The Risk Analysis problem does not accept a unique line of thought and development, but highly depends on the nature of the problem. Typical example is the food processing. It is clear in such cases, and not only, that the chemical hazards can be either naturally occurring (mycotoxins, pyrrolizidine, alkaloids etc) or added chemical hazards (pesticides, antibiotics, hormones, heavy metals etc), see Kitsos,¹⁸ for examples. The Cancer problem, admits, so to speak, this data analysis line, and created “models” for the development of tumour, and epidemiological analysis for describing other characteristics, food being one of them, associating with the Relative Risk. This paper is focused to the former case of study and provides a classification of the developed models. The optimal experimental design techniques, Kitsos,¹⁹ usually have as a primary goal to extract the maximum amount of unbiased information regarding the factors affecting the response. Working in this framework, from a “small” number of observations and get the “best” possible estimators, with Statistical criteria. Risk Assessment in a Bioassay, National Research Council (NCR),²⁰ complex experimental design among many factors that influence the response are often regarded as a “nuisance”, and the problem is getting worse in the neighbourhood of zero, as all the assumed models, appears to be “linear” in that small area, so extrapolation is dangerous for misleading results. We believe the tolerance intervals are more appropriate, as we have already mentioned.

The focus of interest lies in the qualification of the time to tumour, as it depends on various environmental factors, such as chemical substances or radiation. Since the early work of Armitage and Doll²¹ — also Armitage,²²’²³ Doll²⁴’²⁵ — Probability models have been used to describe the process of forming benign and malignant tumours. The cancer problem was eventually the most beloved and impressive statistical problem under consideration and Sir David Cox was providing a number of working examples, Cox,²⁶ Cox and Snell,²⁷ Kitsos,²⁸ with his hazardous function being a fundamental tool for applications. Epidemiological studies were performed under Statistical cover and essential “parameters’’ were evaluating, even without the support of Statistical packages. In principle there are two main reasons for formulating Probability models of carcinogenesis, to:

• provide a framework for evaluating the consequences of the proposed mechanisms of carcinogenesis.

• help to determine allowable concentrations of known carcinogens in the environment, and to estimate the consequences of exceeding them.

This is necessary because animal experiments must be done at concentrations high enough to cause some of the animals to develop tumours, while environmental concentrations must be low enough to produce very few tumours in man. Thus, apart from the animal experiments include under test different doses and exhibit different kinds of injuries, and it is necessary to study low concentrations. Therefore, beyond the Statistical analysis, covered widely with software now, some Probability model theory is needed, which is not always covered from popular software, to extrapolate the dose–response relationships downward from the high doses used in animal experiments to the low doses to be allowed in the environment. In principle there are two different approaches in cancer modelling:

1. Those models that consider the whole organism as the modelling unit and describe the time to overt tumour in this unit.

2. Models that describe the formation of carcinomas at the level of the cell, since knowledge is accumulating about the cellular biological events leading to cancer.

The problems we shall describe in this paper are under the line of thought of Probability and Statistics. As far as the shape of the tumour concerns, based on tissue image analysis, can be studied under (multi)fractal analysis, Stehlik et al.,²⁹ describing different tumour groups. In principle a fractal is a geometric shape containing detailed structure even at arbitrarily small scales, attracting interesting in Biology and Medicine, due to the computational improvements, for their strong Mathematical insight, Losa and Nonnenmacher.³⁰ A particular study based on fractals for mammary cancer, Hermann et al.,³¹ provides evidence that the method can be adopted on case study, by case study as the probability models are devoted for particular cases. Fractal methods, although have developed a strong Mathematical background they haven’t developed an “intimacy”, there is no feedback with users and have not grasped the potential for failure, eventually the existent $P(d,t)\propto P(X\le d|t):=F(d)\text{or}P(d,t)\propto P(T\le t|d):=G(t)$Relevant Risk per case. In section 2 the development of the Probability models, at the early stages are discussed.

2. Probability Models in Carcinogenesis

Let D be the set of random variables, of the dose level of a carcinogen that induces a tumour in an individual agent and let T be the time at which an individual develops a tumour. Then, it is assumed that, an individual can develop tumour, at dose level d, say, at a particular time t, with probability P(d,t), which can be represented either restricted on time or on dose as

• P(d,t) ∝ P(X ≤ d|t) = F(d)
  or

• P(d,t) ∝ P(T ≤ t|d) = G(t)  (2.1)

assumed that the transformed cells are subject to a pure linear birth process and a clone of transformed cells is detected, if it exceeds a certain threshold. The two models for this model could be modified in two different ways:

• If we assume that only one step is needed for the transformation of a normal to a malignant cell.

