- Email: [email protected]
The precision of resting blood pressure measurement
Computers in Biology and Medicine 43 (2013) 900–903
Contents lists available at SciVerse ScienceDirect
Computers in Biology and Medicine journal homepage: www.elsevier.com/locate/cbm
The precision of resting blood pressure measurement Christopher J. Bailey n Department of Biochemistry, Trinity College, Dublin 2, Ireland
art ic l e i nf o
a b s t r a c t
Article history: Received 14 July 2011 Accepted 13 April 2013
By analysis of timed series of blood pressure(BP) measurements from a single individual, it was shown that data-averaging did not usually give a true value of resting systolic or diastolic pressure. Such measurements fitted a pattern of first order decay from an initial pressure towards a resting systolic or diastolic pressure, P. Using non-linear regression analysis it was possible to approach a standard error of 1 mmHg/1 mmHg for P values on a single day; the between-day dispersion, over a period of months, was found to be about 2 mmHg/2 mmHg. Computer analysis is required to give values of resting systolic and diastolic BP accompanied by error estimates. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Blood pressure Non-linear regression Standard error Systolic Diastolic
1. Introduction Blood Pressure (BP) varies with activity. However blood pressure measured at rest (usually indirect measurement of arterial arm pressure of a relaxed subject by means of an inflatable cuff, with results in mmHg, noted as systolic/diastolic), has been used as as an indicator of human health for over 100 years. Studies convincingly demonstrate the link between single BP measurements and factors such as death rate, by comparing subject and control groups of many thousands of subjects; it can be shown that the differences as small as 2 mmHg of systolic pressure correlate with the real outcomes [1]. Data on individual monitored subjects are usually reported to three figures (systolic) or two figures (diastolic). Scientifically, the notable feature about such accounts is the absence of any uncertainty value, although the presentation implies that use of the final figure (quotation to 1 mmHg) is appropriate. In the clinical literature there appears to be some disagreement about accuracy. Many experienced workers have clearly reported individual measurements rounded to 10 mmHg/10 mmHg or 10 mmHg/5 mmHg, scientfically unexceptional if the errors are above 5 mmHg/5 mmHg or 5 mmHg/3 mmHg, but unacceptable to those who consider the overabundant terminal zeros and fives as a ‘bias’ [2]. In collections of data from different eras, some average measures of single-subject between-day BP dispersion (re-estimated by the current author from absolute differences or standard deviation of differences) have standard deviations of 10 mmHg/ 7 mmHg [3], 10 mmHg/6 mmHg [4] and 7 mmHg/5 mmHg [5].
n
Tel.: +35318961608. E-mail address: [email protected]
0010-4825/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compbiomed.2013.04.016
These are population averages: it is clear [6] that the dispersion increases with measured BP. The dispersions imply that it can be scientifically acceptable to round single measurements and quote BP values to 10 mmHg/ 5 mmHg or 10 mmHg/10 mmHg. On the simplest interpretation of dispersion, more exact values for individual subjects would be available by replication. Medical opinion, however, appears to have lost confidence in the validity of professional measurements and endorsed a preference for amateur ‘home BP’ values [7]. This preference seems to result from the need to avoid a bias produced by the clinical situation, in which many individuals appear to remain in an alerted [6] state with consequent raised BP. To determine resting systolic or diastolic BP as conventional scientific parameters, it is necessary to analyse a series of BP measurements. Because the measurements may sample resting state and alerted states, simple data-averaging may not give an exact value of the resting BP parameters. Rather, it seems that an improved data analysis procedure may be required in order to obtain values, and error estimates, of the resting BP parameters. Accordingly extensive self-test progress curves from a single subject (CJB) were analysed to devise a procedure that can give values of the resting BP parameters, together with error estimates. 2. Methods Blood pressure was measured with a recommended [8] automated oscillometric instrument A&D UA-767, in accordance with the manufacturer's instructions. This upper-arm instrument was inflated to an arm pressure of 180 mmHg before obtaining systolic and diastolic values during deflation. The accuracy of the instrument is quoted as 73 mmHg/73 mmHg. Measurements took place when
C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
In a preliminary investigation, 12 consecutive measurements were obtained in a 15 min period on 8 separate days. When the daily BP measurements were plotted against time, three systolic and two diastolic progress curves had significant (p 40.95) negative slopes; the remainder could be interpreted as variation about a mean or having a negative trend; none showed a rising trend. Averaging of such daily data is not generally appropriate. A suitable equation that describes the observed behaviour is BP ¼ A0 expð−λtÞ þ P þ ε
ð1Þ
