Neuropharmacology

Derivation 1: Hill Equation and Cooperative Binding

For a single binding site, the fractional occupancy follows the Langmuir isotherm:

$$\theta = \frac{[L]}{K_D + [L]}$$

For receptors with cooperative binding (e.g., ion channels with multiple subunits), the Hill equation generalizes this:

$$\theta = \frac{[L]^{n_H}}{K_D^{n_H} + [L]^{n_H}} = \frac{1}{1 + \left(\frac{K_D}{[L]}\right)^{n_H}}$$

where $n_H$ is the Hill coefficient: $n_H = 1$ for no cooperativity,$n_H > 1$ for positive cooperativity (steeper curve), $n_H < 1$ for negative cooperativity. The Hill coefficient relates to the slope of the log-logistic curve:

$$\log\frac{\theta}{1-\theta} = n_H \log[L] - n_H \log K_D$$

For GABA$_A$ receptors (pentameric, 2 GABA binding sites), $n_H \approx 1.5\text{--}2$. For NMDA receptors (requiring glycine co-agonist), the effective Hill coefficient for glutamate binding is $n_H \approx 1.4$. Benzodiazepines act as positive allosteric modulators, shifting the GABA dose-response curve leftward (decreasing $K_D$) without changing $n_H$.

Derivation 2: Operational Model (Black and Leff)

The operational model separates binding affinity from efficacy. The response $E$ is:

$$E = \frac{E_{\max} \cdot \tau \cdot [L] / K_D}{1 + \tau \cdot [L] / K_D + [L] / K_D} = \frac{E_{\max} \cdot \tau \cdot [L]}{K_D + (1 + \tau)[L]}$$

where $\tau$ is the transducer ratio (efficacy parameter), $K_D$ is binding affinity, and $E_{\max}$ is the system maximum. The observed EC$_{50}$ and maximum response are:

$$\text{EC}_{50} = \frac{K_D}{1 + \tau}, \quad E_{\text{obs,max}} = \frac{E_{\max} \cdot \tau}{1 + \tau}$$

A full agonist has $\tau \gg 1$ (EC$_{50} \ll K_D$, maximum effect $\approx E_{\max}$). A partial agonist has moderate $\tau$ (cannot achieve full $E_{\max}$). An antagonist has $\tau = 0$. This model explains receptor reserve, where a full agonist can produce maximal response with only partial receptor occupancy.

Derivation 3: Competitive Antagonism (Schild Analysis)

A competitive antagonist $B$ shifts the agonist dose-response curve rightward. In the presence of antagonist at concentration $[B]$:

$$\text{EC}_{50}' = \text{EC}_{50} \cdot \left(1 + \frac{[B]}{K_B}\right)$$

The dose ratio (DR) is the fold-shift: $\text{DR} = \text{EC}_{50}'/\text{EC}_{50}$. The Schild equation linearizes this relationship:

$$\log(\text{DR} - 1) = \log[B] - \log K_B$$

A Schild plot of $\log(\text{DR}-1)$ vs $\log[B]$ yields a straight line with slope 1 for competitive antagonism and x-intercept equal to $\log K_B$ (the pA$_2$value). Deviations from slope 1 indicate non-competitive mechanisms, receptor heterogeneity, or allosteric interactions.

Derivation 4: Two-Compartment Pharmacokinetic Model

For a CNS drug, the plasma ($C_p$) and brain ($C_b$) concentrations follow:

$$\frac{dC_p}{dt} = -k_{el} C_p - k_{in} C_p + k_{out} C_b \cdot \frac{V_b}{V_p} + \frac{D \cdot k_a \cdot e^{-k_a t}}{V_p}$$

$$\frac{dC_b}{dt} = k_{in} C_p \cdot \frac{V_p}{V_b} - k_{out} C_b - k_{met} C_b$$

where $k_{el}$ is renal elimination, $k_{in}$ and $k_{out}$ are BBB transfer rates, $k_{met}$ is brain metabolism, and $k_a$ is absorption rate. The brain-to-plasma ratio at steady state is:

$$\frac{C_b^{ss}}{C_p^{ss}} = \frac{k_{in}}{k_{out} + k_{met}} \cdot \frac{V_p}{V_b}$$

Lipophilic drugs ($\log P > 2$) cross the BBB readily ($k_{in}$ large), but P-glycoprotein efflux pumps can reduce brain exposure. The therapeutic window requires brain concentration above the EC$_{50}$ but below the toxic threshold.

