# Fuzzy Evaluation of Pharmacokinetic Models

Authors: Carlos Sepúlveda; Oscar Montiel; José M. Cornejo Bravo; Roberto Sepúlveda
Journal: Computational Intelligence and Neuroscience
 (2018)

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## Abstract

Population pharmacokinetic (PopPK) models allow researchers to predict and analyze drug behavior in a population of individuals and to quantify the different sources of variability among these individuals. In the development of PopPK models, the most frequently used method is the nonlinear mixed effect model (NLME). However, once the PopPK model has been developed, it is necessary to determine if the selected model is the best one of the developed models during the population pharmacokinetic study, and this sometimes becomes a multiple criteria decision making (MCDM) problem, and frequently, researchers use statistical evaluation criteria to choose the final PopPK model. The used evaluation criteria mentioned above entail big problems since the selection of the best model becomes susceptible to the human error mainly by misinterpretation of the results. To solve the previous problems, we introduce the development of a software robot that can automate the task of selecting the best PopPK model considering the knowledge of human expertise. The software robot is a fuzzy expert system that provides a method to systematically perform evaluations on a set of candidate PopPK models of commonly used statistical criteria. The presented results strengthen our hypothesis that the software robot can be successfully used to evaluate PopPK models ensuring the selection of the best PopPK model.


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## Body

## 1. Introduction

Pharmacokinetics (PK) is a subdivision of pharmacology in which mathematical models are developed to study the processes of absorption, distribution, and elimination of a drug once a given dose has been administered to a given individual [1]. These mathematical models are derived from representing the body as a system of compartments [2] (Figure 1), where the transfer rates of absorption, distribution, redistribution, and elimination between these compartments can be used to determine the parameters of the PK model such as clearance (Cl) and volume of distribution (V) that allow us to predict the drug concentration in plasma (Cp) in a given time [3].Figure 1
Example of the body viewed as (a) one-compartment model or (b) two-compartment model. The central compartment represents the plasma and tissues, and the peripheral compartment represents those tissues that take up the drug at a slower rate.
(a)
(b)Let us consider the one compartment PK model of Figure1, where the rate of elimination of the drug presented in the body decreases in proportion to the Cp, that is, with a first-order elimination process ke and the drug administered as a single intravenous bolus dose (D); the kinetic of Cp in the compartment—the body—at time t>0 is depicted by the following deterministic differential equation:(1)dCptdt=−keCpt,Cp0=DV.In this case, the term ke is given by(2)ke=ClV,where V is the distribution volume of the drug within the body and Cl measures the ability of the liver and the kidney, mainly to extract a drug from the body. In this context, Cl and V are fixed parameters that describe the drug concentration of a given dose D over time t. Equation (1) has the following explicit solution:(3)Cpt=DVe−Cl/V⋅t.The model (3) states a relationship between the independent variable t and the dependent variable Cp [1, 4].PopPK models are designed to analyze drug behavior in a group of individuals; hence, it is possible to generalize the model for similar individuals that were not subject of the study. NLME modeling is the main current approach for PopPK model development; it allows estimation of parameters in the presence of different levels of variability, and it is commonly used when it is not possible to obtain the complete information on repeated measurements of individuals [2]. Most of the time, the process to find the final PopPK model that better represents the data behavior requires the development of several PopPK models. It is common to use interchangeable software programs like R or SAS to apply various evaluation criteria to select which PopPK model is “better” among all the possible PopPK models [5–7]. These evaluation criteria range from graphical analysis to statistical methods [8]. The graphical analysis is limited to expert modelers, and besides that, it can be time-consuming and is subject to human error or misinterpretation.The statistical methods such as the estimated error variance (MSE), the objective function value (OFV), the Akaike’s information criterion (AIC), or the Bayesian information criterion (BIC), also known as Schwarz criterion (CS) offer quantitative terms (values) which are much easier to interpret and for consequence more easily to exclude or accept. However, the respective values may be contrasted between PopPK models when more than one statistical criterion is used complicating in this way ensuring the selection of the best PopPK model, thus becoming a multiple criteria decision making (MCDM) problem [9].Fuzzy expert systems is an area where artificial intelligence and fuzzy logic meet to make robots backing or even replace human experts that want to do automated tasks [10]. The fuzzy logic has been recognized as an essential problem-solving technique when human evaluations are required for interpretation of medical findings [11], as well as in problems where is necessary for managing data with uncertainty such as [12], where a type-2 fuzzy logic methodology is used to help a neural network to handle complex time series data.We contribute with a new software robot to facilitate the selection of the best pharmacokinetic model by an automated process when the comparison of the implemented evaluation criteria values among the models cannot be categorically determined, addressing the problem as a decision problem. We present results using a software robot to select the best PopPK model of tobramycin by an automated process.The organization of this work is as follows: in Section2, an overview of nonlinear mixed-effects modeling is provided. In Section 3, the implementation of the software robot is explained. In Section 4, the parameter estimation results and the outcomes of the software robot are presented. In Section 5, an analysis of the results is exposed. Finally, in Section 6, the conclusions are given.


## 2. Nonlinear Mixed-Effects Model Framework

In mixed-effects models, the data set is longitudinal, i.e., it is composed of repeated samples of the same individuals [13, 14]. This is the case of the PopPK experimental data; it consists of a vector yij of samples of the j−th measurement where,j=1,…,n of Cp in the i−th individual where,i=1,…,n. Therefore, we can obtain the sample yij for the individual i at the time tij by using the model:(4)yij=fxij,ϕi+εij;εij∼0,σ2,where f⋅ is a nonlinear function as Equation (3) relating a vector of known values xij (e.g., dose and time) to the unknown parameter vector ϕi e.g.,Cli,ViT and the variable εij in the second term is the residual error that is assumed to be normally distributed with mean zero and variance σ2. It is sensible to expect that each individual has a different parameter vector ϕi, and this is described in a second stage by the covariate model:(5)ϕi=gzi;θ+ηi;ηi∼0,ω2.Equation (5) describes the variation among different individuals accounted through the individual-specific parameters ϕi.The functiong⋅ relates the specific covariates zi of the individuals to the population parameters θ. ηi is a q×1 vector containing the random effects that have a normal distribution with a covariance matrix ω2; other distributions for random effects exist [14].