In this, each systolic or diastolic BP measurement varies with time t, as set by parameters A0, an initial overpressure; λ, a decay constant; and P, a stationary pressure that may correspond to the resting state. Eq. (1) fits data that oscillates about P according to the value of the random error term ε; that decays during the period of measurement to an asymptotic value of P; or that decays continually without establishing a stationary P. NLR data fitting to minimise the sum value of ε2 provides an objective least-squares choice between the three possible outcomes. When applied to the 12 measurements by 8 days, NLR gave realistic systolic or diastolic P values with data from four of the days, but the four others produced impossibly low, or only very inexact, values of P. It appeared that the time period, or number of measurements, was insufficient for reliable establishment of a stable P value. The repetitive measurements were therefore extended to 30 min (data sets 1–5), and later, after a gap of about two months, to 30–60 min (data sets 6–17). The set numbers correspond to the following dates in 2010: 1, 29-Mar; 2, 30-Mar; 3, 31-Mar; 4, 1-Apr; 5, 2-Apr; 6,28-Jun; 7, 29-Jun; 8, 3-Jul; 9, 13-Jul; 10, 15-Jul; 11, 16Jul; 12, 17-Jul; 13, 19-Jul; 14, 20-Jul; 15, 21-Jul; 16, 22-Jul; 17, 23-Jul. A total of 17 days data was obtained, about 600 measurements each of systolic and diastolic BP. The systolic and diastolic data were initially fitted individually to Eq. (1), but set 9 data proved particularly difficult to fit objectively. As the paired systolic-diastolic BP measurements are highly correlated (paired systolic-diastolic values of A0 and P are also correlated and can supply supplementary constraints), it is appropriate to take the value of decay rate λ as common to a systolic–diastolic pair. This gave a more forceful and universal fitting process in which residual sum of squares (weighted to give equal value to systolic and diastolic members) was minimised to give five parameters, rather than a total of six when diastolic and systolic data are fitted separately (all Figures show the data fits
Systolic Set 6 160 Systolic Blood Pressure(mmHg)
3. Results
from the paired λ procedure). The estimated values of P did not significantly change: the average differences of P between the separate and combined procedures estimates were 0.5 72 mmHg/ 0.3 70.9 mmHg (n ¼ 16 pairs). Twelve of the systolic–diastolic paired records were interpretable as a decay from a substantial initial overpressure to an asymptotic stationary pressure (e.g. Figs. 1 and 2, the respective systolic and diastolic data from set 6), in which the data points are shown with the best fit (stippled line) to Eq. (1). In four other paired records, the initial overpressure effects were so low (a bias of o+. 5 mmHg/+. 5 mmHg) that averaging of the data was also appropriate (illustrated in Figs. 3 and 4, the respective systolic and diastolic data from set 14). The remaining record of the total of seventeen, illustrated by Fig. 5, gave a more complex pattern that did not fit to Eq. (1). During this set, the subject was alerted by his small timing error at 15 min. Within 1.25 min the perturbation produced a transient rise in systolic and diastolic (not shown) BP. The data set was satisfactorily modelled by adding terms for transient overpressure at 16.25 min and subsequently. The values of systolic and diastolic parameter P, in each of the 17 pairs, are presented in sequence in Figs. 6 and 7, which include the four paired cases calculated by NLR and by simple averaging. It seems clear that the fitted daily systolic and diastolic P values did not change systematically over the course of the experiments. Between-days systolic and diastolic P mean values were calculated for weighted (weights proportional to the reciprocal of the square of the standard error) and unweighted conditions: mean values (7standard deviations) were 134.672.4 mmHg/80.872.0 mmHg (n¼17), and 132.37 4.7 mmHg/80.472.8 mmHg for weighted and unweighted estimates respectively. Inclusion of the four day-averaged values from time-
155 150 145 140 135 130 125 120 0
5
10
15
20
25
30
35
40
Elapsed Time(min) Fig. 1. Systolic pressure measured in set 6. The measured data (circles) are graphed together with a line calculated from the best fit to Eq. (1).
Diastolic Set 6 95 Diastolic Blood Pressure(mmHg)
the untreated subject (male, aged 68 y, in general good health) was alone in a domestic situation. Data was collected between 10 am and 12 pm, with the subject seated in a reproducible position, after at least 15 min rest and at least 30 min after eating/drinking. The instrument requires about 50 s to register systolic and diastolic pressures and pulse. The measurements were typed into a spreadsheet immediately after registration. Repetition at 1.25 min intervals was found suitable to accumulate data whilst maintaining the subject in a relaxed state. As a further aid to postural relaxation, the arm-cuff was released and reset after every four readings. All measurements were obtained by self-test of the author and there was therefore no further need for ethics approval. Data were analysed by Excel spreadsheet programmes, notably Solver (fitted with the Solveraid macro [9] to determine standard errors). Results of the non-linear regression (NLR) procedure were confirmed using the econometric package, Gretl [10]. This latter procedure was also used for time-series analysis [11], to calculate autocorrelation functions of BP data streams.
901
90
85
80
75
70
0
5
10
15
20
25
30
Elapsed Time(min)
Fig. 2. Diastolic pressure measured in set 6.
35
40
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C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
Systolic Set 14
135
130
0
10
20
30
40
50
60
Elapsed Time(min)
Fitted Stationary Systolic Pressure(mmHg)
Systolic Blood Pressure(mmHg)
140
125
125
120 NLR value ± S.E.E. Mean
0
5
10
15
20
Set Number
85
Stationary Diastolic Pressures 80
75
70
0
10
20
30
40
50
60
Elapsed Time(min) Fig. 4. Diastolic pressure measured in set 14. The NLR-fitted line indicates a small, transient overpressure, but the asymptotic value of pressure is indistinguishable from the simple mean of the data.
