335 

© Published by College of Education for Pure Science (Ibn Al-Haitham), University of Baghdad. This 

is an open-access article distributed under the terms of the Creative Commons Attribution 4.0 International License 

 

Estimating the Median Lethal Dose of Breast Cancer with Modified 

Weibull Statistical Model 

 

Iden H. Hussein1 , Nadia Hashim Al-Noor2*  and Shaima Abbas Jasim3    
1Department of Mathematics, College of Science for Women, Baghdad University, Baghdad, Iraq. 

2,3Department of Mathematics, College of Science, Mustansiriyah University, Baghdad, Iraq. 

*Corresponding Author.  

 

 Received:25 April 2023                            Accepted:12 June 2023                       Published:20 July 2024  

 doi.org/10.30526/37.3.3442 

 
Abstract   

     In this paper, based on a linear relationship between the natural logarithms of the scale 

parameter and dose, nineteen models are constructed using the modified Weibull statistical model 

to describe the relationship of dose-response and time for multivariate dual-response life 

experiments with two-replicate. The real biological data set is considered to evaluate the response 

rates of breast cancer cells treated with the therapeutic zinc selenide prepared in two different 

ways (physically and environmentally/organically). The unknown parameters are estimated 

using two estimation methods. The mean square error is used to select the best model. The median 

lethal dose is then determined on the basis of a new formula at successive times. The best models 

for each estimation method show that the experiment's replications are unimportant and that the 

median lethal dose estimates exhibit a decreasing dose-time relationship over the days. 

Keywords: biological experiment, median lethal dose, modified Weibull, traditional estimation 

methods. 

 

1. Introduction 

     The standard Weibull model with two parameters [1] is used in numerous disciplines. Various 

applications of the Weibull model include tidal heights, treatment efficiency, temperature 

fluctuations, discharge inference, wind speed, brittle material, reliability growth, raindrop size, 

latent failures of electronic products, and damage in laminated composites. Interested readers can 

find further applications in [2,3]. However, only increasing, decreasing, or constant risk functions 

are possible for the Weibull distribution. Therefore, it cannot be used to simulate lifetime data 

with a bathtub-shaped hazard function, such as human mortality and machine life cycles [4]. For 

many years, several researchers (see [5–7]), have developed various modifications and 

extensions of the Weibull model by adding additional parameters. Additionally, for more recent 

references, one can see [8-12]. In this paper, the modified version proposed by [5] is considered 

to construct nineteen models describing the relationship of dose, response, and time for 

multivariate dual-response life experiments with two replicates. 

https://creativecommons.org/licenses/by/4.0/
https://orcid.org/0000-0002-6492-8196
mailto:Iden_alkanani@yahoo.com
https://orcid.org/0000-0002-4433-9044
mailto:nadialnoor@uomustansiriyah.edu.iq
https://orcid.org/0000-0002-7879-7668
mailto:najassim22@gmail.com


 

IHJPAS. 2024, 37( 3 ) 
 

336 
 

The probability density function (PDF) and cumulative function (CF) of the modified Weibull 

(MW) model, with scale parameter (α) and shape parameters (𝛽, 𝜆), are given respectively by 

[5]  

𝑓(𝑡; 𝛼, 𝛽, 𝜆) = 𝛼(𝛽 + 𝜆𝑡) 𝑡𝛽−1 𝑒𝜆𝑡−𝛼𝑡𝛽𝑒𝜆𝑡  ;  𝑡 ≥ 0, 𝛼 > 0 , 𝛽, 𝜆 ≥ 0                                          (1) 

𝐹(𝑡; 𝛼, 𝛽, 𝜆) = 1 − 𝑒−𝛼𝑡𝛽𝑒𝜆𝑡
        ;  𝑡 ≥ 0, 𝛼 > 0 , 𝛽, 𝜆 ≥ 0                                                             (2) 

For a biological experiment, suppose there is a linear relationship between the natural–logarithms 

of the scale parameter and dose, i.e. ln(𝛼) = 𝛾 + 𝛿 ln(𝑑) (see [13,14]). So, the scale parameter 

is equivalent to 

𝛼 = 𝑒𝛾+𝛿 ln(𝑑) , −∞ < 𝛾 < ∞ , 𝛿 ≠ 0                                                                                                 (3) 

After substituting (3) in (1) and (2), the PDF and CF of a random sample of response times 

𝑡𝑗(𝑡1, … , 𝑡𝑛) taken from 𝑀𝑊 with doses 𝑑𝑖(𝑑1, … , 𝑑𝑘) are given by 

𝑓(𝑡𝑗; 𝛾, 𝛿, 𝛽, 𝜆) =  𝑒𝛾+𝛿 ln(𝑑𝑖)(𝛽 + 𝜆𝑡𝑗) 𝑡𝑗
𝛽−1

 𝑒
𝜆𝑡𝑗− 𝑡𝑗

𝛽
𝑒

𝛾+𝛿 ln(𝑑𝑖)+𝜆𝑡𝑗
;  𝑡 ≥ 0,                                    (4) 

                                                                       −∞ < 𝛾 < ∞, 𝛿 ≠ 0 , 𝛽, 𝜆 ≥ 0
 

𝐹(𝑡𝑗; 𝛾, 𝛿, 𝛽, 𝜆) = 1 − 𝑒
−𝑡𝑗

𝛽
𝑒

𝛾+𝛿 ln(𝑑𝑖)+𝜆𝑡𝑗
 ;  𝑡 ≥ 0,−∞ < 𝛾 < ∞, 𝛿 ≠ 0 , 𝛽, 𝜆 ≥ 0                       (5) 

In statistics, the evaluation of the nature, constitution, or potency of a material/or of a process 

using the reaction that results from its application to living matter is known as a statistical analysis 

of bioassays. An assay is thus a type of biological experiment, but the focus is on comparing the 

magnitudes of different treatments' effects. There are two types of biological experiments: 

univariate quantal responses, which are reliant on the dose-response relationship, and 

multivariate quantal responses, which are dependent on the dose-response relationship over time. 

The calculation of effective doses, particularly the median lethal dose (LD50), is one of the most 

important applications of this type of experiment.  

