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The effect model law states that a natural relationship exists between the frequency (observation) or the probability (prediction) of a morbid event without any treatment and the frequency or probability of the same event with a treatment. This relationship is called the effect model. It applies to a single individual, individuals within a population, or groups. In the latter case, frequencies or probabilities are averages of the group. The relationship is specific to a therapy, a disease or an event, and a period of observation. If one single disease is expressed through several distinct events, a treatment will be characterized by as many effect models. Empirical evidence, simulations with models of diseases and therapies and virtual populations, as well as theoretical derivation support the existence of the law. The effect model could be estimated through statistical fitting or mathematical modelling. It enables the prediction of the (absolute) benefit of a treatment for a given patient. It thus constitutes the theoretical basis for the design of practical tools for personalized medicine.

In 1987, L'Abbe, Detsky and O'Rourke recommend to include a graphical representation of the various trials when designing a meta-analysis. For each trial, on the x-axis, the frequency (risk) of the studied criterion in the control group (Rc) should be represented, and on the y-axis, the risk in the treated group (Rt) [

The Rt, Rc plane according to L'Abbe

The shape of the resulting scatter plot illustrates some important aspects of the information concerning the effect of the treatment.

In 1989, Lubsen and Tijssen used this kind of representation and suggested that a treatment with “an average” benefit may be harmful in patients at low risk [

Without prior knowledge of Lubsen and Tijssen’s article [

The effect model of the class 1 antiarrhythmic drugs during the year following a myocardial infarction [

This equation gives the treatment net mortality reduction. By fitting this equation to the available data through a statistical regression technique, the authors estimated the parameter values and inferred the value of the threshold. In theory, only patients whose risk without treatment is above this threshold should be treated (

This figure illustrates the simulation approach (

Building on these previous ideas and introducing their own personal experience, in 1995 Glasziou and Irwig analysed the relationship between Rt, Rc built upon results of randomized clinical trials [

In 1998, in an editorial accompanying the publication of a study about the risk of bleeding with aspirin therapy, Boissel applied what would become the effect model law to this medication used in cardiovascular prevention. He showed that for subjects with low risk of cardiovascular events, aspirin is probably harmful [

At the end of the 90s, a set of studies dealing with the generalization of this relationship was published [

The relation between Rt and Rc is defined for a triplet Disease, Event, Treatment (DET) with

“For a given treatment, a given disease and a group of subjects, there is a relationship between the course of the disease in untreated subjects and the course of disease in the same treated subjects. This relationship is usually represented in the Rt, Rc plane where Rc and Rt are the rate or probability of the disease outcome in, respectively, untreated and treated subjects. The relationship can be empirically approached—at least in some cases—by a function. This functional representation is important for it enables the computation of the rate or the prediction of the probability of the disease outcome (Rt) in the treated subjects and their absolute benefit. In other instances, Rt and the absolute benefit can be derived by simulation.”

The first empirical case was the study of antiarrhythmic drugs in post-myocardial infarction. The effect model was built from published results of randomized clinical trials (summarized data, see

It consists in modelling mathematically the disease and the treatment, in generating virtual individuals (virtual population: realistic or not) and in applying the disease model and then the therapeutic model (treatment acting on the disease) to every virtual subject [

In order to illustrate this approach, let us take a sham case that Boissel

Using fundamentals of pharmacology (Hill’s model) and an extremely simplified representation of the probability of a morbid event (logistic function), Wang

Note that in this simple theoretical setting, the Rt, Rc relationship is curvilinear.

This is another confirmation of its existence because, as with the simulation approach, the relationship is not explicitly introduced into the equations expressing the phenomena.