Systolic Set 13
Fitted Stationary Diastolic Pressure(mmHg)
Diastolic Blood Pressure(mmHg)
130
Fig. 6. Combined results (closed circles) for systolic P on 17 days, determined by NLR. Averaged results (open circles), shown for 4 sets in which overpressure effects were negligible or transient, have errors indistinguishable from those of the NLR procedure.
Diastolic Set 14
90
135
115
Fig. 3. Systolic pressure measured in set 14. The NLR fitting procedure gives a solution identical to the simple mean of the data.
88 86 84 82 80 78 76 74 NLR value ± S.E.E.
72
Mean values
70 0
150 Systolic Blood Pressure(mmHg)
Stationary Systolic Pressures
140
145
2
4
6
8
10
12
14
16
18
Set Number Fig. 7. Combined results (closed diamonds) for diastolic P on 17 days, determined by NLR. Averaged results (open diamonds), shown for 4 sets in which overpressure effects were negligible or transient, have errors indistinguishable from those of the NLR procedure.
145 140 135 130 125
0
10
20
30
40
50
60
Elapsed Time(min) Fig. 5. Systolic pressure of set 13. Note the alerting reaction at 16.25 min that appears to produces a BP ‘spike’ of 12 mmHg. Data from 16.25 min onwards was analysed by including an extra parameter, A1, that accounts for the spike.
invariant data, highlighted on Figs. 6 and 7, confirms one important result: NLR estimation gives a value of P equal to the simple BP average when there are no overpressure effects. Thus the stationary pressure P is equal to resting BP. Although the respective between-day values of P are quite stable, the precision of the daily estimates of P is seen to vary greatly (Figs. 6 and 7). Of the 17 daily values of each, 8 systolic and 10 diastolic have standard errors for P of about 1 mmHg, around the practical limit with conventional instruments that truncate data to 1 mmHg; five values, all systolic, have standard errors 45 mmHg. Data that can be averaged is seen to be among the most precise. But this is not a necessary condition of exactitude, for datasets that starts from a substantial overpressure (e.g.
Figs. 1 and 2) can supply an accurate value of the asymptotic P. The important condition is that the measurements are extended for such a period that BP approaches the asymptotic P value closely. The time required is governed by decay rate λ, most easily considered in terms of half-life. For the 17 sets, half-lives ranged from o1–26 min. In Figs. 1 and 2, from set 6, collection took place over ten half-lives. In contrast, the data of set 10 give less exact values of P, even though collection continued for a longer time, because the experiment only continued for 2.4 half-lives. In practical terms, 15 min experiments seem insufficient to yield enough data, 30 min data collection may be satisfactory, but 60 min data was required in some cases. The reasons for the huge variation in overpressure decay rate remain unknown and may be very subjective. For the results of this paper, the averages and standard deviations of initial values of BP measurements were 147710 mmHg/ 8774 mmHg (n¼17). If these determinations are treated as values of systolic and diastolic resting pressures, there are average systematic errors of about +15 mmHg/+7 mmHg. The initial value dispersions of 10 mmHg/4 mmHg are quite similar to those of whole-population data quoted in the Introduction. The autocorrelation functions (data not shown) of BP measurements, after removal of trend as appropriate, rarely reached values
C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
significantly different from random and thus the individual daysets may be interpreted as composed of measurements with independent random error. The autocorrelation results also confirm that the regular cuff-release procedure did not produce systematic effects.
4. Discussion Partitioning individual BP systolic or diastolic measurements into components of an overpressure and a resting BP, was introduced [12] by Smirk and colleagues. In their studies, BP was repetitively measured (approx 1 per min) over 30 min, starting from an initial ‘casual’ measure, to obtain the ‘basal’ pressure, equal to the lowest level maintained for three consecutive readings [13], or the average of the two lowest readings [14]. In these early studies, the two BP parameters were obtained without progress curve analysis; most of the measurements were ignored, so that there is no estimate of uncertainty; the selection of lowest value introduces a systematic error; the protocol used passive subjects, most of whom became somnolent, some slept, so that ‘basal’ values may not sample a resting state. However the studies did demonstrate that each BP measurement should be considered as a sum of two components. The present study improves data analysis, but uses a single subject, which has some advantage when investigating the equation that fits a data stream. A potential disadvantage of single-subject data is that the results could be untypical. In the huge literature about resting BP measurement two theories are prominent: that the subject BP values do not vary with time or that subject BP values may be elevated by subjective factors. Data in this paper shows examples of both behaviours and demonstrates that initially elevated BP values will settle to a steady value indistinguishable from the subject's result in a time-invariant data set. Thus Eq. (1) has the generality required to explain the usual situations in all subjects. In the clinical literature there is an overwhelming preference to treat measured BP as a sampling of the resting pressure values. This one-parameter solution is simple, associated with exact values if conditions are appropriate (as in Figs. 3 and 4), and allows the averaging of data from many subjects. There are numerous suggestions for mitigating the potential problem of initial overpressure. For example, it is recommended that three readings are taken, then the initial or outlier measurements are rejected [15] or that the last three of five readings are accepted [16]. Consideration of the data presented here suggests that such stratagems have limited value. A professional opinion, referring to measurements in a clinic, is that the (overpressure) error ‘is almost unpredictable in any given individual’ [6]. The experience of the subject of the present paper, who had negligible ability to predict his own initial overpressure, or its decay rate, is consistent with that multi-subject report. Overpressures, or ‘white coat’ effects, are likely to be even greater in clinical situations. The results of the present report confirm that an initial overpressure effect cannot be avoided by standardising conditions of data collection, and so must be analysed in order to obtain exact resting BP parameters. Without a demonstration that the initial BP measurement is sampling the resting state, a stream of BP data must be fitted by Eq. (1). In which case, at least four measurements of BP are formally required to obtain the value of a resting BP accompanied by an error estimate. Much of the BP literature is based on less than four measurements from a single individual; hence the inadequacy of individual error estimates. Interpretation of the vast comparative literature that shows diet or drug intake, etc., producing small BP changes, presents a further potential difficulty. For there may remain a suspicion that the treatments affect the mood of the subjects and thus change the initial overpressure, rather than resting BP.