As a pioneering work, Trevan (see [15]) developed the LD50 test to determine the dose of a test 

substance that results in 50% mortality in a particular species of animals. Before conducting 

additional toxicity studies on a chemical, this test is typically the first one to be performed. It is 

employed to calculate the possible risks that chemicals may pose to people. Depending on the 

chemical being tested, the non-lethal acute effects may appear as signs of toxicity even though 

death is its primary endpoint. Nowadays, numerous studies have been introduced to do so. For 

example, in 2018, [16] used probit-log(dose) regression models and the maximum likelihood 

method to calculate lethal doses of toxicants at various significance levels, as well as the lethal 

dose ratio for two toxicants. In 2019, [17] investigated probation and the impact of a chemical 

compound using dose-response studies and non-linear regression to assess efficacy metrics. They 

presented a Bayesian inference framework for analyzing and comparing dose-response 

experiments. In 2020, [18] determined the LD50 of Medemia argun seed’s crude ethanolic and 

the dose-response curve of several doses of the extract against carbon tetrachloride-treated 

animals to assess its hepatotoxicity. In 2021, [19] determined LD50 and acute toxicity of the 

formulation Cytoreg, an ionic mixture of strong and weak acids. Also, in 2021 [20] used the 

exponential model to assess the dose-response relationship and calculated the LD50 of the poison 

for disinfectant jungles (Roanstar) on a group of fish. Further, in 2021 [21] determined the LD50 

of zinc chloride administered intraperitoneally to albino rats at concentrations ranging from 

10(10)100 mg/kg body weight of the experimental animals. In 2022, [22] investigated the toxic 

effect of a mixture of three pesticides (cypermethrin, mancozeb, and metalaxyl) on reproduction 

and oxidative stress parameters in male Wistar rats. In 2023, [23] examined the 25, 50, and 75% 



 

IHJPAS. 2024, 37( 3 ) 
 

337 
 

lethal doses of gamma radiation on the survival rate, leaves, shoots, and root morphometric 

features after gamma irradiation. The statistical significance of the differences between the mean 

values of four replications was investigated using a one-way analysis of variance. Further, by 

revising [24-28], one can see more details about dose-response and lethal dose. This work is 

focused on modeling the multivariate quantal response by using the modified Weibull statistical 

model to predict LD50 of breast cancer based on cumulative function. The remainder of this paper 

is organized as follows: Section 2 introduces parameter estimators based on two estimating 

methods. Sections 3 and 4 provide the initial values of the parameters as well as the cumulative 

affected numbers. The best model and median lethal dose estimation are found in Section 5. 

Section 6 discusses real biological applications and their outcomes.  

2 Parameters Estimators 

2.1 Ordinary Least Squares Parameters Estimators  

Let  𝑡1, … , 𝑡𝑛 be a random sample from MW model with doses 𝑑𝑖(𝑑1, … , 𝑑𝑘), then the 

ordinary least squares (OLS) procedure minimizes ∑ 𝜖𝑗
2𝑛

𝑗=1 = ∑ (�̂� (𝑡𝑗) − 𝐹(𝑡𝑗))
2𝑛

𝑗=1 , with 

estimated and empirical CF for  j = 1, … , n respectively given as 

�̂�(𝑡𝑗) = 𝐹(𝑡𝑗; 𝛾, 𝛿, �̂�, �̂�) = 1 − 𝑒
−𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
 and 𝐹(𝑡𝑗) = 𝐹𝑗 =

𝑗− 0.5

𝑛
. Now ∑ 𝜖𝑗

2𝑛
𝑗=1 =

∑ (1 − 𝑒
−𝑡𝑗

�̂�
 𝑒

�̂�+�̂� 𝑙n(𝑑𝑖)+�̂�𝑡𝑗  
− 𝐹𝑗)

2
𝑛
𝑗=1 and then 

∑𝜖𝑗
2

𝑛

𝑗=1

= ∑(𝑆𝑗 − 𝑒
−𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗  )
2𝑛

𝑗=1

                                                                                            (6) 

   where 𝑆𝑗 = 1 − 𝐹𝑗 =
𝑛−𝑗+ 0.5

𝑛
 . 

Let 𝑧1(𝛾),  𝑧2(𝛿), 𝑧3(𝛽) and 𝑧4(𝜆) represent the partial derivatives of ∑ 𝜖𝑗
2𝑛

𝑗=1  in (6) to 𝛾, 𝛿, 𝛽, 𝜆 

and set it equal to zero, as follows  

𝑧1(𝛾) = 2∑𝑡𝑗
�̂�

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(𝑆𝑗 − 𝑒

− 𝑡𝑗
�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) = 0                      (7) 

 𝑧2(𝛿) = 2 ln(𝑑𝑖)∑𝑡𝑗
�̂�

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(𝑆𝑗 − 𝑒

− 𝑡𝑗
�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) = 0        (8) 

𝑧3(𝛽) = 2∑𝑡𝑗
�̂�

ln(𝑡𝑗)

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(𝑆𝑗 − 𝑒

− 𝑡𝑗
�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) = 0         (9) 

𝑧4(𝜆) = 2∑𝑡𝑗
�̂�+1

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(𝑆𝑗 − 𝑒

− 𝑡𝑗
�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) = 0               (10) 

Noticing that Equations (7) to (10) are non–linear and difficult to solve in traditional methods. 

By iterative processes, the four (𝛾, 𝛿, �̂�, �̂�) OLS estimates can be obtained respectively. With 

the iterative Newton-Raphson method, the Jacobin matrix is used with iteration (s), as shown 

below 

[

𝛾𝑠+1

𝛿𝑠+1

𝛽𝑠+1

𝜆𝑠+1

] = [

𝛾𝑠

𝛿𝑠

𝛽𝑠

𝜆𝑠

] − 𝐽𝑠
−1  

[
 
 
 
𝑧1(𝛾)

𝑧2(𝛿)

𝑧3(𝛽)

𝑧4(𝜆)]
 
 
 

                                                                                                            (11)    



 

IHJPAS. 2024, 37( 3 ) 
 

338 
 

The OLS estimated values can be obtained iteratively from (11) until convergence occurs, that 

is, the absolute value for the difference between two successive iterations (𝑠, 𝑠 + 1) is less than 

the assumed small error tolerance, ε > 0. When convergence occurs, the current estimates 

represent the estimates of parameters. We need to mention that with s = 0, the γ0, δ0, β0, λ0 

represent the initial values, and the Jacobin matrix must be a non-singular symmetric matrix  Js 

to obtain its inverse, where 

Js =

[
 
 
 
 
 
 
∂z1(γ)

∂γ

∂z1(γ)

∂δ

∂z1(γ)

∂β

∂z1(γ)

∂λ

∂z2(δ)

∂γ

∂z3(β)

∂γ

∂z4(λ)

∂γ

∂z2(δ)

∂δ

∂z2(δ)

∂β

∂z2(δ)

∂λ

∂z3(β)

∂δ
∂z4(λ)

∂δ

∂z3(β)

∂β

∂z4(λ)

∂β

∂z3(β)

∂λ
∂z4(λ)

∂λ ]
 
 
 
 
 
 

, and the partial derivatives of Equations (7) to 

(10) concerning unknown parameters are  

∂z1(γ)

∂γ
= 2∑Sj tj

β̂

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
 eγ̂+δ̂ ln(di)+λ̂tj)    

      −2∑  tj
β̂

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
 eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z1(γ)

∂δ
=

∂z2(δ)

