference [1–3].	Not a universal individual MAP or optimal clinical target.
ICP 10 mmHg	Literature-informed default	Common normal teaching value; maps CPP 80 to MAP 90.	ICP varies with posture, pathology and measurement site.
CVP 5 mmHg	Teaching default	Keeps venous pressure below ICP in the default Starling-resistor state [17,18].	No respiratory or right-heart dynamics.
PaCO₂ 38 mmHg	Literature-informed calibration	Close to resting human values and near the midpoint of the sigmoid CBF–CO₂ response [6].	PaCO₂ and end-tidal CO₂ are not interchangeable in every subject.
CBF 50 mL/100 g/min	Teaching normalization	A familiar whole-brain reference used to set k and compare territories.	Grey/white matter and regional baseline flow differ.
CMRO₂ 4.0; CaO₂ 0.20	Literature-informed defaults	Their ratio at preferred OEF 0.40 gives target CBF 50 [9–12].	Units and values are representative, not subject-specific.
Hct 0.42	Teaching normalization	Reference for normalized apparent viscosity [16].	Sex, age and microvascular discharge haematocrit differ.

A2. Pressure and resistance network

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
Three fixed MAP boundary sources	Mechanistic abstraction	Separates L ICA, R ICA and basilar contributions and allows CoW redistribution.	No cardiac output or measured source-flow waveforms.
Seven proximal nodes / eight beds	Structural abstraction	Minimum topology that preserves bilateral ACA/MCA/PCA/VB interactions.	No border-zone or grey/white-matter subbeds.
BASE source .18, segment .12, terminal .05, bed .83	WillisWorks calibration	Places most healthy pressure control distally while preserving meaningful proximal losses.	Internal normalized units; ratios strongly influence redistribution.
Flow clamped at zero below outflow pressure	Physical safeguard	Prevents nonphysical negative sink flow in the reduced tissue-bed equation.	Does not model venous reflux or vascular collapse in detail.
Pout=max(ICP,CVPlocal)	Literature-informed abstraction	Captures a vascular-waterfall / Starling-resistor concept [17,18].	No explicit venous sinus/collateral network.
Venous obstruction adds up to 15 mmHg	Teaching calibration	Produces visible regional outflow limitation over the slider range.	Not a conversion from stenosis grade or sinus pressure measurement.
Spatial ICP offsets	Experimental abstraction	Allows left/right and posterior pressure heterogeneity to be explored.	True intracranial pressure fields are continuous and coupled to compliance.

A3. Lesions, geometry and rheology

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
3.2s² stenosis term	Heuristic	Provides gradual resistance growth at moderate severity.	Not Poiseuille flow through the measured residual lumen.
Cubic rise above 95%	Numerical/teaching calibration	Creates a transition to near-occlusion without singular disconnection.	Results near 95–100% are highly sensitive to lesion mapping.
Jet term with cap 85	Reduced-order heuristic	Adds flow-separation/inertial loss and prevents overflow.	No lesion length, eccentricity or Reynolds-number calculation.
R∝μL/D⁴	Established relation used as abstraction	Preserves the dominant geometry dependence of laminar tube flow.	Cerebral networks are branching, compliant and nonuniform.
38-μm representative microvessel	Teaching calibration	Places the apparent-viscosity function in a small-vessel regime [16].	One diameter cannot represent arterioles, capillaries and venules.
Pressure strain coefficient 0.035	Heuristic wall-mechanics coupling	Adds modest pressure-dependent calibre within the calibrated CPP range without creating a second passive tail beyond the active limits.	Not a measured pressure–area curve; the CPP clamp is a structural model choice.
Diameter clamp 0.60–1.45	Numerical safeguard	Prevents D⁴ from producing implausible singular resistance.	Clipping can dominate extreme slider combinations.

