Gólczewski et al. [1] argue that the FEV1/FVC ratio is age dependent because the FEV1 and FVC are mathematically linked as follows: FEV1 = A¿FVC + C, `without respect to the other factors, including age and height¿. As FVC is age dependent, the authors expect the FEV1/FVC ratio to be also age dependent. Although the FEV1 and FVC are undeniably closely related in healthy subjects, their reasoning is a simplification of the biological determinants of the relationship between FEV1, FVC and the FEV1/FVC ratio. The FVC is determined by a host of factors, including body dimensions, mechanical properties of the thorax, and muscular force, and is therefore age and height dependent. The FEV1 is a portion of the vital capacity that can be inhaled in one second. Whereas it is undoubtedly related to the size of the vital capacity, important independent determinants are the mechanical properties of intrathoracic airways and lung elastic recoil [2]. It has been shown in numerous publications that FEV1 varies with age and height, and that there are biological reasons for this, but that the coefficients are not the same as those for FVC [3]. This implies that the FEV1/FVC ratio is a function of both height and age. This can be illustrated as follows. Both FEV1 and FVC have often been shown to be a power function of height and age. For example, Brändli et al. [4] provide the following prediction equations for adult males:
ln(FEV1) = 1.9095 * ln(Height) - 0.0037 * Age - 0.000033 * Age² - 8.240.
ln(FVC) = 2.1685 * ln(Height) + 0.0030 * Age - 0.000075 * Age² - 9.540.
where ln is natural logarithm, Height is in cm, and Age in years. If all coefficients could be estimated without errors, it would follow that ln(FEV1/FVC), which equals ln(FEV1) ¿ ln(FVC), would equal:
ln(FEV1/FVC) = -0.259*ln(Height) - 0.0067 * Age + 0.00042* Age² + 1.3.
This comes fairly close to the estimated relationship quoted by Brändli et al.:
ln(FEV1/FVC) = -0.3144 * ln(Height) - 0.0033 * Age + 1.526.
This shows that the FEV1/FVC ratio is non-linearly related to both age and height. The latter was recently corroborated in a study by the Global Lung Function Initiative of >74,000 healthy non-smokers [5]. The same group previously showed that in children and adolescents the FEV1/FVC ratio declines far from linearly: a steep fall until the start of the adolescent growth spurt, followed by a rise before a non-linear fall in adults [6].
1. Gólczewski T, Lubi¿ski W, Chcia¿owski A. A mathematical reason for FEV1/FVC dependence on age. Respiratory Research 2012, 13: 57.
2. Babb TG, Rodarte JR. Mechanism of reduced maximal expiratory flow with aging. J Appl Physiol 2000; 89 :505-511.
3. http://www.lungfunction.org/publishedreferencevalues.html.
4. Brändli O, Schindler Ch, Leuenberger PH, Baur X, Degens P, Künzli N, Keller R, Perruchoud AP. Letters to the editor. Re-estimated equations for 5th percentiles of lung function variables. Thorax 2000; 55: 172.
5. Quanjer PH, Stanojevic S, Cole TJ et al. Multi-ethnic reference values for spirometry for the 3-95 year age range: the Global Lung Function 2012 equations. Eur Respir J published 27 June 2012 as doi: 10.1183/09031936.00080312.
6. Quanjer PH, Stanojevic S, Stocks J, et al. Changes in the FEV1/FVC ratio during childhood and adolescence: an intercontinental study. Eur Respir J 2010; 36: 1391¿1399.

Competing interests

No competing interests.

Mathematical versus biological reasons for FEV1/FVC dependence on age - a reply to Philip Quanjer

Tomasz Golczewski, Institute of Biocybernetics and Biomed.Eng. PAS

15 August 2012

Philip Quanjer is right that the ratio FEV1/FVC - like other lung function variables - is nonlinearly dependent on age. It has to be noted that the same 'nonlinearity' can be described mathematically with different formulas. Recently, the exponential formula with the quadratic form in the exponent is 'trendy' but the piecewise linear forms, as those proposed by Quanjer et al in the past [1] and recently by Lubinski and Golczewski [2], is sufficiently nonlinear to describe age-dependence with the same precision as this trendy formula or even better - see Fig.2 in the commented article (particularly if the change point is determined with regression methods together with the other coefficients separately for each lung function variable like in [2]). For this reason, Quanjer's piecewise linear prediction equations seem to be still one of the most appropriate, in average, among the most known equations [3]. As it has been shown mathematically, only MEF25 changes with age cannot be described precisely with such a piecewise linear form [2].