Derivation 5: Receptor Occupancy Time Course

Combining pharmacokinetics with receptor binding, the time-dependent receptor occupancy is:

$$\theta(t) = \frac{C_b(t)^{n_H}}{K_D^{n_H} + C_b(t)^{n_H}}$$

The duration of action $T_{\text{dur}}$ depends on how long $\theta(t)$ exceeds the threshold for clinical effect $\theta_{\min}$. For first-order elimination:

$$T_{\text{dur}} \approx t_{1/2} \cdot \log_2\left(\frac{C_{\max}}{K_D \cdot (\theta_{\min}/(1-\theta_{\min}))^{1/n_H}}\right)$$

This explains why drugs with the same half-life can have different durations of action based on their efficacy and receptor affinity. Long-acting drugs may have either slow elimination or very high affinity (slow off-rate, $k_{\text{off}} = K_D \cdot k_{\text{on}}$).

Derivation 1: SERT Occupancy Model

The serotonin transporter (SERT) occupancy by an SSRI follows a simple competitive binding model. If $[\text{Drug}]$ is the free drug concentration at the transporter and IC$_{50}$ is the concentration producing 50% inhibition of serotonin reuptake:

$$\text{Occupancy} = \frac{[\text{Drug}]}{\text{IC}_{50} + [\text{Drug}]}$$

This is derived from the competitive inhibition model. The transporter has a binding site for serotonin with $K_m$. The SSRI competes for a distinct site on SERT. The apparent inhibition constant $K_i$ relates to IC$_{50}$ by the Cheng-Prusoff equation:$K_i = \text{IC}_{50} / (1 + [\text{5-HT}]/K_m)$.

The 80% occupancy threshold: Clinical PET studies show that therapeutic response requires $\geq 80\%$ SERT occupancy. Setting occupancy $= 0.8$:

$$0.8 = \frac{[\text{Drug}]}{\text{IC}_{50} + [\text{Drug}]} \implies 0.8 \cdot \text{IC}_{50} + 0.8 \cdot [\text{Drug}] = [\text{Drug}]$$

$$0.8 \cdot \text{IC}_{50} = 0.2 \cdot [\text{Drug}] \implies [\text{Drug}] = 4 \times \text{IC}_{50}$$

For fluoxetine (IC$_{50} \approx 0.8$ nM at SERT), 80% occupancy requires brain free concentration $\geq 3.2$ nM. For sertraline (IC$_{50} \approx 0.3$ nM), only 1.2 nM is needed, explaining its potency.

Derivation 2: One-Compartment PK Model for SSRIs

After oral administration, the plasma concentration of an SSRI follows a one-compartment model with first-order absorption and elimination. The concentration $C(t)$ after a single dose is:

$$C(t) = \frac{F \cdot \text{Dose}}{V_d} \cdot \frac{k_a}{k_a - k_{el}} \left(e^{-k_{el} t} - e^{-k_a t}\right)$$

where $F$ is oral bioavailability, $V_d$ is volume of distribution, $k_a$ is absorption rate constant, and $k_{el} = \ln 2 / t_{1/2}$ is elimination rate constant. When $k_a \gg k_{el}$(rapid absorption), this simplifies to the monoexponential decay:

$$C(t) \approx \frac{F \cdot \text{Dose}}{V_d} \cdot e^{-k_{el} t}$$

Steady-state concentration: With repeated dosing at interval $\tau$, the average steady-state concentration is derived from the accumulation factor. The total drug exposure per interval (AUC$_\tau$) at steady state equals $F \cdot \text{Dose} / CL$ where$CL = k_{el} \cdot V_d$ is clearance. The average concentration is:

$$C_{ss,\text{avg}} = \frac{F \cdot \text{Dose}}{CL \cdot \tau}$$

For fluoxetine 20 mg/day: $F \approx 0.72$, $CL \approx 40$ L/h (including CYP2D6/3A4 metabolism),$\tau = 24$ h. This gives $C_{ss} \approx 15$ ng/mL, consistent with clinical therapeutic levels of 120-500 ng/mL (including active metabolite norfluoxetine).

