Pharmacokinetics 8/29: Ke, t1/2, AUC, Bayes, and Steady State Basics

Pharmacokinetics notes: ke, half-life, AUC, Bayes, and steady state

• Key idea: first-order elimination governs how concentrations decay over time and how regimens accumulate drugs in the body.

Calculating k_e and half-life from concentration data

• Two-point approach to k_e:
  • k_e =
    ( \dfrac{\ln 2}{\Delta t} ) where (\Delta t) is the time between two measured concentrations.
  • Example from the transcript: 100 at time 0 and 50 at time 10 h → ( \Delta t = 10 \text{ h} ).
  • (\text{ln}(2) = 0.693 )
  • ( k_e = 0.693 / 10 = 0.0693 \ \text{h}^{-1} )
• The half-life relation:
  • ( t{1/2} = \dfrac{0.693}{ke} )
  • In the example above, with (ke = 0.0693), ( t{1/2} = 10 \text{ h} ).
• General exponential decay equation for a single IV bolus (first-order elimination):
  • ( C(t) = C0 \exp(-ke t) )
• Practical takeaway: you first determine (k_e) from observed data, then compute the half-life, then use those to predict future concentrations.

Predicting future concentrations using k_e

• Example: starting concentration (C0 = 10) with (ke = 0.05\ \text{h}^{-1}), predict 24 h later:

  • ( C(24) = 10 \exp(-0.05 \times 24) \approx 3.0 ).

• Another example (given in the transcript): starting from 30 with (k_e = 0.057), predict at 24 h (t' = 24):

  • ( C(24) = 30 \exp(-0.057 \times 24) \approx 3.0 ).

• Conceptual check using half-life visualization (no calculation): if (t_{1/2} \approx 12) h, then in two half-lives (24 h) a starting 30 would move toward 7.5, etc., illustrating why simple intuition can align with the math.

• Example: two-point calculation to get k_e from a 12 h interval:

  • If (C0 = 30) at time 0 and (C = 15) at 12 h, then: ( ke = \dfrac{\ln(2)}{12} \approx 0.057 \text{ h}^{-1} ).

• Takeaway: you can use either the two-point method or the half-life method to reason about concentrations over time.

Area Under the Curve (AUC): concept and methods

• AUC stands for Area Under the Curve – a measure of overall drug exposure.
• Why AUC matters: exposure correlates with therapeutic/toxic effects for many drugs; not just peaks (Cmax) and troughs (Cmin).
• The trapezoidal rule (discrete data):
  • For successive samples at times (ti) with concentrations (Ci):
  • ( \text{AUC}{0\to t} \approx \sumi \frac{(Ci + C{i+1})}{2} (t{i+1} - ti) )
• Hand calculation with two data points (peak and trough):
  • ( \text{AUC}{0\to t} \approx \dfrac{C1 + C2}{2} (t2 - t_1) )
• Bayesian method (pharmacokinetic forecasting):
  • Based on Bayes' theorem: update probabilities as new patient data become available.
  • Uses population data plus individual patient information to tailor predictions without extensive serial sampling.
  • Practical implementation: software (e.g., DoseMe and other PK software) that computes AUC or other PK metrics using Bayesian methods and patient-specific data.
  • Advantages: reduces the number of blood samples, can adapt to organ function (renal/hepatic), improves patient comfort and satisfaction.
  • Real-world considerations: many Bayes-based PK programs exist, with varied costs and capabilities; cost examples discussed (e.g., about \$50,000/year for a hospital system).
• Hand-calculation caveat: when done by hand, AUC typically requires two measured points; software enables more complex fittings with more data points and population priors.

Bayes theorem in pharmacy and PK forecasting

• Bayes’ theorem: updating the probability of an outcome as new information becomes available.
• In pharmacy: used for PK forecasting, decision support, and individualized dosing.
• Applications:
  • Population PK to individual PK: adapt population parameters to patient-specific data.
  • Decision support systems for dosing and AUC estimation without extensive sampling.
• Practical takeaway: Bayesian methods can reduce sampling burden while maintaining accurate exposure estimates, but implementation varies across platforms and institutions.

Practical Pharmacokinetics: IV bolus, Cmax, Cmin, and dosing intervals

• IV bolus principle: a single IV dose increases blood concentration immediately (no absorption delay);
  • If multiple boluses are given at regular intervals ((\tau)) and the drug hasn’t fully cleared, subsequent peak (Cmax) levels increase relative to prior doses.
• Key relationships for repeated IV bolus dosing with first-order elimination:
  • Cmax after the second dose can be approximated by
  • ( C{\max,2} = C{\max,1} + C_{min,1} ) if the same dose is given and there’s residual drug from the first dose.
  • Cmin after the second dose can be calculated as
  • ( C{\min,2} = C{\max,2} \exp(-k_e \tau) ), where (\tau) is the dosing interval.
• Steady state in multiple dosing (first-order kinetics):
  • It typically takes about 4–5 half-lives to reach steady state.
  • Fraction of steady-state reached after n half-lives: (1 - (1/2)^n) → after 1: 50%, after 2: 75%, after 3: 87.5%, after 4: 93.75%, after 5: 96.875% (approaching 100%).
• Example premise from the transcript:
  • If the half-life is 12 h and there is a 6 h dosing interval with steady-state peak concentration of 10, the fifth dose (at 24 h) corresponds to two half-lives; about 75% of steady-state exposure is reached by then, so predicted peak ≈ 7.5. After seven doses (36 h, three half-lives), ≈ 87.5% of steady-state exposure ≈ 8.75, and so on.