A4. Autoregulation and PaCO₂

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
CPP points 50/80/140	Literature-informed calibration	Define the fixed grey 60/90/150 MAP reference at ICP 10 [1–3].	They are reference anchors, not imposed territorial limits; individual limits and plateau slope vary.
CBF 47.5/50/52.5	WillisWorks calibration	Defines the gently sloped fixed reference across its lower, preferred and upper points.	Not a biological threshold or confidence interval.
Territorial knees from Rmin/Rmax	Mechanistic model rule	Pressure-passive limbs emerge when the resistance required for target flow exceeds the available active range.	Knee position depends on pressure loss, collateral support, PaCO₂, compliance, metabolism and calibration.
Hypertension CPP 70/100/165	Literature-informed calibration	Represents a right-shifted operating range described in severe hypertension [4].	Not a universal chronic-hypertension profile.
Impaired efficiency 0.28	Teaching calibration	Produces a visibly pressure-passive but not completely unregulated curve.	No direct clinical conversion.
Preferred OEF 0.40	Literature-informed abstraction	Links demand and content to flow through Fick balance [9–12].	Regional OEF varies.
Target-flow clamp 35–95	Numerical/teaching bound	Prevents unlimited metabolic flow request while allowing strong NVC/hypoxic compensation.	The upper bound is a model safeguard rather than a physiological maximum.
CO₂ midpoint 38, slope 6	Literature-informed calibration	Produces a human-like sigmoid with bounded hypocapnic and hypercapnic limbs [6–8].	Not fit to a specific subject or gas-delivery protocol.
Raw CO₂ range 0.55–1.45	WillisWorks calibration	Scales acute target flow and shifts the usable resistance bounds so pressure limits can move during challenge.	Absolute CVR and knee position depend on endothelial function, stiffness, reserve and network state.

A5. Compliance and source-flow reserve

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
Windkessel τ coefficient 0.18	Teaching calibration on established model family	Creates visible but compact pulse damping [13].	Not physical compliance units or a distributed arterial tree.
Systemic pulse pressure 40 mmHg; 1 Hz	Teaching references	Provide a stable reference for relative distal pulse transmission.	No heart-rate or waveform control.
Stiffness couplings 20%, 18%, 30%	Heuristic empirical coupling	Separates passive damping from modest loss of active regulation/challenge capacity [14,15].	Human associations are variable and confounded.
Source headroom 1.75× healthy rest	WillisWorks calibration	Leaves substantial healthy reserve but permits capacity-limited steal at high demand [8].	Not measured ICA or basilar maximum flow.
Soft knee at 65% acute reserve	Teaching calibration	Allows pressure drop to emerge before a hard capacity limit.	Steal magnitude is sensitive to this knee.
drop terms 8 / 35 / 60	Heuristic nonlinear mapping	Creates gradual pre-capacity and steep post-capacity pressure loss.	No direct vascular-pressure validation.
drop cap 78%	Numerical safeguard	Prevents source pressure from falling below a small margin above outflow pressure.	Extreme states are clamp limited.
Proximal resistance default 100%	Neutral scale	Preserves the calibrated healthy state; slider explores upstream pressure loss.	Global scale cannot reproduce segment-specific geometry.

A6. Circle of Willis, pial collaterals and bypass

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
ACom 50%; PCom 35% defaults	Teaching anatomy defaults	Allow low resting cross-flow but useful recruitment during asymmetry.	Not prevalence- or diameter-based.
R=0.16+20(1−c)²	Heuristic conductance mapping	Gives high sensitivity near low capacity and finite resistance at 100%.	Slider percentage is functional capacity, not lumen diameter.
Pial onset 3.5 mmHg; span 12 mmHg	Teaching recruitment rule	Makes secondary collateral recruitment depend on a meaningful pressure gradient [22,23].	No universal human pressure threshold exists.
Pial edge factors 1/1.1/2.5/1.35	Anatomical teaching weights	ACA–PCA is made weaker/longer; trans-ACA differs from same-side links.	Not measured path lengths or conductances.
65% committed, 35% responsive	WillisWorks acute calibration	Keeps established collateral support protective while permitting pressure-dependent change.	Key assumption; different split changes challenge steal.
No pial or bypass input to VB	Structural choice	Avoids assigning cortical surface collateral anatomy to posterior-fossa beds.	Posterior-fossa collaterals are underrepresented.
Bypass MCA/ACA/PCA weights .34/.94/1.02	Teaching distribution	Makes distal MCA support dominant while allowing weaker ipsilateral spread.	Not graft diameter, measured flow or anastomotic location.
Bypass pressure ≈MAP−2	Teaching pressure source	Represents a high-pressure external supply without exceeding systemic MAP.	No donor ECA limitation or hyperperfusion syndrome model.

