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Multiple factor indices of protection or risk towards disease
Medical Hypotheses (2001) 56(2), 200–206 © 2001 Harcourt Publishers Ltd doi: 10.1054/mehy.2000.1141, available online at http://www.idealibrary.com on
Multiple factor indices of protection or risk towards disease F. Belfiore, S. Iannello Institute of Medicina Interna e Specialità Internistiche, Chair of Internal Medicine, University of Catania Medical School, Ospedale Garibaldi, Catania, Italy
Summary In order to combine several factors entailing protection or risk towards disease and to calculate a Protection Multiple Factor Index (PMFI) or, conversely, a Risk Multiple Factor Index (RMFI), we propose the following formulae: (1) PMFI = 2/[(mF)2+1] and (2) RMFI = 2/[(imF)2+1], where mF is the mean value of the factors considered and imF is the inverse (or reciprocal) of mF. In calculating mF, the value of each ‘risk factor’ observed in the patient under study (Vp) is expressed by taking the mean normal value (Vmn) as the unit, i.e. by calculating the ratio Vp/Vmn, whereas each ‘protection factor’ is expressed as the reciprocal of this ratio, i.e. as Vmn/Vp. The ‘weight’ of the various factors can be changed through multiplication by a number >1 or <1. Values of both PMFI and RMFI are always close to 1 in normal subjects, with extreme variations among patients between 0 and 2. The sum of the values of PMFI and RMFI is always equal to 2, so that one index can be deduced from the other. When factors are only two (F1 and F2), the formulae may be simplified as follows: PMFI = 2/[F1 × F2)+1] and RMFI = 2/[(iF1 × iF2)+1], where iF = 1/F, with only minimal changes in results. © 2001 Harcourt Publishers Ltd
INTRODUCTION From pathophysiological studies or epidemiological data, it is well established that deviation from the normal value of several metabolic parameters may favor or oppose the development of some metabolic diseases. On this basis, several ‘risk factors’ and ‘protection factors’ have been identified for various diseases. The best example may perhaps be the CHD1, for which some metabolic parameters (LDLc, ApoBI, TG, FNG, etc.) are regarded as risk factors whereas others (HDLc, ApoA, etc.) are considered as protection factors (1–2). It is clinically relevant to be able to combine the various risk and protection factors into one ‘index’ which reflects the global status of a subject, as it results from the cumulative effects of the various risk/protection factors present. Such an ‘index’ can be directed to measure the ‘protection’ towards a disease, Received 19 November 1999 Accepted 18 April 2000 Published online 3 January 2001 Correspondence to: Professor F. Belfiore, Institute of Medicina Interna e Specialità Internistiche, Chair of Internal Medicine, University of Catania, Ospedale Garibaldi, 95123 Catania, Italy. Fax: +39 095 1310899; E-mail: [email protected]
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i.e., it can be a protection index, in which instance it will increase with the elevation of the protection factors and the lowering of the risk factors, or it may be directed to measure the ‘risk’ towards a disease, i.e., it can be a risk index, in which instance it will increase with the elevation of risk factors and with the lowering of protection factors. In this paper we propose two formulae which allow us to calculate the Protection Multiple Factor Index (PMFI) and the Risk Multiple Factor Index (RMFI), as defined above, which enable us to define the protection/risk status of a given patient. THE PROPOSED FORMULAE The various factors that may confer protection or risk towards a disease can be combined into one ‘index’ through various formulae, in which the combination of the various factors can be obtained through addition or multiplication. Among possible formulae, we propose the following two for the calculation of PMFI and RMFI, 1 Abbreviations used in the text: CHD = coronary heart disease; TC = total cholesterol; LDLc = low density lipoprotein cholesterol; HDLc = high density lipoprotein cholesterol; TG = triglycerides; ApoA = apoproteinA; ApoBI = aprotein BI; FNG = fibrinogen.
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which possess some unique characteristic, outlined below: (1) PMFI = 2/[mF)2+1] and (2) RMFI = 2/[iF)2+1], where mF is the mean value of the factors considered and imF is the inverse (or reciprocal) of mF. In calculating mF, the value of each ‘risk factor’ observed in the patient under study (Vp) is expressed by taking the mean normal value (Vmn) as the unit, i.e. by calculating the ratio Vp/Vmn, whereas each ‘protection factor’ is expressed as the reciprocal of this ratio, i.e. as Vmn/Vp. By applying our formulae to hypothetical patients with mF ranging from subnormal to 3-fold increased values we obtained the results shown in Fig. 1 (left panel). It is apparent that the results obtained with our formula decrease (PMFI) or increase (RMFI) in a slightly curvilinear fashion within wide ranges of physiological variations of the involved factors.