Derivation 3: Time to Steady State and Delayed Onset

After initiating chronic dosing, the approach to steady state follows an exponential:

$$C(t) = C_{ss} \cdot \left(1 - e^{-k_{el} t}\right) = C_{ss} \cdot \left(1 - 2^{-t/t_{1/2}}\right)$$

The fraction of steady state achieved after $n$ half-lives is $1 - 2^{-n}$: after 1 half-life (50%), 2 half-lives (75%), 3 half-lives (87.5%), 4 half-lives (93.75%), 5 half-lives (96.875%). By convention, 4-5 half-lives are required to reach "steady state" (>90%).

Fluoxetine complication: Fluoxetine ($t_{1/2} = 2\text{-}6$ days) is metabolized to norfluoxetine, an active metabolite with $t_{1/2} = 4\text{-}16$ days. The effective half-life is dominated by the slower component. The time to steady state for the combined active species is:

$$t_{\text{steady}} \approx 4 \times t_{1/2,\text{norfluoxetine}} \approx 4 \times 10 \text{ days} = 40 \text{ days}$$

This PK explanation accounts for part of the 2-4 week delay in clinical effect. However, SERT occupancy reaches 80% within days, so the additional delay is attributed to downstream neuroadaptations: desensitization of 5-HT$_{1A}$ autoreceptors (removing negative feedback), increased BDNF expression, and hippocampal neurogenesis — all processes requiring weeks to mature.

Derivation 4: PK/PD Link Model with Hysteresis

The relationship between plasma concentration and clinical effect shows hysteresis (the effect lags behind concentration). This is modeled using an effect compartment with equilibration rate constant $k_{e0}$:

$$\frac{dC_e}{dt} = k_{e0} \cdot (C_p - C_e)$$

where $C_e$ is the effect-site concentration. The pharmacodynamic response is then linked to $C_e$ via a sigmoid $E_{\max}$ model:

$$\text{Effect}(t) = E_{\max} \cdot \frac{C_e(t)^n}{EC_{50}^n + C_e(t)^n}$$

The equilibration half-life $t_{1/2,e0} = \ln 2 / k_{e0}$ represents the delay between plasma peak and effect peak. For SSRIs, this is not a simple distributional delay but represents the time course of neuroadaptive changes. The effective $t_{1/2,e0}$ for antidepressant effect is approximately 1-2 weeks, reflecting:

• 5-HT$_{1A}$ autoreceptor desensitization ($t_{1/2} \approx 7$ days)
• Downstream gene expression changes ($t_{1/2} \approx 10$ days)
• Neuroplasticity and synaptogenesis ($t_{1/2} \approx 14$ days)
This composite $k_{e0}$ explains why even drugs with short PK half-lives still require weeks for full clinical effect.

Derivation 1: D2 Occupancy from PET Imaging

PET imaging with [$^{11}$C]raclopride measures D2 receptor availability. The occupancy by an antipsychotic is derived from the reduction in radioligand binding:

$$\text{Occupancy} = \frac{B_{\max} \cdot [\text{Drug}]}{K_d + [\text{Drug}]} = 1 - \frac{BP_{\text{drug}}}{BP_{\text{baseline}}}$$

where $B_{\max}$ is total receptor density, $K_d$ is the drug's equilibrium dissociation constant at D2 receptors, and $BP$ is binding potential measured by PET. The therapeutic window boundaries are:

$$\text{At 60% occupancy: } [\text{Drug}]_{60} = \frac{0.6 \cdot K_d}{1 - 0.6} = 1.5 \cdot K_d$$

$$\text{At 80% occupancy: } [\text{Drug}]_{80} = \frac{0.8 \cdot K_d}{1 - 0.8} = 4.0 \cdot K_d$$

The therapeutic window spans a 2.67-fold range in brain drug concentration ($1.5 K_d$ to$4.0 K_d$). For haloperidol ($K_d \approx 1$ nM), this corresponds to brain free concentrations of 1.5-4.0 nM. The hyperbolic shape of the occupancy curve means that doubling the dose from 80% to near 90% provides minimal additional efficacy but dramatically increases EPS risk.

