💉 CSF Bolus Study Calculator

Marmarou Mono-Exponential Model: Pressure-Volume Index (PVI), Outflow Resistance (Rout) & Time Constant of Pressure Decay (τ)
Developed by Dr G Narenthiran MB ChB BSc(MedSci)(Hons) FEBNS FRCS(SN), Neurosurgery Research Listserv, UK • g_narenthiran@hotmail.com
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Disclaimer: The CSF-bolus calculator should not be used for clinical purposes and the user should verify the calculations.
⚙️ Bolus Infusion Parameters & Patient Setup
Marmarou Exponential Decay Protocol
⚡ Clinical Presets:
1
🎯 1) Pressure-Volume Index (PVI)
Normal
-- mL
PVI = ΔV / log₁₀(Ppeak / Pb)
Volume required to raise ICP 10-fold. Characterizes intracranial mechanical compliance slope. Normal: > 18 mL (adult) or > 13 mL (elderly).
2
🎯 2) Outflow Resistance (Rout)
Normal
-- mmHg·min/mL
Rout = ∫[P(t)-Pb]dt / Vinf = AUC / Vinf = τ / C
Calculated via cumulative pressure-time integral divided by bolus volume. Normal: < 10 – 12. Pathological: > 12 – 18 mmHg·min/mL.
3
🎯 3) Time Constant of Decay (τ)
Decay Rate
-- min
τ = Rout × C = AUC / (Ppeak - Pb)
Rate of mono-exponential pressure decay. Slope of ln[P(t) - Pb] vs time equals -1/τ. Prolonged τ signifies delayed CSF resorption.
Intracranial Compliance (C) Buffering
-- mL/mmHg
C = ΔV / (Ppeak - Pb). Immediate volumetric reserve.
Area Under Curve (AUC) Integral
-- mmHg·min
Cumulative excess pressure ∫[P(t) - Pb] dt.
Exponential Fit (R²) Mono-Exp
--
Linear regression goodness-of-fit for ln[P(t) - Pb] vs t.
Decay Half-Time (t½) 50% Fall
-- min
t½ = τ × ln(2) = 0.693 × τ.
Safety Threshold Status Safe
OK
Configured threshold: 30.0 mmHg.
📊 Pressure Decay Time Series
10 Readings
Record time t (minutes post-bolus) and observed ICP as pressure decays back to baseline Pb.
Time (min) ICP (mmHg) P(t) - Pb ln[P - Pb] Act
📌 Marmarou Bolus Mathematical Formulation:
• Intracranial Compliance (C): ΔV / (Ppeak - Pb)
• PVI: ΔV / log₁₀(Ppeak / Pb)
• AUC: ∫₀∞ [P(t) - Pb] dt = (Ppeak - Pb) × τ
• Time Constant (τ): Slope of ln[P(t) - Pb] vs t = -1/τ
• Rout: AUC / Vinf = τ / C
• Mono-Exponential Model: P(t) = Pb + (Ppeak - Pb) · e-t / τ

📈 Graph 1: Pressure Decay Curve P(t) vs Time

Mono-exponential model P(t) = Pb + (Ppeak - Pb)e-t/τ with τ marker and safety threshold.
Peak: 26.0 mmHg τ = -- min Safety: 30 mmHg

📉 Graph 2: Rate of Pressure Decay (dP/dt)

First derivative: dP/dt = -[(Ppeak - Pb) / τ] · e-t/τ (instantaneous recovery velocity).
Initial Rate: -- mmHg/min

🔄 Graph 3: Curvature & Decay Deceleration (d²P/dt²)

Second derivative: d²P/dt² = [(Ppeak - Pb) / τ²] · e-t/τ (decay stabilization).
Curvature: Stabilizing

📏 Graph 4: Logarithmic Linearization Plot (ln[P(t) - Pb] vs Time)

Semi-logarithmic plot demonstrating mono-exponential decay where linear slope = -1/τ.
Slope = -1/τ
🩺 Diagnostic Thresholds & Clinical Interpretation
1. Pressure-Volume Index (PVI)
• Normal Adult: > 18 mL (mean ~25 mL)
• Borderline / Elderly: 13 – 18 mL
• Pathological (< 13 mL): Indicates severe cranial volume exhaustion or stiff intracranial vault.
2. CSF Outflow Resistance (Rout)
• Normal: < 10 – 12 mmHg·min/mL
• Borderline: 10 – 12 mmHg·min/mL
• Pathological (> 12 – 18 mmHg·min/mL): Impaired CSF absorption capacity; supports diagnosis of communicating hydrocephalus or NPH.
3. Time Constant of Decay (τ)
• Definition: Product of outflow resistance and intracranial compliance (τ = Rout × C).
• Significance: Prolonged τ signifies sluggish CSF resorption kinetics.