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The construction of these formulae can be distinguished into several steps. Step 1 (making different factors comparable to each other) In order to make factors which differ as far as their nature and unit of measure are concerned, all the factors considered in the calculation of our index are expressed by taking the mean normal value (Vmn) as the unit. This also entails that in the normal subjects all factors are close to 1. Step 2 (combining the different factors into one value) This can be accomplished through multiplication of the various factors by each other, or through addition of the various factors to each other. The multiplication method has the disadvantage that, with the increase of the various factors, the resulting value increases in an
Fig. 1 LEFT PANEL. Behavior of PMFI (Protection Multiple Factor Index) and of RMFI (Risk Multiple Factor Index) with changes in the mean value of the considered factor (mF) from 0.33 (1/3 of Vmn) up to 3 (3-fold the Vmn). BOTTOM PANEL. This table shows that, with the increase in the mF value, PMFI and RMFI change in a complementary manner (PMFI decreasing and RMFI increasing) so that the sum of the PMFI and RMFI is always equal to 2. RIGHT PANEL. Behavior of PMFI and of a ‘simplified index’ (usable when the factors considered are only two) in the range of mF from 0.67 (1/1.5 of Vmn) to 2 (2-fold the Vmn). The two curves show the ‘maximum possible difference’ between PMFI and the ‘simplified index’, which occurs in the rather unlikely instance when the change (either increase or decrease) affects only one of the two factors, while the other remains unchanged. Even in this extreme instance, the values of the two Indices are close to each other. © 2001 Harcourt Publishers Ltd
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exponential manner (i.e. the changes are progressively amplified ), reaching extremely high values. Concerning the addition method, it actually consists of the addition of the various factors followed by the division of the sum by the number of the factors, thus obtaining the mean value of the various factors (which in normal subjects will be always close to 1). This means to calculate the component of our formulae which we called mF. In the proposed formulae the mF is placed in the denominator of a fraction, which entails that the mF loses ‘weight’ as far as the result is concerned. To increase the weight of mF, in our formulae we use the squared value of mF in the case of PMFI, and of imF in the case of RMFI, i.e. (mF)2 or (imF)2. Thus, in our formulae the combination of the various factors is obtained both by addition (calculation of mF or imF) and by multiplication (calculation of the squared value of mF or imF). Step 3 (making the index values to change between 0 and 2) By adding 1 to (mF)2 and putting the result in the denominator of a fraction in which 2 is the numerator we get the complete formula of PMFI. By using 1/(mF)2 instead of (mF)2 the RMFI formula is obtained. The rationale for these operations is the following. The value of (mF)2 in the normal condition is 1. Therefore, by adding 1 to the (mF)2, the value of 2 is obtained. By putting this value of 2 in the denominator of a fraction in which 2 is the numerator (as is done in our formula), the result of 1 is obtained in the normal subjects. On the other hand, as the value of (mF)2 changes towards the extremely high or low values, the result of the formula approaches 0 or 2 for PMFI and 2 or 0 for RMFI. However, the values of (mF)2, which reflect the mean changes of the considered factors, progressively loses weight as the values move away from the normal value of 1 towards high/extreme values. This is shown by the curvilinear behavior of both PMFI and RMFI with progressive changes in (mF)2, as outlined in Fig. 1 (left panel).
similar but specular curve (Fig. 1, left panel), so that from a given value of PMFI (or RMFI) it is possible to know the mean change of the involved factors. 4. Small changes in the mean value of the various factors (mF), i.e. changes in mF close to 1, have greater ‘weight’ than extreme values, as shown by the curves in Fig. 1 (left panel), which progressively flatten as the value of mF (or imF) increases. 5. PMFI and RMFI are correlated to each other in such a way that their sum is always equal to 2, so that one index can be deduced from the other (Fig. 1, bottom panel). This makes it possible to choose whether to express the status of a given patient by a protection or a risk index.
EXAMPLES OF APPLICATION An example of utilization of the proposed formulae may be the calculation of the CHD risk index based on the well known risk and protection factors. In Table 1 we present data referring to four parameters including three risk factors (LDLC, TG and FNG) and one protection factor (HDLC ) in three hypothetical patients. The mean normal values given in Table 1 for the parameters considered are rounded figures calculated from recently published works (3–8). In one of the hypothetical patients in Table 1, all four factors considered are normal (top panel of the table), and therefore both PMFI and RMFI are equal to 1 (Table 1, top panel). In the second patient (Table 1, middle panel), the three risk factors are increased whereas the protecting factor is decreased, so that PMFI is reduced (= 0.68) and RMFI elevated (= 1.32). In the third patient (Table 1, bottom panel), the reverse condition occurs, with low levels of the risk factors and enhanced value of the protecting factor, so that PMFI is high (= 1.18) and RMFI lowered (= 0.82).
DISCUSSION The main proposed formulae
CHARACTERISTICS OF THE PROPOSED FORMULAE The proposed formulae possess the following unique characteristics. 1. In normal subjects, the value of both PMFI and RMFI is always close to 1. 2. Among patients, the extreme variations lie always between 0 and 2. 3. With the increase in the mF value (i.e., with increase in risk factors and decrease in protection factors), PMFI decreases according to a slightly curvilinear fashion whereas RMFI increases according to a Medical Hypotheses (2001) 56(2), 200–206
Despite the interest in assessing the protection/risk factors both in the clinical setting and in epidemiological studies, a simple method suitable for reaching this goal is not yet available. With our formulae, we attempt to introduce a simple method to measure the protection/risk status as it results from the effects of any number of protection/risk factors. To better understand the proposed formulae, the following aspects should be further discussed. A point to be underlined is that, in our formulae, the risk factors considered are expressed by making the mean normal value (Vmn) equal to 1, i.e. by dividing the value observed in the patient or person under study (Vp) by the Vmn, whereas the protective factors are expressed as the © 2001 Harcourt Publishers Ltd
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Table 1 Calculation of PMFI and RMFI and their components (mF and imF) based on 3 risk factors (LDLc, TG and FNG) and 1 protection factor (HDLc) Risk factors
Protection factors
‘Normal’ subject LDLc TG FNG mF imF PMFI RMFI Subject at risk LDLc TG FNG mF imF PMFI RMFI
HDLc
Conventional units Vmn Vp
110 45 120 300
Vmn as unit rFs & pFs Calculations Results
110 45 120 300
PMFI or RMFI Calculations Results
110/110 = 1.00 45/45 = 1.00 120/120 = 1.00 300/300 = 1.00 1.00 1/1 = 1.00 2/((1.00)2+1) = 1.00 2/((1.00)2+1) = 1.00
HDLc
Subject ‘protected’ LDLc HDLc TG FNG mF imF PMFI RMFI
110 45 120 300
160 35 180 400
160/110 = 1.45 35/45 = 1.29 180/120 = 1.50 400/300 = 1.33 1.39 1/1.39 = 0.72 2/((1.39)2+1) = 0.68 2/((0.72)2+1) = 1.32
110 45 120 300
100 65 110 250
100/110 = 0.91 65/45 = 0.69 110/120 = 0.92 250/300 = 0.83 0.84 1/0.84 = 1.19 2/((0.84)2 + 1) = 1.18 2/((1.19)2 + 1) = 0.82
Risk factors = factors entailing risk towards a disease. Protection factors = factors entailing protection towards a disease. Note that data in this table are merely examples used to explain our formulae, and that the possile different “weight” of the various factors has not been considered. Vmn = mean normal value, expressed in conventional units. Vp = value observed in the patient (or person) under study, expressed in conventional units. rFs & pFs = risk factors & protection factors expressed by considering the Vmn as Unit. mF = Mean of the factors considered (each expressed by taking the Vmn as Unit). imF = the inverse value of mF (i.e., 1/mF). PMFI = Protection Multiple Factor Index. RMFI = Risk Multiple Factor Index. LDLc = LDL-Cholesterol. HDLc = HDL-Cholesterol. TG = Triglycerides. FNG = Fibrinogen.
