2010 dose-effect modeling of experimental data full

8/3/2019 2010 Dose-Effect Modeling of Experimental Data Full

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Journal of Information, Control and Management Systems, Vol. 8, (2010), No.3 257

DOSE-EFFECT MODELING OF EXPERIMENTAL DATA

Kaloyan YANKOV

Medical Faculty, Trakia University,

Stara Zagora, Bulgaria

[email protected]

Abstract

This paper proposes a mathematical model for estimation of the dose-response

relationship of experimental data. modified logistic function is used as

analytical model for determining the dose-effect. Parameters of the model are

identified with the optimization procedure based on the cyclic coordinatedescent method. Formulas are derived to calculate the effective dose level and

the standard deviation of doses. The approach is implemented in the computer

program KORELIA-Dynamics.

Keywords: dose-time-response, toxicity, effective dose level, lethal dose, modeling,coordinate descent method.

1 INTRODUCTIONTo analyze the effects of a drug, the dose-response relationship is studied in

pharmacology and toxicology. Dose-response experiments are routinely conducted in

preclinical and clinical trials to study the relationship between the dose level of a drug

and the probability of a response, be it cured or poisoned. Therefore the response

of these experiments is binary or quantal, which means either the drug takes effect and

the subject has a reaction or not. The underlying assumption for quantal bioassay is

that a subject is administered a stimulus at a dose level x, and that the response Yis abinary random variable with success probabilityp(x), i.e.

Y ~ Bin(1,p(x)).

The function ]1,0[: p is called a dose response curve and is usuallyassumed to be strictly monotone. This curve describes the probability of success of the

analyzed drug.

For a given ))1(),0(( pp the parameter of interest is the effective dose level:

ED = p1()


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Dose-Effect Modeling of Experimental Data258

This is the dose, where 100 % of the subjects reacts. In case the main effect is

death, a lethal dose LD is defined.

Depending on the value of

several doses of practical importance are defined. = 0. Effective/lethal dose ED0/LD0 - the estimated dose at which none of

the population is expected to react or die.

= 1. One hundred percent effective/lethal dose ED100/LD100 the lowestpossible dose that causes effects in all studied cases.

= 0.5. Median effective/lethal dose ED50/LD50 - the dose of a drug predicted to produce a characteristic effect in 50 percent of the subjects towhom it is given. The ED50 is the most frequently used standardized dose by

means of which the potencies of drugs are compared. An ED50 can be

determined only from data involving all or none (quantal) response.Further in the text the term used is effective dose, being understood that the

results will apply also to the lethal dose.

Experimental animals are treated with the drugs in order to study the effects and

to determine the value of the base dose. The experiment is conducted under n dose

levels. For each dose levelxi (i = 1,2,n),Nianimals are examined. They are observed

over a period of time Tsufficient for the investigated drug to demonstrate its action

and at the end of the experiment the number of reactived animals zi is noted. The

probability that an individual animal on dose levelxiresponds by time Tis defined as:

pi = zi/ NiIt is obvious that 0 = p0 p1 pn pn+1 = 1.

The function presented by arranged pairs of points (xi , pi) is cumulative

distribution function. It belongs to the class of the so-called S-shaped curves [1].

Many publications deal with the problem of calculating the effective dose. The

approaches for calculation of dose-effect based on the cumulative curve used in most

of them are described below.

1.1 Linear regression models.

Linear interpolation. The effects determined during the experiment of theapplied doses are associated with line segments. The desired effective dose is

calculated taking into account the coordinates in the relevant line segment.

In case of explicit non-linearity a logarithmic scaling is performed upon thedata. Thus the graph acquires a relatively linear form, and then linear

interpolation is applied. This approach uses the assumption that there is a

logarithmic correlation between doses and effect of doses. For the entire range

of applied doses that is not true, but in a narrow range around the relevant dose

the method is appropriate.

These two approaches were used before the era of modern computers when the

most powerful computational tools were a pocket calculator and graph paper for


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drawing graphs. Publications dealing with these methods are available from the 1970s

till now [2,3,4].

1.2 Nonlinear regression models. Interpolation of the data with cubic spline [5,6,7]. Piece-wise spline

inperpolation is a convenient tool for interpolation of data being continuously

differentiable to the second derivative. When appropriate software is

available, this approach is very convenient and has great precision..

Modeling techniques based on the analytical form of mathematical function.They assume a functional relationship between dose and response,

according to a pre-specified parametric model, such as a linear, quadratic,

logarithmic, exponential, logistic and so on [8]. By the application of

minimizing numerical procedure, parameters of the function are determined,i.e. data identification is implemented. This approach provides the greatest

accuracy due to the use of computers and specialized software. Another

benefit of this approach is the additional parameters that the software can

produce and the possibility for graphical illustration of the results. The most

commonly used optimization methods are the method of least squares, the

method of weighting by least squares or the maximum likelihood method [9].