### 2.1. Estimation of Population Parameters

Since the PopPK data are longitudinal, we can assume that the individuals are sampled randomly from the population, and therefore it is practical to assume that theηi’s are also randomly sampled (even though they are not observable) to conduct the estimations of the parameters of (4) and (5) by optimizing the likelihood function defined by the following equation:(6)Lθ,Ω,Σy≜pyβ,Ω,Σ,where L⋅ is the likelihood function, Ω is the variance-covariance matrix of all the ηi’s, and Σ the residual variance-covariance matrix of all the εi’s where pyβ,Σ,η is the conditional probability density of all measurements [15]. If pyβ,Σ,η is explicit, the likelihood function L⋅ is explicit too, and the exact maximum likelihood estimators can be applied [16], otherwise the estimation of model parameters is made using linear approximations for (6) [17]. Either way, using an explicit likelihood function or an approximation, the precision in the estimated vector θ^ depends not only in the residual variability but also in the estimated variance-covariance matrix Ω^, which is calculated using the inverse Hessian matrix of the −logL⋅.


### 2.2. Nonlinear Mixed-Effects Model Evaluation Criteria

In the literature referring on the analysis and development of NLME models and PopPK models, the researchers frequently highlighted three statistical criteria to carry out the different evaluations of the model and thus decide which PopPK model they should select within a collection of PopPK models they are considering.The first two evaluation criteria that we are going to present in this work are the Akaike information criterion (ACI) [18] and the Schwarz’s information criterion (SC) [19]. These evaluation criteria are derived from information theory and a Bayesian approach and are commonly used for evaluating nonlinear mixed-effects models when the parameters of the model were estimated by the maximum likelihood method. The AIC criterion provides a balance between the goodness of the fit of the model and the number of parameters required to obtain the fit [20] as is shown in the following equation:(7)AIC=N⋅logOFV+2⋅NP,where N is the number of observations and NP the number of parameters. The final and optimal model is the one with the smallest AIC value. Unlike the AIC criterion, the SC criterion accepts equal probability for each model and for every possible parameter value under the model as is shown in the following equation:(8)SC=N⋅logOFV+NP⋅logNP.The third evaluation criterion is the estimated error variance also known as the mean square error (MSE). This is based on the simple variance estimator shown in the following equation:(9)σ^i2=∑​i=1myi−y^n.The use of the MSE is due to the residuals, and it also contains essential information on the quality of the model. Specifically, the estimated variance in the responses, wherey is a vector of observed values and y^i, represents a vector of n predictions.The version of Equation (9) depends on the estimation method used to estimate the PopPK parameters, for example, the variance estimator when the restricted maximum estimation likelihood (REML) is used is shown in the following equation:(10)σ^i2=∑​i=1myi−y^REMLin−p.Equation (10) takes into account the degrees of freedom by subtracting the total of parameters p used in the estimation process [14] where y^REMLi is a vector of predicted values applying REML.


## 2.1. Estimation of Population Parameters

Since the PopPK data are longitudinal, we can assume that the individuals are sampled randomly from the population, and therefore it is practical to assume that theηi’s are also randomly sampled (even though they are not observable) to conduct the estimations of the parameters of (4) and (5) by optimizing the likelihood function defined by the following equation:(6)Lθ,Ω,Σy≜pyβ,Ω,Σ,where L⋅ is the likelihood function, Ω is the variance-covariance matrix of all the ηi’s, and Σ the residual variance-covariance matrix of all the εi’s where pyβ,Σ,η is the conditional probability density of all measurements [15]. If pyβ,Σ,η is explicit, the likelihood function L⋅ is explicit too, and the exact maximum likelihood estimators can be applied [16], otherwise the estimation of model parameters is made using linear approximations for (6) [17]. Either way, using an explicit likelihood function or an approximation, the precision in the estimated vector θ^ depends not only in the residual variability but also in the estimated variance-covariance matrix Ω^, which is calculated using the inverse Hessian matrix of the −logL⋅.


## 2.2. Nonlinear Mixed-Effects Model Evaluation Criteria

In the literature referring on the analysis and development of NLME models and PopPK models, the researchers frequently highlighted three statistical criteria to carry out the different evaluations of the model and thus decide which PopPK model they should select within a collection of PopPK models they are considering.The first two evaluation criteria that we are going to present in this work are the Akaike information criterion (ACI) [18] and the Schwarz’s information criterion (SC) [19]. These evaluation criteria are derived from information theory and a Bayesian approach and are commonly used for evaluating nonlinear mixed-effects models when the parameters of the model were estimated by the maximum likelihood method. The AIC criterion provides a balance between the goodness of the fit of the model and the number of parameters required to obtain the fit [20] as is shown in the following equation:(7)AIC=N⋅logOFV+2⋅NP,where N is the number of observations and NP the number of parameters. The final and optimal model is the one with the smallest AIC value. Unlike the AIC criterion, the SC criterion accepts equal probability for each model and for every possible parameter value under the model as is shown in the following equation:(8)SC=N⋅logOFV+NP⋅logNP.The third evaluation criterion is the estimated error variance also known as the mean square error (MSE). This is based on the simple variance estimator shown in the following equation:(9)σ^i2=∑​i=1myi−y^n.The use of the MSE is due to the residuals, and it also contains essential information on the quality of the model. Specifically, the estimated variance in the responses, wherey is a vector of observed values and y^i, represents a vector of n predictions.The version of Equation (9) depends on the estimation method used to estimate the PopPK parameters, for example, the variance estimator when the restricted maximum estimation likelihood (REML) is used is shown in the following equation:(10)σ^i2=∑​i=1myi−y^REMLin−p.Equation (10) takes into account the degrees of freedom by subtracting the total of parameters p used in the estimation process [14] where y^REMLi is a vector of predicted values applying REML.