An empirical context is ill-suited to explore individual effect models. Evidence for individual effect models for a given triplet DET arises from the mechanistic approach. When the global therapeutic model (see

Patient components are more accessible when the simulation approach is used. Two categories of patient components, which are represented by as many patient descriptors, have been identified as determinant of the shape and the situation of the relationship for a group or a patient: some patient characteristics are linked to the disease (genotypic, phenotypic, environmental), and determine the value of Rc. Others express the interactions between the patient and the treatment (for example, for a drug, the determinants of the volume distribution or of the absorption speed). These descriptors determine the y-axis value according to the average Rt value in the population for the given Rc. Some patient descriptors can be common to the two categories. The values of the first category will be called Y and the second X (see

The relation between Rt and Rc is defined for a triplet Disease, Event, Treatment (DET) with

In order to derive operational tools, the effect model law can be expressed symbolically in two ways.

the Rt function:

the absolute benefit function, AB:

The symbolic forms above put forward the variables that are behind the relation: the two types of patient descriptors X and Y, the treatment of interest and time. These forms show that there are as many values in the Rt, Rc plane as there are patients, each one being represented by a dot which is more or less close to the average curve. The expression of the absolute benefit AB has the advantage of leading directly to an individual prediction.

It should be stressed that, except for the statistical approach (as with the antiarrhythmic case in

Consequences of the law that lead to applications in personalized medicine are based on quite a simple derivation from its absolute benefit function expression:

The absolute benefit is the best expression of what a patient can expect from a treatment because this index tells what the patient would gain in terms of morbi-mortality or quality of life by being treated with T. Other indices such as the relative risk or the odds ratio do not carry the same information. Thus, it is the sensible benchmarker for individual decision making in choosing between T_{i} and T_{j}. Further, as explained later in this article, the prediction of the absolute benefit could be compared to a threshold, whether it is community or individually-based.

When summing all predicted ABs of patients in a group or in a population, one computes the number of patients who would have suffered an event had they not been treated, or the number of prevented events for an outcome that cannot recur. This is shown in

Considering the average value of Rt for each Rc, there are five possible representations (

A straight line crossing the x- and y-axes at 0;

A straight line with a threshold crossing the y-axis at Rt >0;

Curvilinear;

Curvilinear with lower and/or upper thresholds.

The Rt, Rc relationship for a treatment is specific of a therapeutic objective and a set of patients. It also depends on the duration of observation. That is why an “instant” form of the symbolic expressions may be preferred, in the same way we talk about instant risks and hazard ratios. It is even more relevant when it appears that the therapeutic benefit for a chronic disease is not necessarily constant [

The effect model law, the Rt, Rc relationship and its representation in the Rt, Rc plane can be considered according to two distinct perspectives: the Rt, Rc frequencies and the Rt, Rc probabilities. The first comes from the statistical paradigm: we are querying backward-looking data. To do so, we rely on data collected during clinical trials. The second is forward-looking. We are here in the prediction paradigm, with the caution such an approach commends. However, there is of course a link between the two perspectives. Predictions rely on the past,

There are two approaches to estimate the true effect model, which are quite different in terms of data needed, modelling and generalizability of the outcomes.

Classic statistical regression approaches apply when working with data from clinical trials, either summarized or individual data. For instance, in the antiarrhythmic case (

Sensitivity analysis did not change these estimates in a material way (e.g.,

These approaches do not rely on clinical trials data, except for the calibration of some model parameters and the validation of these models. They are based on a thorough analysis of available knowledge about the disease and the therapy, which is then processed into formal models (series of mathematical solutions: algebraic equations partial differential equations, partial derivative equations or others, such as multi-agent solutions, or combinations). To be functional, these models are combined with virtual populations, whether realistic or not [

Several examples of this approach have been published [

With the statistical approach, assumptions are needed (e.g., normal distribution). In addition, one should be cautious when estimating the effect model by fitting regression lines on summarized clinical trial data (as in reference [

With the mathematical modelling approach, the two main potential pitfalls stem from the integration of knowledge and its mathematical representation. The former risk arises when insufficient knowledge is accounted for. The main problem here is that knowledge of disease mechanisms is usually incomplete. A particular type of knowledge frequently missing, or at least often imprecise, is the variable distributions and parameter values. The modeller makes assumptions, the strength of which should be carefully reviewed and tested. The latter risk stems from inappropriate mathematical solutions being used.