903
In clinical practice, a single BP estimate is useful for testing whether a BP limit has been exceeded. In individual cases that require a more exact BP value, or expect results to demonstrate the effect of dietary or drug changes, a data analysis procedure is required: undoubtedly a more cumbersome and complex analysis. Notably the mood of the subject is of great importance, because psychophysiological effects may perturb the data stream and prove difficult to analyse, as in Fig. 5. Further clinical research with multiple subjects is required to define conditions that allow subjects to present at, or rapidly tend to, the resting BP state. Optimally, it may be that data should be obtained automatically, from subjects screened from gross external stimuli and relaxed by carrying out simple repetitive tasks during data collection. The arm-cuff procedure for measuring BP indirectly has not changed greatly in principle during the 120 years since the techniques were introduced. At that time, without automated equipment, it was difficult to obtain a sizeable amount of data, and practically impossible to analyse it. In modern times, current automated multi-reading instrumentation (arm-cuff, or newer methods of continuous non-invasive blood pressure monitoring) can obtain sufficient data with little human intervention. The adoption of correct data analysis by computer, as outlined here, will allow simple determination of resting BP parameters accompanied by scientific error estimates. Conflict of interest statement None declared. Acknowledgements I thank Professor John Haslett(Statistics) and Dr Paul Voorheis MD(Biochemistry), of Trinity College Dublin, for extensive helpful discussions. The author received no funding for this study. References [1] S. Lewington, R. Clarke, N. Qizilbash, N. Peto, R. Collins, Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies, Lancet 360 (2002) 1903–1913. [2] B.S. Everitt, C.R. Palmer, Encyclopaedic Companion to Medical Statistics, Hodder Arnold, London107–108. [3] M. Hamilton, G.W. Pickering, J.A. Roberts, G.S. Sowry, The aetiology of essential hypertension, I. The arterial pressure in the general population, Clin. Sci. (Lond.) 13 (1954) 11–35. [4] S. Freestone, J.H. Silas, L.E. Ramsay, Sample size for short-term trials of antihypertensive drugs, Br. J. Clin Pharmacol. 14 (1982) 265–268. [5] M.J. Jamieson, J. Webster, S. Philips, T.A. Jeffers, A.K. Scott, O.J. Robb, H.G. Lovell, J.C. Petrie, The measurement of blood pressure: sitting or supine, once or twice? J. Hypertens. 8 (1990) 635–640. [6] G. Parati, G. Mancia, Ambulatory blood pressure monitoring, in: G. Mancia G (Ed.), Manual of Hypertension, Churchill, London, 2002, pp. 153–171. [7] G. Parati, T.G. Pickering, Home blood-pressure monitoring: US and European consensus, Lancet 373 (2009) 876–878. [8] E. O'Brien, Measurement of Blood Pressure, in: D. Beevers, G. Lip, E. O'Brien (Eds.), ABC of Hypertension, fifth ed, Blackwell Publishing, Oxford, 2007, pp. 17–25. [9] R. De Levie, How to use Excel in Analytical Chemistry and in General Scientific Data Analysis, Cambridge University Press, Cambridge, 1999. [10] A. Cottrell, Gretl User's Guide, available: 〈http://gretl.sourceforge.net/osx. html〉, 2010 (accessed 05.09.10). [11] C. Chatfield, The analysis of Time Series, Chapman and Hall, London, 1989. [12] G. Pickering, High Blood Pressure, second ed., Churchill, London, 1968. [13] G.M. Alam, F.H. Smirk, Casual and basal blood pressures I.-in British and Egyptian men, Br Heart J. 5 (1943) 152–155. [14] F.H. Smirk, High Arterial Pressure, Blackwell, Oxford, 1957. [15] N.R. Campbell, M.G. Myers, D.W. McKay, Is usual measurement of blood pressure meaningful? Blood Press. Mon. 4 (1999) 71–76. [16] A.P. Shapiro, Hypertension and Stress: a Unified Concept, Lawrence Erlbaum Associates, Mahwah, New Jersey31.