However, the above, i.e. the well-established, nonlinear dependence of lung function variables on age caused by various biological factors, has no relation to the main findings of the commented article, i.e. to: (1) in healthy subjects, there is a strong relationship between FEV1 and FVC being independent of any other factors, including age; (2) the ratio of FEV1 to FVC is not any good description of this relationship because it introduces dependence on age not present in the original relationship.
The commented article shows the mathematical reason for age dependence of FEV1/FVC. Explanation of biological or physical reasons would be equivalent to explanation of the intercept C different from zero; but it has not been the aim of this study.
Note, however, that comparison of old ECSC equations with more recent equations and original data in Fig.2 could suggest that FEV1/FVC dependence on age may be recently more significant because of present guidelines related to the forced expiratory time, which cause that the 'present' FVC is more comparable to VC, and thus it is smaller than the 'original' FVC.

It is not clear why height - depending on authors - does not influence or influences insignificantly the FEV1/FVC ratio; however it is an experimental fact, e.g. in our previous study [2] FEV1/FVC dependence on height appeared statistically insignificant in females and very small in males. Certainly, Philip Quanjer is right that height influences the other lung function variables, which is obvious. Indeed, the most critical task of both the respiratory and cardiovascular systems is oxygen delivery to tissues to keep metabolism at a required level. The total rate of metabolism of homeothermic organisms depends on the body surface for purely physical reasons. The body surface depends on the squared value of the linear size of an organism (e.g. on height in the case of the human being). Because of the above, it is clear that respiratory properties, including lung function variables, depend on height to the power of a value approximately equal to 2.
However, the use of a power of the height in prediction equations has two imperfections:
- height depends on age because of both ageing process and differences in generations, and thus age and height are not mathematically independent variables; this makes it impossible to interpret equations' coefficients and may cause difficulties in regression for some forms of equations (e.g. this 'trendy' exponential one; therefore logarithms are used);
- the length of legs influences significantly the standing height while metabolism is mainly related to sizes of the torso.
For these reasons, we have proposed previously to orthogonalize age and height variables [2,3], i.e. to use the deviation of patient's height from the mean height of his/her age group instead of the total height.

[1] Quanjer PH, Tammeling GJ, Cotes JE, et al: Lung volumes and forced ventilatory flows. Report Working Party Standardization of Lung Function Tests, European Community for Steel and Coal. Official Statement of the European Respiratory Society. Eur. Respir. J. Suppl. 1993; 16:5-40
[2] Lubinski W, Golczewski T.: Physiologically interpretable prediction equations for spirometric indices. J Appl Physiol 2010;108:1440-1446
[3] Golczewski T. Spirometry: a comparison of prediction equations proposed by Lubi¿ski for the Polish population with those proposed by the ECSC/ERS and by Falaschetti et al. Pneumonologia i Alergologia Polska 2012;80:29-40

## Age dependence of the FEV1/FVC ratio

Philip Quanjer, none

1 August 2012

Gólczewski et al. [1] argue that the FEV1/FVC ratio is age dependent because the FEV1 and FVC are mathematically linked as follows: FEV1 = A¿FVC + C, `without respect to the other factors, including age and height¿. As FVC is age dependent, the authors expect the FEV1/FVC ratio to be also age dependent. Although the FEV1 and FVC are undeniably closely related in healthy subjects, their reasoning is a simplification of the biological determinants of the relationship between FEV1, FVC and the FEV1/FVC ratio. The FVC is determined by a host of factors, including body dimensions, mechanical properties of the thorax, and muscular force, and is therefore age and height dependent. The FEV1 is a portion of the vital capacity that can be inhaled in one second. Whereas it is undoubtedly related to the size of the vital capacity, important independent determinants are the mechanical properties of intrathoracic airways and lung elastic recoil [2]. It has been shown in numerous publications that FEV1 varies with age and height, and that there are biological reasons for this, but that the coefficients are not the same as those for FVC [3]. This implies that the FEV1/FVC ratio is a function of both height and age. This can be illustrated as follows. Both FEV1 and FVC have often been shown to be a power function of height and age. For example, Brändli et al. [4] provide the following prediction equations for adult males:

ln(FEV1) = 1.9095 * ln(Height) - 0.0037 * Age - 0.000033 * Age² - 8.240.

ln(FVC) = 2.1685 * ln(Height) + 0.0030 * Age - 0.000075 * Age² - 9.540.

where ln is natural logarithm, Height is in cm, and Age in years. If all coefficients could be estimated without errors, it would follow that ln(FEV1/FVC), which equals ln(FEV1) ¿ ln(FVC), would equal:

ln(FEV1/FVC) = -0.259*ln(Height) - 0.0067 * Age + 0.00042* Age² + 1.3.

This comes fairly close to the estimated relationship quoted by Brändli et al.:

ln(FEV1/FVC) = -0.3144 * ln(Height) - 0.0033 * Age + 1.526.

This shows that the FEV1/FVC ratio is non-linearly related to both age and height. The latter was recently corroborated in a study by the Global Lung Function Initiative of >74,000 healthy non-smokers [5]. The same group previously showed that in children and adolescents the FEV1/FVC ratio declines far from linearly: a steep fall until the start of the adolescent growth spurt, followed by a rise before a non-linear fall in adults [6].

1. Gólczewski T, Lubi¿ski W, Chcia¿owski A. A mathematical reason for FEV1/FVC dependence on age. Respiratory Research 2012, 13: 57.

