Love and differential equations: An introduction to modeling

Transcription

1 Nonlinear Models 1 Love and differential equations: An introduction to modeling November 4, 1999 David M. Allen Professor Emeritus Department of Statistics University of Kentucky

2 Nonlinear Models Introduction to modeling 2 What is a model?

3 Nonlinear Models Introduction to modeling 3 A model airplane A model airplane operates, for the most part, on the same principles as an actual airplane.

4 Nonlinear Models Introduction to modeling 4 A lingerie model Yasmeen Ghauri is a Victoria s Secret model.

5 Nonlinear Models Introduction to modeling 5 Definition of a model A model is a physical representation of a system we wish to study. We want to make the model as close to the real thing as our knowledge and resources will allow. In drug and toxicology studies we are often interested in the metabolism of a substance by a human or an animal. Any model we might propose for one of these systems is going to be an oversimplification. Nevertheless, some models do an excellent job describing observations on these systems.

6 Nonlinear Models Introduction to modeling 6 Introduction to compartment models Models Kinetics Benefits of modeling Deterministic and stochastic models

7 Nonlinear Models Introduction to modeling 7 Compartmental model A fish tank separated into three compartments by membranes is an example of a compartment model. The tank is filled with a medium, and a substance is placed in the medium. We are interested in the movement of the substance from compartment to compartment.

8 Nonlinear Models Introduction to modeling 8 A more complicated system Obviously, the fish tank model is a vast oversimplification of this system a. a Drawn by Diane Allen, now age 30, in about the eighth grade

9 Nonlinear Models Introduction to modeling 9 Kinetics Kinetics: a science that deals with the effects of forces upon the motions of material bodies or with changes in physical or chemical systems.

10 Nonlinear Models Introduction to modeling 10 Orders of kinetics We have to make assumptions about the nature of the transfers from compartment to compartment. We now discuss zero and first order kinetics.

11 Nonlinear Models Introduction to modeling 11 First order kinetics This figure depicts the side view of the fish tank. Suppose that, in any small interval of time, the number of molecules of the substance to move from compartment 1 to compartment 2 is approximately proportional to the number of molecules present. This is first order kinetics.

12 Nonlinear Models Introduction to modeling 12 Zero order kinetics This figure depicts a fish swimming in water with mercury pollution. The uptake of mercury by the fish has no material effect on the amount of mercury in the water. The uptake of mercury is at a constant rate. This is zero order kinetics.

13 Nonlinear Models Introduction to modeling 13 The derivative of a function The derivative of X(t) with respect to t is the change in X(t) per unit change in t. Ẋ(t) = d X(t) = lim dt t 0 X(t + t) X(t) t = rise run

14 Nonlinear Models Introduction to modeling 14 The order of a process in terms of a derivative The order of a kinetic process is expressed in terms of the derivative of a function. Let θ be a rate constant. If the process is zero order. If the process is first order. Ẋ(t) = θ Ẋ(t) = θx(t)

15 Nonlinear Models Introduction to modeling 15 Kinetic diagram The first step of the modeling process is to create the kinetic diagram as follows: Determine what components of the system are to be represented by a compartment. Determine the possible transfers between compartments and adopt a symbol to represent the rate constant associated with each transfer.

16 Nonlinear Models Introduction to modeling 16 Pharmacokinetic model diagram A frequently used pharmacokinetic model is θ 2 θ 1 GI Plasma Other θ 3 θ 4

17 Nonlinear Models Introduction to modeling 17 Deterministic and stochastic models There are two sets of premises commonly assumed for compartmental analysis. They share the kinetic diagram but differ in the mechanics of transfer between compartments. Traditionally we assume transfers are determined by a system of linear differential equations. Many believe it is more realistic to assume transfers are in accordance with a probabilistic mechanism. The two sets of premises are called deterministic and stochastic models, respectively.

18 Nonlinear Models Introduction to modeling 18 What can we learn from compartment models? 1. If we find a compartment model that describes our data well, we may have a plausible description of the mechanics underlying data generation.

19 Nonlinear Models Introduction to modeling 19 What can we learn? (continued) 2. Having found a plausible model that fits the data well, we can calculate many interesting things from the model: the uptake rate and steady state level of a heavy metal in animal tissue; the average time a drug stays at its site of action; the relative bioavailability of two drugs; the relationship between drug concentration in a compartment and symptom relief.