## 3. Implementation of the Software Robot

### 3.1. The Software Robot

The software robot contained a fuzzy expert system that takes the advantage of the human knowledge and their expertise in a given field to create linguistic descriptors for variables and create fuzzy sets that allow to control the behavior of phenomena under study or even as a tool for decision making [21]. In this work, we developed a software robot as shown in Figure 2 to evaluate indistinct versions of a PopPK model for an aminoglycoside antibiotic named tobramycin which has been the subject of many studies due to its narrow therapeutic window [22]. In this figure, first, we defined the number N of PopPK models to be developed taking into account the number of parameters and covariates. After that, we proceed to establish one by one the N PopPK models. Three statistical evaluation criteria are performed on each model, and they are AIC, SC, and MSE. The software robot uses them as the inputs for the fuzzy expert system, and it collects each defuzzified output into an array until all the N models have been evaluated. Then, the software robot chooses the best PopPK model which is the one that had the lowest value in the fuzzy evaluation, and this will be selected as the best PopPK model.Figure 2
Block diagram.Thus, the evaluation made by our software robot will guide us to decide if a PopPK model of tobramycin is the best within a group of PopPK models of Tobramycin.The variables are the evaluation criteria AIC, SC, and MSE presented in Equations (7), (8), and (10), respectively, and commonly used to evaluate a PopPK model, and their linguistic descriptors are the ranges of values. For example, the variable AIC has a range of values from 670 to 672.5 considered “Very Low”, from 670 to 675 considered “Low,” from 672.5 to 677.5 considered “Medium,” from 675 to 680 considered “High,” and from 677.5 to 680 considered “Very High.” “Very Low,” “Low,” “Medium,” “High,” and “Very High” are the linguistic descriptors for the sets of values for the variable AIC.As is typical of the fuzzy theory, the sets of values overlap, and in this way, a value may partially belong to a set and have a degree of membership0≤μ≤1 that is any place between zero and one, where μ represents the degree of membership. Thus, the value belongs to several sets with the total membership adding to one. Going back to the above example, the linguistic descriptors of “Very Low” and “Low” are two fuzzy sets for the variable AIC that may overlap, so an AIC of 670.5 could be mostly “Very Low” with μ=0.8 and somewhat “Low” with μ=0.2.Implementing fuzzy sets with linguistic descriptors to perform a generalization of the output evaluation criteria, results of the fitted PopPK model, we may establish relationships between variables to evaluate the PopPK model by extracting fuzzy rules in terms used by a human expert of the formIF−THEN. In other words, a fuzzy system can perform a description of the phenomena under study, based on the antecedents and consequents presented in a fuzzy rule, that is in the form IF condition THEN action rules [23].Using the three evaluation criteria AIC, SC, and MSE, we can set a knowledge base in such way thatIF AIC is mostly “Very Low” AND CS is mostly “Very Low” AND MSE is also mostly “Very Low” THEN Evaluation is “OPTIMAL”. That is, under the fuzzy logic, we are not only going to have linguistic descriptors as the inputs of the fuzzy system but also output variables.In summary, what a fuzzy system does is to transform crisp input values into fuzzy inputs through a fuzzification unit that establishes the degree of membership to the fuzzy sets for previously defined variables. Then, the fuzzy system uses a rulebase designed by the human expert to predict the fuzzy output of the phenomenon under study, whereby the fuzzy system has a defuzzification unit that transforms the fuzzy output into a crisp value. For this work, we created a fuzzy system with three crisp input variables as shown in Figure3, the Mamdani max-min inferences to obtain the membership functions of the system and the centroid method to defuzzify the output.Figure 3
The crisp input values AIC, SC, and MSE are transformed into fuzzy inputs by the fuzzification unit which assigns the degree of membership to the fuzzy sets defined for the variables. The fuzzy system can then make inferences or predict outputs of the system by applying knowledge from the expert. Finally, the defuzzification unit transforms the obtained fuzzy output into a crisp value.


### 3.2. Fuzzification

As we mentioned, the values of the three evaluation criteria for a developed PopPK model are used to perform a comparison between models, so we can finally decide the PopPK model that is “better,” which leads the researchers to deal with an MCDM problem. As an example, a researcher can get to the point in which he needs to decide between two models that have the best evaluation criteria values (Table1).Table 1
Example of the final evaluation results for a PopPK model.
PopPK model
AIC
CS
MSE
1
652.470
665.300
0.14
2
650.604
668.094
0.092
The best results are the values with red color.In the example of Table1, the model 2 has a better result in the AIC than model 1 for 1.866 units and in the MSE for 0.048 units. However, model 1 performs better in CS for a significant difference of almost 3 units (2.794). This type of ambiguity in the statistical results and the pressure to get the optimal PopPK model can result in a misinterpretation of the evaluation criteria to select the optimal PopPK model.In this work, the values of the linguistic variables AIC, CS, and MSE calculated from the development of indistinct versions of a PopPK model of tobramycin were used as our fuzzy input sets. Their maximum and their minimum values are shown in Table2.Table 2
AIC, CS, and MSE values used for the fuzzy expert system.
Variable
Minimum value
Maximum value
AIC
670
680
CS
680
795
MSE
0.095
0.14The membership function for the three input variables is tagged as “Very Low,” “Low,” “Normal,” “Medium,” “High,” and “Very High,” and their corresponding parameter values are shown in Table3.Table 3
Parameters of the membership function values for the input variables.
Linguistic variables
Linguistic term names, type shapes, and parameters
Very Low trapezoidal
Low triangular
Medium triangular
High triangular
Very High trapezoidal
AIC
[670 670 670.3 672.5]
[670 672.5 675]
[672.5 675 677.5]
[675 677.5 680]
[677.5 679.7 680 680]
CS
[676.6 679.6 680.4 683.4]
[680 683.8 687.5]
[683.8 687.5 691.3]
[687.5 691.3 695]
[691 695 695 698]
MSE
[0.092 0.092 0.09397 0.104]
[0.0921 0.104 0.116]
[0.104 0.116 0.128]
[0.116 0.128 0.14]
[0.128 0.1384 0.14 0.14]The membership functions for the output variable “Evaluation” are represented by five membership functions, and the corresponding types and parameter values are shown in Table4.Table 4
Parameters of the membership function values for the output variable.
Linguistic variables
Lingustic term names, type shapes, and parameters
Optimal trapezoidal
High Acceptable
Acceptable triangular
Low Acceptable triangular
Rejected trapezoidal
Evaluation
[0 0 10 30]
[10 30 50]
[30 50 70]
[50 70 90]
[70 90 100 100]


