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Int J Appl Math (Sofia). Author manuscript; available in PMC 2017 September 25. Published in final edited form as: Int J Appl Math (Sofia). 2005 ; 18(4): 487–500.
Predictive models of postural control based on electronic force platform measures in patients with Parkinson’s disease Gregory M. Constantine1, Marius G. Buliga2, Larry S. Ivanco3, Robert Y. Moore3, and Nicolaas I. Bohnen3
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1Department
of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
2Department
of Mathematics, University of Pittsburgh at Bradford, Bradford, PA 16701, USA
3Department
of Neurology, University of Pittsburgh Medical Center, Pittsburgh, PA 15213, USA
Abstract
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The human postural control system is difficult to quantify since it seems to be subject to both deterministic forces as well as stochastic effects. The attempt made in this paper is to study postural control under quiet stance on the one hand, and by engaging the brain through a fluency test, on the other. A Kistler electronic platform is the vehicle by way of which we gather observations in the form of center of pressure (COP) trajectories. From these two-dimensional trajectories we extract several measures that describe various features of the postural control system. Some of the measures are descriptive, while others incorporate physical forces that enter the process. From these measures we then build predictive models and apply them to a set of patients with Parkinson’s disease (PD) and a set of normal control subjects to validate and calibrate them. We further use the measures built out of the center of pressure trajectories to test the significance of the fluency (cognitive-motor dual task) effect on the two groups. The fluency effect is found significant in the parkinsonian group as well as the normal controls. The clinical importance of these findings lies in the fact that the models may be used as a more objective assessment of postural control that may either replace or supplement the more subjective Unified Parkinson’s Disease Rating Scale (UPDRS). The models may also be used as an assessment tool for the evaluation of patients subsequent to pharmacological and surgical treatment.
Keywords Human postural control; center of pressure; logistic regression
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1 Introduction The objective of this paper is to examine whether electronic force platform data obtained on the movement of the center of pressure under quiet stance can be used to produce measures that accurately assess the state of postural control. Models that predict the degree of balance impairment would also be useful to have. They may help to identify subjects with preclinical
Address correspondence to: Dr. G. Constantine, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA, Tel. 412-624-8308, Fax 412-624-8397, [email protected].
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balance impairment, including those at risk of developing PD. At the present time only expensive technology, such as positron emission technology (PET), is able to diagnose subjects in a preclinical or prodromal phase of PD. By contrast, biomathematical models of COP platform measures are easy to implement and inexpensive by comparison in the very early diagnosis of PD.
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We have collected data on 30 subjects with PG and 45 normal controls (of which 29 were first degree relatives of subjects with PD). A subject was asked to stand quietly on the platform for two minutes during which time data on the x- and y-coordinates of the movement of the center of pressure were recorded at a frequency of 50 registrations per second. We took five such readings on each subject. An additional five readings were taken on the patient under identical quiet stance conditions only by additionally exposing the patient to a verbal fluency tesk. Specifically, while on the platform, the subject was asked to generate as many words as possible that start with the same letter, during the alotted 2minute testing period Their brain is thus engaged in a cognitive-motor dual task while the COP data is collected. We thus generated 10 files of data for each subject; five from just quiet standing (to which we refer as baseline) and another five from quiet standing plus the fluency test (referred to as verbal fluency task).
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Section 2 describes the measures we created to obtain an accurate biomathematical of profile of the COP motion of the subjects. Some measures involve physical forces and velocities, while others are descriptive features of the COP trajectory. Taking these measures as input variables, in Section 3 we develop logistic models to rate the postural control of a subject. The models are developed on 80% of the data and their predictive accuracy tested on the remaining 20%. We devote Section 4 to a comparison of the two processes: baseline and fluency. The last section summarizes the findings of the paper and discusses possible clinical applications.
2 Measures associated to the trajectory
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Human postural control is highly complex; it involves the visual, vestibular and somatosensory systems, along with motor components; for a review, see Dietz (1992) and also Bloem et al. (1998). During quiet standing, the center of pressure (COP) under an individual’s feet, continually fluctuates in a stochastic manner. We developed and aim to utilize several numerical measures derived from the COP trajectories to characterize and quantify postural (and, in particular, parkinsonian) motor dysfunction. The measures used as independent variables in the logistic models developed in the next section are described below. Those that prove statistically significant in the models are described in greater detail. On some of these measures we report certain detailed statistics in the baseline framework. We may partition the measures into three kinds: Those based on the autocovariance of the time series of the x- and y-coordinates of the COP trajectories, Fourier transforms of these series, and descriptive measures of the COP trajectories themselves; see Brockwell and Davis (1991).