reverse of this ratio. i.e. by dividing the Vmn by the observed value. Therefore, if a ‘risk’ factor is 1.5-fold the Vmn is considered as equal to 1.5, if the risk factor is equal to 2-fold the Vmn, it is considered as equal to 2, if the risk factor is equal to 2.5-fold the Vmn, it is considered as equal to 2.5, and so on, whereas in the case of a ‘protection’ factor, the corresponding values would be 0.67, 0.50, 0.40, and so on. It follows that a central point is represented by the definition of the mean normal value (Vmn) for each of the factors considered. For the most studied factors, the Vmn is well known from the literature, while for others the definition of the mean normal value may raise some difficulty. However, it should be pointed out that each laboratory should have its own normal reference values for the parameters which are measured. Expressing the factors by considering the Vmn as unit entails the advantage of making the various factors comparable to each other regardless of the units used to measure them (thus allowing the calculation of the average changes, i.e., of mF and imF in our formulae). A © 2001 Harcourt Publishers Ltd
further advantage of taking Vmn as unit is that in the normal subjects the values of PMFI and RMFI are always close to 1. As shown in Fig. 1, PMFI gives results which decrease progressively in a slightly curvilinear fashion with the increase in the mean change of considered factors (mF) within a wide range of variations (from mF = 1/3 to mF = 3-fold the Vmn), and the extreme limits are always between a minimum of 0 (in the case of maximal reduction in ‘protection’) and a maximum of 2 (in the rather improbable instance of extreme, super-normal ‘protection’). On the other hand, RMFI changes in a complementary manner in respect to PMFI, following a symmetrical curve. It is noteworthy that the sum of the value of the two indices for a given values of mF is always equal to 2, which allows one index to be derived from the other (Fig. 1, bottom panel). For instance, if we calculate that a patient has a PMFI = 0.5, then we can deduce that such a patient has a RMFI = 1.5. It is possible, therefore, to choose whether to express the status of a patient by Medical Hypotheses (2001) 56(2), 200–206
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measuring his degree of protection (PMFI) or of risk (RMFI). It is noteworthy that our formulae are such that, starting from ‘normal’ values (always close to 1), as the mean value of the considered factors (mF) increases, it affects ever less the values of PMFI, as shown by the curve in Fig. 1 (left panel) which progressively flattens as it approaches the minimum possible value of 0. The reverse is true for RMFI, whose curve progressively flattens as it approaches the maximum possible value of 2. This means that changes of the various factors close to the normal mean have greater ‘weight’ than changes in the extreme range of values, which may be clinically sound. In fact, for instance, a change in cholesterol level from 200 to 300 mg/dL means the shift from the normal condition to a risk condition, whereas a change from 500 to 600 mg/dL does not affect greatly the status of high risk. It should be pointed out that our indices are based on the parameter mF, which expresses the mean change of the contributing factors. This means that each factor contributes, with its change, to the result of our formulae to an extent which depends on the number of considered factors. If the number of factors is indicated as ‘n’, each factor contributes by 1/n. For instance, if the factors considered are three, each one contributes by 33.33%, if they are 4, each one contributes by 25%, and so on. The clinical relevance of PMFI and RMFI depends upon the known meaning of the factors considered in the calculation of both indices. Indeed, it should be pointed out that through clinical, epidemiological, and logistic regression studies a different ‘weight’ may be assigned to the various risk/protection factor considered. In this regards, it should be pointed out that both PMFI and RMFI can be modulated by assigning to the various factors a higher or lower ‘weight’ by multiplying them, before the calculation of mF, by a number higher or lower than 1. It should also be noted that the index value indicates the degree of risk linked to the factors considered. Therefore if, for instance, in a given patient we calculate the risk index considering only three factors much altered, the risk index will approach the maximum value (i.e. he has almost the maximum possible risk linked to the three factors considered). However, if we include a fourth risk factor, whose value is normal, the index will be lower, indicating that the patient has not the maximum possible risk as he would have if all four factors were maximally altered. Variants of the proposed formulae Variants of the proposed formulae can be constructed. In both PMFI and RMFI, the (mF)2 can be substituted Medical Hypotheses (2001) 56(2), 200–206
with the result of the multiplication of the various contributing factors by each other (9). The index so calculated, however, changes too much with the changes in the contributing factors. On the other hand, it is possible to use mF in place of (mF)2, i.e., it is possible to combine the various factors through addition (actually through the calculation of the mean), avoiding to use the square of mF (10). This, however, results in an index which changes too little with the changes in the contributing factors. The behavior of both PMFI and RMFI is intermediate compared to that of these two variant formulae, as both PMFI and RMFI change roughly proportionately with the changes in the protection/risk factors, at least within a certain range of variation of these factors.
Simplified formulae 1. Formulae based on only two factors. When factors are only two, if we indicate them as F1 and F2, the formulae may be simplified as follows: PMFI = 2/[(F1 × F2) +1] and RMFI = 2/[(iF1 × iF2) +1], where iF = 1/F, with only minimal change in results. Indeed, there is no difference between PMFI or RMFI and the ‘simplified indices’ when the percent change is similar for the two factors involved, whereas a difference appears and progressively increases with the increase in the difference in the percent variation between the two factors. The two curves in Fig. 1 (right panel) show the ‘maximum possible difference’ between PMFI and the ‘simplified index’; such difference occurs in the rather unlikely instance when the change (either increase or decrease) affects only one of the two factors, while the other remains unchanged. It is apparent that, even in this extreme instance, the values of the two indices are close to each other. This simplification entails the omission of two arithmetic operations (a sum and a division), which may make the formulae more suitable for the clinical setting. This is what we recently proposed (11,12) for calculating the insulin sensitivity index (ISI), which can be considered as a ‘risk’ condition depending on two factors, insulin and glucose (or insulin and FFA). Our insulin sensitivity index (ISI) has been recently evaluated as one of the three most useful methods available (13). 2. Formulae consisting just of mF or (mF) 2. Simpler indices consisting just of mF or (mF)2 could be used to measure the risk state of a patient (i.e. in place of RMFI). However, they do not possess all the characteristics of the full formulae, specifically they do not possess the characteristics described in (2), (3), (4) and (5) under the paragraph Characteristics of © 2001 Harcourt Publishers Ltd
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the Proposed Formulae. Moreover, the inverse of mF and (mF)2 cannot be used to measure the protection status, because they change too sharply when values of mF are <1 and too smoothly when values of mF are >1. Fig. 2 shows the behavior of mF (risk index) and that of imF (protection index).