Derivation 2: Plasma-to-Brain Occupancy Relationship

The link between measurable plasma concentration and brain receptor occupancy involves the free (unbound) fraction $f_u$ and the brain partition coefficient $K_{p,\text{brain}}$:

$$C_{\text{brain,free}} = C_{\text{plasma}} \cdot f_u \cdot K_{p,\text{brain}}$$

Substituting into the occupancy equation:

$$\text{Occupancy} = \frac{C_{\text{plasma}} \cdot f_u \cdot K_{p,\text{brain}}}{K_d + C_{\text{plasma}} \cdot f_u \cdot K_{p,\text{brain}}}$$

Define the apparent plasma $K_d$: $K_{d,\text{plasma}} = K_d / (f_u \cdot K_{p,\text{brain}})$. Then:

$$\text{Occupancy} = \frac{C_{\text{plasma}}}{K_{d,\text{plasma}} + C_{\text{plasma}}}$$

For olanzapine: $K_d = 11$ nM, $f_u = 0.07$, $K_{p,\text{brain}} \approx 10$. Thus$K_{d,\text{plasma}} = 11 / (0.07 \times 10) = 15.7$ nM. This predicts 80% occupancy at$C_{\text{plasma}} = 4 \times 15.7 = 62.9$ nM ($\approx 20$ ng/mL), consistent with clinical observations that olanzapine plasma levels >20 ng/mL increase EPS risk.

Derivation 3: Kapur's "Fast-Off" Theory

Kapur and Seeman proposed that EPS risk depends not just on occupancy level but on the dissociation rate $k_{\text{off}}$ from D2 receptors. The residence time at the receptor is:

$$\tau_{\text{res}} = \frac{1}{k_{\text{off}}}$$

Since $K_d = k_{\text{off}} / k_{\text{on}}$, and $k_{\text{on}}$ is relatively constant across antipsychotics (diffusion-limited, $\sim 10^8$ M$^{-1}$s$^{-1}$), $K_d$ is primarily determined by $k_{\text{off}}$. The key insight is that transient, high occupancy is tolerated if the drug dissociates quickly enough for endogenous dopamine to compete:

Clozapine (atypical, low EPS):$K_d \approx 130$ nM, $k_{\text{off}} \approx 13$ s$^{-1}$,$\tau_{\text{res}} \approx 0.08$ s (fast-off)
Haloperidol (typical, high EPS):$K_d \approx 1$ nM, $k_{\text{off}} \approx 0.1$ s$^{-1}$,$\tau_{\text{res}} \approx 10$ s (slow-off)

The fast-off kinetics allow endogenous dopamine surges (phasic release in the striatum) to transiently displace clozapine, preserving physiological dopamine signaling in the nigrostriatal pathway (motor control) while providing sustained blockade in the mesolimbic pathway (where tonic dopamine is elevated in psychosis). This explains clozapine's unique efficacy in treatment-resistant schizophrenia with minimal EPS.

Derivation 4: Population PK/PD with Interindividual Variability

Individual patients show large variability in drug response due to genetic polymorphisms in metabolizing enzymes (CYP2D6, CYP1A2). Population PK models capture this using mixed-effects:

$$CL_i = CL_{\text{pop}} \cdot \exp(\eta_i), \quad \eta_i \sim \mathcal{N}(0, \omega^2)$$

where $CL_i$ is individual clearance, $CL_{\text{pop}}$ is typical population clearance, and$\eta_i$ is the random effect for individual $i$. For olanzapine, $\omega \approx 0.4$ (40% CV), meaning the 95% range of clearance spans a ~5-fold range ($\exp(\pm 2 \times 0.4)$). The individual steady-state concentration is:

$$C_{ss,i} = \frac{F \cdot \text{Dose}}{CL_i \cdot \tau} = \frac{F \cdot \text{Dose}}{CL_{\text{pop}} \cdot \exp(\eta_i) \cdot \tau}$$

The corresponding D2 occupancy distribution explains why a fixed dose produces therapeutic occupancy (60-80%) in some patients but subtherapeutic (<60%) or toxic (>80%) occupancy in others. CYP2D6 poor metabolizers ($\sim 7\%$ of Caucasians) have ~50% lower clearance, achieving 2-fold higher concentrations — potentially pushing them above the 80% EPS threshold. This motivates pharmacogenomic-guided dosing.