2. Babb TG, Rodarte JR. Mechanism of reduced maximal expiratory flow with aging. J Appl Physiol 2000; 89 :505-511.

3. http://www.lungfunction.org/publishedreferencevalues.html.

4. Brändli O, Schindler Ch, Leuenberger PH, Baur X, Degens P, Künzli N, Keller R, Perruchoud AP. Letters to the editor. Re-estimated equations for 5th percentiles of lung function variables. Thorax 2000; 55: 172.

5. Quanjer PH, Stanojevic S, Cole TJ et al. Multi-ethnic reference values for spirometry for the 3-95 year age range: the Global Lung Function 2012 equations. Eur Respir J published 27 June 2012 as doi: 10.1183/09031936.00080312.

6. Quanjer PH, Stanojevic S, Stocks J, et al. Changes in the FEV1/FVC ratio during childhood and adolescence: an intercontinental study. Eur Respir J 2010; 36: 1391¿1399.

## Competing interests

No competing interests.

## Mathematical versus biological reasons for FEV1/FVC dependence on age - a reply to Philip Quanjer

Tomasz Golczewski, Institute of Biocybernetics and Biomed.Eng. PAS

15 August 2012

Philip Quanjer is right that the ratio FEV1/FVC - like other lung function variables - is nonlinearly dependent on age. It has to be noted that the same 'nonlinearity' can be described mathematically with different formulas. Recently, the exponential formula with the quadratic form in the exponent is 'trendy' but the piecewise linear forms, as those proposed by Quanjer et al in the past [1] and recently by Lubinski and Golczewski [2], is sufficiently nonlinear to describe age-dependence with the same precision as this trendy formula or even better - see Fig.2 in the commented article (particularly if the change point is determined with regression methods together with the other coefficients separately for each lung function variable like in [2]). For this reason, Quanjer's piecewise linear prediction equations seem to be still one of the most appropriate, in average, among the most known equations [3]. As it has been shown mathematically, only MEF25 changes with age cannot be described precisely with such a piecewise linear form [2].

However, the above, i.e. the well-established, nonlinear dependence of lung function variables on age caused by various biological factors, has no relation to the main findings of the commented article, i.e. to: (1) in healthy subjects, there is a strong relationship between FEV1 and FVC being independent of any other factors, including age; (2) the ratio of FEV1 to FVC is not any good description of this relationship because it introduces dependence on age not present in the original relationship.

The commented article shows the mathematical reason for age dependence of FEV1/FVC. Explanation of biological or physical reasons would be equivalent to explanation of the intercept C different from zero; but it has not been the aim of this study.

Note, however, that comparison of old ECSC equations with more recent equations and original data in Fig.2 could suggest that FEV1/FVC dependence on age may be recently more significant because of present guidelines related to the forced expiratory time, which cause that the 'present' FVC is more comparable to VC, and thus it is smaller than the 'original' FVC.

It is not clear why height - depending on authors - does not influence or influences insignificantly the FEV1/FVC ratio; however it is an experimental fact, e.g. in our previous study [2] FEV1/FVC dependence on height appeared statistically insignificant in females and very small in males. Certainly, Philip Quanjer is right that height influences the other lung function variables, which is obvious. Indeed, the most critical task of both the respiratory and cardiovascular systems is oxygen delivery to tissues to keep metabolism at a required level. The total rate of metabolism of homeothermic organisms depends on the body surface for purely physical reasons. The body surface depends on the squared value of the linear size of an organism (e.g. on height in the case of the human being). Because of the above, it is clear that respiratory properties, including lung function variables, depend on height to the power of a value approximately equal to 2.

However, the use of a power of the height in prediction equations has two imperfections:

- height depends on age because of both ageing process and differences in generations, and thus age and height are not mathematically independent variables; this makes it impossible to interpret equations' coefficients and may cause difficulties in regression for some forms of equations (e.g. this 'trendy' exponential one; therefore logarithms are used);

- the length of legs influences significantly the standing height while metabolism is mainly related to sizes of the torso.

For these reasons, we have proposed previously to orthogonalize age and height variables [2,3], i.e. to use the deviation of patient's height from the mean height of his/her age group instead of the total height.

[1] Quanjer PH, Tammeling GJ, Cotes JE, et al: Lung volumes and forced ventilatory flows. Report Working Party Standardization of Lung Function Tests, European Community for Steel and Coal. Official Statement of the European Respiratory Society. Eur. Respir. J. Suppl. 1993; 16:5-40

[2] Lubinski W, Golczewski T.: Physiologically interpretable prediction equations for spirometric indices. J Appl Physiol 2010;108:1440-1446

[3] Golczewski T. Spirometry: a comparison of prediction equations proposed by Lubi¿ski for the Polish population with those proposed by the ECSC/ERS and by Falaschetti et al. Pneumonologia i Alergologia Polska 2012;80:29-40

## Competing interests

None declared