European Journal of Epidemiology 14: 23–29, 1998. 1998 Kluwer Academic Publishers. Printed in the Netherlands.

A mathematical model for the determination of the optimum valueof the treatment threshold for a continuous risk factor

Michel Cucherat & Jean-Pierre BoisselUniversite Claude Bernard Lyon, Hospices Civils de Lyon (MC & JPB), Lyon, France

Accepted in revised form 25 September 1997

Abstract. Hypercholesterolemia or hypertension arecontinuous risk factors for coronary heart disease.When a preventive action is carried out against such arisk factor, it is necessary to specify a risk factor levelvalue, named the treatment threshold, above which asubject should be treated. But a non-arbitrary deter-mination of this threshold value is impossible from theepidemiological data. A method for the non-arbitrarydetermination of the treatment threshold value is pre-sented, based upon mathematical modelling of theclinical and economics consequences of the preven-

tion policy in the whole population. In a cost-effec-tiveness approach, the model was used to estimate thecost per coronary event prevented according to thethreshold value for blood cholesterol. It was foundthat a minimum in this outcome proposed as the opti-mum treatment threshold. It is possible, from a publichealth point of view, to determine an optimum, non-arbitrary treatment threshold value in the preventionof coronary heart disease with cholesterol-loweringinterventions. The model proposed here could be use-ful in decision making processes.

Key words: Cost-effectiveness, Decision modeling, Hypercholesterolemia, Hypertension, Primary prevention

Introduction

Two of the main risk factors for coronary heart dis-eases (CHD), hypertension and hypercholesterole-mia, are continuous risk factors. These risk factors arecontinuous variables and their association with therisk is continuous. For the cholesterol level, this factbas been well illustrated by the risk curve observed inthe MRFIT cohort (Figure 1) [1, 2]. The usual distinc-tion between exposed and non exposed subjects doesnot exist in this situation, as all individuals are more orless at risk, and a base risk does not exist.

When a preventive action is carried out againstsuch a continuous risk factor it is necessary to specify aminimum value, the treatment threshold value, abovewhich an individual should be treated. This thresholdvalue cannot be non-arbitrarily determined from theepidemiological data since the increase in risk is astrictly monotonic function of the risk factor. Al-though recommendations exist for the desirable lev-els of cholesterol or blood pressure, no attempt to ob-tain a non-arbitrary value has previously been report-ed [3, 4].

One approach to this problem using mathematicalmodelling is presented in this paper. The determina-tion of the treatment threshold value was based on thestudy of the consequences of the intervention in the

whole population, since the problem of deciding athreshold value exists only at this level. For the indi-vidual, the aim is to have the lowest risk possible, i.e.the lowest risk factor value possible, and thus the con-cept of a threshold value is inconsistent. Since it is notrealistic and practical to treat everyone, it is essentialto determine a treatment threshold value.

From this public health point of view, the economicimplications of treatment must also be considered.The cost of mass prevention is an important limitationand thus the cost and the cost-effectiveness of a pre-vention policy must be verified. The choice of athreshold value must be made with full knowledge ofthe consequences of this choice. In the whole popula-tion, the expected consequences of the preventionpolicy can be assessed by two outcomes: the numberof clinical events specifically prevented; and the costper event prevented. We will illustrate our approachwith the example of blood cholesterol levels and therisk of coronary heart diseases.

Several outcomes could be used to assess the conse-quences of an intervention on a risk factor. However,intermediate outcomes are irrelevant, and surrogateoutcomes are difficult to validate [5, 6] and cannot ap-ply in the problem we are considering. Thus only clin-ical outcomes, i.e. clinical events such as death or myo-cardial infarction, stroke, etc. are relevant. This clin-

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Figure 1. The risk curve for six-year coronary heart disease(CHD) mortality observed in the Multiple Risk Factor In-tervention Trial (MRFIT) cohort (from ref. [2]). Risk is ex-pressed as the mortality rate per 1000 individuals. Eachpoint represents a decile of the cholesterol level distributionamong the 356,222 men of the primary screenees of theMRFIT. The curve represents the fit to an exponential mod-el.