Comparison with the ‘atherogenic index’ It is of interest to compare our mF with the ‘atherogenic indices’ used by several workers (1,2,5), based on some risk (TC, LDLC, ApoB) and protective (HDLC, ApoAI) factors. Such atherogenic indices are constructed by multiplying these factors by each other. In this calculation, as far as the protective factors are concerned, the reciprocal of the measured values are used (1/HDLc, 1/ApoAI), whereas the risk factors are used as such (TC, LDLc, ApoB). Commonly, the calculation of the atherogenic index is based on lipid parameters such as TC and HDLc (TC/HDLc ratio), or on apoproteins B and AI (ApoB/ApoAI ratio) or on both cholesterol and apoproteins ([LDLc × ApoB]/[HDLc × ApoAI] ratio). Focusing on this latter formula, we can write it as follows (in order to make easier the comparison with our mF): LDLc × ApoB × (1/HDLc) × (1/ApoAI). It is apparent that in this formula the various factors are
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combined through multiplication (3 multiplications are performed in such atherogenic index) which entails a marked, exponential increase of the index with the increase in the contributing factors, whereas in our mF the various factors are combined through the calculation of their mean value, whose changes are roughly proportional to the increase in the contributing factors. Fig. 3 compares the behavior of mF and PMFI, shown in the left panel, with that of the atherogenic index ([LDLc × ApoB]/[HDLc × ApoAI] ratio), outlined in the right panel, for mean changes in the considered factors ranging from 0.5 to 2-fold the mean normal value. It is apparent that there is a great difference in the behavior of both the atherogenic index (which is a ‘risk’ index) and the anti-atherogenic index (which is a ‘protection’ index) depending on whether the mean change in the considered factors is >1 or <1. In the first instance, both indices change (the atherogenic index increases and the anti-atherogenic index decreases) in a very sharply, unrealistic manner (there are no known factors that could entail such a sharp increase in risk or decrease in protection), whereas when the mean change in the considered factors is <1 the variation of both indices becomes very small. This contrasts with the more uniform behavior shown by mF and RMFI, closely reflecting the mean changes in the considered factors.
CONCLUSION
Fig. 2 Behavior of mF and imF. It is apparent that, whereas mF is a good ‘risk’ index, directly expressing the mean change of the considered factors, imF is not suitable as ‘protection’ index, because it changes too sharply when the values of mF are <1 and too smoothly when the values of mF are >1. © 2001 Harcourt Publishers Ltd
Our formulae can be regarded as a tool to evaluate the protection/risk caused by any number of factors through a procedure which yields an index whose values are always close to 1 in normal subjects, and which changes with the variations in mF (mean variation of the considered factors) always according to a fixed curve, comprised between 0 and 2, and which progressively flattens as the mF value becomes high or extreme. It should be underlined that our index is a mathematical tool allowing us to combine multiple risk factors into one index value through addition. Therefore, the factors considered must be quantifiable (i.e. they cannot be ‘yes/no’ factors such as sex or race) and their relationship with the risk towards a disease should be roughly linear. This is not a limitation of our index, as any mathematical method entailing summing or combination of multiple factors is meaningful only if the factors considered are of similar significance. On the basis of the above considerations, both PMFI and RMFI appear suitable to assess the protection/risk status in the clinical setting or in epidemiological research. Medical Hypotheses (2001) 56(2), 200–206
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Fig. 3 Behavior of mF and RMFI (left panel) and of the ‘atherogenic’ and ‘anti-atherogenic’ indices (right panel) with mean changes in the considered factors (increase in risk factors – LDLc and ApoB – and decrease in protection factors – HDLc and ApoAI) ranging from 0.5 to 2fold the mean normal value. It is apparent that there is a great difference in the behavior of both the atherogenic and the anti-atherogenic indices depending on whether the mean change in the considered factors is >1 or <1. In the first instance, both indices change (the atherogenic index increases and the anti-atherogenic index decreases) sharply, in an exponential manner, whereas when the mean change in the considered factors is <1 the variation of both indices becomes very small. This contrasts with the more uniform behavior shown by mF and RMFI (left panel), closely reflecting the mean changes in the considered factors. The atherogenic index is calculated as LDLc x ApoB x C1/HDLc) x (I/ApoAI). The anti-atherogenic index is calculated as (I/LDLc) (I/ApoB) HDLcX1/ApoAI.
REFERENCES 1. Cremer P., Nagel D., Mann H. et al. Ten-year follow-up results from the Goettingen Risk, Incidence and Prevalence Study (GRIPS). I. Risk factors for myocardial infarction in a cohort of 5790 men. Atherosclerosis 1997; 129: 221–230. 2. Eggesbo J. B., Hagve T. A., Borsum K., Hostmark A. T., Hjermann I., Kierulf P. Lipid composition of mononuclear cell membranes and serum from persons with high or low levels of serum HDL cholesterol. Scand J Clin Lab Invest 1996; 56: 199–210. 3. Eriksson M., Egberg N., Wamala S., Orth-Gomer K., Mittlemann M. A., Schenck-Gustafsson K. Relationship between plasma fibrinogen and coronary heart disease in women. Arterioscler Thromb Vasc Biol 1999; 19: 67–72. 4. Carter A. M., Ossei-Gerning N., Wilson I. J., Grant P. J. Association of the platelet PI(A) polymorphism of glycoprotein IIb/IIIa and the fibrinogen Bbeta 448 polymorphism with myocardial infarction and extent of coronary artery disease. Circulation 1997; 96: 1424–1431. 5. Pitsavos C., Skoumas J., Dernellis J. et al. Influence of biological factors on lipid and fibrinogen measurements in young men. An epidemiological study in 2009 recruits. Eur Heart J 1998; 19: 1642–1647. 6. National Cholesterol Education Program. Second Report of the Expert Panel on Detection, Evaluation, and Treatment of High Blood Cholesterol in Adults (Adult Treatment Panel II). Circulation 1994; 89: 1333–1445.