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Measures based on the autocovariance
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We take the x–coordinate of the COP trajectory and compute its autocovariance a(t) = Σs(x(s +t)−μ)(x(s)−μ), where μ is the mean of the series. Aspects of the autocovariance are captured in several measures. 1. The general stiffness measure—By developing a mechanical model of the human body, some dynamics of the quiet-standing COP motion were captured by Chow & Collins (1995). To simplify, they only considered the motion in the antero-posterior direction The dynamics of the body in the y-direction is given below. To see this relation, consider a localized impulsive force given to the quietly standing subject:
(1)
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where δ(t) and δ(z) are Dirac delta functions, ε is the ‘amplitude’ of the forcing which corresponds to an impulsive kick at t = 0 and z = 0, and η is the stochastic force. The kick induces a response y(z, t). The trial-averaged normalized response to this kick is given by the expected response function R(z, t) = E(y(z, t)/ε). Taking a Fourier transform in space and time, yields
(2)
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They then related the response to the autocovariance function a(t) to obtain the analytic result found in Chow & Collins (1995) and Lauk et al. (1999): where
(3)
J0(x) is the zeroth-order Bessel function. For 4αβ < 1, J0 is replaced by the zeroth-order modified Bessel function I0. Using a Levenberg-Marquardt algorithm (Press et al., 1992), we estimated the parameters in this equation from the COP trajectory data. The stiffness measure is , as estimated from data fitted to the last equation. We made the assumption that the standard deviations of the single-trial averages are Gaussian variables.
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2. The average residual measure—The COP motion of a subject is registered for two minutes at 50 readings per second; we repeat the process 5 times on the same subject. On each person we therefore have 5 COP time series, and we focus (for explanatory reasons only) on the x-coordinate of the 5 series. Data on each individual consists, therefore, of 5 time series of the x-coordinates of the COP motion. Each series is smoothed over intervals of 2 seconds, that is, we average 100 consecutive readings across the series; the residual vector is, by definition, the difference between the original series and the smoothed series.
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We view the smoothed series as a trend. The trend cannot be modeled formally in any traditional sense, since two series on the same individual can look very dissimilar in terms of trend. But the residuals show significant consistency if the smoothing is done over the same time interval. It is these residuals that we are taking interest in. One of the simplest statistics is the average squared residual, i.e., the squared norm of the residual vector divided by its length. We summarize the results on the average residuals in Table 1. Inspection of the data shows that the variance in the average residual visibly varies from subject to subject; a formal test rejects the hypothesis of equality of variances (the p-value is < 0.0001). A conservative approach (using always the larger estimate of the variance among two groups) yields the result that the parkinsonians have, on average, the average residual significantly greater than the other subjects (blood relatives and normal controls); p-value less than 0.0001 by using a t-test.
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Nonparametric tests offer similar conclusions. A Wilcoxon rank test finds that the median of the Parkinsonian average squared residual is significantly different from that of Nonparkinsonians (p-value < 0.0078). If we highlight the first degree relatives as a group, the PD patients have median squared residual significantly greater that each of the other two groups, but the first degree relatives and normal controls do not have median average residuals that are significantly different. These comparisons were also performed after taking away a smoothed trend over time intervals of five seconds, rather than two, and the results did not show any dramatic qualitative change.
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Table 2 below displays summary information on the correlation of the average residual with some clinical UPDRS scores. As in the case of the stiffness measure study that appeared in Constantine et al (2004), foot agility shows the highest correlation, with bradykinesia and UPDRS motor both at 58 percent. 3. The rate of decay of the autocovariance—The family of derivatives written in (3) does not appear to have antiderivatives that can easily be expressed in closed analytical form. The data indicate differences in the rate of decay of the autocovariance between the parkinsonians and normal controls. In order to study these rates we fitted a simple exponential decay function. More complicated functions were tried, but they offered no substantial gains to the qualitative understanding of the decay rates. Over a time interval of 30 seconds we fit the exponential family a(t) = ae−bt.