• To assume that a tumour arises from a single transformed cell.

In the former case we could assume that a number of these transformed cells must accumulate for a tumour to arise, while in latter a number (how many?) of steps are needed for a transformed cell to arise from a normal cell.

The improvement was from Nordling³⁶ who assumed that k specific mutational events have to occur, for a normal cell to transform into a malignant cell and called this the multi-hit theory. Armitage and Doll²¹ modified the multi-hit theory: they assumed that a certain sequence of irreversible cell alterations has to be followed. Moreover, the quantitative implication of this approach was really masterly investigated. The approach was called multi-stage theory. It was not only accepted but, eventually, found widespread application as the biological plausibility was combined with the applicability to data sets (either from experimental or epidemiological studies).

Now, let us consider the case of a constant, continuously applied dose at level d. Moreover, the transformation rate from each stage, i, say, to the next one, i + 1, is assumed to increase linearly with the dose. In mathematical terms this is equivalent to: the transformation rate from the stage i – 1 to the next stage i is assumed to be equal to aᵢ + bᵢ d,

F(d,t)=1−exp⁡[−(a1+b1d)(a2+b2d)⋯(ak+bkd)tkk!].(2.2)F(d,t) = 1 – \exp\left[-(a_1 + b_1 d)(a_2 + b_2 d)\cdots(a_k + b_k d)\frac{t^k}{k!}\right]. \tag{2.2}F(d,t)=1−exp[−(a1​+b1​d)(a2​+b2​d)⋯(ak​+bk​d)k!tk​].(2.2)

As the main assumption was that the transformation rate, from each stage, to the next one is linear model, then (2.2) can be written as

G(d)=1−exp⁡[−(θ0+θ1d+⋯+θkdk)](2.3)G(d) = 1 – \exp[-(\theta_0 + \theta_1 d + \cdots + \theta_k d^k)] \tag{2.3}G(d)=1−exp[−(θ0​+θ1​d+⋯+θk​dk)](2.3)

where θᵢ, i = 0,1,…,k are defined through the coefficients of the linear transformations assumed between stages, i.e. θᵢ = θᵢ(t) = aᵢ t + bᵢ t. Notice the biological inside on the coefficients of model (2.3), and therefore we shall refer to that probability model, which describes the sequential cell can be transformed, after k distinct stages in order to be a malignant one as the multistage model describes the phenomenon, and not just an extension to a non-linear mathematical model — this is a crucial point for the Statistician to clarify. Same line of thought exists for the coefficients ϑᵢ = θᵢ(t), which are function of time, which is not

λ(t)=c(t−t0)k−1,  c>0,  k≥1,orlog⁡λ(t)=log⁡c+(k−1)log⁡(t−t0)(2.4)\lambda(t) = c(t – t_0)^{k-1},\; c > 0,\; k \ge 1,\quad \text{or}\quad \log \lambda(t) = \log c + (k – 1)\log(t – t_0) \tag{2.4}λ(t)=c(t−t0​)k−1,c>0,k≥1,orlogλ(t)=logc+(k−1)log(t−t0​)(2.4) where t₀ is fixed for the growth of tumour and k is the number of stages.

The above hazard function is the basis of the Armitage-Doll model, which may be considered biologically inappropriate for very old persons. The explanation is based on the fact that the very old cells lose their propensity to divide, and, therefore, are more refractive to new transformations. Thus, the “plateau” at older ages may simply reflect a compensating mechanism. Notice that the Armitage-Doll hazard function corresponds to the density function f(t)=c(t−t0)exp⁡[−ck(t−t0)k].(2.5)f(t) = c(t – t_0)\exp\left[-\frac{c}{k}(t – t_0)^k\right]. \tag{2.5}f(t)=c(t−t0​)exp[−kc​(t−t0​)k].(2.5)

