Sigmoid Emax (Hill) Model¶

Extended Emax model with Hill coefficient (gamma) controlling the steepness of the concentration-response curve.

Model Overview¶

graph LR
    C[Plasma<br/>Concentration] -->|Hill Function| E[Effect]

    subgraph "Sigmoid Emax"
        direction TB
        E0[E0: Baseline]
        EM[Emax: Maximum]
        EC[EC50: Potency]
        G[gamma: Steepness]
    end

Clinical Applications¶

Neuromuscular blocking agents
Ion channel blockers
Drugs with cooperative binding
Threshold effects
All-or-none responses
Antibiotics (Hill-Langmuir model)

When to Use¶

Use When	Don't Use When
Steep dose-response	gamma = 1 (use Direct Emax)
Threshold effects	Time delays needed
Cooperative binding	Complex hysteresis
Switch-like responses	Tolerance development

Mathematical Formulation¶

Parameters¶

Parameter	Symbol	Units	Description	Constraints
Baseline effect	E0	varies	Effect with no drug	any real
Maximum effect	Emax	varies	Maximum drug-induced change	typically > 0
Half-maximal concentration	EC50	mg/L	Concentration at 50% Emax	EC50 > 0
Hill coefficient	gamma (n)	-	Steepness/cooperativity	gamma > 0

Hill Equation¶

\[E(C) = E_0 + \frac{E_{max} \cdot C^\gamma}{EC_{50}^\gamma + C^\gamma}\]

Equivalent formulation (more stable numerically):

\[E(C) = E_0 + \frac{E_{max}}{1 + (EC_{50}/C)^\gamma}\]

Effect of Gamma¶

Gamma Value	Curve Shape	Biological Interpretation
gamma < 1	Shallow, gradual	Negative cooperativity
gamma = 1	Standard Emax (hyperbolic)	Simple binding
gamma > 1	Steep, sigmoidal	Positive cooperativity
gamma >> 3	Near-threshold	All-or-none response

Model Behavior¶

Concentration-Effect Curves by Gamma¶

Effect
   │
Emax├──────────────────────────────
    │         ╭───gamma=3──────────
    │       ╭─╯
    │      ╱  ╭──gamma=1 (Direct)──
    │     ╱  ╱
    │    ╱  ╱   ╭──gamma=0.5───────
    │   ╱  ╱   ╱
E0+½├──╳──╳───╳────────────────────
    │  │
 E0 ├──┴──────────────────────────→ C
     EC50

Steepness at EC50¶

The slope at EC50 (inflection point):

\[\frac{dE}{dC}\bigg|_{C=EC_{50}} = \frac{\gamma \cdot E_{max}}{4 \cdot EC_{50}}\]

Higher gamma = steeper transition around EC50.

Julia API¶

Type Definitions¶

# Model kind
struct SigmoidEmax <: PDModelKind end

# Parameters
struct SigmoidEmaxParams
    E0::Float64      # Baseline effect
    Emax::Float64    # Maximum effect
    EC50::Float64    # Half-maximal concentration (mg/L)
    gamma::Float64   # Hill coefficient
end

Basic Usage¶

using NeoPKPD

# Define Sigmoid Emax parameters
# gamma = 3 gives steep response
pd_params = SigmoidEmaxParams(0.0, 100.0, 2.0, 3.0)

# Create specification
pd_spec = PDSpec(SigmoidEmax(), "response", pd_params)

# Evaluate across concentration range
concentrations = [0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 4.0, 6.0]
effects = evaluate(pd_spec, concentrations)

for (c, e) in zip(concentrations, effects)
    println("C = $c: Effect = $(round(e, digits=1))%")
end

Expected Output (gamma=3):

C = 0.0: Effect = 0.0%
C = 0.5: Effect = 1.5%
C = 1.0: Effect = 11.1%
C = 1.5: Effect = 29.7%
C = 2.0: Effect = 50.0%
C = 2.5: Effect = 66.1%
C = 3.0: Effect = 77.1%
C = 4.0: Effect = 88.9%
C = 6.0: Effect = 96.4%

Comparing Different Gamma Values¶

using NeoPKPD

gamma_values = [0.5, 1.0, 2.0, 3.0, 5.0]
EC50 = 2.0
Emax = 100.0

for gamma in gamma_values
    params = SigmoidEmaxParams(0.0, Emax, EC50, gamma)
    spec = PDSpec(SigmoidEmax(), "test", params)

    # Effect at different concentrations
    e_low = evaluate(spec, [EC50/2])[1]   # Below EC50
    e_mid = evaluate(spec, [EC50])[1]      # At EC50
    e_high = evaluate(spec, [EC50*2])[1]   # Above EC50

    println("gamma=$gamma: E(EC50/2)=$(round(e_low, digits=1)), ",
            "E(EC50)=$(round(e_mid, digits=1)), ",
            "E(2×EC50)=$(round(e_high, digits=1))")
end

Expected Output:

gamma=0.5: E(EC50/2)=41.4, E(EC50)=50.0, E(2×EC50)=58.6
gamma=1.0: E(EC50/2)=33.3, E(EC50)=50.0, E(2×EC50)=66.7
gamma=2.0: E(EC50/2)=20.0, E(EC50)=50.0, E(2×EC50)=80.0
gamma=3.0: E(EC50/2)=11.1, E(EC50)=50.0, E(2×EC50)=88.9
gamma=5.0: E(EC50/2)=3.0, E(EC50)=50.0, E(2×EC50)=97.0

Derived Parameters¶

Concentration at Any Fraction of Emax¶

For fraction f (0 < f < 1):

\[C_f = EC_{50} \cdot \left(\frac{f}{1-f}\right)^{1/\gamma}\]

# Helper function
function concentration_at_fraction(EC50::Float64, gamma::Float64, fraction::Float64)
    return EC50 * (fraction / (1 - fraction))^(1/gamma)
end

# Example: Find EC10 and EC90 for different gamma
EC50 = 2.0
for gamma in [1.0, 2.0, 3.0]
    EC10 = concentration_at_fraction(EC50, gamma, 0.1)
    EC90 = concentration_at_fraction(EC50, gamma, 0.9)
    ratio = EC90 / EC10

    println("gamma=$gamma: EC10=$(round(EC10, digits=2)), ",
            "EC90=$(round(EC90, digits=2)), ",
            "EC90/EC10=$(round(ratio, digits=1))")
end