Contents lists available at SciVerse ScienceDirect
Computers in Biology and Medicine journal homepage: www.elsevier.com/locate/cbm
The precision of resting blood pressure measurement Christopher J. Bailey n Department of Biochemistry, Trinity College, Dublin 2, Ireland
art ic l e i nf o
a b s t r a c t
Article history: Received 14 July 2011 Accepted 13 April 2013
By analysis of timed series of blood pressure(BP) measurements from a single individual, it was shown that data-averaging did not usually give a true value of resting systolic or diastolic pressure. Such measurements fitted a pattern of first order decay from an initial pressure towards a resting systolic or diastolic pressure, P. Using non-linear regression analysis it was possible to approach a standard error of 1 mmHg/1 mmHg for P values on a single day; the between-day dispersion, over a period of months, was found to be about 2 mmHg/2 mmHg. Computer analysis is required to give values of resting systolic and diastolic BP accompanied by error estimates. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Blood pressure Non-linear regression Standard error Systolic Diastolic
1. Introduction Blood Pressure (BP) varies with activity. However blood pressure measured at rest (usually indirect measurement of arterial arm pressure of a relaxed subject by means of an inflatable cuff, with results in mmHg, noted as systolic/diastolic), has been used as as an indicator of human health for over 100 years. Studies convincingly demonstrate the link between single BP measurements and factors such as death rate, by comparing subject and control groups of many thousands of subjects; it can be shown that the differences as small as 2 mmHg of systolic pressure correlate with the real outcomes [1]. Data on individual monitored subjects are usually reported to three figures (systolic) or two figures (diastolic). Scientifically, the notable feature about such accounts is the absence of any uncertainty value, although the presentation implies that use of the final figure (quotation to 1 mmHg) is appropriate. In the clinical literature there appears to be some disagreement about accuracy. Many experienced workers have clearly reported individual measurements rounded to 10 mmHg/10 mmHg or 10 mmHg/5 mmHg, scientfically unexceptional if the errors are above 5 mmHg/5 mmHg or 5 mmHg/3 mmHg, but unacceptable to those who consider the overabundant terminal zeros and fives as a ‘bias’ [2]. In collections of data from different eras, some average measures of single-subject between-day BP dispersion (re-estimated by the current author from absolute differences or standard deviation of differences) have standard deviations of 10 mmHg/ 7 mmHg [3], 10 mmHg/6 mmHg [4] and 7 mmHg/5 mmHg [5].
n
Tel.: +35318961608. E-mail address: [email protected]
0010-4825/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.compbiomed.2013.04.016
These are population averages: it is clear [6] that the dispersion increases with measured BP. The dispersions imply that it can be scientifically acceptable to round single measurements and quote BP values to 10 mmHg/ 5 mmHg or 10 mmHg/10 mmHg. On the simplest interpretation of dispersion, more exact values for individual subjects would be available by replication. Medical opinion, however, appears to have lost confidence in the validity of professional measurements and endorsed a preference for amateur ‘home BP’ values [7]. This preference seems to result from the need to avoid a bias produced by the clinical situation, in which many individuals appear to remain in an alerted [6] state with consequent raised BP. To determine resting systolic or diastolic BP as conventional scientific parameters, it is necessary to analyse a series of BP measurements. Because the measurements may sample resting state and alerted states, simple data-averaging may not give an exact value of the resting BP parameters. Rather, it seems that an improved data analysis procedure may be required in order to obtain values, and error estimates, of the resting BP parameters. Accordingly extensive self-test progress curves from a single subject (CJB) were analysed to devise a procedure that can give values of the resting BP parameters, together with error estimates. 2. Methods Blood pressure was measured with a recommended [8] automated oscillometric instrument A&D UA-767, in accordance with the manufacturer's instructions. This upper-arm instrument was inflated to an arm pressure of 180 mmHg before obtaining systolic and diastolic values during deflation. The accuracy of the instrument is quoted as 73 mmHg/73 mmHg. Measurements took place when
C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
In a preliminary investigation, 12 consecutive measurements were obtained in a 15 min period on 8 separate days. When the daily BP measurements were plotted against time, three systolic and two diastolic progress curves had significant (p 40.95) negative slopes; the remainder could be interpreted as variation about a mean or having a negative trend; none showed a rising trend. Averaging of such daily data is not generally appropriate. A suitable equation that describes the observed behaviour is BP ¼ A0 expð−λtÞ þ P þ ε
ð1Þ