The target of low dose exposure is to estimate effects of low exposure level of agents, known already and to identify if there are hazardous to human health. The interspecies extrapolation problem has been discussed by Luebeck et al.,³⁸ among others and is based mainly to the fact that the “body weight”, say B, is related to the physiological parameter of interest h exponentially as h = pB^q, with p and q parameters (very often q is reported around the value of q ≈ 0.75). Therefore, the experimentation is based on animals and the results are transferred to humans. So, the target to calculate the probability of the occurrence of a tumour during the individual’s lifetime is exposed to an agent of dose d during lifetime is replacing humans with animals. Moreover, the idea of “tolerance dose distribution” was introduced to provide a statistical link and generate the class of dose risk functions. Consider a nutshell – a tumour occurs at dose level x = d if the individuals’ resistance is broken at x, then the excess tumour risk is given by the Probability “model” $$F(d)=P(X\le d)=P(\text{Tolerance}\le d)$$

The above “model” is actually the unknown cumulative distribution function that is eventually modelled, in the sense that it is assumed: there exists a statistical model which approximates F(d), which acts as a cumulative distribution function. Then the dose level “d” is linked with the binary response (success or failure)

Yi={1success with probability F(d)0failure with probability 1−F(d)Y_i = \begin{cases} 1 & \text{success with probability } F(d) \\ 0 & \text{failure with probability } 1 – F(d) \end{cases}Yi​={10​success with probability F(d)failure with probability 1−F(d)​

Technically speaking the researcher only knows that the parameter vector comes from a subspace of the real numbers, θ ∈ Θ ⊆ ℝⁿ, and we try to estimate as well as possible, different to be considered models, for different studies, see also Hartley and Sielken,³⁹ as far as the safe dose concerns. The best known tolerance distribution is proposed by Finney,⁴⁰ in his early work, the probit model of the form

$$P(d)=\Phi (\mu +\sigma d),$$with Φ being, as usual, the cumulative distribution function of the Normal distribution and μ and σ > 0 are location and scale parameters estimated from the data. Practically, the logarithm of dose is used that implies a log normal tolerance distribution.

The most commonly used parametric model for carcinogenesis is the Weibull model P(T≤t)=1−e−(θt)kP(T \le t) = 1 – e^{-(\theta t)^k}P(T≤t)=1−e−(θt)k with t being the time to create a tumour, and with hazard function λ(t)=k θk tk−1(2.6)\lambda(t) = k\,\theta^k\, t^{k-1} \tag{2.6}λ(t)=kθktk−1(2.6)

For the shape and scale parameters θ, k > 0, respectively, it is assumed when the Weibull model can exhibit a dose-response relationship that is either sub-linear (shape parameter k > 1) or supra-linear (k < 1), and has a point of inflection at x=((k−1)/k)1/k.x = ((k – 1)/k)^{1/k}.x=((k−1)/k)1/k.

The Weibull distribution is the fundamental distribution in Survival Analysis and is used in Reliability theory as a lifetime distribution. It is an extreme value distribution and is obtained as the distribution of the minimum of identical exponentially distributed random variables. Hence, if a “system” consists of independent components, each having identical exponential lifetimes, and if the system fails whenever the first component reaches the end of its lifetime, then the lifetime of the system follows a Weibull distribution.

Due to its flexibility, the Weibull model is suited to describe incidence data as they arise in animal experiments and in epidemiological studies. It has been used for common parametric analyses, e.g. comparison between experimental groups, which were treated with different doses of a carcinogenic substance. The Weibull model has been extensively discussed for tumorigenic potency by Dewanji et al.,⁴¹ through the survival functions and the maximum likelihood estimators.

The second derivatives of the log-likelihood l are Iθθ=−κdθ2−κ(κ−1)θκ−2∑tiκ,Iκκ=−dκ2−θκ∑tiκ [log⁡(θti)]2I_{\theta\theta} = -\frac{\kappa d}{\theta^2} – \kappa(\kappa – 1)\theta^{\kappa – 2}\sum t_i^{\kappa}, \quad I_{\kappa\kappa} = -\frac{d}{\kappa^2} – \theta^{\kappa}\sum t_i^{\kappa}\,[\log(\theta t_i)]^2Iθθ​=−θ2κd​−κ(κ−1)θκ−2∑tiκ​,Iκκ​=−κ2d​−θκ∑tiκ​[log(θti​)]2 Iθκ=dθ−θκ−1(1+κlog⁡θ)∑tiκ−κθκ−1∑tiκlog⁡tiI_{\theta\kappa} = \frac{d}{\theta} – \theta^{\kappa – 1}(1 + \kappa \log \theta)\sum t_i^{\kappa} – \kappa \theta^{\kappa – 1}\sum t_i^{\kappa}\log t_iIθκ​=θd​−θκ−1(1+κlogθ)∑tiκ​−κθκ−1∑tiκ​logti​

Therefore, the 2 × 2 Fisher’s information matrix, with diagonal elements Iθθ,IκκI_{\theta\theta}, I_{\kappa\kappa}Iθθ​,Iκκ​ can be evaluated and then the Variance–Covariance matrix is the inverse of Fisher’s matrix.