∂γ
= 2 ln(di)∑𝑆𝑗  𝑡𝑗

�̂�

n

j=1

e
�̂�+�̂� ln(di)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 e

�̂�+�̂� ln(di)+�̂�𝑡𝑗
(1 − 𝑡𝑗

�̂�
 eγ̂+δ̂ ln(di)+λ̂tj)

                  − 2 ln(di)∑𝑡𝑗
�̂�

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗−2𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(1 − 2tj

β̂
 eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z1(γ)

∂β
=

∂z3(β)

∂γ
= 2∑Sj ln(tj)  tj

β̂

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)    

                   − 2∑ln(tj)  tj
β̂

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z1(γ)

∂λ
=

∂z4(λ)

∂γ
= 2∑𝑆𝑗  𝑡𝑗

�̂�+1

𝑛

𝑗=1

e
�̂�+�̂� ln(di)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 e

�̂�+�̂� 𝒍𝒏(𝒅𝒊)+�̂�𝒕𝒋
(1 − 𝑡𝑗

�̂�
𝑒�̂�+�̂� 𝑙𝑛(𝑑𝑖)+�̂�𝑡𝑗)     

                  − 2∑ tj
β̂+1

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2𝑡𝑗

�̂�
𝑒�̂�+�̂� 𝑙𝑛(𝑑𝑖)+�̂�𝑡𝑗)

 

𝜕𝑧2(𝛿)

𝜕𝛿
= 2 (ln(𝑑𝑖))

2 ∑𝑆𝑗  𝑡𝑗
�̂�

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(1 − 𝑡𝑗

�̂�
𝑒�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗)

                  − 2 (ln(𝑑𝑖))
2 ∑ 𝑡𝑗

�̂�

𝑛

𝑗=1

𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗−2𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
(1 − 2𝑡𝑗

�̂�
𝑒�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗)

 



 

IHJPAS. 2024, 37( 3 ) 
 

339 
 

𝜕𝑧2(𝛿)

𝜕𝛽
=

𝜕𝑧3(𝛽)

𝜕𝛿
= 2 ln(𝑑𝑖)∑𝑆𝑗  𝑡𝑗

�̂�

𝑛

𝑗=1

ln(𝑡𝑗) 𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗− 𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
                  

            (1 − 𝑡𝑗
�̂�
𝑒�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗) − 2 ln(𝑑𝑖)∑𝑡𝑗

�̂�

𝑛

𝑗=1

ln(𝑡𝑗) 𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗−2𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗

                                                                                        (1 − 2𝑡𝑗
�̂�
𝑒�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗)

 

∂z2(δ)

∂λ
=

∂z4(λ)

∂δ
= 2 ln(di)∑Sjtj

β̂+1

n

j=1

 e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)  

                  − 2 ln(di)∑  tj
β̂+1

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z3(β)

∂β
= 2∑Sjtj

β̂

n

j=1

(ln(tj))
2
e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

                   − 2∑tj
β̂
(ln(tj))

2
n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z3(β)

∂λ
=

∂z4(λ)

∂β
= 2∑Sj tj

β̂+1

n

j=1

ln(tj)  e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

                   − 2∑  tj
β̂+1

ln(tj)

n

j=1

 e
γ̂+δ̂ ln(di)+λ̂tj−2tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

 

∂z4(λ)

∂λ
= 2∑Sj tj

β̂+2

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj− tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

                − 2∑ tj
β̂+2

n

j=1

e
γ̂+δ̂ ln(di)+λ̂tj−2 tj

β̂
 e

γ̂+δ̂ ln(di)+λ̂tj
(1 − 2tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj)

 

2.2 Maximum Likelihood Parameters Estimators  

Let  t1, … , tn be a random sample from MW model with doses di(d1, … , dk), then the 

likelihood function is given by  

L = ∏(eγ+δ ln(di)(β + λtj)tj
β−1 e

λtj−eγ+δ ln(di)tj
β
e
λtj

) 

n

j=1

                                                             (12) 

Take the natural–logarithm of the likelihood function 

ln L = nγ + nδ ln(di) + ∑ln(β + λtj) + β∑ln(tj)

n

j=1

 

n

j=1

− ∑ln(tj)

n

j=1

+ λ∑tj 

n

j=1

                          

                                                                                           − eγ+δln(di) ∑tj
β
eλtj

n

j=1

                                (13)

 

Let g1(γ),  g2(δ), g3(β) and g4(λ) represent the partial derivatives of ln L in (13) to γ, δ, β, λ 

and set it equal to zero, as follows  



 

IHJPAS. 2024, 37( 3 ) 
 

340 
 

g1(γ) =
∂ ln L 

∂γ
= n − ∑tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj 

n

j=1

= 0                                                                             (14) 

g2(δ) =
∂ ln L 

∂δ
= n ln(di) − ln(di)∑tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

= 0                                                     (15) 

g3(β) =
∂ ln L 

∂β
= ∑

1

β̂ + λ̂tj
+

n

j=1

∑ln(tj)

n

j=1

− ∑tj
β̂
ln(tj) eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

= 0                         (16) 

g4(λ) =
∂ ln L 

∂λ
= ∑

tj

β̂ + λ̂tj
+ ∑tj 

n

j=1

n

j=1

− ∑ tj
β̂+1

eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

= 0                                        (17) 

The maximum likelihood (ML) estimated values of four parameters can be obtained by solving 

the nonlinear Equations (14)  to (17) numerically through the iterative Newton-Raphson method 

as in (18) until convergence occurs,  

[

γs+1

δs+1

βs+1

λs+1

] = [

γs

δs

βs

λs

] − Js
−1  

[
 
 
 
g1(γ)

g2(δ)

g3(β)

g4(λ)]
 
 
 

                                                                                                              (18) 

where Js =

[
 
 
 
 
 
 
∂g1(γ)

∂γ

∂g1(γ)

∂δ

∂g1(γ)

∂β

∂g1(γ)

∂λ

∂g2(δ)

∂γ

∂g3(β)

∂γ

∂g4(λ)

∂γ

∂g2(δ)

∂δ

∂g2(δ)

∂β

∂g2(δ)

∂λ

∂g3(β)

∂δ
∂g4(λ)

∂δ

∂g3(β)

∂β

∂g4(λ)

∂β

∂g3(β)

∂λ
∂g4(λ)

∂λ ]
 
 
 
 
 
 

, and the partial derivatives for unknown parameters 

are 

∂g1(γ)

∂γ
= −∑tj

β̂
eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g1(γ)

∂δ
=

∂g2(δ)

∂γ
= − ln(di) ∑tj

β̂
e γ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g1(γ)

∂β
=

∂g3(β)

∂γ
= − ∑tj

β̂

n

j=1

ln(tj)  eγ̂+δ̂ ln(di)+λ̂tj  

∂g1(γ)

∂λ
=

∂g4(λ)

∂γ
= −∑tj

β̂+1
e γ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g2(δ)