### 3.3. Fuzzy Rulebase

We created the fuzzy rulebase using the human knowledge of an expert who judged “how much more significant one attribute is than the other.” The inference is made on the rulesIF−THEN; one rule for each fuzzy output set. The total of rules stems from the heuristic R=ln, where R is equal to the total of rules, l is equal to the number of linguistic descriptors, and n represents the number of input variables; thus in our case with l=5 and n=3 results in a total of 125 rules (Table 5).Table 5
Fuzzy rules.
Rule number
Linguistic inputs
Linguistic output
AIC
CS
MSE
Evaluation
1
Very Low
Very Low
Very Low
Optimal
2
Very Low
Very Low
Low
High Acceptable
3
Very Low
Very Low
Medium
Acceptable
4
Very Low
Very Low
High
Low Acceptable
5
Very Low
Very Low
Very
Optimal
7
Very Low
Low
Low
High Acceptable
8
Very Low
Low
Medium
Acceptable
9
Very Low
Low
High
Low Acceptable
10
Very Low
Low
Very High
Rejected
11
Very Low
Medium
Very Low
High Acceptable
12
Very Low
Medium
Low
High Acceptable
13
Very Low
Medium
Medium
Low Acceptable
14
Very Low
Medium
High
Low Acceptable
15
Very Low
Medium
Very High
Rejected
16
Very Low
High
Very Low
Acceptable
17
Very Low
High
Low
Acceptable
18
Very Low
High
Medium
Low Acceptable
19
Very Low
High
High
Rejected
20
Very Low
High
Very High
Rejected
21
Very Low
Very High
Very Low
Low Acceptable
22
Very Low
Very High
Low
Low Acceptable
23
Very Low
Very High
Medium
Rejected
24
Very Low
Very High
High
Rejected
25
Very Low
Very High
Very High
Rejected
26
Low
Very Low
Very Low
Optimal
27
Low
Very Low
Low
Optimal
28
Low
Very Low
Medium
High Acceptable
29
Low
Very Low
High
Low Acceptable
30
Low
Very Low
Very High
Rejected
31
Low
Low
Very Low
Optimal
32
Low
Low
Low
High Acceptable
33
Low
Low
Medium
Acceptable
34
Low
Low
High
Low Acceptable
35
Low
Low
Very High
Rejected
36
Low
Medium
Very Low
Optimal
37
Low
Medium
Low
High Acceptable
38
Low
Medium
Medium
Acceptable
39
Low
Medium
High
Low Acceptable
40
Low
Medium
Very High
Rejected
41
Low
High
Very Low
High Acceptable
42
Low
High
Low
Acceptable
43
Low
High
Medium
Low Acceptable
44
Low
High
High
Rejected
45
Low
High
Very High
Rejected
46
Low
Very High
Very Low
Acceptable
47
Low
Very High
Low
Low Acceptable
48
Low
Very High
Medium
Rejected
49
Low
Very High
High
Rejected
50
Low
Very High
Very High
Rejected
51
Medium
Very Low
Very Low
Optimal
52
Medium
Very Low
Low
High Acceptable
53
Medium
Very Low
Medium
Acceptable
54
Medium
Very Low
High
Low Acceptable
55
Medium
Very Low
Very High
Rejected
56
Medium
Low
Very Low
High Acceptable
57
Medium
Low
Low
High Acceptable
58
Medium
Low
Medium
Acceptable
59
Medium
Low
High
Rejected
60
Medium
Low
Very High
Rejected
61
Medium
Medium
Very Low
Acceptable
62
Medium
Medium
Low
Acceptable
63
Medium
Medium
Medium
Low Acceptable
64
Medium
Medium
High
Rejected
65
Medium
Medium
Very High
Rejected
66
Medium
High
Very Low
Acceptable
67
Medium
High
Low
Low Acceptable
68
Medium
High
Medium
Rejected
69
Medium
High
High
Rejected
70
Medium
High
Very High
Rejected
71
Medium
Very High
Very Low
Low Acceptable
72
Medium
Very High
Low
Low Acceptable
73
Medium
Very High
Medium
Rejected
74
Medium
Very High
High
Rejected
75
Medium
Very High
Very High
Rejected
76
High
Very Low
Very Low
Acceptable
77
High
Very Low
Low
Acceptable
78
High
Very Low
Medium
Low Acceptable
79
High
Very Low
High
Rejected
80
High
Very High
Very High
Rejected
81
High
Low
Very Low
Low Acceptable
82
High
Low
Low
Low Acceptable
83
High
Low
Medium
Rejected
84
High
Low
High
Rejected
85
High
Low
Very High
Rejected
86
High
Medium
Very Low
Low Acceptable
87
High
Medium
Low
Low Acceptable
88
High
Medium
Medium
Rejected
89
High
Medium
High
Rejected
90
High
Medium
Very High
Rejected
91
High
High
Very Low
Low Acceptable
92
High
High
Low
Rejected
93
High
High
Medium
Rejected
94
High
High
High
Rejected
95
High
High
Very High
Rejected
96
High
Very High
Very Low
Low Acceptable
97
High
Very High
Low
Low Acceptable
98
High
Very High
Medium
Rejected
99
High
Very High
High
Rejected
100
High
Very High
Very High
Rejected
101
Very High
Very Low
Very Low
Acceptable
102
Very High
Very Low
Low
Low Acceptable
103
Very High
Very Low
Medium
Rejected
104
Very High
Very Low
High
Rejected
105
Very High
Very Low
Very High
Rejected
106
Very High
Low
Very Low
Low Acceptable
107
Very High
Low
Low
Low Acceptable
108
Very High
Low
Medium
Rejected
109
Very High
Low
High
Rejected
110
Very High
Low
Very High
Rejected
111
Very High
Medium
Very Low
Low Acceptable
112
Very High
Medium
Low
Low Acceptable
113
Very High
Medium
Medium
Rejected
114
Very High
Medium
High
Rejected
115
Very High
Medium
Very High
Rejected
116
Very High
High
Very Low
Rejected
117
Very High
High
Low
Rejected
118
Very High
High
Medium
Rejected
119
Very High
High
High
Rejected
120
Very High
High
Very High
Rejected
121
Very High
Very High
Very Low
Rejected
122
Very High
Very High
Low
Rejected
123
Very High
Very High
Medium
Rejected
124
Very High
Very High
High
Rejected
125
Very High
Very High
Very High
Rejected