This paper aims to formulate a mathematical model of dose-response relationship

using experimental data. The goal is to model the curve in the domain of definition and

to minimize the possible error of approximation. The model is derived applying

algorithm and software for system identification developed by the author [10,11,12].

2 MODEL STRUCTUREOne of the possible equations for describing a sigmoidal curve is the Verhulst

Pearl equation:

)()(

1)(

xpK

xpr

dx

xdp

(1)

p(x0) = C0Where: r the rate constant of the process

Kthe asymptotic limit of growth (carrying capacity).

This equation express quantitative changes in the presence of limited resources,

represented by the constant .

The solutionp(x) of Eq.(1) is the logistic function:
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).(

011

)(xr

eC

K

Kxp

(2)

C0 = p(0)

This equation may be revised in view of the goal to identify an effective dose,

thereby makingK=1. o avoid the case C0=0 in Eq.(2), s training logistic function as

proposed in [11] is used:

)*(1

1)(

xrAexp (3)

A>0, r > 0

Three parameters are necessary to specify the curve, A, r and , defined as

follows:

A is the number of times that the initial valuep(0) must grow to reach 100%.

ris the slope or the growth rate parameter that specifies "width" or "steepness" of

the S-curve.

is a dose correction parameter.

These parameters must be identified by fitting the Eq.(3) to the experimental data.

3 DATA IDENTIFICATIONTo fit the functionp(x) to the data, an identification procedure will be performed.

The values pi in n number of points are known as a result of the measurement. They

are used to identify the parameters,r, .

Let us chose to subject to system identification the function

ixrip

AerAxp

i

)*(1

1),,,(

(4)

A > 0, r > 0, i=1,2,,n

The identification vectorQ(A,r,) must be defined from the system (4). But this

system is not well defined. For each pointxi the residual di is:

di(Q) = || pi-p(xi,Q) ||

The aim is to minimize di(Q) for the whole interval of identification [xn-x1]:

D(Q) = inf|| di(Q) || < , >0 (5)

Thus the identification goal is translated into an optimization problem. The

optimization method is defined depending on the way of minimization ofD(Q).

Least square fitting.


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An object of minimization is the functional:

(6)

uniform fitting is aimed at minimizing the maximal deviation in the base

pointspi:

min)(1

2

n

i i

dQD

Uniform fittingThe

min||max)( idQD (7)

The Cyclic coordinate descent (CCD) method is an iterative heuristic search

technique that attempts to minimize the error by varying one parameter at a time [10].

It reduces the multi-criteria optimization to the single-criteria one. A number of passes

are made over the model equation to find the global minimumD for system (5).

4ssary to find the inverse function

p-1

(x). After trivial transformations of Eq.(3):

CALCULATION OF DOSE-EFFECT PARAMETERS

To calculate the effective or lethal dose, it is nece

)(1 pr

As a result of the identif

)(

lnln1

)(p

Ax (9)

ication, the vector Q(A,r,) is known and x() is

calcuFor most commonly used dose ED50 (LD50) atp()=0.5:lated directly from Eq.(9).

Ar

The final result is easy for computing. The

xED ln1

)5.0(50 (10)

main burden is the process of

ident

uencyf(x) of

the effectiveness of doses analytically and graphically (fig.1) is obtained:

ification of these three parameters.

The received analytical model of the sigmoidal curve allows obtaining other

characteristics of dose-effect curve. After differentiation of Eq.(3) the freq

2)(

)(

]1[ Aedx

The extreme point correspo

)()(

rx

rxArexdpxf (11)

nds to ED50/LD50. Thus the dose value of x(0.5) can

be ca 0)( dxxdf .lculated from the equation

Solution of the equation

0)(

2

dx

2

xfd


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gives the coordinates of inflexion pointsIx1, Ix2 :

rx1

AI

32

ln

rx2

AI

32

ln

and they are used to calculate the standard deviation of the frequency of doses (fig.1):

rr

IISD xx

317.1

2

32

32ln

2

12

(12)

It is possible to define three dose ranges: Low response dose. These doses are

n range

dose. Doses greater than

(x(0.5)+2SD)

Figure.1.Dose frequency

5

stand

on linear interpolation to determine LD50

. After identification with KORELIA-Dynamics the

calcu

lower than (x(0.5)-2SD)

Mean response dose. Doses i(x(0.5)-2SD,x(0.5)+2SD)

High response

EXAMPLE

The method is implemented in program KORELIADynamics using its ability for

system identification. A dialog window is appended for calculation of ED50/LD50,

ard deviation and visualization of sigmoidal curve and frequency of doses.