Figure 2. Graphic illustration of the isotropic hypothesis.Under treatment the risk reduction induced by a reductionin the risk-factor level follows the risk curve.

ical outcome is a binary variable, which we will refer toas the event. The usual outcomes used in epidemio-logical studies and clinical trials (such as all-causedeath, death from a coronary event or fatal and nonfatal coronary events) were considered.

Methods

A mathematical model describing the consequencesof the prevention policy in a specific population hasbeen built. The variations in the consequences of thepolicy, according to the different threshold values,were estimated by computer simulations. The optimaltreatment threshold value was defined as the valuethat optimised one or more of the indicators describ-ing the consequences of the prevention policy.

Description of the model

In the following, the term ‘treatment’ designates anyrisk factor lowering intervention (drug, diet, others).

Number of events in the absence of preventionThe total number, N0 , of events related to risk factor xthat occur in a population, without preventive ther-apy, during a time period of reference can be repre-sented by:

No = ∫ R(x) ⋅ f(x) dx (1)b

a

where x represents the initial risk factor level of a sub-ject, measured before the application of the preven-tive policy; R(x) is the risk curve representing the riskof the clinical outcome during the time period of ref-

erence as a function of the risk factor level. R(x) isestimated from unbiased cohort studies in the consid-ered population; f(x) is the natural distribution of therisk factor levels in the same population; and a and brepresent the lowest and the highest value of the riskfactor level, respectively, encountered in the samepopulation. In the absence of this information, thevalues corresponding to the mean ± 4 standard devia-tions can be used.

Number of events prevented by the prevention policyIf a prevention policy is undertaken with a treatmentthreshold value of s, the population is split in twogroups: those with a risk factor level less than s andwho are not treated; and those subjects with a risk fac-tor level greater than s and who are treated. In the firstgroup, the risk is the natural risk R(x), calculated fromepidemiological studies, whereas the risk for the sec-ond group is the risk under treatment Rt(x), where x isthe risk factor level before treatment, and Rt is a newrisk function specific to treated subjects.

The total number of events, Nt(s), occurring despitethe prevention policy carries being in application witha treatment threshold of s (i.e. with all patients whohas a value greater than s are treated), is the sum of theevents occurring in these two groups:

Nt(s) = ∫ R(x) ⋅ f(x) dx + ∫ Rt(x) ⋅ f(x) dx (2)x<s xùs

The number of events prevented, Npr(s), for a giventhreshold value s, is obtained from equations 1 and 2:

Npr(s) = No − Nt(s) (3)

The risk curve under treatment: the isotropic hypothe-sis

Under treatment the risk factor level of an individualchanges from x to x – ∆x, where ∆x is the absolute riskfactor reduction obtained for the subject. The new

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risk function can be written as Rt(x, ∆x). To evaluatethis function we will assume that the isotropic hypo-thesis can be applied in order to simplify the construc-tion of the model (Figure 2) [5, 7]. Under this hypo-thesis, the risk for an individual who has, under treat-ment, a risk factor level of x − ∆x, should be the sameas that for another individual who has this level spon-taneously, without treatment

Rt(x,∆x) = R(x – ∆x)

The non-isotropic situation would be:

Rt(x,∆x) > R(x – ∆x) or Rt(x,∆x) < R(x – ∆x)

The isotropic hypothesis assumes that there is no car-ryover effect from the time spent at the high risk level,i.e. that the risk change follows rapidly the change inthe risk factor level. This also assumes that the treat-ment has no deleterious effects. Despite these restric-tions the isotropic hypothesis is generally consideredas true, and most cardiovascular risk prevention pol-icies and doctors’ decisions at the individual patientlevel are based on this.

