Sebastian Ueckert, Joakim Nyberg, Andrew C. Hooker

Application of Quasi-Newton Algorithms in Optimal Design

Pharmacometrics Research GroupDepartment of Pharmaceutical BiosciencesUppsala UniversitySweden

Outline

1. Optimizing Designs

2. Introduction: Quasi-Newton Methods (QNMs)

3. Performance QNMs

4. Advantages QNMs

5. Laplace Approximation for Global Optimal Design

6. Using QNMs in Laplace Approximation

2

Optimizing a Design

( ) ( , )j x f F x

* arg max ( )x X

x j x

Model

e.g. D-Optimal Design ( ) ,j x F x

Parameters α Design variables x

3

DataEstimation

Optimization

• Interval methodsTrue global optimizersHard to implementStill under development

• Stochastic methods – Simulated Annealing (SA), Ant colony optimization, Genetic Algorithm(GA) Easy to implement (SA)

Marketing effective (GA)SlowNo information about solutionHeuristic

4

Optimization

• Derivative free methods– Downhill Simplex Method

No derivatives necessaryRobustSlowLocal

5

Gradient Based Methods

6

Gradient Based Methods

7

Gradient Based Methods

8

Gradient Based Methods

9

Gradient Based Methods

10

Gradient Based Methods

11

Gradient Based Methods

Mathematically well understoodFast (if OFV calc not too expensive) Only localComplicated to implement

• Steepest Descent• Conjugate Gradient

12

1. Set xk=x0

2. Determine search direction

3. Do line search along p* to find minimal xk+1

4. Set xk=xk+1 and go to 2

Newton Method

212( ) : ( )T T

k k k k kf x p f p f p f p m p 2( ) 0k k km p f f p

* arg max ( )x R

x f x

* 2 1k kp f f

Goal:

Algorithm:

13

1. Set xk=x0

2. Determine search direction

3. Do line search along p* to find minimal xk+1

4. Set xk=xk+1 and go to 2

Newton Method

212( ) : ( )T T

k k k k kf x p f p f p f p m p 2( ) 0k k km p f f p

* arg max ( )x R

x f x

* 2 1k kp f f

Goal:

Algorithm:

Calculate Hessian

14

1. Set xk=x0, Bk=I2. Determine search direction

3. Do line search along p* to find minimal xk+1

4. Set xk=xk+1, Bk=Bk+Uk and go to 2

Quasi-Newton Methods

12( ) : ( )T T

k k k k k kf x p f p f p B f p m p

( ) 0k k km p f B p * 1

k kp B f

Algorithm:

Calculation of Hessian is computationally expensiveProblem:

Approach: Use approx. Hessian and build up during search

15

Quasi-Newton Methods

• Different methods for different updating formulas

– Davidon–Fletcher–Powell (DFP)

– Broyden-Fletcher-Goldfarb-Shanno (BFGS)

1

TTk k k kk k

k k T Tk k k k k

B x B xy yB B

y x x B x

1

T T Tk k k k k k

k kT T T T Tk k k k k k

y x y x y yB I B I

y x y x y x

1k k ky f f 1k k ks x x

16

Constraints

• Experiments usually come with practicality constraints e.g.:– Administered dose has to be smaller than X mg– Sampling times can only be taken until 8 h after dosing

i i iu x b Box Constraints

BFGS-B17

BFGS-B

1. Set xk=x0, Bk=I2. Determine search direction

3. Project search direction vector on feasible region4. Do line search along p* to find minimal xk+1 respecting bounds5. Set xk=xk+1, Bk=Bk+Uk and go to 2

Algorithm:

12( ) : ( )T T

k k k k k kf x p f p f p B f p m p

( ) 0k k km p f B p * 1

k kp B f

18

Comparison

• Test Scenario– Model:

• PKPD (1 cmp oral absorption; IMAX drug effect)• All parameters (ka,CL,V,IC50, E0, IMAX) with log-normal IIV 30% CV• PK parameters fixed• Combined error

– Design:• 3 groups (40,30,30 subjects)• 1 PK and 1 PD sample per subject

• Approach:– Generate random initial values – Optimize with steepest descent and BFGS

19

Results

BFGS Steepest Descent

01

02

03

04

05

06

0

15.03

60.84

Runti

me

[s]

Freq

uenc

y[%

]

Steepest Descent

BFGS 20

OFV0 2 4 6 8

x 1010

0

5

10

15

20

25

30

Design Sensitivity

• Approximate Hessian matrix can be used to assess sensitivity of design (at no additional computational costs)– Diagonal of the inverse of the Hessian– Use approximate efficiency

*

( )( )

j xEff x

j x

* * * *12( ) T Tj x a j a j a B a

* * *12

*( )