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The parameter a is simply the variance of the time series of the mediolateral COP coordinate (which we call mediolateral noise), while b is the rate of decay. The fit to data yields the least squares estimates of the rate of decay and intercept. For a given subject we then obtain an estimate of the variance from the five repeated platform readings. When categorized by presence or absence of PD we obtain an average decay rate of 14.9 with a standard error of 1.3 for PD patients, and 8.6 with a standard error of 1.1 for normals; by standard error we always understand the standard deviation of the estimate of the parameter under discussion. A formal test for equality of the two population means rejects the hypothesis that the means are equal in favor of the alternative that the parkinsonians have a significantly higher rate of decay of the autocovariane function; the p-value is < 0.001.
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With regard to the mediolateral noise of the model, the parkinsonians on average have mediolateral noise equal to 56.5 with a standard error of 7.6 while the normal controls register 17.5 with an standard error of 5.3. The conclusion is that the parkinsonians produce much greater noise in the mediolateral movement than normal subjects. An additional statistic, with potential clinical use, is the significant Spearman correlation of 0.51 that exists between the mediolateral noise and the number of falls that the subject experiences.
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4. The rebound—A measure of how much a person rebounds while trying to keep balance under quiet standing may be measured by the arc length of the autocovariance function per unit time. A long arc indicates significant rebounding efforts, whereas a short arc length is likely to be associated with autocovariance close to a straight line. We attempted to do a Fourier analysis in order to detect presence of cycles. Our experience has met with mixed success at best, since it appears that, in general, there are no clear cyclical effects. The rebounds, when present, have irregular structure with no evident periodicity. It appears that the arc length of the autocovariance function offers the best opportunity to numerically quantify this effect. We computed the arc length by normalizing the two axes to a same average increment, then summed the square roots of the sum of squares of the differenced time and autocovariance series; it is the usual formula for arc length. The rebound measure (or arc length of the autocovariance function) has a significant Spearman correlations with all the clinical variables listed in Table 3, and of about the same magnitude (plus or minus 10 percent). It has a Spearman correlation of 49.3 percent with the number of falls. Unlike the average residual, the rebound measure has the advantage of not being dependent on any smoothing, thus reflecting more objectively on the data.
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5. First crossing time—The waiting time until the autocovariance function a(t) hits zero for the first time is called the first crossing measure. It measures the time needed for the subject to lose postural control, in the following sense: if s denotes the tirst crossing time in seconds, then the position of the COP after s seconds is not related to its current position. Under Gaussian assumptions this means that the COP positions s seconds from now is independent from where it now is. This measure tends to take small values for parkinsonians and larger values for the normal controls, being consistent with the notion that parkinsonians are losing balance control sooner. The correlations with the UPDR scores are similar to those of the previously discussed measures. First crossing time is a measure that proves significant in the logistic model that we develop. Fourier transforms
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By computing the Fourier tansform of the x−coordinate of the COP series and integrating its absolute value over a range of clinical interest yields a Fourier measure. The idea is to capture differences between the groups at various frequency ranges. We selected to integrate up to 12.5 hertz for our specific selection as measure. The measure does not appear to be informative in discriminating parkinsonians from noemal, but it is very useful in detecting the fluency effect in both groups; see Section 4.
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Descriptive measures
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Several descriptive measure have been studied. Arc length of the COP trajectory proves useful and statistically significant in the logistic model. We also looked at the x-range and the y-range (sway amplitude of the COP projections. Though we realize the limitations of these last two measures and their sensitivity to ouliers, we found that the y-range is an important predictor of differences between groups, but not the x-range. Average velocities in the positive and negative directions along the coordinate axes were also studied. The forward velocity (along the y-axis) proves a useful predictor, but not the others.
3 The predictive models
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We develop a model that takes as input the platform measures and yields as output the probability that the subject in question belongs to a specific group. We investigated several predictive classes of models and found the logistic family to suit the task at hand best. There are several models that we developed, and we describe them below. The subjects are coded from 1 to 75, with the first 30 forming the parkinsonian group, and those with codes 31 to 75 being normal controls. Statistical analysis of all data taken on baseline conditions show that the y-axis projection of the COP trajectory data is a better predictor than the x-axis projection. The model we developed is a logistic regression that uses variables rebound, average residual, forward velocity, arc length, and crossing time as independent variables. Specifically, if p denotes the probability that a a (generic) subject is parkinsonian we express the (log-odds) ratio as a multiple regression of the independent variables
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where the β′s are unknown parameters subject to estimation, and the X′s represent the independent variables. In our case k = 5 and (X1,X2,X3,X4,X5) is the vector of the five variables in the order cited above. Maximum likelihood estimation yields estimates for the β ′s and exponentiation of the log-odds function then leads to the estimates of the individual probabilities.