∂δ
= −(ln(di))

2   ∑tj
β̂
eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

  

∂g2(δ)

∂β
=

∂g3(β)

∂δ
= − ln(di) ∑tj

β̂
 ln(tj) eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g2(δ)

∂λ
=

∂g4(λ)

∂δ
= − ln(di) ∑ tj

β̂+1
eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

 



 

IHJPAS. 2024, 37( 3 ) 
 

341 
 

∂g3(β)

∂β
= − ∑

1

(β̂ + λ̂tj)
2

n

j=1

 −  ∑tj
β̂
 (ln(tj))

2
eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g3(β)

∂λ
=

∂g4(λ)

∂β
= −∑

tj

(β̂ + λ̂tj)
2

n

j=1

− ∑tj
β̂+1

ln(tj)  eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

 

∂g4(λ)

∂λ
= −∑

tj
2

(β̂ + λ̂tj)
2

n

j=1

− ∑ tj
β̂+2

 eγ̂+δ̂ ln(di)+λ̂tj

n

j=1

   

  

 

3. The Initial Values of the Parameters 

It is necessary to indicate the possibility of calculating the initial values of parameters 

(𝛾0, 𝛿0, 𝛽0, 𝜆0) by using the CF as follows: 

Consider the response time tj and dose di (𝑗 = 1, … , 𝑛 ; 𝑖 = 1,… , 𝑘 ; 𝑗 = 𝑖), then the CF in 

(5) will be 𝐹(𝑡𝑗) = 𝐹(𝑡𝑗; 𝛾, 𝛿, 𝛽, 𝜆) = 1 − 𝑒
−𝑡𝑗

𝛽
𝑒

𝛾+𝛿 ln(𝑑𝑖)+𝜆𝑡𝑗
, and F(tj) can be attained through 

the mean rank formula of empirical CF. After taking the double natural logarithm of two sides 

and comparing the result with the linear regression model 𝑌 = 𝑏0 + 𝑏1𝑋1 + 𝑏2𝑋2 + 𝑏3𝑋3, 

then 𝑌 = ln (− ln (1 − 𝐹(𝑡𝑗))), X1 = tj,  𝑋2 = ln(tj) ,  𝑋3 = ln(𝑑𝑖) with coefficients 𝑏0 = 𝛾,

𝑏1 = 𝜆, 𝑏2 = 𝛽 and 𝑏3 = 𝛿. Thus, the initial values (𝛾0, 𝛿0, 𝛽0, 𝜆0) can be attained easily now by 

using the OLS method that is available in different statistical programs such as SPSS. 

 

4. The Cumulative Affected Numbers  

    Repeating the biological experiment is vital to confirm that an observed result represents a 

natural occurrence (see [29]). Consider 𝐶𝑖𝑗𝑙 as the cumulative affected numbers that represent the 

influence of dose  𝑖 in response time 𝑗 and replicate 𝑙. After getting the parameter's estimates, the 

estimates of 𝐶𝑖𝑗𝑙 can be obtained based on the CF in (5) by the following formula: 

�̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 �̂�(𝑡𝑗) = 𝑛𝑖𝑙  (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
)                                                                               (19) 

   where nil represents the number of experimental sample units of dose i and replicate l.  

Since the MW contains four parameters, the models' CF may be estimated with and without 

replication using different models shown in Table 1. The additional last three models are used to 

investigate the effectiveness of the experiment without replication at specific values of the 

parameters. The estimated affected numbers (units) can be calculated directly using the 

cumulative affected numbers obtained from the models' cumulative function estimator in Table1.   

 

5. Best Model and Median Lethal Dose Estimator 

Based on the actual number of affected (observed) units (𝑢𝑖𝑗𝑙), estimated affected units 

(�̂�𝑖𝑗𝑙), total number of units (𝑁), and the number of model parameters (𝑝), the best model can 

be chosen with the lowest value of the mean square error (𝑀𝑆𝐸) criterion, where 

𝑀𝑆𝐸(�̂�) =
1

𝑁 − 𝑝
∑ (𝑢𝑖𝑗𝑙 − �̂�𝑖𝑗𝑙)

2

𝑖,𝑗,𝑙
; 𝑖 = 1,… , 𝑘 ; 𝑗 = 1,… , 𝑛 ; 𝑙 = 1,… , 𝑟                         (20) 

The median lethal dose (LD50) for the best model can be estimated by making the CF with 

considered estimated values equal to 0.50, �̂�(𝑡𝑗) = 0.50, i.e. 1 − e
−tj

β̂
e
γ̂+δ̂ ln(d)+λ̂tj

= 0.50. After 



 

IHJPAS. 2024, 37( 3 ) 
 

342 
 

taking the double logarithm for two-sided, we get, 𝛾 + 𝛿 ln(𝑑) + �̂� ln(𝑡𝑗) + �̂�𝑡𝑗 =

ln(− ln(0.50)), and then with xj = ln(d), the LD50 is equal to  

𝐿𝐷50 = 𝑑 = 𝑒𝑥𝑗                                                                                                                                      (21) 

where 

𝑥𝑗 =
−𝛾 − �̂� ln(𝑡𝑗) − �̂�𝑡𝑗 − 0.366513

𝛿
                                                                                           (22) 

 
Table 1. The MW model's cumulative function estimators 

Model Cumulative function estimator 

1 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙  𝑒
�̂�𝑙+�̂�𝑙 ln(𝑑𝑖)+�̂�𝑙 𝑡𝑗

) 

2 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙 𝑒
�̂�𝑙+�̂�𝑙 ln(𝑑𝑖)+�̂�𝑡𝑗

) 

3 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒
�̂�𝑙+�̂�𝑙 ln(𝑑𝑖)+𝜆 ̂𝑙𝑡𝑗

) 

4 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙  𝑒
�̂�𝑙+�̂� ln(𝑑𝑖)+�̂�𝑙 𝑡𝑗

) 

5 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙  𝑒
�̂� +�̂�𝑙 ln(𝑑𝑖)+�̂�𝑙 𝑡𝑗

) 

6 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�𝑙+�̂�𝑙 ln(𝑑𝑖)+�̂�𝑡𝑗
) 

7 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙  𝑒
�̂�𝑙+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗

) 

8 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙 𝑒
�̂� +�̂�𝑙 ln(𝑑𝑖)+�̂�𝑡𝑗

) 

9 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�𝑙+�̂� ln(𝑑𝑖)+�̂�𝑙𝑡𝑗
) 

10 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒
�̂� +�̂�𝑙 ln(𝑑𝑖)+�̂�𝑙𝑡𝑗

) 

11 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑙𝑡𝑗

) 

12 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�𝑙 +�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) 

13 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�+�̂�𝑙 ln(𝑑𝑖)+�̂�𝑡𝑗
) 

14 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡

𝑗

�̂�𝑙𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗

) 