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7. Frost P. H., Davis B. R., Burlando A. J. et al. Serum lipids and incidence of coronary heart disease. Findings from the Systolic Hypertension in the Elderly Program (SHEP). Circulation 1996; 94: 2381–2388. 8. Goldberg R. B., Mellies M. J., Sacks F. M. et al. Cardiovascular events and their reduction with pravastatin in diabetic and glucose-intolerant myocardial infarction survivors with average cholesterol levels: subgroup analyses in the cholesterol and recurrent events (CARE) trial. The Care Investigators. Circulation 1998; 98: 2513–2519. 9. Belfiore F., Volpicelli G., Iannello S., Campione R. An index to measure insulin resistance from basal and OGTT-induced insulin, glucose and FFA levels. Diabetologia 1995; 38 (Suppl 1): A126 (abstract). 10. Belfiore F., Volpicelli G., Iannello S., Campione R. Measurement of insulin resistance from basal and OGTT-induced insulin, glucose and free fatty acid levels. Eur J Intern Med 1995; 6 (Suppl 1): A144 (abstract). 11. Belfiore F., Iannello S., Volpicelli G. Insulin sensitivity indices calculated from basal and OGTT-induced insulin, glucose, and FFA levels. Mol Genet Metab 1998; 63: 134–141. 12. Belfiore F., Iannello S. Insulin resistance in obesity: metabolic mechanisms and measurement methods. Mol Genet Metab 1998; 65: 121–128. 13. Matsuda M., DeFronzo R. A. Insulin sensitivity indices obtained from oral glucose tolerance testing: comparison with the euglycemic insulin clamp. Diabetes Care 1999; 22: 1462–1470.
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Multiple factor indices of protection or risk towards disease F. Belfiore, S. Iannello Institute of Medicina Interna e Specialità Internistiche, Chair of Internal Medicine, University of Catania Medical School, Ospedale Garibaldi, Catania, Italy
Summary In order to combine several factors entailing protection or risk towards disease and to calculate a Protection Multiple Factor Index (PMFI) or, conversely, a Risk Multiple Factor Index (RMFI), we propose the following formulae: (1) PMFI = 2/[(mF)2+1] and (2) RMFI = 2/[(imF)2+1], where mF is the mean value of the factors considered and imF is the inverse (or reciprocal) of mF. In calculating mF, the value of each ‘risk factor’ observed in the patient under study (Vp) is expressed by taking the mean normal value (Vmn) as the unit, i.e. by calculating the ratio Vp/Vmn, whereas each ‘protection factor’ is expressed as the reciprocal of this ratio, i.e. as Vmn/Vp. The ‘weight’ of the various factors can be changed through multiplication by a number >1 or <1. Values of both PMFI and RMFI are always close to 1 in normal subjects, with extreme variations among patients between 0 and 2. The sum of the values of PMFI and RMFI is always equal to 2, so that one index can be deduced from the other. When factors are only two (F1 and F2), the formulae may be simplified as follows: PMFI = 2/[F1 × F2)+1] and RMFI = 2/[(iF1 × iF2)+1], where iF = 1/F, with only minimal changes in results. © 2001 Harcourt Publishers Ltd
INTRODUCTION From pathophysiological studies or epidemiological data, it is well established that deviation from the normal value of several metabolic parameters may favor or oppose the development of some metabolic diseases. On this basis, several ‘risk factors’ and ‘protection factors’ have been identified for various diseases. The best example may perhaps be the CHD1, for which some metabolic parameters (LDLc, ApoBI, TG, FNG, etc.) are regarded as risk factors whereas others (HDLc, ApoA, etc.) are considered as protection factors (1–2). It is clinically relevant to be able to combine the various risk and protection factors into one ‘index’ which reflects the global status of a subject, as it results from the cumulative effects of the various risk/protection factors present. Such an ‘index’ can be directed to measure the ‘protection’ towards a disease, Received 19 November 1999 Accepted 18 April 2000 Published online 3 January 2001 Correspondence to: Professor F. Belfiore, Institute of Medicina Interna e Specialità Internistiche, Chair of Internal Medicine, University of Catania, Ospedale Garibaldi, 95123 Catania, Italy. Fax: +39 095 1310899; E-mail: [email protected]
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i.e., it can be a protection index, in which instance it will increase with the elevation of the protection factors and the lowering of the risk factors, or it may be directed to measure the ‘risk’ towards a disease, i.e., it can be a risk index, in which instance it will increase with the elevation of risk factors and with the lowering of protection factors. In this paper we propose two formulae which allow us to calculate the Protection Multiple Factor Index (PMFI) and the Risk Multiple Factor Index (RMFI), as defined above, which enable us to define the protection/risk status of a given patient. THE PROPOSED FORMULAE The various factors that may confer protection or risk towards a disease can be combined into one ‘index’ through various formulae, in which the combination of the various factors can be obtained through addition or multiplication. Among possible formulae, we propose the following two for the calculation of PMFI and RMFI, 1 Abbreviations used in the text: CHD = coronary heart disease; TC = total cholesterol; LDLc = low density lipoprotein cholesterol; HDLc = high density lipoprotein cholesterol; TG = triglycerides; ApoA = apoproteinA; ApoBI = aprotein BI; FNG = fibrinogen.
Multiple factor indices
which possess some unique characteristic, outlined below: (1) PMFI = 2/[mF)2+1] and (2) RMFI = 2/[iF)2+1], where mF is the mean value of the factors considered and imF is the inverse (or reciprocal) of mF. In calculating mF, the value of each ‘risk factor’ observed in the patient under study (Vp) is expressed by taking the mean normal value (Vmn) as the unit, i.e. by calculating the ratio Vp/Vmn, whereas each ‘protection factor’ is expressed as the reciprocal of this ratio, i.e. as Vmn/Vp. By applying our formulae to hypothetical patients with mF ranging from subnormal to 3-fold increased values we obtained the results shown in Fig. 1 (left panel). It is apparent that the results obtained with our formula decrease (PMFI) or increase (RMFI) in a slightly curvilinear fashion within wide ranges of physiological variations of the involved factors.