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The model was developed on 80% of the available data and used to predict the remaining 20%. On the 80% of the data, under the hypothesis that the classification into groups is assigned randomly, we find a p-value less than .001, which reflects the fact that the model classifies correctly 18 out of the 24 parkinsonians selected in the 80% sample. The parameters associated to the variables were
Coefficient
Std. Error
t value
(Intercept)
2.906
1.938
1.499
ycrs.b
0.070
0.036
1.908
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Coefficient
Std. Error
t value
−0.378
0.129
−2.922
yavres.b
0.273
0.156
1.752
yarc.b
−0.405
0.182
−2.224
yvn.b
−6.805
3.044
−2.235
yreb.b
The 24 positions with lowest fitted values (low values are associated to the parkisonian group) are taken by subjects with codes written below 7 29 1 17 23 26 19 35 11 15 59 18 24 27 13 14 65 6 39 30 45 21 40 3
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among which we easily detect the 18 parkinsonians as the code numbers no greater than 30. Furthermore, of the 6 controls misclassified as parkinsonians 5 are family members of parkinsonians (those with codes between 31 and 60, inclusive). This allows the potential study of these subjects further since these subjects may be in a very early (preclinical or prodromal) stage of PD. Selective subjects identified to be at risk of developing PD may then undergo the more expensive brain PET imaging technique to confirm the diagnosis. Once diagnosed with PD, these subjects may be selected for novel preventive or symptomatic anti-PD treatment strategies. Prediction using this model on the 20% of the remaining data leads to correct classification of 4 out of the 6 remaining parkinsonians. On the entire data the model correctly predicts 22 out of the 30 parkinsonians. The model yields,
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Coefficient
Std. Error
t value
(Intercept)
1.959
1.652
1.185
ycrs.b
0.067
0.033
2.040
yreb.b
−0.405
0.118
−3.427
yavres.b
0.321
0.128
2.493
yarc.b
−0.326
0.135
−2.399
yvn.b
−6.255
2.563
−2.440
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To assess the significance of this prediction we note that, if subjects are randomly assigned into the two groups, a hypergeometric distribution results, yielding 12 parkinsonians correctly assigned with a standard error of 2.4; correct classification of 22 parkinsonians under such assumption carries a probability of less than 0.001. We thus conclude that the model is a helpful classification tool.
4 Identification of the verbal fluency effect To study the verbal fluency cognitive-motor dual task effect we perform paired comparisons, that is, fluency - baseline across parkinsonian subjects, then across normals. Since the data Int J Appl Math (Sofia). Author manuscript; available in PMC 2017 September 25.
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consists of measures that are averages, t-tests are appropriate to test the hypothesis that there is no difference between fluency and baseline. Several measures capture significant differences. In the case of the parkinsonians we perceive significant differences in the measures listed in Table 3. We conclude that the hypothesis of a null fluency effect is untenable and should be rejected in favor of a significant non-zero effect for each of the measures listed in Table 3. The associated p-values are for one-sided departures from the null hypothesis. Paired comparisons of the fluency effect across the normal subjects yield the results in Table 4. Data informs that the fluency effect is significantly different from zero in normal subjects as well.
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Measures that perceive differences in both groups of subjects are: the x- and y-Fourier measures, the arc length of the two-dimensional trajectory, and a mediolateral velocity. It is interesting that in parkinsonians all velocities are significantly affected by the fluency test but not in normals. On the other hand, the stiffness measure detects the fluency effect in normals but not in parkinsonians. The main conclusion is that a verbal fluency effect is clearly differentially present in the PD patient and control subject populations. Performance of a verbal fluency task in the PD patients alters the velocity of the COP motion whereas this dual-task performance selectively affects measures of rigidity in the control subjects. Therefore, performance on the verbal fluency task may have diagnostic value in the diagnosis of subjects suspected of having PD.