Derivation 1: Three-Compartment Mammillary Model for Propofol

Propofol disposition follows a three-compartment model with a central compartment (blood, well-perfused organs), fast peripheral (muscle, viscera), and slow peripheral (fat):

$$\frac{dC_1}{dt} = -(k_{10} + k_{12} + k_{13})C_1 + k_{21}C_2 + k_{31}C_3 + \frac{R(t)}{V_1}$$

$$\frac{dC_2}{dt} = k_{12}\frac{V_1}{V_2}C_1 - k_{21}C_2$$

$$\frac{dC_3}{dt} = k_{13}\frac{V_1}{V_3}C_1 - k_{31}C_3$$

where $C_1, C_2, C_3$ are compartment concentrations, $k_{10}$ is elimination from central compartment, $k_{12}, k_{21}$ are intercompartmental transfer rates (central ↔ fast peripheral),$k_{13}, k_{31}$ are transfer rates (central ↔ slow peripheral), $R(t)$ is infusion rate (mg/min), and $V_1$ is central volume.

Typical Marsh model parameters for a 70 kg adult: $V_1 = 15.9$ L,$V_2 = 32.4$ L, $V_3 = 202$ L, $k_{10} = 0.119$ min$^{-1}$,$k_{12} = 0.112$ min$^{-1}$, $k_{21} = 0.055$ min$^{-1}$,$k_{13} = 0.042$ min$^{-1}$, $k_{31} = 0.0033$ min$^{-1}$. The slow peripheral compartment (fat) has a very slow equilibration, explaining prolonged recovery after long infusions (context-sensitive half-time).

Derivation 2: BET Scheme for Target-Controlled Infusion

The BET (Bolus-Elimination-Transfer) scheme calculates the infusion rate $R(t)$ needed to achieve and maintain a target plasma concentration $C_{\text{target}}$. The total infusion must compensate for three processes simultaneously:

$$R(t) = R_{\text{bolus}} + R_{\text{elimination}} + R_{\text{transfer}}$$

Bolus: Instantaneously fill the central compartment:$\text{Bolus} = C_{\text{target}} \times V_1$
Elimination: Replace drug lost via metabolism:$R_{\text{elim}} = C_{\text{target}} \times V_1 \times k_{10}$
Transfer: Replace drug distributing to peripheral compartments:

$$R_{\text{transfer}}(t) = C_{\text{target}} \times V_1 \times \left[k_{12} \cdot e^{-k_{21}t} + k_{13} \cdot e^{-k_{31}t}\right]$$

In practice, TCI pumps (e.g., Marsh or Schnider models) recalculate $R(t)$ every 10 seconds, adjusting for the current state of all three compartments. The Schnider model additionally accounts for age, lean body mass, and height, using an effect-site targeting mode where$k_{e0} = 0.456$ min$^{-1}$ (effect-site equilibration half-life $\approx 1.5$ min).

Derivation 3: GABA$_A$ Receptor Binding and Anesthetic Depth

Propofol potentiates GABA$_A$ receptor currents by binding at the transmembrane domain. The fractional receptor activation follows a Hill equation with cooperativity:

$$f_{\text{active}} = \frac{C_e^{n}}{EC_{50}^{n} + C_e^{n}}$$

For propofol at GABA$_A$ receptors, $n \approx 2.8$ (steep concentration-response) and the EC$_{50}$ depends on the clinical endpoint:

• Loss of verbal response: EC$_{50} \approx 2.0$ $\mu$g/mL
• Loss of consciousness: EC$_{50} \approx 3.0$ $\mu$g/mL
• Surgical anesthesia (no movement to incision): EC$_{50} \approx 5.0$ $\mu$g/mL
• Burst suppression on EEG: EC$_{50} \approx 8.0$ $\mu$g/mL

The steep Hill coefficient ($n = 2.8$) means that the transition from consciousness to unconsciousness occurs over a narrow concentration range. At $C_e = 2$ $\mu$g/mL, only$2^{2.8}/(3^{2.8} + 2^{2.8}) = 6.96/(21.67 + 6.96) \approx 24\%$ receptor activation (awake). At $C_e = 4$ $\mu$g/mL, $4^{2.8}/(3^{2.8} + 4^{2.8}) \approx 74\%$ activation (deeply anesthetized). This 2-fold concentration change produces a 3-fold change in activation.