In this, each systolic or diastolic BP measurement varies with time t, as set by parameters A0, an initial overpressure; λ, a decay constant; and P, a stationary pressure that may correspond to the resting state. Eq. (1) fits data that oscillates about P according to the value of the random error term ε; that decays during the period of measurement to an asymptotic value of P; or that decays continually without establishing a stationary P. NLR data fitting to minimise the sum value of ε2 provides an objective least-squares choice between the three possible outcomes. When applied to the 12 measurements by 8 days, NLR gave realistic systolic or diastolic P values with data from four of the days, but the four others produced impossibly low, or only very inexact, values of P. It appeared that the time period, or number of measurements, was insufficient for reliable establishment of a stable P value. The repetitive measurements were therefore extended to 30 min (data sets 1–5), and later, after a gap of about two months, to 30–60 min (data sets 6–17). The set numbers correspond to the following dates in 2010: 1, 29-Mar; 2, 30-Mar; 3, 31-Mar; 4, 1-Apr; 5, 2-Apr; 6,28-Jun; 7, 29-Jun; 8, 3-Jul; 9, 13-Jul; 10, 15-Jul; 11, 16Jul; 12, 17-Jul; 13, 19-Jul; 14, 20-Jul; 15, 21-Jul; 16, 22-Jul; 17, 23-Jul. A total of 17 days data was obtained, about 600 measurements each of systolic and diastolic BP. The systolic and diastolic data were initially fitted individually to Eq. (1), but set 9 data proved particularly difficult to fit objectively. As the paired systolic-diastolic BP measurements are highly correlated (paired systolic-diastolic values of A0 and P are also correlated and can supply supplementary constraints), it is appropriate to take the value of decay rate λ as common to a systolic–diastolic pair. This gave a more forceful and universal fitting process in which residual sum of squares (weighted to give equal value to systolic and diastolic members) was minimised to give five parameters, rather than a total of six when diastolic and systolic data are fitted separately (all Figures show the data fits
Systolic Set 6 160 Systolic Blood Pressure(mmHg)
3. Results
from the paired λ procedure). The estimated values of P did not significantly change: the average differences of P between the separate and combined procedures estimates were 0.5 72 mmHg/ 0.3 70.9 mmHg (n ¼ 16 pairs). Twelve of the systolic–diastolic paired records were interpretable as a decay from a substantial initial overpressure to an asymptotic stationary pressure (e.g. Figs. 1 and 2, the respective systolic and diastolic data from set 6), in which the data points are shown with the best fit (stippled line) to Eq. (1). In four other paired records, the initial overpressure effects were so low (a bias of o+. 5 mmHg/+. 5 mmHg) that averaging of the data was also appropriate (illustrated in Figs. 3 and 4, the respective systolic and diastolic data from set 14). The remaining record of the total of seventeen, illustrated by Fig. 5, gave a more complex pattern that did not fit to Eq. (1). During this set, the subject was alerted by his small timing error at 15 min. Within 1.25 min the perturbation produced a transient rise in systolic and diastolic (not shown) BP. The data set was satisfactorily modelled by adding terms for transient overpressure at 16.25 min and subsequently. The values of systolic and diastolic parameter P, in each of the 17 pairs, are presented in sequence in Figs. 6 and 7, which include the four paired cases calculated by NLR and by simple averaging. It seems clear that the fitted daily systolic and diastolic P values did not change systematically over the course of the experiments. Between-days systolic and diastolic P mean values were calculated for weighted (weights proportional to the reciprocal of the square of the standard error) and unweighted conditions: mean values (7standard deviations) were 134.672.4 mmHg/80.872.0 mmHg (n¼17), and 132.37 4.7 mmHg/80.472.8 mmHg for weighted and unweighted estimates respectively. Inclusion of the four day-averaged values from time-
155 150 145 140 135 130 125 120 0
5
10
15
20
25
30
35
40
Elapsed Time(min) Fig. 1. Systolic pressure measured in set 6. The measured data (circles) are graphed together with a line calculated from the best fit to Eq. (1).
Diastolic Set 6 95 Diastolic Blood Pressure(mmHg)
the untreated subject (male, aged 68 y, in general good health) was alone in a domestic situation. Data was collected between 10 am and 12 pm, with the subject seated in a reproducible position, after at least 15 min rest and at least 30 min after eating/drinking. The instrument requires about 50 s to register systolic and diastolic pressures and pulse. The measurements were typed into a spreadsheet immediately after registration. Repetition at 1.25 min intervals was found suitable to accumulate data whilst maintaining the subject in a relaxed state. As a further aid to postural relaxation, the arm-cuff was released and reset after every four readings. All measurements were obtained by self-test of the author and there was therefore no further need for ethics approval. Data were analysed by Excel spreadsheet programmes, notably Solver (fitted with the Solveraid macro [9] to determine standard errors). Results of the non-linear regression (NLR) procedure were confirmed using the econometric package, Gretl [10]. This latter procedure was also used for time-series analysis [11], to calculate autocorrelation functions of BP data streams.
901
90
85
80
75
70
0
5
10
15
20
25
30
Elapsed Time(min)
Fig. 2. Diastolic pressure measured in set 6.
35
40
902
C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
Systolic Set 14
135
130
0
10
20
30
40
50
60
Elapsed Time(min)
Fitted Stationary Systolic Pressure(mmHg)
Systolic Blood Pressure(mmHg)
140
125
125
120 NLR value ± S.E.E. Mean
0
5
10
15
20
Set Number
85
Stationary Diastolic Pressures 80
75
70
0
10
20
30
40
50
60
Elapsed Time(min) Fig. 4. Diastolic pressure measured in set 14. The NLR-fitted line indicates a small, transient overpressure, but the asymptotic value of pressure is indistinguishable from the simple mean of the data.
