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What is biophase? Pharmokinetic models of IV drug and pharmacodynamic models of IV drugs.

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biophase pharmacokinetics pharmacodynamics intravenous drug administration effect compartment model

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https://pmc.ncbi.nlm.nih.gov/articles/PMC3099368/

Biophase

Biophase is the actual site where a drug produces its effect, such as the brain for propofol, the neuromuscular junction for a muscle relaxant, or the heart for an antiarrhythmic.
After an IV injection, plasma concentration rises immediately, but the clinical effect may be delayed because the drug must distribute from blood to its site of action, bind receptors, and trigger a response. This produces plasma-effect disequilibrium, often seen as a hysteresis loop when effect is plotted against plasma concentration.
Because the true concentration at the receptor site is rarely measurable, an effect-site compartment is used as a theoretical representation of the biophase.
  • It receives drug conceptually from the central (plasma) compartment.
  • It is assumed to have negligible volume and does not alter the drug's pharmacokinetics.
  • The equilibration rate is described by (k_{e0}).
  • Larger (k_{e0}) means more rapid equilibration and a shorter onset delay.
[ \frac{dC_e}{dt}=k_{e0}(C_p-C_e) ]
Where:
  • (C_p) = plasma concentration
  • (C_e) = effect-site or biophase concentration
  • (k_{e0}) = plasma-to-effect-site equilibration constant
The reciprocal relation is commonly expressed as:
[ t_{1/2,ke0}=\frac{0.693}{k_{e0}} ]
The effect compartment is a model of the time delay in delivery to the true biophase, rather than a literal anatomical compartment. A PK-PD review explains this distinction clearly. Barash, Cullen, and Stoelting’s Clinical Anesthesia, 9e, pp. 770-771.

Pharmacokinetic models for IV drugs

Pharmacokinetics (PK) describes what the body does to the drug: distribution and elimination after administration. With IV administration, bioavailability is 100%, and there is no absorption phase.

1. One-compartment model

The body is treated as one kinetically uniform compartment. Following an IV bolus, the drug is assumed to distribute instantaneously throughout the compartment and then undergoes first-order elimination.
[ C_p(t)=C_0 e^{-kt} ]
Where:
  • (C_0 = \frac{\text{Dose}}{V_d})
  • (k) = elimination rate constant
  • (V_d) = apparent volume of distribution
  • (CL = k \times V_d)
The plasma concentration-time curve is monoexponential.
Example use: drugs whose distribution phase is too rapid to be distinguished clinically.

2. Two-compartment model

This is common for IV anesthetic drugs.
  • Central compartment ((V_1)): blood and highly perfused organs, including brain, heart, liver, and kidneys.
  • Peripheral compartment ((V_2)): less well-perfused tissues, particularly muscle and fat.
After IV bolus:
[ C_p(t)=Ae^{-\alpha t}+Be^{-\beta t} ]
  • (\alpha)-phase: rapid distribution from central to peripheral tissues.
  • (\beta)-phase: slower terminal elimination phase.
The relevant intercompartmental rate constants are:
  • (k_{12}): central to peripheral distribution
  • (k_{21}): peripheral to central redistribution
  • (k_{10}): elimination from central compartment

3. Three-compartment model

Many IV anesthetic agents, including propofol, are better represented by three compartments:
  • Central compartment
  • Rapidly equilibrating peripheral compartment
  • Slowly equilibrating peripheral compartment
[ C_p(t)=Ae^{-\alpha t}+Be^{-\beta t}+Ce^{-\gamma t} ]
This model helps explain why an IV anesthetic can have a rapid onset and early recovery after a single bolus, despite a long terminal elimination half-life. Early recovery often results mainly from redistribution out of the brain, not elimination from the body.

IV infusion PK

For a constant-rate infusion into a one-compartment model:
[ C_p(t)=\frac{R_0}{CL}(1-e^{-kt}) ]
At steady state:
[ C_{ss}=\frac{R_0}{CL} ]
Where (R_0) is infusion rate. Approximate steady state is approached after 4 to 5 elimination half-lives.

Pharmacodynamic models for IV drugs

Pharmacodynamics (PD) describes what the drug does to the body, relating drug concentration to clinical effect.

1. Direct-effect model

Used when plasma concentration and effect are effectively in immediate equilibrium:
[ E=f(C_p) ]
For example, a measurable anticoagulant effect may closely track circulating drug concentration.

2. Effect-compartment or biophase model

Used when effect lags behind plasma concentration, such as with IV propofol, opioids, and neuromuscular blockers.
[ C_p \rightarrow C_e \rightarrow E ]
The PK model predicts (C_p), the effect-site model predicts (C_e), and the PD model relates (C_e) to effect. The effect compartment is conventionally assumed to receive too little drug to affect plasma concentrations. Miller’s Anesthesia, 10e, pp. 3026-3027.

3. Linear model

Effect rises in direct proportion to concentration:
[ E=E_0+mC ]
Where (E_0) is baseline effect and (m) is the slope. This is usually valid only over a limited concentration range.

4. Maximum-effect or (E_{\max}) model

Most useful for receptor-mediated drug effects:
[ E=E_0+\frac{E_{\max}C}{EC_{50}+C} ]
Where:
  • (E_0) = baseline effect
  • (E_{\max}) = maximum possible drug effect
  • (EC_{50}) = concentration producing 50% of maximum effect
At low concentration, effect rises substantially with a concentration increase. At high concentration, the effect plateaus near (E_{\max}).

5. Sigmoid (E_{\max}), or Hill model

This is commonly used for IV anesthetics because it can model a steep concentration-response relationship:
[ E=E_0+\frac{E_{\max}C_e^\gamma}{EC_{50}^\gamma+C_e^\gamma} ]
Where (\gamma), the Hill coefficient, determines curve steepness.
  • High (\gamma): small concentration changes near (EC_{50}) cause large effect changes.
  • In anesthesia, this helps explain why a modest increase in propofol effect-site concentration may cause a marked increase in hypnosis.

6. Indirect-response or turnover model

Used when the drug does not directly produce the measured effect but alters the formation or loss of an endogenous mediator.
[ \frac{dR}{dt}=k_{in}-k_{out}R ]
A drug may:
  • inhibit mediator production, for example reducing clotting-factor synthesis
  • stimulate mediator loss
  • stimulate mediator production
  • inhibit mediator loss
These models are appropriate when the delay reflects physiological turnover or downstream signaling, not merely drug distribution to the biophase.

Integrated PK-PD model for an IV drug

A clinically useful sequence is:
[ \text{IV dose/infusion} \rightarrow C_p \rightarrow C_e \rightarrow \text{effect} ]
  1. PK model predicts plasma concentration, (C_p).
  2. Effect-site model accounts for distributional delay and predicts (C_e).
  3. PD model, often sigmoid (E_{\max}), predicts the clinical response.
Example: propofol
[ \text{IV propofol} \rightarrow C_p \rightarrow C_e \rightarrow \text{decreased BIS / hypnosis} ]
There may be a delay between plasma peak and maximal hypnotic effect because propofol must equilibrate with the brain biophase. This is why target-controlled infusion systems can target either plasma concentration or effect-site concentration. Barash, Cullen, and Stoelting’s Clinical Anesthesia, 9e, pp. 770-771.
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