BioSB Conference 2016

Quantification of variability and uncertainty in systems medicine models

April 20, 2016Natal van RielEindhoven University of Technology, the NetherlandsDepartment of Biomedical EngineeringSystems Biology and Metabolic Diseasesn.a.w.v.riel@tue.nl

@nvanriel

Computational modelling

• Explaining the data & understanding the biological system

2Wolkenhauer, Front Physiol. 2014; 5:21.

TOP-DOWN

BOTTOM-UP

Developing models of dynamical systems

Explaining the data & understanding the system• Estimating models

• Comparing alternative hypotheses (differences in model structure)

• Given a fixed model structure, find sets of parameter values that accurately describe the data

• Evaluate the capability of the model to reproduce the measured data and the complexity of the model

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arg min Description of Data Penalty on FlexibilityModelClass

Model

Model complexity / granularity

Model Errors

The error in an estimated model has two sources: 1. Too much constraints and restrictions; “too simple model sets". This

gives rise to a bias error or systematic error. 2. Data is corrupted by noise, which gives rise to a variance error or

random error.

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^

arg min Description of Data Penalty on FlexibilityModelClass

Model

Adapted from Ljung & Chen, 2013

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Model calibration

Parameter identification• Maximum likelihood techniques

• Implemented using nonconvex optimization

• Error model

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i i

i k ik

d k y k

2

ˆ 0

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Quantitative and Predictive Modelling

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Bias – Variance trade-off

• To minimize the MSE is a trade off in constraining the model: A flexible model gives small bias (easier to describe complex behavior) and large variance (with a flexible model it is easier to get fooled by the noise), and vice versa

• This trade-off is at the heart of all modelling that aims to explain data

Zero biasHigh variance(overfitting)

Adequate Bias - Variance trade-off

Fitting elephants

• Famous aphorism: ‘‘With four parameters I can fit an elephant, and with five I can make him wiggle his trunk’’

• Estimating dynamic models of networks is not equivalent to curve fitting• The interconnected structure of biological systems imposes strong

constraints

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http://en.wikiquote.org/wiki/John_von_Neumann

“Even with a thousand parameters I cannot fit the biological network in a single cell of an elephant. Let alone to make him blink his eye”

Information-rich data

It is often not trivial to find a mechanistic (mechanism-based) model that can describe information-rich data of an interconnected system

• If the measurements provide sufficient coverage of the system components (details)

• Under (multiple) physiological, in vivo conditions (operational context)

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measurements

No.

of c

ompo

nent

sNo. of observations per component

Rethinking Maximum Likelihood Estimation

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• The bias - variance trade-off is often reached for rather large bias

• Typically, we are far away from the asymptotic situation in which Maximum Likelihood Estimation (MLE) provides the best possible estimates

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Tiemann et al. (2011) BMC Syst Biol, 5:174Van Riel et al, Interface Focus 3(2): 20120084, 2013Tiemann et al. (2013) PloS Comput Biol, 9(8):e1003166

Room for more flexibility

• Instead of increasing structural complexity (increasing model size)• Introduce more freedom in model parameters to compensate for

bias (‘undermodelling’) in the original model structure• Increasing model flexibility using time-varying parameters

•ADAPTAnalysis of Dynamic Adaptations in Parameter Trajectories

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Disease progression and treatment of T2DM

• 1 year follow-up of treatment-naïve T2DM patients (n=2408)• 3 treatment arms: monotherapy with different hypoglycemic agents

– Pioglitazone – insulin sensitizer• enhances peripheral glucose uptake• reduces hepatic glucose production

– Metformin - insulin sensitizer• decreases hepatic glucose production

– Gliclazide - insulin secretogogue• stimulates insulin secretion by the pancreatic beta-cells

FPG

[mm

ol/L

]

Schernthaner et al, Clin. Endocrinol. Metab. 89:6068–6076 (2004)Charbonnel et al, Diabetic Med. 22:399–405 (2004)

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Glucose-insulin homeostasis model

• Pharmaco-Dynamic model • 3 ODE’s, 15 parameters

hepatic glucose production

glucose utilization

insulin secretion

glucose (FPG)

insulinsensitivity (S)

insulin (FSI)HbA1c

beta-cell function (B)

OHA(insulin sensitizer)

