Drug Concentration, Plasma Profiles, Reaction Rates & Half-Life – Comprehensive Study Notes

Introduction & Learning Objectives

After completing this lesson you should be able to:

1. Discuss methodologies for measuring drug concentration in biological fluids (milk, saliva, plasma, urine, etc.).

2. Explain the relationship between plasma drug concentration and time.

3. Describe the parts and parameters of a plasma‐level concentration–time curve (MEC, MTC, C_max, T_max, AUC, therapeutic window, etc.).

4. Distinguish zero-, first-, and mixed-order (Michaelis–Menten) kinetics and calculate/interpret half-life for each.

Biological Matrices Used for Drug-Concentration Measurement

• Drugs can be quantified in milk (important for lactating mothers), saliva, plasma/serum, urine, feces, synovial fluid, spinal fluid, expired air, and tissue biopsies.
• The effect of a drug—therapeutic or toxic—ultimately depends on the amount that reaches its receptor site and forms the drug–receptor complex (pharmacodynamic “R” step of LADMER (\rightarrow) response).

Analytical Techniques

• Highly sensitive, accurate, and precise methods are available, most commonly:
– Chromatographic methods (HPLC, GC)
– Mass-spectrometric methods (LC–MS/MS)
• Choice of matrix & technique depends on the drug’s chemical properties, expected concentration, and clinical question (e.g., therapeutic-drug monitoring).

Specimen Collection: Invasive vs Non-Invasive

Ethical/Practical Note: Non-invasive sampling is preferred where feasible to reduce patient risk and cost, but may give less direct information about systemic exposure.

Blood-Based Matrices & Their Pharmacokinetic Utility

Whole Blood (with Anticoagulant)

• Contains plasma + RBCs + WBCs + platelets.
• Used when the drug partitions significantly into blood cells (e.g., immunosuppressants).
• Yields total drug concentration.
Anticoagulants: EDTA, heparin, citrate prevent clotting so all components remain suspended.

Red Blood Cells (RBC Fraction)

• Obtained by centrifuging anticoagulated whole blood.
• Separate analysis quantifies the amount of drug inside RBCs vs plasma (\rightarrow) informs distribution characteristics.

Plasma (Anticoagulated, Non-Clotted)

• Supernatant after centrifugation; clotting factors (e.g., fibrinogen) retained.
• Preferred matrix for most PK measurements requiring intact clotting proteins.

Serum (No Anticoagulant, Clotted)

• Blood allowed to clot; clot traps cells & clotting proteins.
• After centrifugation, clear fluid = serum (no clotting factors).
• Measures free + protein-bound drug in the liquid portion; widely used when clotting factors would interfere with the assay.

Visual Reminder: Plasma = Serum + Clotting Factors.

Plasma Drug Concentration–Time Curve (Single-Dose)

A set of plasma samples taken at defined time intervals are plotted (concentration vs time) on rectilinear coordinates, yielding the classic curve.

Key Regions & Terms

• Sub-Therapeutic Zone: [Concentration] < MEC → no clinical effect. • Minimum Effective Concentration (MEC): lowest plasma level that produces desired therapeutic effect (analogous term in antibiotics: MIC – Minimum Inhibitory Concentration). • Therapeutic Window: \text{MEC} < C < \text{MTC} – Wide window ⇒ safer; narrow window ⇒ requires careful monitoring. • Minimum Toxic Concentration (MTC): just-detectable level of toxicity. • Toxic Zone: concentrations > MTC.

Temporal Parameters

• Onset Time: time from drug administration until plasma level first reaches MEC.
• Intensity of Effect: proportional to the number of receptor sites occupied; correlates with plasma concentration up to a maximum.
• Duration of Action: time between first crossing MEC and final fall below MEC.
• Elimination Phase: downward slope after peak; reflects metabolism + excretion.

Pharmacokinetic Parameters Extracted

1. $C_{\max}$ (Maximum or Peak Concentration) – highest measured therapeutic level; unit $\mu g\,mL^{-1}$.

2. $T<em>{\max}$ – time after administration at which $C</em>{\max}$ occurs; unit = hours.

3. $\text{AUC}$ (Area Under the Curve) – overall systemic exposure; units $\mu g\,\text{hr}\,mL^{-1}$.
   – Larger AUC ⇒ more total drug in body over time.

Graph Interpretation Cheat-Sheet:

             Cmax
              •                 (Therapeutic Range)
             /|\                ------------------ MTC
            / | \              |
Absorption /  |  \ Elimination  |
----------/   |   \------------  MEC ------------
   ^          |                 |     Sub-therapeutic
Administration|<-- Duration --> |
              |                 |     Toxic Zone above MTC
              |<-- Tmax -->

Reaction Rate (Kinetic) Models

1. Zero-Order Kinetics (Saturation Kinetics)

• Elimination rate is constant, independent of concentration.
$\frac{dC}{dt} = -k<em>0$ → $C = C</em>0 - k<em>0 t$ • Occurs when metabolic/transport enzymes are saturated (e.g., ethanol). • Linear decline on concentration–time plot. Example: Starting $C</em>0 = 100\,\text{mg}$, k_0 = 5\,\text{mg·hr}^{-1}
After 1 h → $95\,\text{mg}$, 2 h → $90\,\text{mg}$, etc.
• No true half-life (t_{1/2} varies with concentration).