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The construction of these formulae can be distinguished into several steps. Step 1 (making different factors comparable to each other) In order to make factors which differ as far as their nature and unit of measure are concerned, all the factors considered in the calculation of our index are expressed by taking the mean normal value (Vmn) as the unit. This also entails that in the normal subjects all factors are close to 1. Step 2 (combining the different factors into one value) This can be accomplished through multiplication of the various factors by each other, or through addition of the various factors to each other. The multiplication method has the disadvantage that, with the increase of the various factors, the resulting value increases in an
Fig. 1 LEFT PANEL. Behavior of PMFI (Protection Multiple Factor Index) and of RMFI (Risk Multiple Factor Index) with changes in the mean value of the considered factor (mF) from 0.33 (1/3 of Vmn) up to 3 (3-fold the Vmn). BOTTOM PANEL. This table shows that, with the increase in the mF value, PMFI and RMFI change in a complementary manner (PMFI decreasing and RMFI increasing) so that the sum of the PMFI and RMFI is always equal to 2. RIGHT PANEL. Behavior of PMFI and of a ‘simplified index’ (usable when the factors considered are only two) in the range of mF from 0.67 (1/1.5 of Vmn) to 2 (2-fold the Vmn). The two curves show the ‘maximum possible difference’ between PMFI and the ‘simplified index’, which occurs in the rather unlikely instance when the change (either increase or decrease) affects only one of the two factors, while the other remains unchanged. Even in this extreme instance, the values of the two Indices are close to each other. © 2001 Harcourt Publishers Ltd
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exponential manner (i.e. the changes are progressively amplified ), reaching extremely high values. Concerning the addition method, it actually consists of the addition of the various factors followed by the division of the sum by the number of the factors, thus obtaining the mean value of the various factors (which in normal subjects will be always close to 1). This means to calculate the component of our formulae which we called mF. In the proposed formulae the mF is placed in the denominator of a fraction, which entails that the mF loses ‘weight’ as far as the result is concerned. To increase the weight of mF, in our formulae we use the squared value of mF in the case of PMFI, and of imF in the case of RMFI, i.e. (mF)2 or (imF)2. Thus, in our formulae the combination of the various factors is obtained both by addition (calculation of mF or imF) and by multiplication (calculation of the squared value of mF or imF). Step 3 (making the index values to change between 0 and 2) By adding 1 to (mF)2 and putting the result in the denominator of a fraction in which 2 is the numerator we get the complete formula of PMFI. By using 1/(mF)2 instead of (mF)2 the RMFI formula is obtained. The rationale for these operations is the following. The value of (mF)2 in the normal condition is 1. Therefore, by adding 1 to the (mF)2, the value of 2 is obtained. By putting this value of 2 in the denominator of a fraction in which 2 is the numerator (as is done in our formula), the result of 1 is obtained in the normal subjects. On the other hand, as the value of (mF)2 changes towards the extremely high or low values, the result of the formula approaches 0 or 2 for PMFI and 2 or 0 for RMFI. However, the values of (mF)2, which reflect the mean changes of the considered factors, progressively loses weight as the values move away from the normal value of 1 towards high/extreme values. This is shown by the curvilinear behavior of both PMFI and RMFI with progressive changes in (mF)2, as outlined in Fig. 1 (left panel).
similar but specular curve (Fig. 1, left panel), so that from a given value of PMFI (or RMFI) it is possible to know the mean change of the involved factors. 4. Small changes in the mean value of the various factors (mF), i.e. changes in mF close to 1, have greater ‘weight’ than extreme values, as shown by the curves in Fig. 1 (left panel), which progressively flatten as the value of mF (or imF) increases. 5. PMFI and RMFI are correlated to each other in such a way that their sum is always equal to 2, so that one index can be deduced from the other (Fig. 1, bottom panel). This makes it possible to choose whether to express the status of a given patient by a protection or a risk index.
EXAMPLES OF APPLICATION An example of utilization of the proposed formulae may be the calculation of the CHD risk index based on the well known risk and protection factors. In Table 1 we present data referring to four parameters including three risk factors (LDLC, TG and FNG) and one protection factor (HDLC ) in three hypothetical patients. The mean normal values given in Table 1 for the parameters considered are rounded figures calculated from recently published works (3–8). In one of the hypothetical patients in Table 1, all four factors considered are normal (top panel of the table), and therefore both PMFI and RMFI are equal to 1 (Table 1, top panel). In the second patient (Table 1, middle panel), the three risk factors are increased whereas the protecting factor is decreased, so that PMFI is reduced (= 0.68) and RMFI elevated (= 1.32). In the third patient (Table 1, bottom panel), the reverse condition occurs, with low levels of the risk factors and enhanced value of the protecting factor, so that PMFI is high (= 1.18) and RMFI lowered (= 0.82).
