Dose-Effect Modeling of Experimental Data

11 Pages·1.409 MB·English

by  Yankov K.

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. А 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 coordinate
descent 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 INTRODUCTION
To 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 Y is a
binary random variable with success probability p(x), i.e.
Y ~ Bin(1,p(x)).
The function p:[0,1]is called a dose response curve and is usually
assumed to be strictly monotone. This curve describes the probability of success of the
analyzed drug.
For a given (p(0),p(1)) the parameter of interest is the effective dose level:
ED = p−1(α)
α
258 Dose-Effect Modeling of Experimental Data
This is the dose, where 100α % of the subjects reacts. In case the main effect is
death, a lethal dose LD is defined.
Depending on the value of α several doses of practical importance are defined.
 α = 0. Effective/lethal dose ED0/LD0 - the estimated dose at which none of
the population is expected to react or die.
 α = 1. One hundred percent effective/lethal dose ED100/LD100 – the lowest
possible dose that causes effects in all studied cases.
 α = 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 to
whom 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 level x (i = 1,2,…n), N animals are examined. They are observed
i i
over a period of time T sufficient for the investigated drug to demonstrate its action
and at the end of the experiment the number of reactived animals z is noted. The
i
probability that an individual animal on dose level x responds by time T is defined as:
i
p = z/ N
i i i
It is obvious that 0 = p ≤ p ≤···≤ p ≤ p = 1.
0 1 n n+1
The function presented by arranged pairs of points (x , p) is cumulative
i i
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 the
applied 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 the
data. 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
Journal of Information, Control and Management Systems, Vol. 8, (2010), No.3 259
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 STRUCTURE
One of the possible equations for describing a sigmoidal curve is the Verhulst –
Pearl equation:
dp(x)  p(x)
 r1 p(x) (1)
dx  K 
p(x ) = C
0 0
Where: r – the rate constant of the process
K –the asymptotic limit of growth (carrying capacity).
This equation express quantitative changes in the presence of limited resources,
represented by the constant К.
The solution p(x) of Eq.(1) is the logistic function:
260 Dose-Effect Modeling of Experimental Data
K
p(x)
 K  (2)
11 e(r.x)
 
C
 
0
C = p(0)
0
This equation may be revised in view of the goal to identify an effective dose,
thereby making K=1. Тo avoid the case C =0 in Eq.(2), аs training logistic function as
0
proposed in [11] is used:
1
p(x) (3)
1 Ae(r*x)
A>0, r > 0
Three parameters are necessary to specify the curve, A, r and β, defined as
follows:
A is the number of times that the initial value p(0) must grow to reach 100%.
r is the slope or the growth rate parameter that specifies "width" or "steepness" of
the S-curve.
β is a dose correction parameter.
These parameters must be identified by fitting the Eq.(3) to the experimental data.
3 DATA IDENTIFICATION
To fit the function p(x) to the data, an identification procedure will be performed.
The values p in n number of points are known as a result of the measurement. They
i
are used to identify the parameters А,r, β.
Let us chose to subject to system identification the function
1
p(x,A,r,)  p (4)
i 1 Ae(r*xi) i
A > 0, r > 0, i=1,2,…,n

The identification vector Q(A,r,β) must be defined from the system (4). But this
system is not well defined. For each point x the residual d is:
i i
d(Q) = || p-p(x,Q) ||
i i i
The aim is to minimize d(Q) for the whole interval of identification [x -x ]:
i n 1
D(Q) = inf|| d(Q) || < , >0 (5)
i
Thus the identification goal is translated into an optimization problem. The
optimization method is defined depending on the way of minimization of D(Q).
 Least square fitting.
Journal of Information, Control and Management Systems, Vol. 8, (2010), No.3 261
An object of minimization is the functional:
n
D(Q)d2 min (6)
i
i1
 Uniform fitting
The uniform fitting is aimed at minimizing the maximal deviation in the base
points p:
i
D(Q)max|d |min (7)
i
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 minimum D for system (5).
4 CALCULATION OF DOSE-EFFECT PARAMETERS
To calculate the effective or lethal dose, it is necessary to find the inverse function
p-1(x). After trivial transformations of Eq.(3):
1 p() 
x() lnAln  (9)
r  1 p() 
As a result of the identification, the vector Q(A,r,β) is known and x(α) is
calculated directly from Eq.(9).
For most commonly used dose ED50 (LD50) at p(α)=0.5:
1
 
ED50 x(0.5) lnA (10)
r
The final result is easy for computing. The main burden is the process of
identification 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 frequency f(x) of
the effectiveness of doses analytically and graphically (fig.1) is obtained:
dp(x) Are(rx)
f(x)  (11)
dx [1 Ae(rx)]2
The extreme point corresponds to ED50/LD50. Thus the dose value of x(0.5) can
be calculated from the equation df(x) dx 0.
Solution of the equation
d2f(x)
0
dx2
262 Dose-Effect Modeling of Experimental Data
gives the coordinates of inflexion points I , I :
x1 x2
2 3 2 3
ln  ln 
A A
   
I  I 
x1 r x2 r
and they are used to calculate the standard deviation of the frequency of doses (fig.1):
2 3
ln 
I I 2 3 1.317 (12)
SD  x2 x1  
2 2r r

It is possible to define three dose ranges:
 Low response dose. These doses are
lower than (x(0.5)-2SD)
 Mean response dose. Doses in range
(x (0.5)-2SD, x (0.5)+2SD)
 High response dose. Doses greater than
(x (0.5)+2SD)

Figure.1. Dose frequency
5 EXAMPLE
The method is implemented in program KORELIA–Dynamics using its ability for
system identification. A dialog window is appended for calculation of ED50/LD50,
standard 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 Кoerber (cit.
Кoerber G., Arch.exper. Path. Pharm., 162, 480, 1931). The experimental data are
scattered on fig.2. Both methods are based on linear interpolation to determine LD50
and standard deviation SD. Cited data are:
 Behrens method: LD50±SD = 172.3 ±9.5 [mg/kg].
 Кoerber method: LD50±SD =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. After identification with KORELIA-Dynamics the
calculated model parameters are:
A = 927.3; r = 0.1855; β = -25.13
Journal of Information, Control and Management Systems, Vol. 8, (2010), No.3 263
Identification errors are: absolute = 0.0346; relative = 0.0834; quadratic =
0.00453.
The identified equation for data is:
1
p(x)
1927.3e(0.1855*x25.13)
Thus, the LD50 and SD according to equations (10) and (12) are
LD50±SD = 172.3034 ±7.0997 [mg/kg].
On fig.2 the experimental data, identification curve and frequency of doses are
presented.

Figure.2. Graph of the identification curve and frequency of doses
6 CONCLUSIONS
This work proposes an analytical model for determining the dose-effect. А
modified logistic function is used for data identification. Parameters of the model are
calculated with the optimization procedure that targets minimal residual using least-
square or uniform criteria. From the analytical model formulas for effective dose and
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 reflects the
nonlinearity of the process by selecting the appropriate function;
 Shows the behavior of the system throughout the range of tested doses;
 Drops some requirements as strictly regulated modification of doses to
conduct the experiment at pre-selected law;