15 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
𝑒

�̂�+�̂� ln(𝑑𝑖)+�̂�𝑙𝑡𝑗
) 

16 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
 𝑒

 �̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗
) 

17 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒
−𝑡𝑗

�̂�
 𝑒

�̂�+�̂� ln(𝑑𝑖)+𝑡𝑗
) ; 𝜆 = 1 

18 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒−𝑡𝑗 𝑒
�̂�+�̂� ln(𝑑𝑖)+�̂�𝑡𝑗

) ;  𝛽 = 1 

19 �̂�𝑖𝑗𝑙 = 𝑛𝑖𝑙 (1 − 𝑒−𝑡𝑗 𝑒
�̂�+�̂� ln(𝑑𝑖)+𝑡𝑗

) ; 𝜆 = 1 ; 𝛽 = 1 

 

 



 

IHJPAS. 2024, 37( 3 ) 
 

343 
 

6. Real Biological Application 

In this section, the nineteen models listed in Table 1 are evaluated, and the median lethal 

dose at various period of the best model is determined using two-replicate multivariate dual-

response biological experiments. The real application is focused on using biologically accurate 

data to treat breast cancer, one of the most important threats to public health, with the use of the 

therapeutic zinc selenide (ZnSe), which was produced in two different ways: first, physically 

(physical treatment) using plasma and second, environmentally (green treatment) using plant 

extract (Kalgan plant) (for more details see [30]). Tables 2 and 3 represent the response rates of 

breast cancer cells for the two ways of applying ZnSe on cells for exposure times 24 and 72 hours 

with different concentrations of ZnSe (mg/ml) ranging between (12.5 − 100) percent.  

 

Table 2. Response rates of breast cancer cells treated physically 

Rep. 
Time   

(Days) 

Dose/Concentration 

12.5 25 75 100 

24 hours 

1 0.3774 0.6604 0.7547 0.9057 

2 0.3208 0.5660 0.7925 0.9811 

3 0.1887 0.5472 0.8113 0.9434 

4 0.3019 0.5849 0.7925 0.9434 

72 hours 

1 0.5077 0.6923 0.8923 0.9692 

2 0.4615 0.7077 0.8769 0.9538 

3 0.4769 0.6923 0.8769 0.9846 

4 0.4769 0.6923 0.8769 0.9692 
 

Table 3. Response rates of breast cancer cells treated environmentally 

Rep. 
Time  

(Days) 

Dose/Concentration 

12.5 25 75 100 

24 hours 

1 0.4737 0.7895 0.8772 0.9298 

2 0.5088 0.8772 0.9123 0.9825 

3 0.4912 0.7368 0.8421 0.9649 

4 0.4912 0.8070 0.8772 0.9649 

72 hours 

1 0.6528 0.8194 0.9444 0.9861 

2 0.6250 0.8611 0.9167 0.9583 

3 0.6667 0.8889 0.9306 0.9722 

4 0.6528 0.8611 0.9306 0.9722 

By using the least squares method as described in section 3, the initial values for the four 

parameters are γ0 = −1.671, δ0 = −0.081, β0 = 0.734, and λ0 = 0.375. Now, based on the 

initial values and ε = 0.001, with a program written by MATLAB (R2018b), the iterative 

Newton-Raphson method with Equations (11) and (18) are used to calculate the OLS and ML 

estimates of four parameters. Tables 4–7 summarized the obtained estimates for each replicate 

and the entire experiment, as well as the obtained estimates corresponding to each model and 

each estimation method. 

 
  

Table 4. The OLS estimates for each replicate and entire experiment  

Rep. �̂� �̂� �̂� �̂� 

1 -1.1454 -0.1827 0.5234 0.3825 

2 -1.1454 -0.1827 0.5234 0.3825 

Experiment -2.1507   0.0298 0.6183 0.3787 
 

 



 

IHJPAS. 2024, 37( 3 ) 
 

344 
 

Table 5. The OLS estimates of 2nd experiment corresponding to each model 

Models  �̂� �̂� �̂� �̂� 

1 -1.1454 -0.1827 0.5234 0.3825 

2 -1.1454 -0.1827 0.5234 0.3787 

3 -1.1454 -0.1827 0.6183 0.3825 

4 -1.1454 0.0298 0.5234 0.3825 

5 -2.1507 -0.1827 0.5234 0.3825 

6 -1.1454 -0.1827 0.6183 0.3787 

7 -1.1454 0.0298 0.5234 0.3787 

8 -2.1507 -0.1827 0.5234 0.3787 

9 -1.1454 0.0298 0.6183 0.3825 

10 -2.1507 -0.1827 0.6183 0.3825 

11 -2.1507 0.0298 0.5234 0.3825 

12 -1.1454 0.0298 0.6183 0.3787 

13 -2.1507 -0.1827 0.6183 0.3787 

14 -2.1507 0.0298 0.5234 0.3787 

15 -2.1507 0.0298 0.6183 0.3825 

16 -2.1507 0.0298 0.6183 0.3787 

17 -2.1507 0.0298 0.6183 --- 

18 -2.1507 0.0298 --- 0.3787 

19 -2.1507 0.0298 --- --- 

 
Table 6. The ML estimates for each replicate and entire experiment  

Rep. �̂� �̂� �̂� �̂� 

1 - 0.3095 - 0.6942 1.6349 0.3374 

2 - 0.3095 - 0.6942 1.6349 0.3374 

Experiment -13.2077 2.4845 1.6349 0.3374 

 
Table 7. The ML estimates of 2nd experiment corresponding to each model  

Models  �̂� �̂� �̂� �̂� 

1 - 0.30950 - 0.6942 1.6349 0.3374 

2 - 0.30950 - 0.6942 1.6349 0.3374 

3 - 0.30950 - 0.6942 1.6349 0.3374 

4 - 0.30950 2.48450 1.6349 0.3374 

5 -13.2077 - 0.6942 1.6349 0.3374 

6 - 0.30950 - 0.6942 1.6349 0.3374 

7 - 0.30950 2.48450 1.6349 0.3374 

8 -13.2077 - 0.6942 1.6349 0.3374 

9 - 0.30950 2.48450 1.6349 0.3374 

10 -13.2077 - 0.6942 1.6349 0.3374 

11 -13.2077 2.48450 1.6349 0.3374 

12 - 0.30950 2.48450 1.6349 0.3374 

13 -13.2077 2.48450 1.6349 0.3374 

14 -13.2077 2.48450 1.6349 0.3374 

15 -13.2077 2.48450 1.6349 0.3374 

16 -13.2077 2.48450 1.6349 0.3374 

17 -13.2077 2.48450 1.6349 --- 

18 -13.2077 2.48450 --- 0.3374 

19 -13.2077 2.48450 --- --- 

Further, based on formula (19) and estimated values of parameters in Tables 5 and 7, the 

cumulative and estimated response rates for the nineteen models corresponding to OLS and ML 

estimation methods with respect to the different doses and replicates are calculated and listed in 

Tables 8 and 9. 