2. First-Order Kinetics (Linear PK)

• Elimination rate proportional to concentration.
$\frac{dC}{dt} = -k C$ → $C = C<em>0 e^{-kt}$ • Most drugs; enzymes not saturated. • On ordinary linear coordinates: exponential decline; on semilog plot: straight line. • Constant half-life: $t</em>{1/2} = \frac{0.693}{k}$.

Example (10 % eliminated per hour):
$C_0 = 100\,\text{mg}$ → 1 h ⇒ $90\,\text{mg}$ → 2 h ⇒ $81\,\text{mg}$ …

3. Mixed-Order / Michaelis–Menten Kinetics

• Follows first order at low concentrations, shifts to zero order once saturation occurs.
• Rate equation: $v = \frac{V<em>{\max} C}{K</em>m + C}$.
– $V<em>{\max}$ = maximum velocity (when fully saturated). – $K</em>m$ = concentration at which rate is $\frac{V_{\max}}{2}$.
• Clinically relevant for drugs like phenytoin; small dose increases near saturation can cause disproportionate concentration rises.

Graphical Behavior: Curve resembles first-order initially, then approaches linear zero-order segment as C → high.

Half-Life (t_{1/2}) Concepts

• Definition: time required for plasma concentration to fall to one-half of its value at the beginning of the interval.

First-Order t_{1/2}

• Constant irrespective of dose.
• Formula: $t_{1/2} = \frac{0.693}{k}$ where $k$ is the first-order elimination constant (hr⁻¹).

Example: Half-life = 4 h, $C_0 = 100\,\text{mg}$
After 4 h → $50\,\text{mg}$; after 8 h → $25\,\text{mg}$; after 12 h → $12.5\,\text{mg}$.

Zero-Order “Half-Life”

• Not constant; depends on current concentration because a fixed amount, not percentage, is removed per time.
Example: k0 = 10\,\text{mg·hr}^{-1}, $C</em>0 = 100\,\text{mg}$
1 h → 90 mg, 2 h → 80 mg, 3 h → 70 mg (so apparent t_{1/2} lengthens as concentration falls).

Clinical Pearl: For zero-order drugs, accumulation and toxicity risk increase disproportionately with dose escalation.

Comparative Summary: First vs Zero Order

Example Calculations & Practice Scenarios

1. First-Order: A drug has $k = 0.2\,\text{hr}^{-1}$.
   – $t<em>{1/2} = \frac{0.693}{0.2} = 3.47\,\text{h}$. – If C0 = 50\,\text{mg·L}^{-1}, what is C after 7 h?
   C = 50 e^{-0.2\times7} \approx 50 e^{-1.4} \approx 50(0.247) \approx 12.3\,\text{mg·L}^{-1}.

2. Zero-Order: Elimination rate k0 = 8\,\text{mg·hr}^{-1}, initial C = 64 mg. How many hours until concentration is ≤ 40 mg? $t = \frac{C</em>0 - C}{k_0} = \frac{64-40}{8} = 3\,\text{h}$.

3. Mixed-Order (Conceptual): Phenytoin therapeutic range = 10–20 µg/mL. A patient at 15 µg/mL receives a 15 % dose increase; concentration jumps to 22 µg/mL (near MTC).
   – Explanation: enzyme saturation; kinetics shift toward zero order.

Real-World & Ethical Considerations

• Narrow-therapeutic-window drugs (e.g., digoxin, warfarin, lithium) demand therapeutic drug monitoring (TDM) to minimize toxicity.
• Non-invasive matrices (saliva, urine) improve patient comfort but require validation to correlate with plasma levels.
• Over-the-counter agents (e.g., acetaminophen) are first order at therapeutic doses but can saturate (mixed/zero order) in overdose → hepatic failure; underscores public-health education on safe dosing.

Quick Reference Formulae

• First-Order: $C = C<em>0 e^{-kt}$, $t</em>{1/2} = \frac{0.693}{k}$.
• Zero-Order: $C = C<em>0 - k</em>0 t$.
• Mixed/Michaelis–Menten: $v = \frac{V<em>{\max} C}{K</em>m + C}$.
• AUC (trapezoidal): $\text{AUC}<em>{0\rightarrow t} = \sum \frac{(C</em>i + C<em>{i+1})(t</em>{i+1}-t_i)}{2}$.

Study Tips & Connections

• Always assess whether enzymes may saturate at therapeutic or supra-therapeutic doses; this dictates which kinetic model applies.
• Memorize typical MEC/MTC values only for clinically critical drugs; for others, focus on the concept of a window.
• Practice plotting curves from raw plasma data to internalize where C_max, T_max, AUC, onset, and duration appear.
• Relate reaction-rate laws to basic chemistry (zero/first order) to reinforce conceptual continuity across courses.
• Use half-life to design dosing intervals: maintain C above MEC but below MTC.

End of Notes – re-read with curve diagrams and work additional practice problems for mastery.