Under the isotropic hypothesis Equation 2 be-comes:

Nt(s) = ∫ R(x) ⋅ f(x) dx + ∫ R(x – ∆x) ⋅ f(x) dx (4)x<s xùs

Distribution of risk factor changes

To allow for the fact that the changes in the risk factorlevels are not the same for all individuals, we intro-duced the distribution of these changes h(∆x) into themodel. Equation 2 thus becomes:

Nt(s) = ∫ R(x) ⋅ f(x) dx (5)x<s

+ ∫ ∫ R(x – ∆x) ⋅ f(x) dx ⋅ h(∆x) d∆x dxxùs

Economic considerations

In a cost effectiveness approach, the cost per eventprevented (kpr(s)) as a function of the treatmentthreshold s is calculated as follows:

kpr(s) = Kscre + Ktreat(s)

Npr(s)

where Kscre is the global cost of the screening neededto identify the subjects with a risk factor level abovethe treatment threshold value, and Ktreat(s) is the glob-al cost of treating all the subjects with a risk factor lev-el above the treatment threshold value s and Npr(s) isas defined in Equation 3.

The following writing convention for the costs hasbeen used: the capital K indicates a global cost overthe whole population, and the small k indicates a costper individual.

Kscre is independent of the threshold value and onlydepends on the size of population and the unitary costof risk factor assay. The global cost of the treatmentKtreat varies with the threshold value and depends onthe cost of treatment and the risk factor level distribu-tion.

Variability and imprecision

Usually in decision making oriented modelling, thevariability of parameters and variables are not takeninto consideration and only punctual estimations aregiven. In this model the risk curve, the risk factor pop-ulation parameters and the quantification of the bi-ological effect of intervention are subject to statisticalimprecision, because they are estimated from obser-vational or experimental studies. However the othercomponents are not subject to variability (costs, forinstance, are fixed and perfectly known). The confi-dence interval for kpr(s) can be built using a Monte-Carlo simulation. A large number of estimation ofkpr(s) are computed with random values for the par-ameters of the risk curve and for the change in choles-terol level. The 5th and 95th percentils of the generat-ed distribution of the kpr(s) are taken as lower andhigher limits of the 95% confidence interval.

The full model

In the full model, the population is split into groupswith similar risk functions to allow for the differencein risk between men and women, between middle-aged and elderly individuals, and for other covariatesof the risk. Each group has a specific risk curve. Thesize of the group is determined from the distributionof the population in terms of age, gender and othercovariates.

Sensitivity analysis

The influence of the main model parameters on therelation between the treatment threshold value andthe outcomes was studied by sensitivity analysis, byvarying the parameters of the model.

Results

As an example we chose serum cholesterol concentra-tion, which is a continuous risk factor for death andcoronary events; cholesterol-lowering intervention as

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Figure 3. Number of coronary events prevented as a func-tion of the treatment threshold value. The parameter valuesused are those described in the methods section.

Figure 4. Cost per coronary events prevented according totreatment threshold. The dashed lines describe the 95%confidence interval obtained by Monte-Carlo simulation.

the prevention policy; and the combined criteria: fataland non-fatal coronary events. This outcome is themost precise and relevant because, in terms of preven-tion, fatal and non fatal events have the same value,i.e. the value of a prevention failure.

The following results regarding primary preven-tion of coronary heart diseases (CHD) in CHD symp-tom-free individuals, by a cholesterol-lowering inter-vention were obtained with a model based on param-eters specific to the French population and Frenchcosts. The distribution of the serum cholesterol con-centration was assumed to be Gaussian and was esti-mated from the French data from the MONICAstudy [8, 9] (mean = 219 mg/dl, SD = 46 mg/dl). Thespecific French risk curve was estimated using the da-ta from the ‘Etude Prospective Parisienne’ [10, 11],and the mean follow-up of 6.6 years (observed in thisstudy) was used as the reference time period. The riskof fatal and non-fatal coronary events (r) was ex-pressed, with a univariate exponential model, as afunction of the serum cholesterol concentration (x)(r = 1