The example from page 44 of [2] will be considered. The author discusses a

toxicity of antibiotic tubazid. The calculations are performed using the Behrens

approach (cit. Behrens B., Arch.exper. Path. Pharm., 140, 237, 1929) and oerber (cit.

oerber G., Arch.exper. Path. Pharm., 162, 480, 1931). The experimental data are

scattered on fig.2. Both methods are based

and standard deviation SD. Cited data are:

Behrens method: LD50SD = 172.3 9.5 [mg/kg]. oerber method: LD50SD =172.5 15 [mg/kg].

The scatterplot on fig.2 should clearly indicate the appropriateness of using a

logistic model (3) to fit this data

lated model parameters are:

A = 927.3; r = 0.1855; = -25.13


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Identification errors are: absolute = 0.0346; relative = 0.0834; quadratic =

0.00453.

The identified equation for data is:

)13.25*1855.0(

1)(

3.9271

xxp

e

LD50SD = 172.3034 7.0997 [mg/kg].

On fig.2 the experimental data, identification curve and frequency of doses are

presented.

Thus, the LD50 and SD according to equations (10) and (12) are

Figure.2. Graph of the identification curve and frequency of doses

ure that targets minimal residual using least-squa ulas for effective dose and

ts the

t the range of tested doses;

Drops some requirements as strictly regulated modification of doses toconduct the experiment at pre-selected law;

6 CONCLUSIONSThis work proposes an analytical model for determining the dose-effect.

modified logistic function is used for data identification. Parameters of the model arecalculated with the optimization proced

re or uniform criteria. From the analytical model form

standard deviation of doses are derived.

The advantages of the proposed analytical model are:

Explicit provides the frequency of doses and standard deviation. Can provide the smallest error of approximation, because it reflec

nonlinearity of the process by selecting the appropriate function;

Shows the behavior of the system throughou


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A mathematical model can easily be embedded in more complex models,including the use of computers and also to broaden the community of research

and modeled processes.

The computer program Korelia-Dynamics offers friendly user interface for data

input and graphical output of results. The program is easy and rapid to use, requiring

minimum computer knowledge.

REFERENCES:

[1] KUCHARAVY, D., R. DE GUIO. Application of S-Shaped Curves, in 7th ETRIATRIZ Future Conference. Kassel University Press GmbH, Kassel: Frankfurt,

Germany 2007.

[2] BELENKII, M.A., Elements of Quantitative Evaluation of PharmacologicalEffects. GIML, Leningrad 1963 (russian).

[3] SEPETLIEV, D., Statistical methods for data processing from medical researchstudies. dicina I fizkultura, Sofia, 1965 (bulgarian).

[4] BRENNER D.J, L.R. HLATKY, P.J. HAHNFELDT, Y.HUANG, R.K.SACHS.The Linear-Quadratic Model and Most Other Common Radiobiological Models

Result in Similar Predictions of Time-Dose Relationships. RADIATION

RESEARCH 150, 83-91, 1998.

[5] REMMENGA M. D., G. A. Milliken, D. Kratzer, J. R. Schwenke, H. R. Rolka.Estimating the maximum effective dose in a quantitative dose-response

experiment. Journal of Animal Science, Vol 75: 2174-2183, 1997.

[6] YANKOV, K., Software Utilities for Investigation of Regulating Systems, NinthNat. Conf. "Modern Tendencies in the Development of Fundamental and Applied

Sciences". June, 5-6, 1998, Stara Zagora, Bulgaria, 401-408.

[7] YANKOV, K. Evaluation of Some Dynamics Characteristics of TransientProcesses. Proc. 12-th Int.Conf. SAER'98. St.Konstantin resort, sept.19-20, 1998,

Varna, Bulgaria. 113-117.

[8] BRETZ F., Jason Hsu , Jose Pinheiro, Yi Liu. Dose Finding A Challenge inStatistics. Biometrical Journal 50 (4), 480504, 2008.

[9] HUANG Y. Robustness of choice of number of doses for maximum likelihoodestimation of the ED(50) in bioassay. Stat Med. Aug 15;21(15):2215-23, 2002.[10] YANKOV, K. System Identification f Biological Processes. Proc. 20-thInt.Conf. "Systems for Automation of Engineering and Research (SAER-2006).

St.St. Constantine and Elena resort, sept.23-24, Varna, Bulgaria, 144-149, 2006.

[11] YANKOV, K Recognition and Function Association of Experimental Data.Proc. Int. Conference on Information Technologies (InfoTech-2009). Constantine

and Elena resort, sept.17-20, Varna, Bulgaria, 131-140, 2009.

[12] YANKOV, K Decision Planning of System Identification. Proc. Int. Conferenceon Information Technologies (InfoTech-2010). Constantine and Elena resort,

sept.16-18, Varna, Bulgaria, 229-238,2010.
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