A7. Oxygen exchange and metabolism

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
1.34·Hb·SaO₂ +0.30	Established approximation	Represents haemoglobin-bound plus fixed dissolved oxygen [9].	PaO₂ and dyshemoglobins are not modeled.
OEF ceiling 0.85 default	Literature-informed capacity choice	Allows marked compensatory extraction before failure.	Not a universal measured maximum.
CTH penalty exponent 1.25, coefficient .48	Heuristic based on CTH concept	Produces nonlinear extraction loss as heterogeneity increases [19,20].	Not a fit to capillary transit distributions.
Shunt penalty 0.55 and exchange-flow removal	Heuristic	Represents both unavailable flow and reduced extraction efficiency.	Functional shunt is not an anatomic AV-shunt model.
Diffusivity reserve 50%	Teaching default	Provides finite recruitable transport capacity in Flow–diffusion mode [10,11].	Normalized D is not a measured tissue diffusivity.
Mito-PO₂ proxy weights	Heuristic output	Combines delivery margin, extraction stress and CTH into an intuitive oxygen-tension direction.	Not tissue PO₂ in mmHg despite the displayed 0–45 scale.
NVC demand +25%; flow +30%	Literature-informed teaching amplitudes	Separates metabolic and feed-forward vascular components [21].	Whole-territory activation is simplified.
Endothelial function as common multiplier	Mechanistic abstraction	Provides a shared vascular-response impairment across CO₂ and NVC.	Endothelial pathways are not identical across stimuli.

A8. Staging rules

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
Stage CBF 45 and hyperemia 57	Teaching rules	Create readable low/high-flow categories around 50.	Small changes can cross labels without abrupt physiology.
Reserve thresholds .55 and .18	Teaching rules	Separate early reserve use from near exhaustion.	Not PET or clinical CVR cutoffs.
Stage-II OEF +0.08	Literature-inspired rule	Represents increased extraction in hemodynamic failure [25,26].	Not a quantitative O-15 PET diagnostic criterion.
CMRO₂ failure <95% demand	Teaching rule	Flags inability to meet represented demand.	No tissue-injury validation.

A9. Simulation and visualization choices

Implementation	Status	Why chosen / link to physiology	Limit or sensitivity
Suzuki anchor values	Literature-informed teaching calibration	Reproduces increasing ICA/branch disease, peak basal channels and later ECA support [27–29].	No progression rate or prognosis.
Stenosis simulation to 98%	Teaching endpoint	Approaches severe disease without singular occlusion.	Depends on lesion mapping.
Collateral adaptation trajectory	Simulation-specific heuristic	Demonstrates compensation during progressive disease.	Not a biological remodeling law.
Playback 12/24/40 s	Interface choice	Supports teaching and observation.	No biological time meaning.
43-point full /17-point preview curves	Performance choice	Maintains smooth final curves and responsive interaction.	Preview is temporarily lower resolution.
Six painted / eight numerical territories	Visualization compromise	Preserves established brain graphic while retaining VB physiology numerically.	VB changes are not visible on the brain map.
Source fractions by superposition	Explanatory calculation	Shows relative source support in the fixed-resistance solved state.	Not a tracer-validated flow territory measurement.

Appendix B: Default values and ranges

Startup settings in the v21j code. Percentages are relative model scales unless otherwise stated.