 

IHJPAS. 2024, 37( 3 ) 
 

345 
 

Table 8. The cumulative and estimated response rates for  OLS Method  

Model 1 

Rep. 
Time 

(days) 
Death 

Dose (Concentration) 

12.5 25 75 100 

1, 2 

1 
Cumulative 1.018741 0.912659 0.763774 0.728472 

Estimate 1.018741 0.912659 0.763774 0.728472 

2 
Cumulative 1.846860 1.682244 1.440435 1.381251 

Estimate 0.828119 0.769585 0.676662 0.652778 

3 
Cumulative 2.698293 2.512333 2.219169 2.143833 

Estimate 0.851433 0.830089 0.778734 0.762582 

4 
Cumulative 3.409524 3.258638 2.992711 2.919015 

Estimate 0.711232 0.746305 0.773541 0.775182 

Model 2 

1, 2 

1 Cumulative 1.015415 0.909625 0.761172 0.725977 

 Estimate 1.015415 0.909625 0.761172 0.725977 

2 Cumulative 1.836739 1.672648 1.431769 1.372839 

 Estimate 0.821324 0.763023 0.670597 0.646862 

3 Cumulative 2.681622 2.495560 2.202759 2.127608 

 Estimate 0.844883 0.822912 0.770990 0.754770 

4 Cumulative 3.392235 3.239545 2.971537 2.897467 

 Estimate 0.710613 0.743986 0.768778 0.769859 

Model 3 

1, 2 

1 Cumulative 1.018741 0.912659 0.763774 0.728472 

 Estimate 1.018741 0.912659 0.763774 0.728472 

2 Cumulative 1.935649 1.766663 1.516964 1.455598 

 Estimate 0.916909 0.854004 0.753191 0.727125 

3 Cumulative 2.849363 2.665547 2.370688 2.294012 

 Estimate 0.913714 0.898884 0.853724 0.838415 

4 Cumulative 3.548790 3.415071 3.170255 3.100665 

 Estimate 0.699427 0.749524 0.799567 0.806652 

Model 4 

1, 2 

1 Cumulative 1.580575 1.605830 1.646378 1.657101 

 Estimate 1.580575 1.605830 1.646378 1.657101 

2 Cumulative 2.613282 2.643605 2.691556 2.704084 

 Estimate 1.032707 1.037776 1.045178 1.046984 

3 Cumulative 3.413634 3.436668 3.472244 3.481367 

 Estimate 0.800352 0.793062 0.780688 0.777283 

4 Cumulative 3.848304 3.858318 3.873225 3.876937 

 Estimate 0.434669 0.421650 0.400981 0.395569 

Model 5 

1, 2 

1 Cumulative 0.407931 0.361676 0.298427 0.283703 

 Estimate 0.407931 0.361676 0.298427 0.283703 

2 Cumulative 0.811194 0.724058 0.602907 0.574371 

 Estimate 0.403263 0.362381 0.304479 0.290668 

3 Cumulative 1.347535 1.214705 1.025206 0.979759 

 Estimate 0.536340 0.490648 0.422300 0.405389 

4 Cumulative 2.013813 1.841337 1.585098 1.521888 

 Estimate 0.666278 0.626631 0.559891 0.542128 

Model 6 

1, 2 

1 Cumulative 1.015415 0.909625 0.761172 0.725977 

 Estimate 1.015415 0.909625 0.761172 0.725977 

2 Cumulative 1.925285 1.756786 1.507984 1.446868 

 Estimate 0.909870 0.847162 0.746812 0.720891 



 

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3 Cumulative 2.832997 2.648838 2.354016 2.277454 

 Estimate 0.907712 0.892051 0.846032 0.830586 

4 Cumulative 3.533690 3.397859 3.150331 3.080188 

 Estimate 0.700693 0.749021 0.796315 0.802735 

Model 7 

1, 2 

1 Cumulative 1.575957 1.601164 1.641639 1.652342 

 Estimate 1.575957 1.601164 1.641639 1.652342 

2 Cumulative 2.602115 2.632453 2.680439 2.692978 

 Estimate 1.026158 1.031289 1.038800 1.040636 

3 Cumulative 3.400732 3.424011 3.459987 3.469218 

 Estimate 0.798617 0.791557 0.779548 0.776239 

4 Cumulative 3.840628 3.850995 3.866449 3.870301 

 Estimate 0.439896 0.426984 0.406462 0.401084 

Model 8 

1, 2 

1 Cumulative 0.406465 0.360368 0.297339 0.282666 

 Estimate 0.406465 0.360368 0.297339 0.282666 

2 Cumulative 0.805718 0.719101 0.598702 0.570348 

 Estimate 0.399253 0.358733 0.301363 0.287682 

3 Cumulative 1.335155 1.203255 1.015205 0.970125 

 Estimate 0.529437 0.484154 0.416502 0.399777 

4 Cumulative 1.992726 1.821157 1.566644 1.503924 

 Estimate 0.657572 0.617903 0.551440 0.533799 

Model 9 

1, 2 

1 Cumulative 1.580575 1.605830 1.646378 1.657101 

 Estimate 1.580575 1.605830 1.646378 1.657101 

2 Cumulative 2.709651 2.739763 2.787285 2.799682 

 Estimate 1.129076 1.133933 1.140907 1.142582 

3 Cumulative 3.525174 3.545830 3.577552 3.585649 

 Estimate 0.815523 0.806067 0.790266 0.785967 

4 Cumulative 3.904245 3.911420 3.921970 3.924570 

 Estimate 0.379071 0.365590 0.344418 0.338921 

Model 10 

1, 2 

1 Cumulative 0.407931 0.361676 0.298427 0.283703 

 Estimate 0.407931 0.361676 0.298427 0.283703 

2 Cumulative 0.859958 0.768235 0.640433 0.610285 

 Estimate 0.452027 0.406558 0.342006 0.326582 

3 Cumulative 1.464610 1.323310 1.120448 1.071580 

 Estimate 0.604653 0.555075 0.480015 0.461296 

4 Cumulative 2.200008 2.020661 1.750509 1.683226 

 Estimate 0.735398 0.697351 0.630061 0.611646 

Model 11 

1, 2 

1 Cumulative 0.672187 0.684941 0.705597 0.711097 

 Estimate 0.672187 0.684941 0.705597 0.711097 

2 Cumulative 1.285414 1.307288 1.342520 1.351859 

 Estimate 0.613227 0.622347 0.636922 0.640762 

3 Cumulative 2.018883 2.047723 2.093776 2.105902 

 Estimate 0.733470 0.740435 0.751256 0.754042 

4 Cumulative 2.792088 2.821900 2.868865 2.881099 

 Estimate 0.773204 0.774177 0.775089 0.775198 

Model 12 

1, 2 

1 Cumulative 1.575957 1.601164 1.641639 1.652342 

 Estimate 1.575957 1.601164 1.641639 1.652342 

2 Cumulative 2.698551 2.728694 2.776278 2.788693 

 Estimate 1.122593 1.127530 1.134639 1.136351 



 