Systolic Set 13
Fitted Stationary Diastolic Pressure(mmHg)
Diastolic Blood Pressure(mmHg)
130
Fig. 6. Combined results (closed circles) for systolic P on 17 days, determined by NLR. Averaged results (open circles), shown for 4 sets in which overpressure effects were negligible or transient, have errors indistinguishable from those of the NLR procedure.
Diastolic Set 14
90
135
115
Fig. 3. Systolic pressure measured in set 14. The NLR fitting procedure gives a solution identical to the simple mean of the data.
88 86 84 82 80 78 76 74 NLR value ± S.E.E.
72
Mean values
70 0
150 Systolic Blood Pressure(mmHg)
Stationary Systolic Pressures
140
145
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Set Number Fig. 7. Combined results (closed diamonds) for diastolic P on 17 days, determined by NLR. Averaged results (open diamonds), shown for 4 sets in which overpressure effects were negligible or transient, have errors indistinguishable from those of the NLR procedure.
145 140 135 130 125
0
10
20
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60
Elapsed Time(min) Fig. 5. Systolic pressure of set 13. Note the alerting reaction at 16.25 min that appears to produces a BP ‘spike’ of 12 mmHg. Data from 16.25 min onwards was analysed by including an extra parameter, A1, that accounts for the spike.
invariant data, highlighted on Figs. 6 and 7, confirms one important result: NLR estimation gives a value of P equal to the simple BP average when there are no overpressure effects. Thus the stationary pressure P is equal to resting BP. Although the respective between-day values of P are quite stable, the precision of the daily estimates of P is seen to vary greatly (Figs. 6 and 7). Of the 17 daily values of each, 8 systolic and 10 diastolic have standard errors for P of about 1 mmHg, around the practical limit with conventional instruments that truncate data to 1 mmHg; five values, all systolic, have standard errors 45 mmHg. Data that can be averaged is seen to be among the most precise. But this is not a necessary condition of exactitude, for datasets that starts from a substantial overpressure (e.g.
Figs. 1 and 2) can supply an accurate value of the asymptotic P. The important condition is that the measurements are extended for such a period that BP approaches the asymptotic P value closely. The time required is governed by decay rate λ, most easily considered in terms of half-life. For the 17 sets, half-lives ranged from o1–26 min. In Figs. 1 and 2, from set 6, collection took place over ten half-lives. In contrast, the data of set 10 give less exact values of P, even though collection continued for a longer time, because the experiment only continued for 2.4 half-lives. In practical terms, 15 min experiments seem insufficient to yield enough data, 30 min data collection may be satisfactory, but 60 min data was required in some cases. The reasons for the huge variation in overpressure decay rate remain unknown and may be very subjective. For the results of this paper, the averages and standard deviations of initial values of BP measurements were 147710 mmHg/ 8774 mmHg (n¼17). If these determinations are treated as values of systolic and diastolic resting pressures, there are average systematic errors of about +15 mmHg/+7 mmHg. The initial value dispersions of 10 mmHg/4 mmHg are quite similar to those of whole-population data quoted in the Introduction. The autocorrelation functions (data not shown) of BP measurements, after removal of trend as appropriate, rarely reached values
C.J. Bailey / Computers in Biology and Medicine 43 (2013) 900–903
significantly different from random and thus the individual daysets may be interpreted as composed of measurements with independent random error. The autocorrelation results also confirm that the regular cuff-release procedure did not produce systematic effects.
4. Discussion Partitioning individual BP systolic or diastolic measurements into components of an overpressure and a resting BP, was introduced [12] by Smirk and colleagues. In their studies, BP was repetitively measured (approx 1 per min) over 30 min, starting from an initial ‘casual’ measure, to obtain the ‘basal’ pressure, equal to the lowest level maintained for three consecutive readings [13], or the average of the two lowest readings [14]. In these early studies, the two BP parameters were obtained without progress curve analysis; most of the measurements were ignored, so that there is no estimate of uncertainty; the selection of lowest value introduces a systematic error; the protocol used passive subjects, most of whom became somnolent, some slept, so that ‘basal’ values may not sample a resting state. However the studies did demonstrate that each BP measurement should be considered as a sum of two components. The present study improves data analysis, but uses a single subject, which has some advantage when investigating the equation that fits a data stream. A potential disadvantage of single-subject data is that the results could be untypical. In the huge literature about resting BP measurement two theories are prominent: that the subject BP values do not vary with time or that subject BP values may be elevated by subjective factors. Data in this paper shows examples of both behaviours and demonstrates that initially elevated BP values will settle to a steady value indistinguishable from the subject's result in a time-invariant data set. Thus Eq. (1) has the generality required to explain the usual situations in all subjects. In the clinical literature there is an overwhelming preference to treat measured BP as a sampling of the resting pressure values. This one-parameter solution is simple, associated with exact values if conditions are appropriate (as in Figs. 3 and 4), and allows the averaging of data from many subjects. There are numerous suggestions for mitigating the potential problem of initial overpressure. For example, it is recommended that three readings are taken, then the initial or outlier measurements are rejected [15] or that the last three of five readings are accepted [16]. Consideration of the data presented here suggests that such stratagems have limited value. A professional opinion, referring to measurements in a clinic, is that the (overpressure) error ‘is almost unpredictable in any given individual’ [6]. The experience of the subject of the present paper, who had negligible ability to predict his own initial overpressure, or its decay rate, is consistent with that multi-subject report. Overpressures, or ‘white coat’ effects, are likely to be even greater in clinical situations. The results of the present report confirm that an initial overpressure effect cannot be avoided by standardising conditions of data collection, and so must be analysed in order to obtain exact resting BP parameters. Without a demonstration that the initial BP measurement is sampling the resting state, a stream of BP data must be fitted by Eq. (1). In which case, at least four measurements of BP are formally required to obtain the value of a resting BP accompanied by an error estimate. Much of the BP literature is based on less than four measurements from a single individual; hence the inadequacy of individual error estimates. Interpretation of the vast comparative literature that shows diet or drug intake, etc., producing small BP changes, presents a further potential difficulty. For there may remain a suspicion that the treatments affect the mood of the subjects and thus change the initial overpressure, rather than resting BP.
