MiniCourse on Mathematical Modeling of Biological Systems

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Mini-Course onMathematical Modeling of
Biological Systems

• Frank Doyle
• Dept. of Chemical Engineering
• University of Delaware
• [phone]
• fdoyle_at_udel.edu

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Approaches to Modeling Biological Systems
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Modeling Biomedical Systems

• Descriptive
• quantitative relationships
• Predictive
• e.g., drug response
• Explanatory
• e.g., analysis of internal parameters

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Sample Applications

• Patient management
• Identification
• Control models
• Parameter estimation
• Diagnostics
• Teaching
• Database
• Instrumentation design
• Improved measurement

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Hurdles

• complexity of physiological systems
• highly nonlinear
• empirical models give little physiological
  insight
• isolation of individual system impossible
• engineering control theory ? biological control
  theory
• time varying behavior -gt difficult equations to
  solve
• observability
• data record lenghts invariably short
• robustness parametric sensitivity
• noise (nature may use to her advantage)
• time varying parameters
• persistent excitation for ID experiments

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Development of a Mathematical Model
Problem
Modeling Purposes
Modeling Formulation
Laws Theories Data
Model Validation
Conceptualization Realization Solution
Modeling Identification
Model
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Approaches to Mathematical Modeling

• Empirical models
• No a priori knowledge
• Black-box models (e.g., time series, ANNs, etc.)
• No extrapolative power
• Requires validation
• Fundamental models
• Chemical/physical laws
• Conservation principles
• Potentially extrapolative
• Requires validation
• Semi-empirical models

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Model Formulation

• Conceptual Model
• Aggregation (degree of lumping)
• Abstraction (degree of coverage)
• Idealization (simplifying assumptions)
• Mathematical Realization
• Lumped deterministic (ODEs)
• Linear
• Nonlinear
• Distributed (PDEs)
• Stochastic
• Model Solution
• Numerical solution
• Standard packages (e.g., SPICE)

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Test Signal Design

• Theoretically rich (persistent excitation)
• Convenient to generate
• Large (S/N ratio) but not too large (nonlinear)
• Minimize duration of experiment
• Should enable/facilitate identification

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Identifiability

• Is it theoretically possible to make unique
  (unbiased) estimates of model parameters from
  data record?
• Combination of model structure and input sequence
  design
• Unique ID possible
• Finite number of feasible parameter values
• Infinite number of parameter values (ill-posed)

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Simple Example
Drug delivery u injected drug ybioassay a1frac
tional rate const. a2inverse distribution
vol. c1proportionality constant
c1 and a2 are not indentifiable
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Second Example Simple Glucose-Insulin Regulation
Bolie model (1961) x1deviation
glucose x2deviation insulin
Unidentifiable 4 eqns, 5 vars
Add additional meaurement (x2)
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General Tests
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Related Concept - Observability

• Can one determine the states of a system from
  measurements, and what is the relationship with
  the identifiability of the parameters?
• Structural requirement (Kalman)
• Observability and Identifiability neither is
  necessary for the other

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System IdentificationTutorial
obtain a good model at a low price
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Preliminaries

• Impulse Response
• complete characterization of linear dynamic
  system
• Sampling
• uniform sampling instants (kT, k1,2, )
• zero order hold (ZOH) u(t)u(k)
  kTlttlt(k1)T
• resulting impulse response
• (Stochastic) Disturbance

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General Problem
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Step Testing

• Hold inputs constant to establish steady-state
• Step change in input u(t)u0Du (tgt1)
• Sample y(t) at uniform intervals
• Step response coefficients

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Issues

• Sampling period size
• Given T(sampling time), NT (time to reach
  steady-state) t (time constant), and q
  (deadtime)
• Aim for 30-40 model coefficients
• Heuristic

• Input step size
• too small (S/N ratio)
• too large (nonlinearity, dangerous regimes)
• Number of experiments
• minimize unmeasured disturbance effects (e.g.,
  noise) leading to reduced variance in model
  parameter estimates
• validation purposes

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Issues, contd

• Determination of equilibrium
• Step testing, in general is good for lo-frequency
  vs. hi-frequency model ID

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Pulse Test

• Use pulse of widthT, heightdu
• Calculate pulse response coefficients
• Equivalence to step response coeffs.

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Issues

• Sample time, model length as before
• Generally du gtgt Du (S/N ratio)
• Repeated experiments to minimize noise effects
• Equilibrium determination
• Quicker return to desired equilibrium

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Stochastic Input

• Random Binary Sequence (RBS)
• 11sign(rand(100,1)-.5)
• Random Sequence
• 12(rand(100,1)-.5)
• Pseudo-RBS (PRBS)
• periodic RBS

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Regression Approach

• Correlation approach Consider
  y(k)h(1)u(k-1)h(2)u(k-2)n(k)
• u zero-mean stationary input sequence
• n zero-mean stationary noise sequence
• u n uncorrelated

Goes to zero in limit large M
autocorrelation
cross-correlation
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• simple problem
• general problem
• note independent sequence

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Issues

• Random input sequence more effective than
  deterministic input (across frequency range)
• FIR model correlation method LS method
• Unbiased estimates if residual uncorrelated with
  u
• Estimates improve as M increases (to infinity)
• Persistent excitation
• input not sufficiently excited (directions,
  frequencies)
• rank failure in Ruu

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• Linear Least Squares Method
• general setup
• least squares problem
• given N points

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Technical Issues

• Will parameter estimates converge to true value?
• Desirable property
• Conditions
• input noise uncorrelated
• persistent excitation

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Parameter Estimate variance

• Ill-conditioning arises because input is not
  excited in all directions
• Modifications
• Partial Least Squares (PLS) - project input to
  lower dimensional space before regresion
• Principle Components Analysis (PCA) - eliminate
  data range with poor signal/noise ratio

covariance of n
nonnegligible for small N
Ill-conditioning may amplify
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Bergman TF Case Study

• Idealized linear model
• TF(-.005)/(100s1)(13s1)
• Mmeasurement record length
• NIR model length
• Tsample time
• vnvariance on signal noise
• udel/- input value
• Heuristic rules
• t140min (settling time 600 min)
• Try T15min
• Settling model N40 coeffs.

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Bergman Model
Definitions Initial conditions g glucose
(mg/dl) go 80 I insulin (?U/min) Io
0 I? external insulin infusion (?U/min) I?o
0 X transition state (1/min) Xo
0 Parameters Rab p1go p1 0.01 p2 1/30 p3
4p2/10,000 p4 1/7 p5 5p4/3,000
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Model Validation

• ID algorithm best model
• Validation Is best model good enough?
• agreement with data
• suitable for end purpose
• description of true system

The screening of variables should never be left
to the sole discretion of any statistical
procedure Draper Smith (1981)
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Validation techniques

• Parameter values
• feasible
• low estimation variance
• Ability to minimize prediction error for fresh
  data
• Model reduction
• Residual analysis
• Frequency domain

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Validity measures

• In general, asymptotic value of PE
• Choices for V
• Akaikes Information Theoretic Criterion (AIC)
• Akaikes Final Prediction-error Criterion (FPE)
• Statistical Hypothesis Tests
• Penalties for Complexity

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Final Practical Issues

• Pretreatment of data
• anti-alias filter (use analog filter before
  sampling)
• spike/outlier removal (exercise judgement)
• Directionality issues in MIMO systems
• gain amplification nonlinear function (input
  direction)
• need to excite output directions uniformly
  (S/N), thus input directions are not
  necessarily uniform
• Closed-loop Identification