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Two new parameters for characterizing ligand binding systems.

Andrew Stein

October 17, 2017

ACoP8 – Mathematical Pharmacology

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Overview

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Differences between small molecules and monoclonal antibodies

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Property	Small Molecules	Monoclonal Antibodies (IgG1)
Molecular Weight	500 Da	150,000 Da
Half-Life (human)	minutes - hours	3 weeks
Affinity for target	Moderate to high	Very high

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Soluble vs Membrane-Bound Targets

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Drug (D)

Target (T)

Complex (DT)

Soluble

Target

Total Target (T+DT)

Accumulation

t1/2 ≈ 21 days

t1/2 ≈ 1h

t1/2 ≈ 21 days

t1/2 ≈ 1 hour

Membrane-bound

Target

Nonlinear PK

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AS

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Characterizing onset of PK nonlinearity for membrane-bound targets

In collaboration with Bert Peletier at Leiden University in The Netherlands

Manuscript in progress. Contact andrew.stein@novartis.com if interested in further details.

t1/2 ≈ 21 days

t1/2 ≈ 1 hour

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Nonlinear pharmacokinetics occurs at “critical concentration” = C_crit

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Ccrit

Burmester, G. R. et al. Mavrilimumab, a human monoclonal antibody targeting gm-csf receptor-α, in subjects with rheumatoid arthritis: a randomised, double-blind, placebo- controlled, phase I first-in-human study. Annals of the rheumatic diseases 70, 1542–1549 (2011).

Choose dose so conc. stays above Ccrit

Maintain target engagement
Reduce PK variability

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Michaelis-Menten approximation of ligand binding model

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C

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T

CT

C_P

CL/V_c

Dose

k_syn

k_eT

k_eCT

K_ss

Ligand Binding Model

(Membrane-Bound)

Approximation derived in: Ma, Peiming. "Theoretical considerations of target-mediated drug disposition models: simplifications and approximations." Pharmaceutical research 29.3 (2012): 866-882.

V_m = k_syn·V_c

K_m = K_ss

k₁₂

k₂₁

C

C_P

CL/V_c + V_m/(C+K_m)

Dose

≈

Michaelis-Menten

Model

k₁₂

k₂₁

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“Derivation” for C_crit

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For large doses (and concentrations), C ≫ K_m

Define C_crit to be where the linear (CL·C) and nonlinear (V_m) components contribute equally to total elimination

A = C·V_c A_p = C_p·V_p

A_tot =

A + A_p

A

A_P

CL/V_c + V_m/(C+K_m)

k₁₂

k₂₁

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Sensitivity analysis for Ccrit

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Ccrit helps in understanding parameter identifiability

Deriving initial parameter estimates: V_m = C_crit·CL_{from NCA}
Understanding which model parameters are identifiable

Vm = ksyn·Vc is often easiest “TMDD” parameter to estimate from nonlin. PK
Km (from steep elimination phase) can be more difficult to identify
R₀ (receptor density) is often impossible to estimate from PK alone [1]

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Month

Concentration (nM)

1. Stein, Andrew. "Practical unidentifiability of receptor density in target mediated drug disposition models can lead to over-interpretation of drug concentration data." bioRxiv (2017): 123240.

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Random effect on Vm is important for describing variable Ccrit.

Some pts exhibited MM kinetics while others appear linear.
Ccrit varied between patients.
PopPK required random effect on Vm to characterize this heterogeneity

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Characterizing soluble target inhibition

Stein, A. M., and Ramprasad Ramakrishna. "AFIR: A dimensionless potency metric for characterizing the activity of monoclonal antibodies." CPT: pharmacometrics & systems pharmacology 6.4 (2017): 258-266.

Additional work done at 2017 Math-to-Industry Bootcamp at the Institute of Mathematics and its Applications at University of Minnesota by Sameed Ahmed, Miandra Ellis, Ngartelbaye Guerngar, Hongshan Li, Luca Pallucchini

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t1/2 ≈ 21 days

t1/2 ≈ 1h

t1/2 ≈ 21 days

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Ligand binding model (soluble target)

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C

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T

CT

C_P

CL/V_c

Dose

k_syn

k_eT

k_eCT

K_d

Total Target (Ttot) Assay

Measures both free and bound target

k₁₂

k₂₁

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Sensitivity analysis. How does inhibition relate to parameters?

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Total Target Conc.

(Ttot = T + CT)

150 mg

28% free

300 mg

14% free

subcutaneous

dosing

Above 150 mg,

total target conc. plateaus

Above 150 mg, doubling dose halves the free target conc.