T Tj a j a B aEff a

j

21

Design Sensitivity - Visual

7 7.5 8 8.5 9 9.50.9

0.92

0.94

0.96

0.98

1

1.02

6 6.5 7 7.5 8 8.50.9

0.92

0.94

0.96

0.98

1

1.02

1 1.5 2 2.5 30.9

0.92

0.94

0.96

0.98

1

1.02

Group 2 PD Group 1 PK Group 1 PD

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Design Sensitivity - Numerical

PK Sample PD Sample

Group 1

7.12 [0.35;13.9]8.38[5.28;11.38]

Group 2

1.26 [0;3.74]1.79[1.03;2.55]

Group 3

9.22 [-1.31E;+1.31E] 0[0;0.0025]

0 20 40 60 80 10080

85

90

95

100

Group 2 PD

0 20 40 60 80 10080

85

90

95

100

0 20 40 60 80 10080

85

90

95

100

Group 1 PK Group 3 PK23

LAPLACE APPROXIMATION

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Global Optimal Design

• Integral has to be evaluated• FIM occurs in integrand• For example ED optimal design:

• Usually evaluated with Monte-Carlo integrationComputationally intensive or imprecise

( ) ( ) ( , )EDj x p F x d

25

Laplace Approximation

,( ) ( ) ( , ) k xEDj x p F x d e d

, : log ( ) ( , )k x p F x

,

2

1

2 ,

mk x

me

k x

arg min ,m k x

26

Laplace Approximation

1. Minimize

2. Calculate the Hessian

3. Evaluate

Algorithm:

, : log ( ) ( , )k x p F x

2 ,mk x

,

2

1

2 ,

mk x

me

k x

B

27

Laplace-BFGS Approximation

1. Minimize using BFGS algorithm

2. Evaluate

Algorithm:

, : log ( ) ( , )k x p F x

,1

2

mk xe

B

28

Laplace-BFGS – Random Effects

arg min ,m k x

( )g e

Problem:

Approach:

For variance parameter α ≥ 0

Perform optimization on log-domain

1. Minimize using BFGS algorithm

2. Rescale approximate Hessian

3. Evaluate

, : log ( ( )) ( ( ), )k x p g F g x

( ),1

2

mk g xe

B

Algorithm:

1TB B g g

29

Comparison

• Comparison of 4 algorithms:1. Monte Carlo integration with random sampling (MC-RS)2. Monte Carlo integration with Latin hypercube sampling (MC-LHS)3. Laplace integral approximation (LAPLACE)4. Laplace integral approximation with BFGS Hessian (LAPLACE-BFGS)

• Testing MC methods with 50 and 500 random samples

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Comparison

• Test Scenario– Model:

• 1 cmp IV bolus• CL,V with log-normal IIV• Additive error

– Design:• 20 subjects• 2 samples per subject

– Parameter distribution:• Log-normal an all parameters (Fixed effect Var=0.05; Random Effect

Var=0.09)

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Results - OFV

Method Mean OFV1021 [95% CI]MC-RS 100,000 3.24MC-RS 50 3.27[2.2-5.0]MC-RS 500 3.33[2.8-3.8]MC-LHS 50 3.24[2.2-4.6]MC-LHS 500 3.22[2.9-3.7]LAPLACE 2.95LAPLACE-BFGS 3.01

Mean OFV and non-parametric confidence intervals for different integration methods from 100 evaluations

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Results - DesignMC-RS 50 MC-LHS 50 LAPLACE

LAPLACE-BFGSMC-RS 500 MC-LHS 500

33

Results – Runtimes

MC-LHS 50 MC-RS 50 LAPLACE-BFGS LAPLACE MC-LHS 500 MC-RS 500

01

23

4

0.35 0.37 0.460.63

3.533.67

Ru

nti

me [

s]

34

Conclusions

• Quasi-Newton methods constitute fast alternative for continuous design variable optimization

• Information about design sensitivity can be obtained with no additional cost

• Global Optimal Design:– Monte-Carlo methods are easy and flexible but need high number of

samples to give stable results– Laplace approximation constitutes fast alternative for priors with

continuous probability distribution function– Laplace integral approximation with BFGS Hessian gave same sampling

times with approx. 30% shorter runtimes

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THANK YOU!

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References

1) C.G. Broyden, “The Convergence of a Class of Double-rank Minimization Algorithms 1. General Considerations,” IMA J Appl Math, vol. 6, Mar. 1970, pp. 76-90.

2) R. Fletcher, “A new approach to variable metric algorithms,” The Computer Journal, vol. 13, 1970, p. 317.

3) D. Goldfarb, “A family of variable-metric methods derived by variational means,” Mathematics of Computation, 1970, pp. 23–26.

4) D.F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Mathematics of Computation, 1970, pp. 647–656.

5) R.H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Comput., vol. 16, 1995, pp. 1190-1208.

6) M. Dodds, A. Hooker, and P. Vicini, “Robust Population Pharmacokinetic Experiment Design,” Journal of Pharmacokinetics and Pharmacodynamics, vol. 32, Feb. 2005, pp. 33-64.

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