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In clinical practice, a single BP estimate is useful for testing whether a BP limit has been exceeded. In individual cases that require a more exact BP value, or expect results to demonstrate the effect of dietary or drug changes, a data analysis procedure is required: undoubtedly a more cumbersome and complex analysis. Notably the mood of the subject is of great importance, because psychophysiological effects may perturb the data stream and prove difficult to analyse, as in Fig. 5. Further clinical research with multiple subjects is required to define conditions that allow subjects to present at, or rapidly tend to, the resting BP state. Optimally, it may be that data should be obtained automatically, from subjects screened from gross external stimuli and relaxed by carrying out simple repetitive tasks during data collection. The arm-cuff procedure for measuring BP indirectly has not changed greatly in principle during the 120 years since the techniques were introduced. At that time, without automated equipment, it was difficult to obtain a sizeable amount of data, and practically impossible to analyse it. In modern times, current automated multi-reading instrumentation (arm-cuff, or newer methods of continuous non-invasive blood pressure monitoring) can obtain sufficient data with little human intervention. The adoption of correct data analysis by computer, as outlined here, will allow simple determination of resting BP parameters accompanied by scientific error estimates. Conflict of interest statement None declared. Acknowledgements I thank Professor John Haslett(Statistics) and Dr Paul Voorheis MD(Biochemistry), of Trinity College Dublin, for extensive helpful discussions. The author received no funding for this study. References [1] S. Lewington, R. Clarke, N. Qizilbash, N. Peto, R. Collins, Age-specific relevance of usual blood pressure to vascular mortality: a meta-analysis of individual data for one million adults in 61 prospective studies, Lancet 360 (2002) 1903–1913. [2] B.S. Everitt, C.R. Palmer, Encyclopaedic Companion to Medical Statistics, Hodder Arnold, London107–108. [3] M. Hamilton, G.W. Pickering, J.A. Roberts, G.S. Sowry, The aetiology of essential hypertension, I. The arterial pressure in the general population, Clin. Sci. (Lond.) 13 (1954) 11–35. [4] S. Freestone, J.H. Silas, L.E. Ramsay, Sample size for short-term trials of antihypertensive drugs, Br. J. Clin Pharmacol. 14 (1982) 265–268. [5] M.J. Jamieson, J. Webster, S. Philips, T.A. Jeffers, A.K. Scott, O.J. Robb, H.G. Lovell, J.C. Petrie, The measurement of blood pressure: sitting or supine, once or twice? J. Hypertens. 8 (1990) 635–640. [6] G. Parati, G. Mancia, Ambulatory blood pressure monitoring, in: G. Mancia G (Ed.), Manual of Hypertension, Churchill, London, 2002, pp. 153–171. [7] G. Parati, T.G. Pickering, Home blood-pressure monitoring: US and European consensus, Lancet 373 (2009) 876–878. [8] E. O'Brien, Measurement of Blood Pressure, in: D. Beevers, G. Lip, E. O'Brien (Eds.), ABC of Hypertension, fifth ed, Blackwell Publishing, Oxford, 2007, pp. 17–25. [9] R. De Levie, How to use Excel in Analytical Chemistry and in General Scientific Data Analysis, Cambridge University Press, Cambridge, 1999. [10] A. Cottrell, Gretl User's Guide, available: 〈http://gretl.sourceforge.net/osx. html〉, 2010 (accessed 05.09.10). [11] C. Chatfield, The analysis of Time Series, Chapman and Hall, London, 1989. [12] G. Pickering, High Blood Pressure, second ed., Churchill, London, 1968. [13] G.M. Alam, F.H. Smirk, Casual and basal blood pressures I.-in British and Egyptian men, Br Heart J. 5 (1943) 152–155. [14] F.H. Smirk, High Arterial Pressure, Blackwell, Oxford, 1957. [15] N.R. Campbell, M.G. Myers, D.W. McKay, Is usual measurement of blood pressure meaningful? Blood Press. Mon. 4 (1999) 71–76. [16] A.P. Shapiro, Hypertension and Stress: a Unified Concept, Lawrence Erlbaum Associates, Mahwah, New Jersey31.