600 mg

7% free

T_tot,ss

T₀

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Estimating the free target compared to baseline

Three key parameters that determine target inhibition

Kd: Binding affinity (Kd)
Tacc: Target accumulation at steady state
Cavg: Average drug concentration at steady state

Doubling dose or halving Kd reduces free target by 50%

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* L₅₀ was recently derived in: Gabrielsson et al., Pharmacology & Therapeutics, https://doi.org/10.1016/j.pharmthera.2017.10.011

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AFIR can also be applied to a target tissue (in progress)

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D₁

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T₁

DT₁

D₂

Dose

D₃

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T₃

DT₃

Target

Tissue (3)

Peripheral (2)

Central (1)

B = tissue biodistribution coefficient,
≈ 30% is often assumed
Target accumulation (Tacc) in tissue is difficult to measure. Often assumed = 1

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Using AFIR, sensitivity to assumptions is easy to report

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Development of oral drug with same target as subcutaneous biologic

Challenges

Oral biologics denatured in stomach
Small molecules will have poorer affinity (Kd) and higher elimination (CL)

How will new drug compare to first generation drug?

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Applying AFIR to 2^nd generation drugs

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F	Bioavailability
fu	Fraction Unbound (small molecule)
CL	Clearance
τ	Dosing Interval

Kd	Binding Affinity
T_acc	Target Accumulation
C_avg	Avg. Free Drug Conc.

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Using AFIR, other drug candidates can be rapidly assessed

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			subcutaneous mAb (150 kDa)	Oral biologic (50 kDa)	Oral, small molecule (500 Da)
Affinity	Kd	nM	0.2	0.002	100
Accumulation	Tacc	-	200	200	1
Clearance	CL	L/day	0.17	1	100
Dose Interval	τ	day	30	1	1
Bioavailability	F	-	0.76	0.002	0.1
Free fraction	fu	-	1	1	0.1
Dose at 300 mg	Dose	nmol	2000	6000	600,000
AFIR		-	14%	14%	≈100%

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Greater efficacy is observed at 2x dose, even though total target has plateaued.

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Total Target Data

Similar for X mg and 2X mg

Efficacy Metric

Superior for 2X dosing

Week

Conc. (nM)

2x dose

x dose

AFIR goes from 28% to 14% after doubling dose.

It is the 200x accumulation of target (T_acc) that necessitates high doses

T_acc = 200x

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Identifiability –target inhibition can be estimated without baseline levels

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Varying T0,

keeping T0/Kd fixed

LOQ

Same % inhibition

for different baseline T0

See poster T-087 for application to 4 compounds

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Two new parameters for characterizing ligand binding systems

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Acknowledgements

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Mick Looby
Bert Peletier
Prasad Ramakrishna
Henning Schmidt
Jean-Louis Steimer
Phil Lowe
Karthik Subramanian
Bruce Gomes
Wenping Wang

Kostas Biliouris
Alison Margolskee
Jaeyeon Kim
Siyan Xu
Brian Stoll
Gerard Bruin
Irina Koroleva
Frank Kolbinger
Max Woisetschlaeger

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Backups

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Calculating Free Target vs Total Target

Algebraic manipulation of quasi-equilibrium equation

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1) Free Target

vs Total Target

When Dtot >> Ttot, Kd

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The Quasi-Equilibrium Assumption

Assumption: binding is rapid such that drug, target, and complex stay in equilibrium.

This assumption allows us to reduce number of state variables from four [D, T, DT, D_P] to three [D_tot, T_tot, D_P] plus an algebraic set of equations

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Analysis of total target dynamics before dosing

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At steady state: dT_tot/dt = 0

Before Dose, DT=0

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Analysis of total target dynamics �after dosing

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T_acc

At steady state: dT_tot/dt = 0

For large drug conc.,

most target is bound,

T ≈ 0, and DT ≈ T_tot

2) Total Target

accumulation

T_tot

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Is there an optimal Kd? It depends on definition of “optimal”

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1. Tiwari, Abhinav, et al. "Optimal Affinity of a Monoclonal Antibody: Guiding Principles Using Mechanistic Modeling." The AAPS journal 19.2 (2017): 510-519.

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Tiwari defines “optimal” Kd with respect to dose reduction

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Questions for TMDD models

What % of drug gets into the tumor (B)?
Can soluble targets accumulate in tumor? Or other tissue?
Future research

Account for both membrane-bound and soluble ligands + shedding
Account for competitive endogenous ligand
Account for spatial heterogeneity in the tumor

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