5 Discussion and Conclusions Author Manuscript Author Manuscript
Langston and Koller (1991) noted that PD can be viewed as a disease having two phases. The first phase is a preclinical period that covers the period from disease inception to the time when the disease becomes symptomatic. The second phase represents the symptomatic period where the classical symptoms of PD such as bradykinesia, tremor and rigidity occur. Before the appearance of the classic parkinsonian signs and symptoms, there may be a prodromal stage where signs and symptoms occur but do not specifically indicate PD (Horstink and Morrish, 1999). The clinical diagnosis of PD is most difficult early in the disease when the signs and symptoms are subtle (Koller and Montgomery, 1997). Montgomery et al. (1999), using a test battery evaluating motor function, found that 22.5 percent of first degree relatives of patients with PD had abnormal test scores compared to 9 percent of normal control subjects of the general population. None of the subjects who had an abnormal test score had neurological signs or symptoms of PD. The increased frequency of abnormal test results in first degree relatives of patients with PD may represent a genetic or shared environmental factor in the pathogenesis of PD (or a combination of both). The causes of PD are being unravelled and neuroprotective therapy is close to reality. Neuroprotective therapies are likely more effective when given early in the course of the disease rather than late. Therefore, identification of very early disease has become crucial to select subjects for treatments that may have the potential of secondary prevention of PD. A biomarker, to be useful in screening large populations to identify preclinical disease, should
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be inexpensive, easily administered, and sufficiently sensitive and specific to avoid unacceptable false negatives or positives (Tetrud, 1991). Measures of dopamine nerve terminal integrity with positron emission tomography (PET) have allowed preclinical disease to be detected in relatives of patients with PD (Bernheimer et al., 1973, Brooks, 2000). However, PET imaging is very expensive and not suitable for mass screening of at-risk subjects. In this paper we focused on the development of measures of postural stiffness and motor dysfunction by making use of a Kistler electronic platform. There is preliminary evidence that some of these measures are significantly correlated with descriptive clinical assessments, such as those used in the Unified Parkinson’s Disease Rating Scale, particularly the Motor Sub-scale (Lauk et al., 1999). The platform measures are more objective measures than those offered by the UPDRS. The predictive model that we developed here has the potential to identify subjects with or who are at risk of developing PD. Although PET imaging can be used for the diagnosis of PD, the high costs prohibit the use of PET imaging for screening purposes of large numbers of subjects. Therefore, the above described platform measures can be used as an inexpensive and easy-to-use screening tool in a medical doctor’s office. Subjects with abnormal screening results can then undergo the more expensive PET imaging technique to confirm the diagnosis of PD and then be selected for novel preventive treatment strategies. We are now in the process of collecting covariate data on combined platform and PET imaging testing in first degree relatives of patients with PD in an on-going study at the University of Pittsburgh Medical Center.
Acknowledgments This work was supported by grants from the Scaife Family Foundation - Pittsburgh, the NIH, and the Department of Veterans Affairs.
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References
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Bernheimer H, Birkmayer W, Hornykiewicz O, Jellinger K, Seitelberger F. Brain dopamine and the syndromes of Parkinson and Huntington. Clinical, morphological and neurochemical correlations. J Neurol Sci. 1973; 20:415–55. [PubMed: 4272516] Bloem BR, Beckley DJ, van Hilten BJ, Roos RA. Clinimetrics of postural instability in Parkinson’s disease. J Neurol. 1998; 245:669–73. [PubMed: 9776467] Brockwell, PJ., Davis, RA. Time Series: Theory and Methods. New York: Springer–Verlag; 1991. Brooks DJ. Morphological and functional imaging studies on the diagnosis and progression of Parkinson’s disease. J. Neurol. 2000; 247(Suppl 2):11–18. Chow CC, Collins JJ. Pinned polymer model of posture control. Physical Review E. 1995; 52:907– 912. Constantine GM, Bohnen NI, Chow CC. Electronic platform measures of balance impairment in parkinsonians and first degree relatives, I. Journ of Pure and Applied Mathematics. 2004; 13:259– 273. Dietz V. Human neuronal control of automatic functional movements: interactions between central programs and afferent input. Physiological Reviews. 1992; 72:33–69. [PubMed: 1731372] Horstink, MWIM., Morrish, PK. Preclinical diagnosis of Parkinson’s disease. In: Stern, GM., editor. Parkinson’s disease: Advances in Neurology. Vol. 80. Philadelphia: Lippincott Williams Wilkins; 1999. p. 327-33. Koller W, Montgomery E. Issues in the early diagnosis of Parkinson’s disease. Neurology. 1997; 49(Suppl 1):S10–S25. Langston JW, Koller WC. The next frontier in Parkinson’s disease: presymptomatic detection. Neurology. 1991; 41(Suppl 2):8–13. [PubMed: 2041599]
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Lauk M, Chow CC, Lipsitz LA, Mitchell SL, Collins JJ. Assessing muscle stiffness from quiet stance: applicability to Parkinson’s disease. Muscle & Nerve. 1999; 22:635–639. [PubMed: 10331364] Montgomery EBJ, Baker KB, Lyons K, Koller WC. Abnormal performance on the PD test battery by asymptomatic first-degree relatives. Neurology. 1999; 52:757–62. [PubMed: 10078723] Press, W., Flannery, B., Saul, S., Vetterling, W. Numerical Recipes. 2. Cambridge: Cambridge University Press; 1992. Tetrud JW. Preclinical Parkinson’ disease: Detection of motor and non-motor manifestations. Neurology. 1991; 41(Suppl 2):69–72.