20 Nonlinear Models Differential equations 20 Differential equations The solution of systems of linear, first order, homogeneous, differential equations with constant coefficients is the main tool for deriving and fitting compartmental models to data.

21 Nonlinear Models Differential equations 21 Derivatives and Integrals In the following display, c and d are constants. f(t) ct d exp(ct) exp(ct)x(t) sin(ct) cos(ct) g(t)dt d dt f(t) cdt d 1 c exp(ct) ) exp(ct) (Ẋ(t) + cx(t) c cos(ct) c sin(ct) g(t) Read from left to right for derivatives and from right to left for integerals.

22 Nonlinear Models Differential equations 22 Characteristic Polynomials The characteristic polynomial of the 2 2 matrix K = k 11 k 12 k 21 k 22 is k 11 λ k 12 k 21 k 22 λ. Expanding the determinant gives λ 2 (k 11 + k 22 )λ + k 11 k 22 k 12 k 21.

23 Nonlinear Models Differential equations 23 Characteristic Equation Setting the characteristic polynomial equal to zero gives the characteristic equation. The solutions to the characteristic equation are λ = k 11 + k 22 2 ± (k 11 k 22 ) 2 /4 + k 21 k 12. The quantity under the radical is called the discriminant. The solutions to the characteristic equation are called the characteristic roots or eigenvalues of K.

24 Nonlinear Models Differential equations 24 Solving 2 2 Systems Let c = (k 11 k 22 ) 2 /4 + k 21 k 12 denote the discriminant of the characteristic equation. The solutions of Ẋ1(t) = k 11 k 12 X 1(t) Ẋ 2 (t) k 21 k 22 X 2 (t) take a different form for each of the cases: c > 0, c = 0, and c < 0.

25 Nonlinear Models Differential equations 25 Case c > 0 Without loss of generality, we can assume k 12 = 0, and k 11 k 22. For illustration, we assume X 1 (0) = D, and X 2 (0) = 0. Solving for X 1 (t) Ẋ 1 (t) = k 11 X 1 (t) Ẋ 1 (t) k 11 X 1 (t) = 0 exp( k 11 t)(ẋ1(t) k 11 X 1 (t)) = 0 exp( k 11 t)x 1 (t) = d X 1 (t) = d exp(k 11 t) X 1 (t) = D exp(k 11 t)

26 Nonlinear Models Differential equations 26 Solving for X 2 (t) Ẋ 2 (t) = Dk 21 exp(k 11 t) + k 22 X 2 (t) exp( k 22 t)(ẋ2(t) k 22 X 2 (t)) = Dk 21 exp((k 11 k 22 )t) exp( k 22 t)x 2 (t) = D k 21 k 11 k 22 exp((k 11 k 22 )t) + d X 2 (t) = D k 21 k 11 k 22 exp(k 11 t) + d exp(k 22 t) X 2 (t) = D k 21 k 11 k 22 (exp(k 11 t) exp(k 22 t))

27 Nonlinear Models Differential equations 27 Case c = 0 Without loss of generality, we can assume k 12 = 0, and k 11 = k 22. For illustration, we assume X 1 (0) = D, and X 2 (0) = 0. The solution for X 1 (t) is the same as for the previous case.

28 Nonlinear Models Differential equations 28 Solving for X 2 (t) Ẋ 2 (t) = Dk 21 exp(k 11 t) + k 22 X 2 (t) exp( k 22 t)(ẋ2(t) k 22 X 2 (t)) = Dk 21 exp((k 11 k 22 )t) exp( k 22 t)(ẋ2(t) k 22 X 2 (t)) = Dk 21 exp( k 22 t)x 2 (t) = Dk 21 t + d X 2 (t) = (Dk 21 t + d) exp(k 22 t) X 2 (t) = Dk 21 t exp(k 22 t)

29 Nonlinear Models Differential equations 29 Case c < 0 Let m = (k 11 + k 22 )/2 and M = 1 k 11 m k 12 c k 21 k 22 m and I is the identity matrix. The solutions are X 1(t) = ( cos( c t)i + sin( c t)m ) X 1(0) X 2 (t) X 2 (0) exp(mt).

30 Nonlinear Models Pharmacokinetic modeling 30 A pharmacokinetic example The kinetics of drug absorption, distribution, and elimination is called pharmacokinetics. A knowledge of pharmacokinetics is needed to administer drugs optimally.

