experimentaldesigninvalidatingmodelsof infectiousdiseases · institution-logo introduction model...

institution-logo

IntroductionModel Selection

Experimental DesignBacteremia

Summary

Experimental Design in Validating Models ofInfectious Diseases

D. M. Bortz

Department of Applied MathematicsUniversity of ColoradoBoulder, CO 80309

April 26, 2008ACSMSBB, Raleigh, NC

D. M. Bortz Experimental Design in Validating Models of Infectious Diseases

institution-logo



Summary

Outline

1 Introduction

2 Model Selection

3 Experimental Design

4 Bacteremia

5 Summary


institution-logo



Summary

Motivating Ideas

Many proposed models for pathogenesis of infectious diseasesQuantitative metric to decide best modelExperimental data to support/refute

modelselection ranking


institution-logo



Summary

Conclusions

1 Model selection in HIV1 DMB & PWN, Bull. Math. Biol., 2006.2 DMB & JC, in progress.

2 Experimental Design1 DMB, et. al., Bull. Math. Biol., 2008.2 DMB, JGY, et. al., in progress.

3 Bacteremia1 Model in Chung, Cartwright, DMB, et.al., Shock, 2008.2 Take blood data early (before 5 hours).


institution-logo



Summary

Function and Structure Selection

Ho, et.al. Nature 1995Perelson, et.al., Science1996

Model Feature: Viral ProdExponential Decay ~109

Infected T ∗ Cells ~1010

Density Dependence ~1010

Latently Infected Cells ~1010

Eclipse Phase (Delay) ~1011

T T ∗

VI

A


institution-logo



Summary








T T ∗

VI

A

L


institution-logo



Summary








T T ∗

VI

A

L

VNI


institution-logo



Summary

Model Selection Criteria

Stochastic Complexity

SIC = − log(L(Θ|Data)) +K2log

M2π

+12log |I (Θ)|

= Fit penalty + Complexity penalty

Grünwald, 2007; Hansen & Yu, JASA, 2001.

Rissanen’s Minimum Description Length principleShortest possible code length, given a class of modelsNumber of distinguishable distributionsAIC, BIC for large sample sizesgzip

Mass-spring oscillator : x(t) = A cos(ωt + φ); ω =√

km


institution-logo



Summary

HIV Experiment

0 2 4 6 8 10 12 14 16 18 20 22

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

Days

Pla

sma

HIV

RN

A c

opie

s/m

l

Patient 201 on Reverse Transcriptase Inhibitor Therapy

Given finite resources, what sampling strategy should be employed?


institution-logo



Summary

Numerical Results

DMB & PWN, Bull. Math. Biol., 2006; DMB & JC, in progress.


institution-logo



Summary

Experimental Design

Toy example

Quadratic model, linear parametersy = f (x) + ε = β0 + β1x + β2x2 + ε with ε ∼ N(0, σ2)

Given (x1, x2, . . . , xM), the design matrix in linear regression is

F =

1 x1 x2

11 x2 x2

2...

......

1 xM x2M

=

f (x1)T

f (x2)T

...f (xM)T

Variances

var(β) = σ2(FTF )−1

var{y(x)} = σ2f (x)T (FTF )−1f (x)

d(x ,P) = Mf (x)T (FTF )−1f (x)


institution-logo



Summary

3 and 4 Point Designs

|(FTF )−1| = 1/4

L∞-norm=3

|(FTF )−1| = 1/7.0233

L∞-norm=3.8

Minimize |(FTF )−1| (parameter estimate variance)OR L∞-norm (model prediction variance).


institution-logo



Summary

Existing Mathematical Framework

General Measure of Imprecision Ψ{P} for P a data observationdistribution on X .

D-optimal Ψ{P} = − log |I (P)|, where I (P) =RX f (x)f (x)TdP(x)

G-optimal Ψ{P} = maxx∈X Mf (x)T (FTF )−1f (x)

General Equivalency Theorem (Kiefer & Wolfowitz)Compact domain for XConvexity and Differentiability of ΨTFAE (asymptotic) for

φ(x ,P) = limh→0+

1h

(Ψ{I (P)− h(I (Px)− I (P))} −Ψ{I (P)})

1 P∗ minimizes Ψ{I (P)}2 minx∈X φ(x ,P∗) ≥ 03 φ(x ,P∗) achieves its minimum at points in the design.


institution-logo



Summary

Model Selection

Complexity (statistical regret) term

K2log

N2π

+12log |I (Θ)|

contains D-optimality condition!Gradient in direction of pdf with one point at x (think Gateauxdifferentiability)

φ(x ,P) = limh→0+

1h

(Ψ{I (P)− h(I (Px)− I (P))} −Ψ{I (P)})

= K − d(x ,P)

= K −Mf (x)T (FTF )−1f (x)


institution-logo



Summary

Generalized Sensitivity

For which data “locations” is the parameter variance most dramaticallyreduced?

gsinc(i) =

M1M2∑

j=1

F (j)F (j)T

−1

f (i)

• f (i) , (1)

Tomaseth & Cobelli, An. of Biomed. Eng., 1999.


institution-logo



Summary

Generalized Sensitivity

Flow Cytometry


institution-logo



Summary

Finite Resources

Given M points, where do we place them?Select P to minimize − log |I (P)|


institution-logo



Summary

Central Venous Catheter


institution-logo



Summary

Disseminated Bacterial Infections

Dislodgement from biofilm, followed by establishment of newinfection.Can lead to sepsis (widespread activation of inflammation andcoagulation pathways).The leading cause of death in non-coronary ICU patients, and thetenth most common cause of death overall (CDC).Opportunity: Think of heart disease understanding in 1950.


institution-logo



Summary

Staphylococcus epidermidis


institution-logo



Summary

Mouse model


institution-logo



Summary

Mouse equation

LungdLdt

= (aL − dL)L +qL

vLB − qL

vLpLL

SpleendSdt

= (aS − dS)S +qS

vSB

LiverdHdt

= (aH − dH)H +qH

vHB +

qS

vSpSS − qH + qS

vHpHH

BlooddBdt

= (aB − dB)B +qH + qS

vHpHH +

qL

vLpLL−

(qH

vH+

qS

vS+

qL

vL

)B


institution-logo



Summary

Mouse Simulation


institution-logo



Summary

Identify Partitioning Coefficients pL, pH , & pS


institution-logo



Summary

Mouse imaging

Chung, Cartwright, Bortz, Jackson, & Younger, Shock, 2008.D. M. Bortz Experimental Design in Validating Models of Infectious Diseases

institution-logo



Summary

Conclusions

1 Model selection in HIV1 DMB & PWN, Bull. Math. Biol., 2006.2 DMB & JC, in progress.

2 Experimental Design1 DMB, et. al., Bull. Math. Biol., 2008.2 DMB, JGY, et. al., in progress.

3 Murine Bacteremia1 Model in Chung, Cartwright, DMB, et.al., Shock, 2008.2 Take blood data early (before 5 hours).


institution-logo



Summary

Future Work

Relationship between luminescence and bacterial loadSufficiencyDistinguishabilityMaterial costConvergence toward “correct” ordering


institution-logo



Summary

Acknowledgments

NIH NIGMSNSF MCTPJ. G. Younger (Michigan)J. Corcoran (CU)K. Wilson (CU/NWU)M. J. Solomon (Michigan)D. Hohne (Michigan)M. Cartwright (Michigan)A. J. Christlieb (Michigan State)


experimentaldesigninvalidatingmodelsof infectiousdiseases · institution-logo introduction model...

Documents