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TABLE 1
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Average squared residual findings Number of Subjects
Average Squared Residual (ASR)
Standard Deviation of ASR
Parkinsonians
30
14.27
1.05
Nonparkinsonians
45
3.78
0.22
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TABLE 2
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Spearman correlations between UPDRS measures and the average squared residual Clinical measure
Spearman correlation coefficient
Rigidity
0.58 (P < 0.0009)
Bradykinesia
0.55 (P < 0.0010)
Posture
0.63 (P < 0.0001)
Foot agility (right)
0.52 (P < 0.0002)
Foot agility (left)
0.68 (P < 0.0000)
UPDRS motor score
0.58 (P < 0.0006)
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2.4517
0.0102
t-value
p-value
xf
0.0044
−2.8105
xrate
0.0191
2.1713
arc
0.0013
3.2998
yf
0.0262
2.0227
yreb
0.0071
2.6111
xp
0.0108
2.4267
xvp
0.0081
−2.5553
xvn
0.0030
2.9591
yvp
0.0052
−2.7362
yvn
Fluency effects for parkinsonians: the t-values and associated p-values for each Measure
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TABLE 3 Constantine et al. Page 13
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3.3206
0.0009
t-value
p-value
xf
0.0097
−2.4260
xarea
0.0162
2.2084
xaveres
0.0026
2.9474
arc
0.0134
2.2909
yf
0.0013
−3.1964
yarea
0.0162
2.2084
yaveres
0.0080
2.5068
xvb
Fluency effects for normal subjects: the t-values and associated p-values for each measure
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TABLE 4 Constantine et al. Page 14
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Int J Appl Math (Sofia). Author manuscript; available in PMC 2017 September 25. Published in final edited form as: Int J Appl Math (Sofia). 2005 ; 18(4): 487–500.
Predictive models of postural control based on electronic force platform measures in patients with Parkinson’s disease Gregory M. Constantine1, Marius G. Buliga2, Larry S. Ivanco3, Robert Y. Moore3, and Nicolaas I. Bohnen3
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1Department
of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA
2Department
of Mathematics, University of Pittsburgh at Bradford, Bradford, PA 16701, USA
3Department
of Neurology, University of Pittsburgh Medical Center, Pittsburgh, PA 15213, USA
Abstract
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The human postural control system is difficult to quantify since it seems to be subject to both deterministic forces as well as stochastic effects. The attempt made in this paper is to study postural control under quiet stance on the one hand, and by engaging the brain through a fluency test, on the other. A Kistler electronic platform is the vehicle by way of which we gather observations in the form of center of pressure (COP) trajectories. From these two-dimensional trajectories we extract several measures that describe various features of the postural control system. Some of the measures are descriptive, while others incorporate physical forces that enter the process. From these measures we then build predictive models and apply them to a set of patients with Parkinson’s disease (PD) and a set of normal control subjects to validate and calibrate them. We further use the measures built out of the center of pressure trajectories to test the significance of the fluency (cognitive-motor dual task) effect on the two groups. The fluency effect is found significant in the parkinsonian group as well as the normal controls. The clinical importance of these findings lies in the fact that the models may be used as a more objective assessment of postural control that may either replace or supplement the more subjective Unified Parkinson’s Disease Rating Scale (UPDRS). The models may also be used as an assessment tool for the evaluation of patients subsequent to pharmacological and surgical treatment.