#### 3.3.1. Defuzzifier

The fuzzified functions obtained from fuzzy inference are converted into numeric values, for example:IF theAIC is Low (677)ANDCS is Low (686)ANDMSE is High (.12)THEN the PopPK model is Low Acceptable(75.2)The above is achieved by applying the method of center of gravity defuzzifier forN rules using the formula:(11)C=∑​i=1Nbi∫​μi∑​i=1N∫​μi,where C represents the crisp output value, bi represents the center of the membership function of the consequent of the rule i, and ∫​μi is the area under the membership function ∫​μi of the consequent in the rule i. In other words, the center of gravity method calculates the center of mass from the output membership functions.Figure4 shows the global behavior of the fuzzy system for the relation between the input variables CS and AIC, MSE and CS, and MSE and AIC considering all their rules. For example, if the value of CS is “Very Low,” that is 681 and AIC is “High,” that is 678.5, then the PopPK model obtains a bad evaluation, or if the value of MSE is “Low,” that is 0.1 and the value of CS is also “Low”, that is 684, the model obtains an excellent evaluation. Another example will be a scenario where the value of MSE is “Medium,” that is 0.12, and the value of AIC is “Low,” that is 674.5, and the model obtains a good evaluation.Figure 4
The figure shows the visualization of the system surface when the inputs AIC, CS, and MSE are interacting with each other. (a) The inputs AIC and CS are interacting. (b) The inputs MSE and CS are interacting. (c) The inputs MSE and AIC are interacting.
(a)
(b)
(c)If it was required, the search for the lowest value could be done according to Algorithm1. The crisp output value can be stored in an array together with other output values taken from other PopPK models that were evaluated previously and then compare them in an automated way to find the lowest value which will determine the optimal PopPK model.Algorithm 1: Find the lowest value algorithm.
(1):
i
←
0
(2):
l
o
w
←
i
(3):
for
j = i + 1 to total of candidate models–1 do
(4):
ifarray[j] is less than array[low] then
(5):
Set low toj
(6):
end if
(7):
end for


## 3.1. The Software Robot

The software robot contained a fuzzy expert system that takes the advantage of the human knowledge and their expertise in a given field to create linguistic descriptors for variables and create fuzzy sets that allow to control the behavior of phenomena under study or even as a tool for decision making [21]. In this work, we developed a software robot as shown in Figure 2 to evaluate indistinct versions of a PopPK model for an aminoglycoside antibiotic named tobramycin which has been the subject of many studies due to its narrow therapeutic window [22]. In this figure, first, we defined the number N of PopPK models to be developed taking into account the number of parameters and covariates. After that, we proceed to establish one by one the N PopPK models. Three statistical evaluation criteria are performed on each model, and they are AIC, SC, and MSE. The software robot uses them as the inputs for the fuzzy expert system, and it collects each defuzzified output into an array until all the N models have been evaluated. Then, the software robot chooses the best PopPK model which is the one that had the lowest value in the fuzzy evaluation, and this will be selected as the best PopPK model.Figure 2
Block diagram.Thus, the evaluation made by our software robot will guide us to decide if a PopPK model of tobramycin is the best within a group of PopPK models of Tobramycin.The variables are the evaluation criteria AIC, SC, and MSE presented in Equations (7), (8), and (10), respectively, and commonly used to evaluate a PopPK model, and their linguistic descriptors are the ranges of values. For example, the variable AIC has a range of values from 670 to 672.5 considered “Very Low”, from 670 to 675 considered “Low,” from 672.5 to 677.5 considered “Medium,” from 675 to 680 considered “High,” and from 677.5 to 680 considered “Very High.” “Very Low,” “Low,” “Medium,” “High,” and “Very High” are the linguistic descriptors for the sets of values for the variable AIC.As is typical of the fuzzy theory, the sets of values overlap, and in this way, a value may partially belong to a set and have a degree of membership0≤μ≤1 that is any place between zero and one, where μ represents the degree of membership. Thus, the value belongs to several sets with the total membership adding to one. Going back to the above example, the linguistic descriptors of “Very Low” and “Low” are two fuzzy sets for the variable AIC that may overlap, so an AIC of 670.5 could be mostly “Very Low” with μ=0.8 and somewhat “Low” with μ=0.2.Implementing fuzzy sets with linguistic descriptors to perform a generalization of the output evaluation criteria, results of the fitted PopPK model, we may establish relationships between variables to evaluate the PopPK model by extracting fuzzy rules in terms used by a human expert of the formIF−THEN. In other words, a fuzzy system can perform a description of the phenomena under study, based on the antecedents and consequents presented in a fuzzy rule, that is in the form IF condition THEN action rules [23].Using the three evaluation criteria AIC, SC, and MSE, we can set a knowledge base in such way thatIF AIC is mostly “Very Low” AND CS is mostly “Very Low” AND MSE is also mostly “Very Low” THEN Evaluation is “OPTIMAL”. That is, under the fuzzy logic, we are not only going to have linguistic descriptors as the inputs of the fuzzy system but also output variables.In summary, what a fuzzy system does is to transform crisp input values into fuzzy inputs through a fuzzification unit that establishes the degree of membership to the fuzzy sets for previously defined variables. Then, the fuzzy system uses a rulebase designed by the human expert to predict the fuzzy output of the phenomenon under study, whereby the fuzzy system has a defuzzification unit that transforms the fuzzy output into a crisp value. For this work, we created a fuzzy system with three crisp input variables as shown in Figure3, the Mamdani max-min inferences to obtain the membership functions of the system and the centroid method to defuzzify the output.Figure 3
The crisp input values AIC, SC, and MSE are transformed into fuzzy inputs by the fuzzification unit which assigns the degree of membership to the fuzzy sets defined for the variables. The fuzzy system can then make inferences or predict outputs of the system by applying knowledge from the expert. Finally, the defuzzification unit transforms the obtained fuzzy output into a crisp value.


## 3.2. Fuzzification

As we mentioned, the values of the three evaluation criteria for a developed PopPK model are used to perform a comparison between models, so we can finally decide the PopPK model that is “better,” which leads the researchers to deal with an MCDM problem. As an example, a researcher can get to the point in which he needs to decide between two models that have the best evaluation criteria values (Table1).Table 1
Example of the final evaluation results for a PopPK model.
PopPK model
AIC
CS
MSE
1
652.470
665.300
0.14
2
650.604
668.094
0.092
The best results are the values with red color.In the example of Table1, the model 2 has a better result in the AIC than model 1 for 1.866 units and in the MSE for 0.048 units. However, model 1 performs better in CS for a significant difference of almost 3 units (2.794). This type of ambiguity in the statistical results and the pressure to get the optimal PopPK model can result in a misinterpretation of the evaluation criteria to select the optimal PopPK model.In this work, the values of the linguistic variables AIC, CS, and MSE calculated from the development of indistinct versions of a PopPK model of tobramycin were used as our fuzzy input sets. Their maximum and their minimum values are shown in Table2.Table 2
AIC, CS, and MSE values used for the fuzzy expert system.
Variable
Minimum value
Maximum value
AIC
670
680
CS
680
795
MSE
0.095
0.14The membership function for the three input variables is tagged as “Very Low,” “Low,” “Normal,” “Medium,” “High,” and “Very High,” and their corresponding parameter values are shown in Table3.Table 3
Parameters of the membership function values for the input variables.
Linguistic variables
Linguistic term names, type shapes, and parameters
Very Low trapezoidal
Low triangular
Medium triangular
High triangular
Very High trapezoidal
AIC
[670 670 670.3 672.5]
[670 672.5 675]
[672.5 675 677.5]
[675 677.5 680]
[677.5 679.7 680 680]
CS
[676.6 679.6 680.4 683.4]
[680 683.8 687.5]
[683.8 687.5 691.3]
[687.5 691.3 695]
[691 695 695 698]
MSE
[0.092 0.092 0.09397 0.104]
[0.0921 0.104 0.116]
[0.104 0.116 0.128]
[0.116 0.128 0.14]
[0.128 0.1384 0.14 0.14]The membership functions for the output variable “Evaluation” are represented by five membership functions, and the corresponding types and parameter values are shown in Table4.Table 4
Parameters of the membership function values for the output variable.
Linguistic variables
Lingustic term names, type shapes, and parameters
Optimal trapezoidal
High Acceptable
Acceptable triangular
Low Acceptable triangular
Rejected trapezoidal
Evaluation
[0 0 10 30]
[10 30 50]
[30 50 70]
[50 70 90]
[70 90 100 100]