Example 2.1:

For a working example on the above, see Kitsos and Limniakopoulou.⁴²

Example 2.2:

Simulation Study for the One Hit Model.

The One Hit model was one of the basic models studied by Gaddum, in 1940. For this case study under the imposed framework we face this $$T(x;\theta )=P(y=1|x,\theta )=\exp (-\theta x)=1-P(y=0|u,\theta )$$The corresponding log-likelihood function is ℓ(θ)=−∑xiyi−∑(1−yi)[xiexp⁡(−θxi)]/[1−exp⁡(−θxi)]\ell(\theta) = -\sum x_i y_i – \sum(1 – y_i)[x_i \exp(-\theta x_i)] / [1 – \exp(-\theta x_i)]ℓ(θ)=−∑xi​yi​−∑(1−yi​)[xi​exp(−θxi​)]/[1−exp(−θxi​)] while the summation $\sum$ runs from 1 to n. In probability terms the value
xopt=1.59/θx_{\text{opt}} = 1.59 / \thetaxopt​=1.59/θ corresponds to p = 0.2 – notice the dependence on the unknown parameter θ. For the binomial model with success probability p = P(y = 1 | x, θ) it seems reasonable in practice to keep probability levels within the interval [0.025, 0.975]. Notice that the optimum design point depends on the unknown parameter, therefore a prior guess is needed. The unknown parameter is estimated sequentially.

3. On the Michaelis–Menten Model

A general theory for enzyme kinetics was firstly developed by Michaelis and Menten⁴⁴ in their pioneering work. This is discussed very briefly below:

When an enzyme, say E, combines reversibly with a substrate, say S, to form a quite complex, say ES, which can dissociate or proceed to the product, say P, the following scheme is assumed: E+S    ⇌k2k1    ES    →k3    E+PE + S \;\; \xrightleftharpoons[k_2]{k_1} \;\; ES \;\; \xrightarrow{k_3} \;\; E + PE+Sk1​k2​​ESk3​​E+P with k₁, k₂, k₃ the associated rate constants. We let KM=k2+k3k1K_M = \frac{k_2 + k_3}{k_1}KM​=k1​k2​+k3​​ be the Michaelis–Menten (MM) constant, Vmax⁡=k3CTOTV_{\max} = k_3 C_{TOT}Vmax​=k3​CTOT​, CTOTC_{TOT}CTOT​ = the total enzyme concentration.
In principle Kₘ is the concentration at one half Vmax⁡V_{\max}Vmax​, with Vmax⁡V_{\max}Vmax​ being the maximum metabolic rate constant. As far as the interspecies extrapolation concerns the MM constant is assumed to remain constant. Then a plot of the initial velocity of reaction u, against the concentration of substrate Cₛ, will provide the MM rectangular hyperbola of the form u=KM​+CS​Vmax​CS​​(3.1)

Notice that in practice we only have readings of the form (3.1) with $f(x,\theta )=u$ being a non-linear function as above with the parameter vector to be θ=(Vmax⁡,KM),  x=CS.\theta = (V_{\max}, K_M),\; x = C_S.θ=(Vmax​,KM​),x=CS​. The deterministic relation (3.1) is linked with the experimental error. And in practice only readings for the stochastic non-linear model of the form yi=ui+ei,i=1,2,…,ny_i = u_i + e_i,\quad i = 1,2,\ldots,nyi​=ui​+ei​,i=1,2, is obtained. That is, readings yiy_iyi​ are associated with noise, with mean zero and variance constant, with the Normality assumption valid when inference is needed. For the optimal design consideration and evaluation of the parameters in Michaelis–Menten Model, see the early work of Endrenyi and Chan,⁴⁵ while Currie⁴⁶ have discussed the heteroscedasticity in the Michaelis–Menden Model and Gilberg et al.⁴⁷ provides an extensive discussion, while for all the possible ways the experimentalist wishes to fit the model, see Toulias and Kitsos.⁴⁸ As far as the construction of the confidence intervals, here a special consideration is needed. An extra statistic is defined: the supremum value of Beale’s measure of nonlinearity, Bates and Watts,⁴⁹ for models with two or more parameters, can be reduced to