OHA(insulin secretagogue)

1 2

1 2

1 2

1

2

compensation phase: hyperinsulinemiaexhaustion phase: disease onsettreatment effects

De Winter et al. (2006) J Pharmacokinet Pharmcodyn, 33(3):313-343

FPG: fasting plasma glucoseFSI: fasting serum insulinHbA1c: glycosylated hemoglobin A1c

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T2DM disease progression model

• Fixed parameters

• Adaptive changes in -cell function B(t) and insulin sensitivity S(t)

• Parameter trajectories

Nyman et al, Interface Focus. 2016 Apr 6;6(2): 20150075

Reducing bias while controlling variance

• The common way to handle the flexibility constraint is to restrict / broaden the model class

• If an explicit penalty is added, this is known as regularization

14 Cedersund & Roll (2009) FEBS J 276: 903

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Regularization approaches in statistics

• Multivariable regression

• Lasso (least absolute shrinkage and selection operator) solves the l1-penalized regression problem of finding the parameters to minimize

• l1-penalty accomplishes:– Shrinkage of parameters values– Selection of parameters (0)

• It enforces sparsity in models that have too many degrees of freedom

• Regularization has not been used so much in dynamic system modelling

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1

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y x

i i iy x

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y x

Ljung, Annual Reviews in Control 34 (2010) 1–12 van Riel & Sontag. Syst Biol (Stevenage) 153: 263-274, 2006

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Regularization of parameter trajectories

[ ]

ˆ[ ] arg min Fit to Data Penalty on Parameters Changesn

n

r

r

• Shrinkage of changes in parameters values• Selection of parameters that change

Progressive changes in lipoprotein metabolism

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Rader & Daugherty, Nature 451,2008

Lipolysis

• Lipoprotein distribution (LPD) codetermines metabolic and cardio-vascular disease risks

• Liver X Receptor (LXR, nuclear receptor),induces transcription of multiple genes modulating metabolism of fatty acids, triglycerides, and lipoproteins

• LXR agonists increase plasma high density lipoprotein cholesterol (HDLc)

• LXR as target for anti-atherosclerotic therapy?

Levin et al, (2005) Arterioscler Thromb Vasc Biol. 25(1):135-42

Progressive changes in lipoprotein metabolism after pharmacological intervention

• LXR activation in C57Bl/6J mice leads to complex time-dependent perturbations in cholesterol and triglyceride metabolism

• Dynamic model of lipid and lipoprotein metabolism• ADAPT: time-varying metabolic parameters to accommodate

regulation not included in the metabolic model

• Hepatic steatosis: Increased influx of free fatty acids from plasma is the initial and main contributor to hepatic triglyceride accumulation

18Tiemann et al., PLOS Comput Biol 2013 9(8):e1003166

Hijmans et al. (2015) FASEB J. 29(4):1153-64

Model: the darker the more likely

Quantification of Identifiability and Uncertainty

Verification, Validation, and Uncertainty Quantification (VVUQ)

• Profile Likelihood Analysis (PLA)

• Prediction Uncertainty Analysis (PUA)– Ensemble modelling

• Uncertainty quantification: the elephant in the room

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Raue.et al 2009 Bioinformatics, 25(15): 1923-1929Vanlier et al. 2012 Bioinformatics, 28(8):1130-5

“Uncertainty quantification is an underdeveloped science, emerging from real-life problems.” Bassingthwaighte JB. Biophys J. 2014 Dec 2;107(11):2481-3

Vanlier et al. Math Biosci. 2013 Mar 25Vanlier et al. Bioinformatics. 2012, 28(8):1130-5

Conclusions

• The network structure of the biological systems imposes strong constraints on possible solutions of a model

• The bias - variance trade-off is often reached for rather large bias, not favoring MLE

• Systems Biology / Systems Medicine is entering an era in which dynamic models, despite their size and complexity, are not flexible enough to correctly describe all data

• Computational techniques to introduce more degrees of freedom in models, but simultaneously enforcing sparsity if extra flexibility is not required (ADAPT)

• Model estimation tools are complemented with ‘regularization’ methods to reduce the error (bias) in models without escalating uncertainties (variance)

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Systems Biology of Disease Progression - ADAPT modelinghttp://www.youtube.com/watch?v=x54ysJDS7i8