1000 ⋅ (0.2740 e0.0109⋅x)). Fenofibrate (Lipanthylt)was taken to be the reference treatment because thislipid lowering agent of fibrate family is the most pre-scribed in France. The daily cost of treatment by feno-fibrate is 2.15 FF in France and the cost of one screen-ing test is 50 FF. The empirical distribution of serumcholesterol concentration changes obtained underthese treatment was estimated from the relativechange reviewed in a monograph about fenofibrate[12], which was represented by a Gaussian distribu-tion with mean 27% and standard deviation 5%.

The values for several parameters of the full modelwere not available for the French population (riskcurve by gender and age, for instance). The resultspresented here were obtained from a reduced modelwhere these missing parameters and their relatedfunctions were omitted.

Number of coronary events prevented

The curve representing the evolution of the numberof coronary event prevented under treatment, whenthe value of the treatment threshold value varied from150 mg/dl to 350 mg/dl is strictly monotonic, and doesnot allow a non-arbitrary determination of a treat-ment threshold (Figure 3).

Cost per coronary events prevented

The results of the cost effectiveness approach is repre-sented by the curve of cost per coronary event pre-vented as a function of the treatment threshold (Fig-ure 4). The minimum observed on this curve provides

a means to fix a non-arbitrary treatment thresholdvalue.

Sensitivity analysis

The influence of the model parameters on the mini-mum of the previous curve were studied in the sensi-tivity analysis. As an example, Figure 5 shows thecurves obtained for several values of biological effica-cy of treatment. It appears that the minimum value(i.e. the treatment threshold) is not affected by the bi-ological efficacy, but the cost of a prevented event atthe minimum is.

After studying the effects it was possible to classifythem as either having a notable (more than 5% of rel-ative change in the threshold value between the low-est and the highest value possible of the parameterstudied) or a mild (less than 1% of relative change)influence on the threshold value. Those with a notableinfluence were: the cost of screening (marked positiveinfluence, the threshold increased when the costrose), the cost of treatment (slight positive influence),

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Figure 5. Relation between the cost of coronary events pre-vented and the treatment threshold value as a function ofthe biological efficacy (mean change in serum cholesterolconcentration under treatment).

the mean of the cholesterol concentration distribu-tion (positive influence). Those with a milder in-fuence were: the biological efficacy and the mean risklevel of the population. In the case of serum choles-terol and death or coronary heart disease, the overalllikely range of threshold variation is narrow.

Discussion

The main objective of this work is not to resolve defin-itively the problem of the treatment threshold for cho-lesterol-lowering drugs, but to propose a non arbi-trary, general method to determine an optimumthreshold and to identify the data needed for its appli-cation. The illustration of the process proposed hereuses very optimistic hypotheses that do not totallyhold true in real life (e.g. optimum decrease in choles-terol level obtained by the drugs, isotropic hypothe-sis). We placed the simulation in the most favourablesituation of the efficacy of the prevention, to estimatethe lowest threshold. Despite that, it appears that thethreshold values obtained (275–300 mg/dl) are higherthat those recommended by current guidelines (250mg/dl in France). Another point of view was also con-ceivable, taking more realistic approach where themodel was based on less ideal assumptions (e.g. a low-er biological efficacy). In this case the threshold valueobtained would be still greater than that of the currentguidelines.

The problem is more acute since the publication ofthe WOSCOPS results [13] suggest that, for examplein France, 1 in 4 middle aged men should be treated.The ethical and economical consequences of this arehuge, making a rational and evidence-based approachimperative.