## 3.3. Fuzzy Rulebase

We created the fuzzy rulebase using the human knowledge of an expert who judged “how much more significant one attribute is than the other.” The inference is made on the rulesIF−THEN; one rule for each fuzzy output set. The total of rules stems from the heuristic R=ln, where R is equal to the total of rules, l is equal to the number of linguistic descriptors, and n represents the number of input variables; thus in our case with l=5 and n=3 results in a total of 125 rules (Table 5).Table 5
Fuzzy rules.
Rule number
Linguistic inputs
Linguistic output
AIC
CS
MSE
Evaluation
1
Very Low
Very Low
Very Low
Optimal
2
Very Low
Very Low
Low
High Acceptable
3
Very Low
Very Low
Medium
Acceptable
4
Very Low
Very Low
High
Low Acceptable
5
Very Low
Very Low
Very
Optimal
7
Very Low
Low
Low
High Acceptable
8
Very Low
Low
Medium
Acceptable
9
Very Low
Low
High
Low Acceptable
10
Very Low
Low
Very High
Rejected
11
Very Low
Medium
Very Low
High Acceptable
12
Very Low
Medium
Low
High Acceptable
13
Very Low
Medium
Medium
Low Acceptable
14
Very Low
Medium
High
Low Acceptable
15
Very Low
Medium
Very High
Rejected
16
Very Low
High
Very Low
Acceptable
17
Very Low
High
Low
Acceptable
18
Very Low
High
Medium
Low Acceptable
19
Very Low
High
High
Rejected
20
Very Low
High
Very High
Rejected
21
Very Low
Very High
Very Low
Low Acceptable
22
Very Low
Very High
Low
Low Acceptable
23
Very Low
Very High
Medium
Rejected
24
Very Low
Very High
High
Rejected
25
Very Low
Very High
Very High
Rejected
26
Low
Very Low
Very Low
Optimal
27
Low
Very Low
Low
Optimal
28
Low
Very Low
Medium
High Acceptable
29
Low
Very Low
High
Low Acceptable
30
Low
Very Low
Very High
Rejected
31
Low
Low
Very Low
Optimal
32
Low
Low
Low
High Acceptable
33
Low
Low
Medium
Acceptable
34
Low
Low
High
Low Acceptable
35
Low
Low
Very High
Rejected
36
Low
Medium
Very Low
Optimal
37
Low
Medium
Low
High Acceptable
38
Low
Medium
Medium
Acceptable
39
Low
Medium
High
Low Acceptable
40
Low
Medium
Very High
Rejected
41
Low
High
Very Low
High Acceptable
42
Low
High
Low
Acceptable
43
Low
High
Medium
Low Acceptable
44
Low
High
High
Rejected
45
Low
High
Very High
Rejected
46
Low
Very High
Very Low
Acceptable
47
Low
Very High
Low
Low Acceptable
48
Low
Very High
Medium
Rejected
49
Low
Very High
High
Rejected
50
Low
Very High
Very High
Rejected
51
Medium
Very Low
Very Low
Optimal
52
Medium
Very Low
Low
High Acceptable
53
Medium
Very Low
Medium
Acceptable
54
Medium
Very Low
High
Low Acceptable
55
Medium
Very Low
Very High
Rejected
56
Medium
Low
Very Low
High Acceptable
57
Medium
Low
Low
High Acceptable
58
Medium
Low
Medium
Acceptable
59
Medium
Low
High
Rejected
60
Medium
Low
Very High
Rejected
61
Medium
Medium
Very Low
Acceptable
62
Medium
Medium
Low
Acceptable
63
Medium
Medium
Medium
Low Acceptable
64
Medium
Medium
High
Rejected
65
Medium
Medium
Very High
Rejected
66
Medium
High
Very Low
Acceptable
67
Medium
High
Low
Low Acceptable
68
Medium
High
Medium
Rejected
69
Medium
High
High
Rejected
70
Medium
High
Very High
Rejected
71
Medium
Very High
Very Low
Low Acceptable
72
Medium
Very High
Low
Low Acceptable
73
Medium
Very High
Medium
Rejected
74
Medium
Very High
High
Rejected
75
Medium
Very High
Very High
Rejected
76
High
Very Low
Very Low
Acceptable
77
High
Very Low
Low
Acceptable
78
High
Very Low
Medium
Low Acceptable
79
High
Very Low
High
Rejected
80
High
Very High
Very High
Rejected
81
High
Low
Very Low
Low Acceptable
82
High
Low
Low
Low Acceptable
83
High
Low
Medium
Rejected
84
High
Low
High
Rejected
85
High
Low
Very High
Rejected
86
High
Medium
Very Low
Low Acceptable
87
High
Medium
Low
Low Acceptable
88
High
Medium
Medium
Rejected
89
High
Medium
High
Rejected
90
High
Medium
Very High
Rejected
91
High
High
Very Low
Low Acceptable
92
High
High
Low
Rejected
93
High
High
Medium
Rejected
94
High
High
High
Rejected
95
High
High
Very High
Rejected
96
High
Very High
Very Low
Low Acceptable
97
High
Very High
Low
Low Acceptable
98
High
Very High
Medium
Rejected
99
High
Very High
High
Rejected
100
High
Very High
Very High
Rejected
101
Very High
Very Low
Very Low
Acceptable
102
Very High
Very Low
Low
Low Acceptable
103
Very High
Very Low
Medium
Rejected
104
Very High
Very Low
High
Rejected
105
Very High
Very Low
Very High
Rejected
106
Very High
Low
Very Low
Low Acceptable
107
Very High
Low
Low
Low Acceptable
108
Very High
Low
Medium
Rejected
109
Very High
Low
High
Rejected
110
Very High
Low
Very High
Rejected
111
Very High
Medium
Very Low
Low Acceptable
112
Very High
Medium
Low
Low Acceptable
113
Very High
Medium
Medium
Rejected
114
Very High
Medium
High
Rejected
115
Very High
Medium
Very High
Rejected
116
Very High
High
Very Low
Rejected
117
Very High
High
Low
Rejected
118
Very High
High
Medium
Rejected
119
Very High
High
High
Rejected
120
Very High
High
Very High
Rejected
121
Very High
Very High
Very Low
Rejected
122
Very High
Very High
Low
Rejected
123
Very High
Very High
Medium
Rejected
124
Very High
Very High
High
Rejected
125
Very High
Very High
Very High
Rejected