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3 Cumulative 3.513564 3.534491 3.566649 3.574862 

 Estimate 0.815013 0.805797 0.790371 0.786169 

4 Cumulative 3.898699 3.906180 3.917195 3.919914 

 Estimate 0.385135 0.371689 0.350546 0.345052 

Model 13 

1, 2 

1 Cumulative 0.406465 0.360368 0.297339 0.282666 

 Estimate 0.406465 0.360368 0.297339 0.282666 

2 Cumulative 0.854198 0.763012 0.635992 0.606033 

 Estimate 0.447732 0.402644 0.338653 0.323368 

3 Cumulative 1.451473 1.311094 1.109700 1.061211 

 Estimate 0.597275 0.548081 0.473708 0.455178 

4 Cumulative 2.178195 1.999543 1.730892 1.664061 

 Estimate 0.726722 0.688449 0.621192 0.602850 

Model 14 

1, 2 

1 Cumulative 0.669864 0.682578 0.703172 0.708655 

 Estimate 0.669864 0.682578 0.703172 0.708655 

2 Cumulative 1.277435 1.299207 1.334279 1.343577 

 Estimate 0.607570 0.616629 0.631108 0.634922 

3 Cumulative 2.003042 2.031785 2.077694 2.089783 

 Estimate 0.725607 0.732577 0.743414 0.746206 

4 Cumulative 2.770071 2.799974 2.847106 2.859389 

 Estimate 0.767029 0.768189 0.769413 0.769606 

Model 15 

1, 2 

1 Cumulative 0.672187 0.684941 0.705597 0.711097 

 Estimate 0.672187 0.684941 0.705597 0.711097 

2 Cumulative 1.356028 1.378775 1.415387 1.425087 

 Estimate 0.683841 0.693835 0.709790 0.713990 

3 Cumulative 2.166090 2.195697 2.242874 2.255275 

 Estimate 0.810062 0.816922 0.827487 0.830188 

4 Cumulative 2.979255 3.007940 3.052922 3.064597 

 Estimate 0.813165 0.812243 0.810047 0.809321 

Model 16 

1, 2 

1 Cumulative 0.669864 0.682578 0.703172 0.708655 

 Estimate 0.669864 0.682578 0.703172 0.708655 

2 Cumulative 1.347727 1.370374 1.406827 1.416486 

 Estimate 0.677863 0.687796 0.703656 0.707831 

3 Cumulative 2.149807 2.179341 2.226413 2.238789 

 Estimate 0.802080 0.808967 0.819586 0.822303 

4 Cumulative 2.958007 2.986854 3.032114 3.043867 

 Estimate 0.808200 0.807513 0.805701 0.805078 

Model 17 

1, 2 

1 Cumulative 1.156191 1.176367 1.208905 1.217539 

 Estimate 1.156191 1.176367 1.208905 1.217539 

2 Cumulative 3.036551 3.064754 3.108910 3.120356 

 Estimate 1.880360 1.888387 1.900005 1.902817 

3 Cumulative 3.972285 3.975017 3.978900 3.979832 

 Estimate 0.935734 0.910263 0.869990 0.859475 

4 Cumulative 3.999999 3.999999 3.999999 3.999999 

 Estimate 0.027714 0.024983 0.021100 0.020168 

Model 18 

1, 2 
1 Cumulative 0.669864 0.682578 0.703172 0.708655 

 Estimate 0.669864 0.682578 0.703172 0.708655 



 

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2 Cumulative 1.658078 1.684097 1.725837 1.736867 

 Estimate 0.988214 1.001519 1.022666 1.028212 

3 Cumulative 2.761836 2.791772 2.838963 2.851262 

 Estimate 1.103758 1.107674 1.113125 1.114395 

4 Cumulative 3.592248 3.611224 3.640244 3.647628 

 Estimate 0.830411 0.819452 0.801281 0.796365 

Model 19 

1, 2 

1 Cumulative 1.156191 1.176367 1.208905 1.217540 

 Estimate 1.156191 1.176367 1.208905 1.217540 

2 Cumulative 3.373990 3.397759 3.434537 3.443982 

 Estimate 2.217799 2.221392 2.225631 2.226442 

3 Cumulative 3.997921 3.998225 3.998627 3.998718 

 Estimate 0.623931 0.600466 0.564090 0.554736 

4 Cumulative 4.000000 4.000000 4.000000 4.000000 

 Estimate 0.002079 0.001775 0.001373 0.001282 

Table 9. The cumulative and estimated response rates for ML Method  

Models 1,2,3,7 

Rep. 
Time 

(days) 
Death 

Dose (Concentration) 

12.5 25 75 100 

1, 2 

1 
Cumulative 0.652535 0.416912 0.200175 0.164694 

Estimate 0.652535 0.416912 0.200175 0.164694 

2 
Cumulative 2.157267 1.522434 0.800904 0.668858 

Estimate 1.504733 1.105522 0.600730 0.504164 

3 
Cumulative 3.513791 2.912584 1.821188 1.567891 

Estimate 1.356524 1.390149 1.020284 0.899033 

4 
Cumulative 3.964571 3.784533 2.975964 2.689493 

Estimate 0.450780 0.871950 1.154776 1.121602 

Models 4,8,10,13 

1, 2 

1 Cumulative 4.000000 4.000000 4.000000 4.000000 

 Estimate 4.000000 4.000000 4.000000 4.000000 

2 Cumulative 4.000000 4.000000 4.000000 4.000000 

 Estimate 0.000000 0.000000 0.000000 0.000000 

3 Cumulative 4.000000 4.000000 4.000000 4.000000 

 Estimate 0.000000 0.000000 0.000000 0.000000 

4 Cumulative 4.000000 4.000000 4.000000 4.000000 

 Estimate 0.000000 0.000000 0.000000 0.000000 

Models 5,9,11,14 

1, 2 

1 Cumulative 0.000002 0.000001 0.000001 0.000000 

 Estimate 0.000002 0.000001 0.000001 0.000000 

2 Cumulative 0.000008 0.000005 0.000002 0.000002 

 Estimate 0.000006 0.000004 0.000002 0.000001 

3 Cumulative 0.000021 0.000013 0.000006 0.000005 

 Estimate 0.000013 0.000008 0.000004 0.000003 

4 Cumulative 0.000047 0.000029 0.000014 0.000011 

 Estimate 0.000026 0.000016 0.000008 0.000006 

Models 6,12,15,16 

1, 2 

1 Cumulative 0.005464 0.030485 0.442526 0.852252 

 Estimate 0.005464 0.030485 0.442526 0.852252 

2 Cumulative 0.023726 0.130984 1.598594 2.590079 



 

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Table 8 shows that, in relation to the OLS method, all models provided cumulative values for 

response rates of breast cancer cells treated physically and environmentally that were less than 

4, with the exception of model 19, which produced values that were equal to 4.  