DISCUSSION The main proposed formulae
CHARACTERISTICS OF THE PROPOSED FORMULAE The proposed formulae possess the following unique characteristics. 1. In normal subjects, the value of both PMFI and RMFI is always close to 1. 2. Among patients, the extreme variations lie always between 0 and 2. 3. With the increase in the mF value (i.e., with increase in risk factors and decrease in protection factors), PMFI decreases according to a slightly curvilinear fashion whereas RMFI increases according to a Medical Hypotheses (2001) 56(2), 200–206
Despite the interest in assessing the protection/risk factors both in the clinical setting and in epidemiological studies, a simple method suitable for reaching this goal is not yet available. With our formulae, we attempt to introduce a simple method to measure the protection/risk status as it results from the effects of any number of protection/risk factors. To better understand the proposed formulae, the following aspects should be further discussed. A point to be underlined is that, in our formulae, the risk factors considered are expressed by making the mean normal value (Vmn) equal to 1, i.e. by dividing the value observed in the patient or person under study (Vp) by the Vmn, whereas the protective factors are expressed as the © 2001 Harcourt Publishers Ltd
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Table 1 Calculation of PMFI and RMFI and their components (mF and imF) based on 3 risk factors (LDLc, TG and FNG) and 1 protection factor (HDLc) Risk factors
Protection factors
‘Normal’ subject LDLc TG FNG mF imF PMFI RMFI Subject at risk LDLc TG FNG mF imF PMFI RMFI
HDLc
Conventional units Vmn Vp
110 45 120 300
Vmn as unit rFs & pFs Calculations Results
110 45 120 300
PMFI or RMFI Calculations Results
110/110 = 1.00 45/45 = 1.00 120/120 = 1.00 300/300 = 1.00 1.00 1/1 = 1.00 2/((1.00)2+1) = 1.00 2/((1.00)2+1) = 1.00
HDLc
Subject ‘protected’ LDLc HDLc TG FNG mF imF PMFI RMFI
110 45 120 300
160 35 180 400
160/110 = 1.45 35/45 = 1.29 180/120 = 1.50 400/300 = 1.33 1.39 1/1.39 = 0.72 2/((1.39)2+1) = 0.68 2/((0.72)2+1) = 1.32
110 45 120 300
100 65 110 250
100/110 = 0.91 65/45 = 0.69 110/120 = 0.92 250/300 = 0.83 0.84 1/0.84 = 1.19 2/((0.84)2 + 1) = 1.18 2/((1.19)2 + 1) = 0.82
Risk factors = factors entailing risk towards a disease. Protection factors = factors entailing protection towards a disease. Note that data in this table are merely examples used to explain our formulae, and that the possile different “weight” of the various factors has not been considered. Vmn = mean normal value, expressed in conventional units. Vp = value observed in the patient (or person) under study, expressed in conventional units. rFs & pFs = risk factors & protection factors expressed by considering the Vmn as Unit. mF = Mean of the factors considered (each expressed by taking the Vmn as Unit). imF = the inverse value of mF (i.e., 1/mF). PMFI = Protection Multiple Factor Index. RMFI = Risk Multiple Factor Index. LDLc = LDL-Cholesterol. HDLc = HDL-Cholesterol. TG = Triglycerides. FNG = Fibrinogen.
reverse of this ratio. i.e. by dividing the Vmn by the observed value. Therefore, if a ‘risk’ factor is 1.5-fold the Vmn is considered as equal to 1.5, if the risk factor is equal to 2-fold the Vmn, it is considered as equal to 2, if the risk factor is equal to 2.5-fold the Vmn, it is considered as equal to 2.5, and so on, whereas in the case of a ‘protection’ factor, the corresponding values would be 0.67, 0.50, 0.40, and so on. It follows that a central point is represented by the definition of the mean normal value (Vmn) for each of the factors considered. For the most studied factors, the Vmn is well known from the literature, while for others the definition of the mean normal value may raise some difficulty. However, it should be pointed out that each laboratory should have its own normal reference values for the parameters which are measured. Expressing the factors by considering the Vmn as unit entails the advantage of making the various factors comparable to each other regardless of the units used to measure them (thus allowing the calculation of the average changes, i.e., of mF and imF in our formulae). A © 2001 Harcourt Publishers Ltd
further advantage of taking Vmn as unit is that in the normal subjects the values of PMFI and RMFI are always close to 1. As shown in Fig. 1, PMFI gives results which decrease progressively in a slightly curvilinear fashion with the increase in the mean change of considered factors (mF) within a wide range of variations (from mF = 1/3 to mF = 3-fold the Vmn), and the extreme limits are always between a minimum of 0 (in the case of maximal reduction in ‘protection’) and a maximum of 2 (in the rather improbable instance of extreme, super-normal ‘protection’). On the other hand, RMFI changes in a complementary manner in respect to PMFI, following a symmetrical curve. It is noteworthy that the sum of the value of the two indices for a given values of mF is always equal to 2, which allows one index to be derived from the other (Fig. 1, bottom panel). For instance, if we calculate that a patient has a PMFI = 0.5, then we can deduce that such a patient has a RMFI = 1.5. It is possible, therefore, to choose whether to express the status of a patient by Medical Hypotheses (2001) 56(2), 200–206
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measuring his degree of protection (PMFI) or of risk (RMFI). It is noteworthy that our formulae are such that, starting from ‘normal’ values (always close to 1), as the mean value of the considered factors (mF) increases, it affects ever less the values of PMFI, as shown by the curve in Fig. 1 (left panel) which progressively flattens as it approaches the minimum possible value of 0. The reverse is true for RMFI, whose curve progressively flattens as it approaches the maximum possible value of 2. This means that changes of the various factors close to the normal mean have greater ‘weight’ than changes in the extreme range of values, which may be clinically sound. In fact, for instance, a change in cholesterol level from 200 to 300 mg/dL means the shift from the normal condition to a risk condition, whereas a change from 500 to 600 mg/dL does not affect greatly the status of high risk. It should be pointed out that our indices are based on the parameter mF, which expresses the mean change of the contributing factors. This means that each factor contributes, with its change, to the result of our formulae to an extent which depends on the number of considered factors. If the number of factors is indicated as ‘n’, each factor contributes by 1/n. For instance, if the factors considered are three, each one contributes by 33.33%, if they are 4, each one contributes by 25%, and so on. The clinical relevance of PMFI and RMFI depends upon the known meaning of the factors considered in the calculation of both indices. Indeed, it should be pointed out that through clinical, epidemiological, and logistic regression studies a different ‘weight’ may be assigned to the various risk/protection factor considered. In this regards, it should be pointed out that both PMFI and RMFI can be modulated by assigning to the various factors a higher or lower ‘weight’ by multiplying them, before the calculation of mF, by a number higher or lower than 1. It should also be noted that the index value indicates the degree of risk linked to the factors considered. Therefore if, for instance, in a given patient we calculate the risk index considering only three factors much altered, the risk index will approach the maximum value (i.e. he has almost the maximum possible risk linked to the three factors considered). However, if we include a fourth risk factor, whose value is normal, the index will be lower, indicating that the patient has not the maximum possible risk as he would have if all four factors were maximally altered. Variants of the proposed formulae Variants of the proposed formulae can be constructed. In both PMFI and RMFI, the (mF)2 can be substituted Medical Hypotheses (2001) 56(2), 200–206
with the result of the multiplication of the various contributing factors by each other (9). The index so calculated, however, changes too much with the changes in the contributing factors. On the other hand, it is possible to use mF in place of (mF)2, i.e., it is possible to combine the various factors through addition (actually through the calculation of the mean), avoiding to use the square of mF (10). This, however, results in an index which changes too little with the changes in the contributing factors. The behavior of both PMFI and RMFI is intermediate compared to that of these two variant formulae, as both PMFI and RMFI change roughly proportionately with the changes in the protection/risk factors, at least within a certain range of variation of these factors.