Altogether, these limitations impose a rigorous approach to model validation. For instance, data and/or knowledge utilized to test the model should not have been used to design the model.

Since mathematical form, nature and number of parameters and variables depend on the disease (D), treatment (T), and clinical event of interest (E), there is no general mathematical expression for

We will limit here the overview to the consequences in terms of public health assessment and treatment decision-making. It should be emphasized that there is a number of applications ranging from discovery (e.g., new target identification [

According to the effect model law, the individual absolute benefit of a therapy varies from one patient to another. Also, the number of prevented events, all other things being equal, varies from one group of patients to another, and from one population to another. These values are determined by the distributions of X and Y in these sets. Therefore, the number of subjects to treat, which is the inverted function of the absolute benefit, varies in the same way. If its value was inferred from the result of a clinical trial or a meta-analysis, it could not be considered characteristic of a therapy [

As shown in the example of antiarrhythmic drugs, a therapy can be beneficial for some patients and increase the risk for others. In addition to these thresholds, called “natural” because they are implicit in the studied phenomenon (the triplet: disease, clinical event, therapy), we can apply constraints that are external to the system; for example, a risk, whether constant or not, of side effects that does not express through the same event than what the therapy is supposed to avoid, or the amount allocated to the reimbursement of prescriptions [

In reference [

Regarding individually-tailored thresholds, an application will need further practical developments, such as scoring the side effect impairment of patient wellbeing and quality of life with the same scale to the outcome of interest.

The second expression of the effect model law enables the prediction of the individual benefit (AB). If several treatments are competing to achieve the same therapeutic objective, the different predicted ABs can be compared for a given patient (same X, Y and different functions). So, in theory, this formulation of the effect model law allows us to personalize treatment decisions. It has been shown, first by a logical demonstration and then by simulation in a real case, that an approach of individual therapeutic decision-making based on the effect model law would usually be more efficient and more ethical (for individuals as well as from a collective point of view) than a decision-making approach based on clinical practice guidelines [

For a given triplet DET, the effect model is specific (by construction) to each competitive treatment. Actually, although the disease model is the same, the therapeutic model varies with drugs since the pharmacokinetics and pharmacodynamics and often the target in the disease model change with each drug. Thus, one can compare for a given patient defined by Y and X the absolute benefits expected with the available competitive treatments.

Summing up the individual benefits (AB) across the virtual population of patients yields the number of prevented events (NPE) for a fixed period of time. This is a measure of the public health impact when the virtual population is realistic,

Along similar lines, the effect model law opens up avenues for the exploration of the heterogeneity of treatment effects (HTE), over and above traditional methods reviewed in the document published by the PCORI (Patent-Centered Outcome Research Initiative) initiative [

The effect model law says that there is a relationship between the risk of event (or, depending on the context, rate of event, size of a continuous outcome such quality of life, concentration of a chemical,

For a given subject, from the knowledge of the effect model and the valuation of the X and Y descriptors, for every available therapeutic, the estimation or prediction of the absolute benefit enables the selection of the most efficient treatment for this patient. The individualization of the threshold, by taking into account predictable adverse effects for each one of these therapeutics, for every subject, should enable the individualization of the prediction of the benefit/risk ratio. In pharmaco-economic terms, the determination of the threshold, taking into account the amount of spending that the community has decided to devote to a treatment or a particular disease’s prevention, effectively yields efficiency in the medical decision. The necessary algorithm can be installed on the practitioner’s computer and can be connected to the electronic health record for the valuation of the X and Y descriptors.

Jean-Pierre Boissel is Chief Scientific Officer, Riad Kahoul is Senior R&D Scientist, Draltan Marin is Senior Technology Solutions Developer and François-Henri Boissel is Chief Executive Officer of Novadiscovery.