Keywords Human postural control; center of pressure; logistic regression
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1 Introduction The objective of this paper is to examine whether electronic force platform data obtained on the movement of the center of pressure under quiet stance can be used to produce measures that accurately assess the state of postural control. Models that predict the degree of balance impairment would also be useful to have. They may help to identify subjects with preclinical
Address correspondence to: Dr. G. Constantine, Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA, Tel. 412-624-8308, Fax 412-624-8397, [email protected].
Constantine et al.
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balance impairment, including those at risk of developing PD. At the present time only expensive technology, such as positron emission technology (PET), is able to diagnose subjects in a preclinical or prodromal phase of PD. By contrast, biomathematical models of COP platform measures are easy to implement and inexpensive by comparison in the very early diagnosis of PD.
Author Manuscript
We have collected data on 30 subjects with PG and 45 normal controls (of which 29 were first degree relatives of subjects with PD). A subject was asked to stand quietly on the platform for two minutes during which time data on the x- and y-coordinates of the movement of the center of pressure were recorded at a frequency of 50 registrations per second. We took five such readings on each subject. An additional five readings were taken on the patient under identical quiet stance conditions only by additionally exposing the patient to a verbal fluency tesk. Specifically, while on the platform, the subject was asked to generate as many words as possible that start with the same letter, during the alotted 2minute testing period Their brain is thus engaged in a cognitive-motor dual task while the COP data is collected. We thus generated 10 files of data for each subject; five from just quiet standing (to which we refer as baseline) and another five from quiet standing plus the fluency test (referred to as verbal fluency task).
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Section 2 describes the measures we created to obtain an accurate biomathematical of profile of the COP motion of the subjects. Some measures involve physical forces and velocities, while others are descriptive features of the COP trajectory. Taking these measures as input variables, in Section 3 we develop logistic models to rate the postural control of a subject. The models are developed on 80% of the data and their predictive accuracy tested on the remaining 20%. We devote Section 4 to a comparison of the two processes: baseline and fluency. The last section summarizes the findings of the paper and discusses possible clinical applications.
2 Measures associated to the trajectory
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Human postural control is highly complex; it involves the visual, vestibular and somatosensory systems, along with motor components; for a review, see Dietz (1992) and also Bloem et al. (1998). During quiet standing, the center of pressure (COP) under an individual’s feet, continually fluctuates in a stochastic manner. We developed and aim to utilize several numerical measures derived from the COP trajectories to characterize and quantify postural (and, in particular, parkinsonian) motor dysfunction. The measures used as independent variables in the logistic models developed in the next section are described below. Those that prove statistically significant in the models are described in greater detail. On some of these measures we report certain detailed statistics in the baseline framework. We may partition the measures into three kinds: Those based on the autocovariance of the time series of the x- and y-coordinates of the COP trajectories, Fourier transforms of these series, and descriptive measures of the COP trajectories themselves; see Brockwell and Davis (1991).
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Measures based on the autocovariance
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We take the x–coordinate of the COP trajectory and compute its autocovariance a(t) = Σs(x(s +t)−μ)(x(s)−μ), where μ is the mean of the series. Aspects of the autocovariance are captured in several measures. 1. The general stiffness measure—By developing a mechanical model of the human body, some dynamics of the quiet-standing COP motion were captured by Chow & Collins (1995). To simplify, they only considered the motion in the antero-posterior direction The dynamics of the body in the y-direction is given below. To see this relation, consider a localized impulsive force given to the quietly standing subject:
(1)
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where δ(t) and δ(z) are Dirac delta functions, ε is the ‘amplitude’ of the forcing which corresponds to an impulsive kick at t = 0 and z = 0, and η is the stochastic force. The kick induces a response y(z, t). The trial-averaged normalized response to this kick is given by the expected response function R(z, t) = E(y(z, t)/ε). Taking a Fourier transform in space and time, yields
(2)
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They then related the response to the autocovariance function a(t) to obtain the analytic result found in Chow & Collins (1995) and Lauk et al. (1999): where
(3)
J0(x) is the zeroth-order Bessel function. For 4αβ < 1, J0 is replaced by the zeroth-order modified Bessel function I0. Using a Levenberg-Marquardt algorithm (Press et al., 1992), we estimated the parameters in this equation from the COP trajectory data. The stiffness measure is , as estimated from data fitted to the last equation. We made the assumption that the standard deviations of the single-trial averages are Gaussian variables.
