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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
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Outline
1 Introduction
2 Model Selection
3 Experimental Design
4 Bacteremia
5 Summary
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Motivating Ideas
Many proposed models for pathogenesis of infectious diseasesQuantitative metric to decide best modelExperimental data to support/refute
modelselection ranking
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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).
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
VNI
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
VNI
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
L
VNI
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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?
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Numerical Results
DMB & PWN, Bull. Math. Biol., 2006; DMB & JC, in progress.
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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)
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
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).
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
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.
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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)
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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.
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Generalized Sensitivity
Flow Cytometry
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Finite Resources
Given M points, where do we place them?Select P to minimize − log |I (P)|
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
Summary
Central Venous Catheter
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
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.
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
Summary
Staphylococcus epidermidis
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
Summary
Mouse model
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
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IntroductionModel Selection
Experimental DesignBacteremia
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
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Mouse Simulation
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Identify Partitioning Coefficients pL, pH , & pS
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Mouse imaging
Chung, Cartwright, Bortz, Jackson, & Younger, Shock, 2008.D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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).
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
Summary
Future Work
Relationship between luminescence and bacterial loadSufficiencyDistinguishabilityMaterial costConvergence toward “correct” ordering
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases
institution-logo
IntroductionModel Selection
Experimental DesignBacteremia
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)
D. M. Bortz Experimental Design in Validating Models of Infectious Diseases