### 3.3.1. Defuzzifier

The fuzzified functions obtained from fuzzy inference are converted into numeric values, for example:IF theAIC is Low (677)ANDCS is Low (686)ANDMSE is High (.12)THEN the PopPK model is Low Acceptable(75.2)The above is achieved by applying the method of center of gravity defuzzifier forN rules using the formula:(11)C=∑​i=1Nbi∫​μi∑​i=1N∫​μi,where C represents the crisp output value, bi represents the center of the membership function of the consequent of the rule i, and ∫​μi is the area under the membership function ∫​μi of the consequent in the rule i. In other words, the center of gravity method calculates the center of mass from the output membership functions.Figure4 shows the global behavior of the fuzzy system for the relation between the input variables CS and AIC, MSE and CS, and MSE and AIC considering all their rules. For example, if the value of CS is “Very Low,” that is 681 and AIC is “High,” that is 678.5, then the PopPK model obtains a bad evaluation, or if the value of MSE is “Low,” that is 0.1 and the value of CS is also “Low”, that is 684, the model obtains an excellent evaluation. Another example will be a scenario where the value of MSE is “Medium,” that is 0.12, and the value of AIC is “Low,” that is 674.5, and the model obtains a good evaluation.Figure 4
The figure shows the visualization of the system surface when the inputs AIC, CS, and MSE are interacting with each other. (a) The inputs AIC and CS are interacting. (b) The inputs MSE and CS are interacting. (c) The inputs MSE and AIC are interacting.
(a)
(b)
(c)If it was required, the search for the lowest value could be done according to Algorithm1. The crisp output value can be stored in an array together with other output values taken from other PopPK models that were evaluated previously and then compare them in an automated way to find the lowest value which will determine the optimal PopPK model.Algorithm 1: Find the lowest value algorithm.
(1):
i
←
0
(2):
l
o
w
←
i
(3):
for
j = i + 1 to total of candidate models–1 do
(4):
ifarray[j] is less than array[low] then
(5):
Set low toj
(6):
end if
(7):
end for


## 3.3.1. Defuzzifier

The fuzzified functions obtained from fuzzy inference are converted into numeric values, for example:IF theAIC is Low (677)ANDCS is Low (686)ANDMSE is High (.12)THEN the PopPK model is Low Acceptable(75.2)The above is achieved by applying the method of center of gravity defuzzifier forN rules using the formula:(11)C=∑​i=1Nbi∫​μi∑​i=1N∫​μi,where C represents the crisp output value, bi represents the center of the membership function of the consequent of the rule i, and ∫​μi is the area under the membership function ∫​μi of the consequent in the rule i. In other words, the center of gravity method calculates the center of mass from the output membership functions.Figure4 shows the global behavior of the fuzzy system for the relation between the input variables CS and AIC, MSE and CS, and MSE and AIC considering all their rules. For example, if the value of CS is “Very Low,” that is 681 and AIC is “High,” that is 678.5, then the PopPK model obtains a bad evaluation, or if the value of MSE is “Low,” that is 0.1 and the value of CS is also “Low”, that is 684, the model obtains an excellent evaluation. Another example will be a scenario where the value of MSE is “Medium,” that is 0.12, and the value of AIC is “Low,” that is 674.5, and the model obtains a good evaluation.Figure 4
The figure shows the visualization of the system surface when the inputs AIC, CS, and MSE are interacting with each other. (a) The inputs AIC and CS are interacting. (b) The inputs MSE and CS are interacting. (c) The inputs MSE and AIC are interacting.
(a)
(b)
(c)If it was required, the search for the lowest value could be done according to Algorithm1. The crisp output value can be stored in an array together with other output values taken from other PopPK models that were evaluated previously and then compare them in an automated way to find the lowest value which will determine the optimal PopPK model.Algorithm 1: Find the lowest value algorithm.
(1):
i
←
0
(2):
l
o
w
←
i
(3):
for
j = i + 1 to total of candidate models–1 do
(4):
ifarray[j] is less than array[low] then
(5):
Set low toj
(6):
end if
(7):
end for


## 4. Experimental Results for the Evaluation of Several PopPK Models of Tobramycin

This section describes the experiments performed to evaluate and compare 21 PopPK models based on the tobramycin database. We developed the PopPK Model of tobramycin in Matlab applying the single compartment model, as it is shown in Figure1(a). The covariate model for both V and Cl were defined as linear.The estimation of the parameters for all the generated PopPK models was made by applying REML. The optimization process was conducted with the quasi-Newton algorithm using the initial set values for fixed effects of 0.01 forCl and 0.01 for V and with a maximum of 100 iterations. The same residual error model was applied to all the experiments. The only variations in the developed PopPK models are the covariate type and the number of parameters included in each model.