Derivation 4: BIS as Pharmacodynamic Endpoint

The Bispectral Index (BIS) is an EEG-derived measure of anesthetic depth (100 = fully awake, 0 = isoelectric EEG). The relationship between effect-site concentration and BIS follows a sigmoid $E_{\max}$ model:

$$\text{BIS}(C_e) = \text{BIS}_0 - (\text{BIS}_0 - \text{BIS}_{\min}) \cdot \frac{C_e^{\gamma}}{EC_{50,\text{BIS}}^{\gamma} + C_e^{\gamma}}$$

where $\text{BIS}_0 = 97$ (awake baseline), $\text{BIS}_{\min} = 0$ (isoelectric),$EC_{50,\text{BIS}} \approx 3.5$ $\mu$g/mL (concentration for BIS = 50), and$\gamma \approx 3$ (steep Hill coefficient). The target BIS range for general surgery is 40-60. Inverting for the required concentration:

$$C_e = EC_{50,\text{BIS}} \cdot \left(\frac{\text{BIS}_0 - \text{BIS}_{\text{target}}}{\text{BIS}_{\text{target}} - \text{BIS}_{\min}}\right)^{1/\gamma}$$

For BIS = 50: $C_e = 3.5 \times (47/50)^{1/3} = 3.5 \times 0.98 = 3.43$ $\mu$g/mL. For BIS = 40: $C_e = 3.5 \times (57/40)^{1/3} = 3.5 \times 1.13 = 3.94$ $\mu$g/mL. The narrow concentration range (3.4-3.9 $\mu$g/mL) for the entire BIS 40-50 surgical window underscores the precision required of TCI systems.

Derivation 1: Operational Model of Agonism (Black & Leff)

The operational model provides a quantitative framework for understanding full vs partial agonism. In its general form with Hill coefficient $n$:

$$\text{Response} = \frac{[A]^n \cdot \tau^n \cdot E_{\max}}{[A]^n \cdot \tau^n + ([A] + K_A)^n}$$

where $[A]$ is agonist concentration, $K_A$ is the equilibrium dissociation constant,$\tau$ is the efficacy parameter (transducer ratio), and $n$ is the transducer slope. The parameter $\tau = [R_T] / K_E$ where $[R_T]$ is total receptor density and $K_E$ is the concentration of agonist-receptor complex producing half-maximal system response.

How $\tau$ determines agonist type: At maximal receptor occupancy ($[A] \to \infty$), the observed maximum is:

$$E_{\text{obs,max}} = \frac{\tau^n \cdot E_{\max}}{\tau^n + 1}$$

For $\tau = 100, n = 1$: $E_{\text{obs,max}} = 99\%$ of $E_{\max}$ (full agonist).
For $\tau = 1, n = 1$: $E_{\text{obs,max}} = 50\%$ of $E_{\max}$ (partial agonist).
For $\tau = 0.3, n = 1$: $E_{\text{obs,max}} = 23\%$ of $E_{\max}$ (weak partial agonist).
The critical insight is that $\tau$ is a property of the drug-receptor-tissue system — the same drug can be a full agonist in a tissue with high receptor density (large $[R_T]$) but a partial agonist in a tissue with low receptor density.

Derivation 2: Buprenorphine — Ceiling Effect on Respiratory Depression

Buprenorphine is a $\mu$-opioid partial agonist with $\tau \approx 0.8$ at respiratory centers (brainstem) but $\tau \approx 2$ for analgesia (spinal cord, where receptor density is higher). The dose-response for respiratory depression:

$$\text{Resp. depression} = \frac{E_{\max} \cdot \tau_{\text{resp}} \cdot [\text{Bup}]}{K_A + (1 + \tau_{\text{resp}}) \cdot [\text{Bup}]}$$

At maximal occupancy, the ceiling is:$E_{\text{ceiling}} = E_{\max} \cdot \tau_{\text{resp}} / (1 + \tau_{\text{resp}}) = E_{\max} \cdot 0.8 / 1.8 \approx 44\%$of maximal respiratory depression. Compare with morphine ($\tau \approx 30$):$E_{\max,\text{morph}} = 30/31 \approx 97\%$ — nearly full respiratory depression.