B1. Core and Advanced model

Parameter	Default	Range / choices	Unit or note
MAP	90	45–180	mmHg
ICP	10	2–35	mmHg
CVP	5	0–25	mmHg
Lateral ICP gradient	0	−10 to +10	mmHg
Posterior ICP gradient	0	−5 to +15	mmHg
Proximal network resistance	100	50–250	%
Source flow reserve	100	40–200	%
Vascular compliance	100	25–150	%
Vasodilatory capacity	100	20–150	%
PaCO₂	38	20–65	mmHg
Haematocrit	42	20–60	%
Diameter scale	100	75–125	%
Path-length scale	100	75–150	%
CMRO₂ demand	4.0	2.0–5.5	mL O₂/100 g/min
Arterial O₂ content	0.20	0.08–0.24	mL O₂/mL blood
OEF ceiling	0.85	0.60–0.95	fraction
Diffusivity reserve	50	0–150	%
CTH	100	50–250	%
Functional shunt	0	0–30	%
Neural activation	0	0–100	%
NVC efficiency	100	0–150	%
Endothelial function	100	0–100	%

B2. Anatomy and support

Parameter	Default	Range / choices
Primary / second lesion	None	None; L/R ICA, M1, A1, P1; basilar
Severity	0	0–100%
ACom capacity	50%	0–100%
Left / right PCom	35% / 35%	0–100%
CoW variant	Complete	Absent ACom; L/R A1 hypoplasia; absent L/R PCom; fetal L/R PCA; reduced VB inflow
Leptos	Off	Off/on
Lepto pattern	Ipsilateral	Ipsilateral ACA–MCA–PCA; transhemispheric ACA bridge
Lepto strength	65%	0–100%
L/R EC–IC bypass	Off	Off/on
Bypass potency	100%	0–100%
Metabolic model	Classic	Classic; Flow–diffusion
Autoregulation	On	On/off

B3. Simulation defaults

Simulation	Default configuration
Selected simulation	Suzuki progression; Off; duration 24 s.
Suzuki	Left I→VI; right unaffected; EC–IC conversion 100%.
Autoregulation	Normal profile; MAP 45→165 mmHg.
Stenosis	Left M1; collateral adaptation 65%; endpoint 98%.
Oxygen	Normal; anaemia target Hb 8 g/dL; hypoxaemia target SaO₂ 80%; demand target 4.0.
Dynamics	Autoregulation view by default; selecting the pressure/autoregulation simulation opens this view automatically, after which view and metric remain user selectable.

Appendix C: Glossary

Definitions as used in WillisWorks.

Term	Definition
Available reserve	Reserve after considering current external or collateral support; differs from intrinsic arteriolar capacity.
CaO₂	Arterial oxygen content.
CBF	Cerebral blood flow, displayed in mL/100 g/min.
Compliance	Relative vascular volume-storage and pulse-damping parameter.
CPP	Perfusion pressure across the territory: distal arterial pressure minus effective outflow pressure.
CTH	Capillary transit-time heterogeneity; a normalized exchange-efficiency modifier.
CVR	Cerebrovascular response/reactivity; context dependent. In the app, PaCO₂ challenge response can be positive or negative.
Distal pressure	Solved arterial pressure immediately upstream of a tissue bed.
DO₂	Oxygen delivery = CBF × CaO₂.
Intrinsic reserve	Unused portion of the bed’s active vasodilatory resistance range.
OEF	Oxygen extraction fraction.
PaCO₂	Arterial carbon-dioxide partial pressure; unified challenge variable.
Pial collateral	Secondary leptomeningeal pathway between distal cortical arterial territories.
Source reserve	Additional acute inflow capacity above the current resting source flow before substantial pressure drop.
Steal	A fall in a territory’s CBF during vasodilation elsewhere, usually because the recipient is exhausted and shared supply is limited.
VB territory	Left or right vertebrobasilar distal bed; represented numerically but not painted on the cortical brain map.
WillisWorks calibration	A deliberate educational parameter selected for stability or interpretability rather than a universal physiological constant.

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End of manual · WillisWorks v21j · July 2026