### 4.1. Experiment 1

After having evaluated 21 PopPK models using the evaluations criteria (7), (8), and (10), we end up with the 8 best PopPK models shown in Table6.Table 6
Results of the 8 best PopPK model of tobramicyn. The covariates used are weight (WT), age, height (HT), and body mass index (BMI).
PopPK
# of estimated
Model
Parameters
Covariates
AIC
CS
MSE
1
6
(AGE/V)/(WTCl)
682.04
698.2
0.0936
2
6
(SEX,AGE/V)
679.9
695
0.093
3
7
(WT,AGE/V)/(WT/Cl)
683.6
701.1
0.093
4
6
(SEX,AGE/V)
673.1
689.2
0.09336
5
9
(SEX,AGE/V)/(SEX,AGE,WT/Cl)
670.4
693
0.0936
6
7
(SEX, AGE/V) (SEX/Cl)
672.7
691
0.093
7
7
(WT, AGE, SEX/Cl)
675.3
691.3
0.139
8
10
(WT, HT/V)/(WT, AGE, HT, BMI/Cl)
682.7
707.5
0.0928Once we performed a simple analysis of the 8 PopPK models, we can easily decide that the PopPK models 1 and 2 have the worst values. However, considering the rest of the six PopPK models, it is hard to determine which model is the best given that some of their evaluations values are countered.


### 4.2. Experiment 2

In this experiment, we used the results of the evaluation criteria shown in Table6 of the PopPK models 2, 4, 5, 6, 7, and 8, as input variables in our software robot as described in Section 3. The results of the evaluations made for these PopPK models are shown in Table 7, where now the fuzzy system evaluation (FSE) criteria results are presented for the six selected PopPK models.Table 7
Summary of all evaluation criteria applied. Model 4 is the best evaluated by the fuzzy system.
PopPK
# of estimated
Model
Parameters
Covariates
AIC
CS
MSE
FSE
2
6
(WT,AGE/V)
679.9
695
0.093
88.4
4
6
(SEX,AGE/V)
673.1
689.2
0.09336
31.6
5
9
(SEX,AGE/V)/(SEX,AGE,WT/Cl)
670.4
693
0.0936
55.6
6
7
(SEX, AGE/V) (SEX/Cl)
672.7
691
0.093
34.4
7
7
(WT, AGE, SEX/Cl)
675.3
691.3
0.139
88.9
8
10
(WT, HT/V)/(WT, AGE, HT, BMI/Cl)
682.7
707.5
0.0928
88.7


## 4.1. Experiment 1

After having evaluated 21 PopPK models using the evaluations criteria (7), (8), and (10), we end up with the 8 best PopPK models shown in Table6.Table 6
Results of the 8 best PopPK model of tobramicyn. The covariates used are weight (WT), age, height (HT), and body mass index (BMI).
PopPK
# of estimated
Model
Parameters
Covariates
AIC
CS
MSE
1
6
(AGE/V)/(WTCl)
682.04
698.2
0.0936
2
6
(SEX,AGE/V)
679.9
695
0.093
3
7
(WT,AGE/V)/(WT/Cl)
683.6
701.1
0.093
4
6
(SEX,AGE/V)
673.1
689.2
0.09336
5
9
(SEX,AGE/V)/(SEX,AGE,WT/Cl)
670.4
693
0.0936
6
7
(SEX, AGE/V) (SEX/Cl)
672.7
691
0.093
7
7
(WT, AGE, SEX/Cl)
675.3
691.3
0.139
8
10
(WT, HT/V)/(WT, AGE, HT, BMI/Cl)
682.7
707.5
0.0928Once we performed a simple analysis of the 8 PopPK models, we can easily decide that the PopPK models 1 and 2 have the worst values. However, considering the rest of the six PopPK models, it is hard to determine which model is the best given that some of their evaluations values are countered.


## 4.2. Experiment 2

In this experiment, we used the results of the evaluation criteria shown in Table6 of the PopPK models 2, 4, 5, 6, 7, and 8, as input variables in our software robot as described in Section 3. The results of the evaluations made for these PopPK models are shown in Table 7, where now the fuzzy system evaluation (FSE) criteria results are presented for the six selected PopPK models.Table 7
Summary of all evaluation criteria applied. Model 4 is the best evaluated by the fuzzy system.
PopPK
# of estimated
Model
Parameters
Covariates
AIC
CS
MSE
FSE
2
6
(WT,AGE/V)
679.9
695
0.093
88.4
4
6
(SEX,AGE/V)
673.1
689.2
0.09336
31.6
5
9
(SEX,AGE/V)/(SEX,AGE,WT/Cl)
670.4
693
0.0936
55.6
6
7
(SEX, AGE/V) (SEX/Cl)
672.7
691
0.093
34.4
7
7
(WT, AGE, SEX/Cl)
675.3
691.3
0.139
88.9
8
10
(WT, HT/V)/(WT, AGE, HT, BMI/Cl)
682.7
707.5
0.0928
88.7


## 5. Analysis of Results

The results of experiment 1 of Table6 represent an example of the type of problem the researcher can face when working in the development of any PopPK model. In this particular case, Table 6 shows that the selection of the best PopPK model is not a trivial decision due to the time consumed to determine which PopPK model obtained the most significant evaluations in comparative with the rest of the models. For example, in Table 6, it can be seen that PopPK model 5 has best AIC evaluation than the PopPK models 4, 6, and 7, but worst evaluation in CS as well as worst evaluation in MSE than the PopPK model 6. The results of experiment 2 of Table 7 show that the fuzzy system gives the worst FSE value to PopPK model 7 (88.9). The PopPK model 7 has the worst evaluation of AIC, CS, and MSE than the PopPK model 4, which is the model with the best FSE value (31.6). It can be seen that the fuzzy system considered the trade-offs among the rest of standard statistical evaluation criteria: AIC, CS, and MSE provided a clearer picture of the order of the worst PopPK model to the best PopPK model.


## 6. Conclusions

Our results bolster the hypothesis that the software robot can be successfully implemented to evaluate PopPK models ensuring the selection of the best PopPK model when the choice of this becomes an MCDC problem. Given that the FSE criteria is built taking into account the classical evaluation criteria (AIC, CS, and MSE), we are able to only use the FSE as our unique automated evaluation criteria. This reduced the time and the error in the selection of the best PopPK model.Another advantage is that if we want to use the same fuzzy system methodology for the evaluation of a different case study, for example, a PopPK model that involves another type of drug as phenobarbital, or the amount of individuals in the study is different, we only need to evaluate a certain amount of models to perform adjustments in the membership function ranges.Our software robot performs fuzzy evaluations offering a stronger alternative to increase the efficiency in the selection of the best PopPK from a set. It can help the pharmaceutical scientist or the expert researchers in the area of computer intelligence to incorporate, expand, and improve the implementation of this software either in new versions of commercial software or in the development of new software to incorporate human expertise.The proposed software robot can become a strong support to further studies regarding this novel approach by helping the pharmaceutical scientist or the expert researchers in the area of computer intelligence to incorporate, expand, and improve this development.



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*Source: 1983897-2018-11-01.xml*