Simplified formulae 1. Formulae based on only two factors. When factors are only two, if we indicate them as F1 and F2, the formulae may be simplified as follows: PMFI = 2/[(F1 × F2) +1] and RMFI = 2/[(iF1 × iF2) +1], where iF = 1/F, with only minimal change in results. Indeed, there is no difference between PMFI or RMFI and the ‘simplified indices’ when the percent change is similar for the two factors involved, whereas a difference appears and progressively increases with the increase in the difference in the percent variation between the two factors. The two curves in Fig. 1 (right panel) show the ‘maximum possible difference’ between PMFI and the ‘simplified index’; such difference occurs in the rather unlikely instance when the change (either increase or decrease) affects only one of the two factors, while the other remains unchanged. It is apparent that, even in this extreme instance, the values of the two indices are close to each other. This simplification entails the omission of two arithmetic operations (a sum and a division), which may make the formulae more suitable for the clinical setting. This is what we recently proposed (11,12) for calculating the insulin sensitivity index (ISI), which can be considered as a ‘risk’ condition depending on two factors, insulin and glucose (or insulin and FFA). Our insulin sensitivity index (ISI) has been recently evaluated as one of the three most useful methods available (13). 2. Formulae consisting just of mF or (mF) 2. Simpler indices consisting just of mF or (mF)2 could be used to measure the risk state of a patient (i.e. in place of RMFI). However, they do not possess all the characteristics of the full formulae, specifically they do not possess the characteristics described in (2), (3), (4) and (5) under the paragraph Characteristics of © 2001 Harcourt Publishers Ltd
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the Proposed Formulae. Moreover, the inverse of mF and (mF)2 cannot be used to measure the protection status, because they change too sharply when values of mF are <1 and too smoothly when values of mF are >1. Fig. 2 shows the behavior of mF (risk index) and that of imF (protection index).
Comparison with the ‘atherogenic index’ It is of interest to compare our mF with the ‘atherogenic indices’ used by several workers (1,2,5), based on some risk (TC, LDLC, ApoB) and protective (HDLC, ApoAI) factors. Such atherogenic indices are constructed by multiplying these factors by each other. In this calculation, as far as the protective factors are concerned, the reciprocal of the measured values are used (1/HDLc, 1/ApoAI), whereas the risk factors are used as such (TC, LDLc, ApoB). Commonly, the calculation of the atherogenic index is based on lipid parameters such as TC and HDLc (TC/HDLc ratio), or on apoproteins B and AI (ApoB/ApoAI ratio) or on both cholesterol and apoproteins ([LDLc × ApoB]/[HDLc × ApoAI] ratio). Focusing on this latter formula, we can write it as follows (in order to make easier the comparison with our mF): LDLc × ApoB × (1/HDLc) × (1/ApoAI). It is apparent that in this formula the various factors are
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combined through multiplication (3 multiplications are performed in such atherogenic index) which entails a marked, exponential increase of the index with the increase in the contributing factors, whereas in our mF the various factors are combined through the calculation of their mean value, whose changes are roughly proportional to the increase in the contributing factors. Fig. 3 compares the behavior of mF and PMFI, shown in the left panel, with that of the atherogenic index ([LDLc × ApoB]/[HDLc × ApoAI] ratio), outlined in the right panel, for mean changes in the considered factors ranging from 0.5 to 2-fold the mean normal value. It is apparent that there is a great difference in the behavior of both the atherogenic index (which is a ‘risk’ index) and the anti-atherogenic index (which is a ‘protection’ index) depending on whether the mean change in the considered factors is >1 or <1. In the first instance, both indices change (the atherogenic index increases and the anti-atherogenic index decreases) in a very sharply, unrealistic manner (there are no known factors that could entail such a sharp increase in risk or decrease in protection), whereas when the mean change in the considered factors is <1 the variation of both indices becomes very small. This contrasts with the more uniform behavior shown by mF and RMFI, closely reflecting the mean changes in the considered factors.
CONCLUSION
Fig. 2 Behavior of mF and imF. It is apparent that, whereas mF is a good ‘risk’ index, directly expressing the mean change of the considered factors, imF is not suitable as ‘protection’ index, because it changes too sharply when the values of mF are <1 and too smoothly when the values of mF are >1. © 2001 Harcourt Publishers Ltd
Our formulae can be regarded as a tool to evaluate the protection/risk caused by any number of factors through a procedure which yields an index whose values are always close to 1 in normal subjects, and which changes with the variations in mF (mean variation of the considered factors) always according to a fixed curve, comprised between 0 and 2, and which progressively flattens as the mF value becomes high or extreme. It should be underlined that our index is a mathematical tool allowing us to combine multiple risk factors into one index value through addition. Therefore, the factors considered must be quantifiable (i.e. they cannot be ‘yes/no’ factors such as sex or race) and their relationship with the risk towards a disease should be roughly linear. This is not a limitation of our index, as any mathematical method entailing summing or combination of multiple factors is meaningful only if the factors considered are of similar significance. On the basis of the above considerations, both PMFI and RMFI appear suitable to assess the protection/risk status in the clinical setting or in epidemiological research. Medical Hypotheses (2001) 56(2), 200–206
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Fig. 3 Behavior of mF and RMFI (left panel) and of the ‘atherogenic’ and ‘anti-atherogenic’ indices (right panel) with mean changes in the considered factors (increase in risk factors – LDLc and ApoB – and decrease in protection factors – HDLc and ApoAI) ranging from 0.5 to 2fold the mean normal value. It is apparent that there is a great difference in the behavior of both the atherogenic and the anti-atherogenic indices depending on whether the mean change in the considered factors is >1 or <1. In the first instance, both indices change (the atherogenic index increases and the anti-atherogenic index decreases) sharply, in an exponential manner, whereas when the mean change in the considered factors is <1 the variation of both indices becomes very small. This contrasts with the more uniform behavior shown by mF and RMFI (left panel), closely reflecting the mean changes in the considered factors. The atherogenic index is calculated as LDLc x ApoB x C1/HDLc) x (I/ApoAI). The anti-atherogenic index is calculated as (I/LDLc) (I/ApoB) HDLcX1/ApoAI.
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