Related to the ML method, Table 9 shows that six models (4,8,10,13,17, and 19) produced 

cumulative values equal to 4, whereas other models produced cumulative values below 4. 

Now, the values of MSE for each model can be determined by using the formula (20) with the 

observed values for response rates in Tables 2 and 3 and estimated values in Tables 8 and 9. The 

results according to each estimating method are given in Table 10, where the bold numbers 

represent the associated values of the best model. 

 

 

 

 

 

 

 

 

 Estimate 0.018262 0.100499 1.156067 1.737827 

3 Cumulative 0.064185 0.346210 3.001090 3.765209 

 Estimate 0.040459 0.215226 1.402496 1.175130 

4 Cumulative 0.142521 0.735019 3.821890 3.993077 

 Estimate 0.078336 0.388809 0.820800 0.227868 

Model 17 

1, 2 

1 Cumulative 0.010593 0.058923 0.813696 1.486949 

 Estimate 0.010593 0.058923 0.813696 1.486949 

2 Cumulative 0.088551 0.471013 3.413573 3.920940 

 Estimate 0.077958 0.412090 2.599877 2.433991 

3 Cumulative 0.445495 1.934278 3.999000 4.000000 

 Estimate 0.356944 1.463266 0.586268 0.079060 

4 Cumulative 1.606939 3.774333 4.000000 4.000000 

 Estimate 1.161445 1.840055 0.000160 4.11E-09 

Model 18 

1, 2 

1 Cumulative 0.005464 0.030485 0.442526 0.852252 

 Estimate 0.005464 0.030485 0.442526 0.852252 

2 Cumulative 0.015295 0.084851 1.120239 1.956280 

 Estimate 0.009831 0.054366 0.677713 1.104028 

3 Cumulative 0.032083 0.176269 1.995046 3.024897 

 Estimate 0.016787 0.091418 0.874807 1.068617 

4 Cumulative 0.059734 0.323026 2.899416 3.713771 

 Estimate 0.027652 0.146758 0.904370 0.688874 

Model 19 

1, 2 

1 Cumulative 0.010593 0.058923 0.813696 1.486949 

 Estimate 0.010593 0.058923 0.813696 1.486949 

2 Cumulative 0.057252 0.310047 2.838349 3.680374 

 Estimate 0.046659 0.251124 2.024654 2.193425 

3 Cumulative 0.228352 1.121340 3.974145 3.999866 

 Estimate 0.171099 0.811292 1.135795 0.319492 

4 Cumulative 0.767534 2.785921 4.000000 4.000000 

 Estimate 0.539182 1.664581 0.025855 0.000134 



 

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For the OLS method, Table 10 demonstrates that the MSE values for the green treatment are 

lower than those for the physical treatment with all models except four models, which are 5, 8, 

10, and 13. The same holds for the MSE values related to the ML method for nine models which 

are 1, 2, 3, 4, 7, 8, 10, 13, and 17. Further, it shows that the MSE values of OLS with all models 

except 7, 12, and 19 (physical treatment) and 7 and 19 (green treatment) are lower than those of 

the ML. For the OLS method related to the green treatment, model 16 is the best model with a 

value of MSE equal to 0.035413, and model 18 is the best for the ML method related to the 

physical treatment with a value of MSE equal to 0.151080. The best models indicate that the 

experiment's replications are not important. Generally, in comparing the MSE values for the best 

model related to OLS and ML methods, model 16 has the lowest MSE value. This indicates that 

the OLS method with green treatment has outperformed the ML method with physical treatment 

in providing the best model for the breast cancer experiment. Further, the estimates of the LD50 

for the best model are then determined using the formula (21), the results are shown in Table 

11. 

 

According to Table 11, the LD50 estimates of the green and physical treatments related to OLS 

and ML methods exhibit a decreasing dose-time relationship over four days. 

 

 

 

Table 10. The MSE value with the OLS and ML methods   

Model 
OLS method ML method 

Physical Treatment Green Treatment Physical Treatment Green Treatment 

1 0.111641 0.064690 0.396101 0.303643 

2 0.106413 0.062718 0.380257 0.291497 

3 0.115889 0.058925 0.380257 0.291497 

4 0.390278 0.310393 3.978605 3.936748 

5 0.204173 0.255596 0.701694 0.888094 

6 0.110147 0.056561 0.231475 0.333524 

7 0.370256 0.293933 0.365632 0.280286 

8 0.198639 0.249542 3.825582 3.785335 

9 0.410069 0.328924 0.674705 0.853936 

10 0.178690 0.214900 3.825582 3.785335 

11 0.061955 0.046661 0.674705 0.853936 

12 0.389466 0.311650 0.249280 0.359180 

13 0.173773 0.210110 3.683894 3.645137 

14 0.059495 0.046282 0.649716 0.822309 

15 0.064128 0.036096 0.240048 0.345877 

16 0.061036 0.035413 0.240048 0.345877 

17 0.644657 0.556320 0.797178 0.764556 

18 0.130385 0.063516 0.151080 0.270285 

19 0.860582 0.774939 0.488225 0.507734 

Table 11. The estimates of the LD50 for the best model 

Days 
  Model 16  Model 18 

OLS (Green Treatment)   ML (Physical Treatment) 

1 3.04E+20 153.3492 

2 5.23E+08 101.2839 

3 0.351173 75.10801 

4 2.72E-09 58.40109 



 

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7. Conclusion  

     For the relationship of response with dose and time, nineteen models are constructed based 

on the MW model. The unknown parameters are estimated using the OLS and ML estimation 

methods. The real experiment is focused on using real data to cure breast cancer by employing 

the therapeutic zinc selenide, which was physically created using plasma and environmentally 

using plant extract. The cumulative and estimated response rates are computed for each model. 

With the lowest MSE, the best models are determined, and then the estimated LD50 is introduced. 

Models 16 and 18 (where the experiment's replications are not important) represent respectively 

the best model with the OLS and ML estimation methods, and over four days, the LD50 estimates 

related to the OLS and ML methods exhibit a decreasing dose-time relationship. Finally, in 

providing the best model for the breast cancer experiment, the OLS method has outperformed the 

ML method. 

 

Acknowledgment 

     The authors are very grateful to the executive manager and editorial board members of the 

Ibn AL-Haitham Journal of Pure and Applied Sciences. 

 

Conflict of Interest  

     The authors declare that they have no conflicts of interest. 

 

Funding 

     There is no funding for the article. 

 

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