The minimum

The most important result obtained with this modelwas the minimum seen with the curve representingthe cost per coronary event prevented since this pro-vides a means to set the treatment threshold valuenon-arbitrarily. This risk factor level for which thecurve reaches its minimum represent an optimal treat-ment threshold value because it corresponds to thelowest cost possible. At this point, the number ofevents prevented is sufficient to make the investmentin screening worthwhile. Below this threshold value,the treatment cost (depending directly on the numberof individual concerned by the preventive therapy) in-creases more rapidly than the number of events pre-vented, leading to an increase in the cost per coronaryevent prevented.

The existence of this minimum and the correspond-ing threshold value is due to the general model, and isnot specific to the cholesterol example. No other ap-proach has lead to a scientifically defined thresholdvalue for a continuous risk factor in a prevention pol-icy. Our model, which takes into account both the costof screening and the cost of treatment would be usefulto those who elaborate policies for prevention whichcould then be based on clear, well-defined, decision-making components. It can be applied to any contin-uous risk factor providing sufficient epidemiologicaldata is available.

Isotropic hypothesis

The model used here is based on the assumption of anisotropic situation, although there is no reliable evi-dence suggesting that this is true. In the cholesterolexample, some authors investigated this by comparingthe epidemiological data with that from clinical trialsof cholesterol-lowering interventions. Their conclu-sions were consistent with an isotropic hypothesis, butthis cannot be taken as a reliable demonstration be-cause of some methodological limitations (relativescale and external reference). For example, Tyroler[14] conclude that a 1% decrease in cholesterol levelgives a 2% decrease in coronary risk, correspondingto the value predicted from the epidemiological dataunder the isotropic hypothesis. Collins et al. in theirmeta-analysis of antihypertensive drugs showed thatthe isotropic hypothesis held for stroke reduction [15].Despite these results, the problem of the validity ofthe isotropic hypothesis is crucial and will have to bestudied further. However, the model could be usedwhen the isotropy hypothesis does not hold, providedthat Rt(x,∆x) is known from clinical trials [6, 16].

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Representativity

The only reliable data for the risk curve available forthe French population is for middle-aged men, andthus the validity of these results is restricted to thisnarrow subpopulation. The use of an exogenous riskcurve (from north American population for instance)would be possible, and although this would lead tomore robust results it would also lead to the lost of theFrench specificity [17].

Multifactorial prevention

The consequence of the treatment of the other riskfactors can be take into account through the meanrisk. When a preventive action is undertaken againstthe other risk factors, the mean risk will decrease butthe multiplicative influence of the ‘cholesterol factor’will be unchanged. From the sensitivity analysis, it ap-pears that the value of the threshold varies slightlywhen the mean risk decreases. This result suggeststhat the optimum threshold could be the same wheth-er or not the other risk factors are treated.

Conclusion

From a public health point of view (society and/or thepayer) our model enabled an optimum, non-arbitrarytreatment threshold value to be determined for theprevention of risk predicted by a continuous risk fac-tor by taking into account economic considerations.The optimal treatment threshold value was deter-mined in a cost-effectiveness approach, as the valuethat minimised the cost per event prevented. This ap-proach can be applied to all situations with continuousrisk factors.

Sensitivity analysis by simulated variations of theparameters in the model, helped identify the likelyrange of such a threshold value. If the range is narrow,a fixed treatment threshold value could be proposedfor the prevention policy.

This type of model, with the use of specific param-eters for a given population, could be useful in deci-sion-making processes, in which the consequences ofthe choice of a particular treatment threshold value orparticular prevention options need to be known.

Acknowledgements

This study was supported in part by a grant from‘Cnamts/Inserm’ and by a grant awarded by the ‘Asso-ciation pour le Développement de la Pharmacologie

Clinique’. We are indebted to Margaret Haugh for herhelpful editorial assistance.

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Address for correspondence: M. Cucherat, Service de Phar-macologie Clinique, Hôpital cardiologique, BP3041, 69394LYON Cedex 03, FrancePhone: +33 (0)4 72 11 52 49; Fax: +33 (0)4 78 53 10 30E-mail: MCU6upcl.univ-lyon1.fr