Steady state: dose, dosing interval, and clearance

• Css,avg concept:
  • For extravascular dosing with constant bioavailability, steady-state average concentration is roughly
  • ( C_{ss,avg} = \dfrac{F \cdot D}{CL \cdot \tau} )
  • For IV ((F = 1)), this reduces to ( C_{ss,avg} = D / (CL \cdot \tau) ).
• Key controllable factors to reach a target Css,avg:
  • Dose (D) and dosing interval ((\tau)) are controllable.
  • Bioavailability (F) and clearance (CL) can also influence Css,avg; V_d and some absorption factors influence the shape of the curve and time to steady state but not the simple Css,avg formula when dosing is regular and CL is constant.
• Important pharmacokinetic note:
  • Clearance (CL) is a product of elimination rate constant and volume of distribution: ( CL = ke \cdot Vd ).
  • Changes in clearance or volume of distribution change the time to reach steady state and the peak/trough magnitudes, but the steady-state average depends primarily on dosing and clearance.
• Bioavailability considerations:
  • IV administration yields 100% bioavailability ((F = 1)).
  • Oral administration reduces bioavailability ((0 \le F \le 1)); oral AUC = F × AUC_IV, which can affect Cmax, Cmin, and Css,ss.
• Absorption and distribution factors:
  • Absorption rate constant ((ka)) and volume of distribution ((Vd)) influence the shape of the concentration-time profile (e.g., peak height, time to peak) but, if dose and schedule are maintained and CL is constant, Css,ss can be achieved with the same average level.
  • Altered absorption can delay time to peak but does not necessarily prevent reaching steady state if dosing is consistent.
• Volume of distribution (V_d) effects (single-dose perspective):
  • Larger Vd → lower Cmax for the same dose (more drug dispersed into tissues), and smaller Vd → higher Cmax (more drug in the plasma initially).
  • V_d can affect peak levels, but is less determinant of Css,avg than clearance and dose regimen when steady-state is achieved.

Absorption, distribution, and altered pharmacokinetic parameters

• Absorption changes can be due to formulation (immediate vs extended release), pH, and interactions; they shift Cmax and tmax but, with consistent dosing, still allow steady state.
• Bioavailability (F) and steady state:
  • If F is less than 1, the same dose will yield a lower AUC and a lower Css,avg unless dosing is increased to compensate.
• The two most important PK parameters for steady-state concentration are clearance (CL) and dosing interval/dose; distribution and absorption influence the time profile but not the basic Css,avg relation when dosing is regular and CL is constant.

Practical takeaways for exam preparation

• You can calculate ke from two concentrations and the time between them: ( ke = \dfrac{\ln 2}{\Delta t} ).
• You can predict future concentrations with C(t) = C0 e^{-ke t} and relate ke to t{1/2} via ( t{1/2} = \dfrac{0.693}{ke} ).
• AUC (exposure) can be estimated handily with the trapezoidal rule; with two data points, AUC \approx \dfrac{C1 + C2}{2} \Delta t.
• Bayesian/pharmacometric software provides population-based priors plus patient-specific data to estimate AUC and optimize dosing with fewer samples; costs and implementation vary by institution.
• In multiple dosing, Cmax and Cmin evolve toward steady state; after four to five half-lives, Css,avg is established; the percent of Css reached after n half-lives is (1 - (1/2)^n).
• Steady-state concentration depends on dose, dosing interval, and clearance: ( C_{ss,avg} = \dfrac{F D}{CL \tau} ) (IV: F = 1).
• Bioavailability and absorption affect peak/trough magnitudes and time to steady state but regular dosing with stable clearance still leads to steady-state exposure.
• When time allows, practice solving problems that require computing ke, t{1/2}, C(t), Cmax/Cmin for multiple doses, and AUC by hand to build confidence before using software tools.

Quick end-of-section reminders

• Be comfortable transforming between C0, Cmax, Cmin, k_e, t, and tau.
• Always check whether the problem is IV (F = 1) or extravascular (F < 1) and apply the correct Css,avg relation.
• For exams, you may be asked to reason qualitatively about how changes in clearance or V_d affect peak vs trough and time to steady state, not just to compute exact numbers.

Practice pointers from class activities mentioned

• Active-learning scratch-off exercise described as a group problem-solving activity to reinforce pharmacokinetic concepts; emphasizes collaboration, justification of answers, and verbal reasoning.
• Homework and check-in reminders provided for ongoing practice with PK problems.

Summary of symbols used

• (k_e) = elimination rate constant (h^{-1})

• (t{1/2}) = half-life, ( t{1/2} = \dfrac{0.693}{k_e} )

• (C(t) = C0 e^{-ke t}) (concentration-time profile for first-order elimination)

• (AUC = \int_{0}^{t} C(t) dt), approximated by trapezoidal rule with samples

• (F) = bioavailability (IV: F = 1)

• (D) = dose, (\tau) = dosing interval

• (CL) = clearance, (V_d) = volume of distribution

• (CSS,avg) = average steady-state concentration

• (C{max}), (C{min}) = peak and trough concentrations

• Remember: steady state is about consistency in dosing; four to five half-lives typically achieve near-steady-state exposure.

• Final note: always align your calculations with the problem context (IV vs oral, dosing amount and interval, and patient-specific factors).