B=1+nn−21F(3.2)B = 1 + \frac{n}{n – 2}\frac{1}{\sqrt{F}} \tag{3.2}B=1+n−2n​F​1​(3.2) with F being the F distribution with the appropriate degrees of freedom (df). Therefore, the approximate confidence region for the MM model is (θ−θ^)TI(θ^,ξ^)(θ−θ^)≤BFp,s2F(a;p,n−p)(\theta – \hat{\theta})^T I(\hat{\theta}, \hat{\xi}) (\theta – \hat{\theta}) \le B F_{p, s}^2 F(a; p, n-p)(θ−θ^)TI(θ^,ξ^​)(θ−θ^)≤BFp,s2​F(a;p,n−p) where I is the information matrix. They worked out, empirically, that this confidence region includes the classical Wald confidence intervals for 100% or 95% efficient designs. The Optimal Design theory can be applied in experimental Carcinogenesis, Kitsos,⁵¹’⁵¹’⁹ while the Metabolizing Enzymes for Lung Cancer Risk Factors have been discussed by Risch et al..⁵²

There are two lines of thought to approach the MM model as an optimal experimental design:

1. Biological point of view: enzymatic process plays an important role in practice. In cancer studies, the question is whether enzymatic induced interactions have an influence on the production of a carcinogen. This is strongly related with the estimation of Vmax⁡V_{\max}Vmax​ and KMK_MKM​, with such attempts giving more emphasis to practical problems, Currie,⁴⁶ Endrenyi and Chan,⁴⁵ Gilberg et al..⁴⁷

2. Statistical point of view: includes Michaelis–Menten within the class of non-linear models and uses the model within this theoretical framework, Kitsos⁵⁰ among others. Two more very crucial points are also mentioned namely: that this design is optimum for estimating the ratio Vmax⁡/KMV_{\max}/K_MVmax​/KM​ and that the variance σ2\sigma^2σ2 is not constant.

Endrenyi and Chan⁴⁵ discussed and obtained the D-optimal design for the MM model, with the minimization in mind of the information matrix Re. They worked out that only one parameter needs prior information and then applied all more classical terms for 100% or 95% efficient designs. It is clear that in enzyme kinetic studies, with constant variance the one design point might be at the highest possible concentration, and the second at the maximal feasible velocity. The main result, Kitsos⁵⁰ for the Michaelis–Menten Model is that the D-optimal design for the MM model as in (3.1) does not depend on Vmax⁡V_{\max}Vmax​, the “linearly contributed” parameter. That practical means that only for one parameter you need prior information, and the D-optimal design for the extended Michaelis–Menten model of the form

u=Vmax⁡CSKM+CS+θ0CSu = \frac{V_{\max} C_S}{K_M + C_S} + \theta_0 C_Su=KM​+CS​Vmax​CS​​+θ0​CS​ for estimating θ=(Vmax⁡,KM,θ0)\theta = (V_{\max}, K_M, \theta_0)θ=(Vmax​,KM​,θ0​) does not depend on Vmax⁡V_{\max}Vmax​ and θ0\theta_0θ0​, which can be considered as the linear metabolic rate constant. Moreover, the Dₛ-optimal design for estimating θ0\theta_0θ0​ depends only on KMK_MKM​. This comes as a result that an extra linear term does not influence the MM optimal experimental design, Kitsos,⁵⁰ while if U >> K₀ (U is too large comparing K₀) the locally D-optimal design (the one which allocates half observations at two contributed optimal design points) allocates half points at U (the maximum value of the concentration of substrate) and K₀ (the initial guess for the MM constant). Moreover, it is Optimum value of Cₛ ≅ K₀. Notice that the D-optimal designs as above for estimating the parameter vector (Vmax⁡,KM)(V_{\max}, K_M)(Vmax​,KM​) are the D-optimal designs for estimating the ratio φ=Vmax⁡/KM.\varphi = V_{\max}/K_M.φ=Vmax​/KM​. This result is really helpful for the experimentalist, as to get a ratio estimates is a difficult task in Statistics. This result is Statistically essential, as there is, in principle, a difficulty on ratio estimates. The problem is always the initial guess. Therefore, the MMM can be based on two steps of calculations:

A1: Determine a proportion p for your n observations at first stage i.e. allocate np2\frac{np}{2}2np​ observations at U and K₀, or at U and Optimum Cₛ. Get the estimates. Use these estimates to feed the next step A2.

