Time Constant

Definition and Formula

• The time constant of the anesthesia breathing circuit (τ_”circuit” ) is defined as the time required for the concentration of an anesthetic gas within the circuit to reach approximately 63% of its new target or equilibrium value after a step change in the fresh gas concentration.¹

• • VBS: The internal volume of the breathing systemVBS, including all hoses, the reservoir bag, the CO₂ absorber, and the internal components of the machine. This volume typically ranges from 4 to 6 liters in an adult circle system.²
  • FGF: The rate at which fresh is delivered into the circuit, measured in liters per minute (L/min).²
• The resulting time constant (τ_circuit) is expressed in units of time.

The Exponential Process of Wash-In and Wash-Out

• The change in anesthetic concentration in the circuit, whether increasing (wash-in) or decreasing (wash-out), follows an exponential curve. This is a first-order kinetic process where the rate of change is proportional to the difference between the current concentration and the target concentration.¹

1. τ: 63% complete
2. τ: 86% complete
3. τ: 95% complete (sufficient equilibration for the brain)
4. τ: 98% complete
5. τ: Approximately 99% (considered full equilibration for the breathing circuit)

• As fresh gas carrying the anesthetic begins to flow into the air-filled circuit (assuming complete mixing), the concentration in the circuit (FI) will rise according to first-order kinetics:

The specific equation that models this behavior is:⁴

F_I: The fraction of inspired anesthetic in the circuit at time T.
F_FGO: The fraction of inspired anesthetic in the gas leaving the fresh gas outlet (vaporizer setting), which is the maximum or final concentration achieved.
T: The time elapsed since the flow began.
τ: The time constant

Clinical Implications and Examples

The time constant is a practical tool for the anesthesiologist to manage the speed of induction and emergence.⁵

Rapid Induction and Emergence (High Flow)⁵

• During the initial phase of anesthesia (preoxygenation and induction) or at the end of the procedure (emergence), a high FGF rate (e.g., 6-8 L/min for an adult patient) is used.
• This high flow rate minimizes the time constant, allowing the inspired concentration to quickly match the vaporizer setting and facilitating a rapid change in patient anesthetic depth.

Maintenance (Low Flow)⁵

• Once the desired depth is reached, the clinician may reduce the FGF to low-flow anesthesia (e.g., 0.5-2 L/min) to conserve anesthetic agent, maintain patient heat and humidity, and minimize environmental pollution.
• The time constant increases, and any changes in the vaporizer setting will take several minutes to be reflected in the inspired gas and, subsequently, in the patient’s brain concentration.

Patient Factors⁵

• The patient is an integral part of the breathing system volume, specifically their functional residual capacity (FRC). The total VBS to consider for the time constant calculation can actually be considered:

• This means that a patient with a large FRC or a large circuit volume (e.g., an adult circuit on a pediatric patient) will have a longer time constant, slowing the control of anesthesia depth.

Below are some examples to illustrate some common clinical scenarios:

The Role of Low-Flow Anesthesia

• Low-flow anesthesia is defined as the practice of using a fresh gas flow (FGF) of less than 3 L/min, often as low as 0.5 L/min during maintenance.⁶
• This technique offers economic benefits (reduced agent consumption), ecological benefits (less environmental pollution), and physiological benefits (better preservation of patient heat and humidity in the circuit).⁶
• A reduced FGF rate directly increases the time constant of the breathing system.

Clinical Implications of Long-Time Constants

• Slower Equilibration: Changes in vaporizer settings or flow rates take much longer to reach the desired inspired concentration (F_I). It may take longer than desired to reach the target minimum alveolar concentration (MAC).
• Potential for Accumulation: With less gas washout, there is a potential for the accumulation of undesired trace gases degradation products, although this is largely mitigated by modern absorbents and monitoring.

Overcoming the Time Constant in Low-Flow Anesthesia with the Concentration Effect

• While the time constant dictates how quickly a percentage of the final concentration is reached in the breathing circuit, the actual speed of induction is also dependent on the concentration effect.⁷
• The Concentration Effect states that the increase in the rate of rise for F_A (alveolar concentration)/F_I (inspired concentration) as the alveolar concentration is increased, and results subsequently in a more rapid induction.⁸

A clinician can leverage the concentration effect to achieve rapid induction even with relatively low flow rates by using a high concentration/percentage of volatile anesthetic.

• Vaporizer “Overshoot”: Instead of setting the vaporizer to the desired maintenance concentration (e.g., 1 MAC), the anesthesia provider can temporarily set the vaporizer to a much higher concentration (e.g., 2 MAC or 4% sevoflurane).
• Rapid F_I  and F_A rise: This high fresh gas outlet concentration rapidly increases the inspired concentration (F_I) and immediately creates a high partial pressure gradient that drives the agent into the patient’s blood and brain, counteracting the patient’s initial large uptake.
• Monitoring is essential: Because the clinician is intentionally pushing concentrations higher than needed for maintenance, continuous real-time monitoring of inspired (F_I) and end-tidal (F_A) agent concentrations is essential to guide the safe reduction of the vaporizer setting once the desired anesthetic depth is achieved.
• This technique effectively accelerates the initial “wash-in” phase, allowing the clinician to transition to a truly low-flow (e.g., 1 L/min) maintenance phase sooner and more safely, using the time constant principle for stable maintenance.