31 Nonlinear Models Pharmacokinetic modeling 31 Pharmacokinetic modeling t = time from administration of a drug X(t) = amount of drug in tissue at time t

32 Nonlinear Models Pharmacokinetic modeling 32 Modeling tetracycline metabolism For Tetracycline given orally, what are the values of the rate constants, and what is the formula for the amount in the plasma? GI θ 1 θ 2 Plasma Let X 1 (t) and X 2 (t) represent the respective amounts of tetracycline in the two compartments at time t. The equations associated with this diagram are Ẋ 1 (t) = θ 1 X 1 (t) Ẋ 2 (t) = θ 1 X 1 (t) θ 2 X 2 (t).

33 Nonlinear Models Pharmacokinetic modeling 33 Let θ = θ 1 θ 2 Matrix representation of equations, X(t) = K = K(θ) = θ 1 0 θ 1 θ 2 tetracycline equations is X 1(t) X 2 (t), Ẋ(t) = Ẋ1(t) Ẋ 2 (t), and. The matrix representation of the Ẋ(t) = KX(t).

34 Nonlinear Models Pharmacokinetic modeling 34 The solution of the differential equations We assume θ 1 θ 2, X 1 (0) = D, and X 2 (0) = 0. Using results of derivation beginning on slide 25 we obtain: X 1 (t) = D exp( θ 1 t) X 2 (t) = D θ 1 θ 1 + θ 2 (exp( θ 1 t) exp( θ 2 t))

35 Nonlinear Models Pharmacokinetic modeling 35 Lag times Sometimes there is a time delay before the model takes effect. This delay is called a lag time, and we denote it by τ. The response allowing for the possibility of a lag time is X 2(t) X 2 (t τ) if t > τ = 0 otherwise

36 Nonlinear Models Pharmacokinetic modeling 36 The data We will fit the model to the data a : Time Concentration Time Concentration t i Y i t i Y i a Wagner, 1967

37 Nonlinear Models Pharmacokinetic modeling 37 The observational model The observational model is Y i = X 2(t i ) + ɛ i. If the ɛ i terms have equal variances and are uncorrelated, least squares is an efficient method of estimation. The parameter estimates are the values that minimize the least squares criteria: n (Y i X 2(t i )) 2. i=1

38 Nonlinear Models Pharmacokinetic modeling 38 Fit the model to the data conc hours

39 Nonlinear Models Pharmacokinetic modeling 39 Results of model fitting The estimate of τ is The estimated response for t is X 2(t) = exp( (t )) exp( (t ))

40 Nonlinear Models Romeo and Juliet 40 Romeo and Juliet This little story is by Strogatz a. I have added a bit about model fitting. Juliet is in love with Romeo; but in our version of this story, Romeo is a fickle lover. The more Juliet loves him; the more he begins to dislike her. But when she loses interest, his feelings for her warm up. She, on the other hand, tends to echo him. Her love grows when he loves her, and it turns to hate when he hates her. a Steven H. Strogatz, Love Affairs and Differential Equations, Mathematics Magazine, Vol. 61, No. 1, February 1988.

41 Nonlinear Models Romeo and Juliet 41 A model A simple model for their ill-fated romance is Ẋ 1 (t) = θ 1 X 2 (t) Ẋ 2 (t) = θ 2 X 1 (t) where X 1 (t) = Romeo s love/hate for Juliet at time t, X 2 (t) = Juliet s love/hate for Romeo at time t, and θ 1 > 0 and θ 2 > 0 are intensity constants.

42 Nonlinear Models Romeo and Juliet 42 The solution The discriminant is θ 1 θ 2, a negative number. The solution is obtained by applying the formula derived starting on slide 29. The result is X ( 1(t) = cos( θ 1 θ 2 t)i + sin( ) θ 1 θ 2 t)m X 1(0) X 2 (t) X 2 (0) where M = 0 θ 1 /θ 2 θ2 /θ 1 0.

43 Nonlinear Models Romeo and Juliet 43 The data Juliet s nosey aunt looks in once a month to see how the relationship is going. The following are her observations: Month Romeo Juliet Month Romeo Juliet Positive numbers reflect love, and negative numbers reflect hate.

44 Nonlinear Models Romeo and Juliet 44 The fitted response Love 10 5 X 1 (t) = 10 cos(t) X 2 (t) = 5 sin(t) The sad outcome of their affair is, of course, a never ending cycle of love and hate; their governing equations are those of a simple harmonic oscillator. At least they manage to achieve simultaneous love one quarter of the time Month