This ceiling effect provides a critical safety margin. Even at supratherapeutic doses, buprenorphine cannot produce more than ~44% of the maximal respiratory depression that a full agonist like morphine or fentanyl can. The dose-response curves:

• Morphine: steep, reaches ~97% maximal effect, lethal respiratory depression possible
• Buprenorphine: plateaus at ~44%, clinically significant overdose much harder to achieve
• Fentanyl ($\tau \approx 200$): supramaximal efficacy, respiratory arrest at moderate doses

Derivation 3: Competitive Displacement and Precipitated Withdrawal

Buprenorphine has extremely high $\mu$-receptor affinity ($K_d \approx 0.2$ nM) and slow dissociation ($k_{\text{off}} \approx 0.007$ min$^{-1}$, $\tau_{\text{res}} \approx 140$ min). When introduced to a patient on full agonists (heroin $K_d \approx 5$ nM, fentanyl $K_d \approx 1$ nM), competitive displacement occurs:

$$\theta_{\text{full}} = \frac{[\text{Full}]/K_{d,\text{full}}}{1 + [\text{Full}]/K_{d,\text{full}} + [\text{Bup}]/K_{d,\text{bup}}}$$

$$\theta_{\text{bup}} = \frac{[\text{Bup}]/K_{d,\text{bup}}}{1 + [\text{Full}]/K_{d,\text{full}} + [\text{Bup}]/K_{d,\text{bup}}}$$

As buprenorphine concentrations rise, it displaces the full agonist due to its 5-25x higher affinity. The net receptor activation drops from near-maximal (full agonist, $\tau \gg 1$) to the buprenorphine ceiling ($\sim 44\%$). This abrupt decrease in receptor activation precipitates withdrawal symptoms.

Clinical implication: Buprenorphine induction must wait until the patient is in mild-moderate withdrawal (COWS score $\geq 12$), meaning most full agonist has already dissociated. The "micro-dosing" (Bernese method) protocol gives tiny buprenorphine doses (0.5 mg) to gradually replace the full agonist over 7-10 days, avoiding the abrupt displacement that causes precipitated withdrawal.

Derivation 4: Varenicline — Nicotinic Partial Agonist Dual Mechanism

Varenicline targets $\alpha_4\beta_2$ nicotinic acetylcholine receptors (nAChRs) with a dual mechanism derived from its partial agonist properties ($\tau \approx 0.6$):

Mechanism 1 — Withdrawal prevention: Varenicline provides ~45% of maximal receptor activation ($E = E_{\max} \times 0.6/1.6 = 0.375 \times E_{\max}$). This is sufficient to stimulate mesolimbic dopamine release (preventing craving/withdrawal) but at a much lower level than nicotine ($\tau \approx 5$, producing $\sim 83\%$ activation):

$$E_{\text{varenicline}} = \frac{0.6 \cdot E_{\max}}{1.6} \approx 37.5\% \cdot E_{\max}$$

Mechanism 2 — Reward blockade: When the patient smokes, nicotine must compete with varenicline for receptor binding. Since varenicline has higher affinity ($K_d = 0.4$ nM vs nicotine $K_d = 15$ nM) and slow $k_{\text{off}}$, nicotine cannot fully displace it. The combined activation is limited by varenicline's ceiling:

$$E_{\text{combined}} \approx E_{\text{varenicline}} + \Delta E_{\text{nicotine}} \approx 37.5\% + \text{small increment}$$

The reward from smoking is thus blunted (reduced dopamine surge compared to the usual ~83% activation). Over time, the learned association between smoking and reward is extinguished. Varenicline increases smoking cessation rates 2-3x vs placebo by simultaneously reducing withdrawal (agonist effect) and reducing reward (competitive antagonist effect).