A2: Perform sequentially (1−p)n2\frac{(1 – p)n}{2}2(1−p)n​ more runs OR a second static design with the estimates obtained at stage A1 (this case is known as two stage design). The MMM is rather a Statistical model, than a Probability one, but still within the useful models facing the Cancer problem. More elegant, mathematically, models can be constructed and are discussed below.

4. Advanced Probability Models

The mechanistic models have been named so, because they are based on the presumed mechanism of the carcinogenesis and they form a particular class of models. The main and typical models of this class are the dose response models, used in Risk Assessment, based on the following main characteristics:

➤ There is no threshold dose below which the carcinogenic effect will not occur.
➤ Carcinogenic effects of chemicals are induced proportionally to dose (target tissue concentration) at low dose levels.
➤ A tumour originates from a single cell that has been damaged by one of the two reasons: either the parent compound or one of its metabolites.

The mechanistic class of models can be subdivided into those models that describe the process on the level of the organism or on the level of individual cells, see section 1. The following subclasses are referred.

The sub-class of Global models, includes those models that on the level of the whole organism, are closely related to the introduced Probability Models for Cancer, as also describe the time to detectable carcinoma. Typical example is the cumulative damage model (it is mathematically now really visible). This model considers that the environmental factors cause damage to a system, which although does not fail “immediately”, eventually fails whenever the accumulating damage exceeds a threshold. The model adopts a Poisson process generating time points in which damage occurs.

The second sub-class of Mechanistic Models is the Cell Kinetic models.

They attempt to incorporate a number of biological theories, based on the line of thought that the process of carcinogenesis is on the cellular level. There has been a common understanding among biologists that the process of carcinogenesis involves several biological phenomena including mutations and replication of altered cells. Cell kinetic models are subdivided by the method, which is used for their analysis, in Multistage Models and Cell Interaction Models, as follows. The main class of cell kinetic models comprises multistage models, which describe the fate of single cells, but does not take into account interactions

between cells. In these models, mutations and cell divisions are described. These models stay analytically tractable because cells are assumed to act independently to each other. The variables of interest can be derived explicitly, and hence the usual statistical techniques can be used to apply these models to data. The pioneering work of Moolgavkar and Venzon¹¹ formulates a two–stage model with stochastic clonal expansion of both normal and intermediate cells and they introduced a mathematical technique to analyse a two–stage model with deterministic growth of normal cells and stochastic growth of intermediate cells. Moreover, Moolgavkar and Knudson¹² showed how to apply this latter model to data from epidemiological studies. Working on incidence data from animal experiments and epidemiological studies, Kopp–Schneider,¹⁶ is providing the appropriate definitions and understanding for the underlying probabilistic mechanism for applications.

The second sub-class of the cell kinetic models is the Cell interaction models incorporating both the geometrical structures of the tissue and communication between cells are too complicated for analytical results. These models aim to describe the behaviour (described by a number of simulations) of complex tissues in order to test biological hypotheses about the mechanism of carcinoma formation.

The Generalised Multistage Models (GMS) or Cell Interaction Models and the Moolgavkar–Venzon–Knudson Models (MVK) are based on the following assumptions:

➤ Carcinogenesis is a stochastic multistage process on cell level.
➤ Transition between stages is caused by an external carcinogen, but it may also occur spontaneously.

Cell death and division is important in MVK models. The normal, intermediate and malignant cells are depending on time. Intermediate cells arise from a normal cell due to a Poisson process with known rate. A single intermediate cell may die with rate β, divide into two intermediate cells with rate α, or divide into one intermediate and one malignant cell with rate μ. Therefore, the process gives rise to three steps: Initiation, promotion, progression, which are very different from the biological point of view.
If an agent increases the net cell proliferation α – β, the cancer risk will also increase, Luebeck et al.,³⁸ Luebeck and Moolgavkar.¹⁵ These cell kinetic models were used to describe the time to tumour as a function of exposure to a carcinogenic agent. Two objectives guide this research. On one hand, the models can be used to investigate the mechanism of tumour formation by testing biological hypotheses that are incorporated into the models. On the other hand, they are used to describe the dose–response relationship for carcinogens.