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2. The average residual measure—The COP motion of a subject is registered for two minutes at 50 readings per second; we repeat the process 5 times on the same subject. On each person we therefore have 5 COP time series, and we focus (for explanatory reasons only) on the x-coordinate of the 5 series. Data on each individual consists, therefore, of 5 time series of the x-coordinates of the COP motion. Each series is smoothed over intervals of 2 seconds, that is, we average 100 consecutive readings across the series; the residual vector is, by definition, the difference between the original series and the smoothed series.
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We view the smoothed series as a trend. The trend cannot be modeled formally in any traditional sense, since two series on the same individual can look very dissimilar in terms of trend. But the residuals show significant consistency if the smoothing is done over the same time interval. It is these residuals that we are taking interest in. One of the simplest statistics is the average squared residual, i.e., the squared norm of the residual vector divided by its length. We summarize the results on the average residuals in Table 1. Inspection of the data shows that the variance in the average residual visibly varies from subject to subject; a formal test rejects the hypothesis of equality of variances (the p-value is < 0.0001). A conservative approach (using always the larger estimate of the variance among two groups) yields the result that the parkinsonians have, on average, the average residual significantly greater than the other subjects (blood relatives and normal controls); p-value less than 0.0001 by using a t-test.
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Nonparametric tests offer similar conclusions. A Wilcoxon rank test finds that the median of the Parkinsonian average squared residual is significantly different from that of Nonparkinsonians (p-value < 0.0078). If we highlight the first degree relatives as a group, the PD patients have median squared residual significantly greater that each of the other two groups, but the first degree relatives and normal controls do not have median average residuals that are significantly different. These comparisons were also performed after taking away a smoothed trend over time intervals of five seconds, rather than two, and the results did not show any dramatic qualitative change.
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Table 2 below displays summary information on the correlation of the average residual with some clinical UPDRS scores. As in the case of the stiffness measure study that appeared in Constantine et al (2004), foot agility shows the highest correlation, with bradykinesia and UPDRS motor both at 58 percent. 3. The rate of decay of the autocovariance—The family of derivatives written in (3) does not appear to have antiderivatives that can easily be expressed in closed analytical form. The data indicate differences in the rate of decay of the autocovariance between the parkinsonians and normal controls. In order to study these rates we fitted a simple exponential decay function. More complicated functions were tried, but they offered no substantial gains to the qualitative understanding of the decay rates. Over a time interval of 30 seconds we fit the exponential family a(t) = ae−bt.
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The parameter a is simply the variance of the time series of the mediolateral COP coordinate (which we call mediolateral noise), while b is the rate of decay. The fit to data yields the least squares estimates of the rate of decay and intercept. For a given subject we then obtain an estimate of the variance from the five repeated platform readings. When categorized by presence or absence of PD we obtain an average decay rate of 14.9 with a standard error of 1.3 for PD patients, and 8.6 with a standard error of 1.1 for normals; by standard error we always understand the standard deviation of the estimate of the parameter under discussion. A formal test for equality of the two population means rejects the hypothesis that the means are equal in favor of the alternative that the parkinsonians have a significantly higher rate of decay of the autocovariane function; the p-value is < 0.001.
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With regard to the mediolateral noise of the model, the parkinsonians on average have mediolateral noise equal to 56.5 with a standard error of 7.6 while the normal controls register 17.5 with an standard error of 5.3. The conclusion is that the parkinsonians produce much greater noise in the mediolateral movement than normal subjects. An additional statistic, with potential clinical use, is the significant Spearman correlation of 0.51 that exists between the mediolateral noise and the number of falls that the subject experiences.
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4. The rebound—A measure of how much a person rebounds while trying to keep balance under quiet standing may be measured by the arc length of the autocovariance function per unit time. A long arc indicates significant rebounding efforts, whereas a short arc length is likely to be associated with autocovariance close to a straight line. We attempted to do a Fourier analysis in order to detect presence of cycles. Our experience has met with mixed success at best, since it appears that, in general, there are no clear cyclical effects. The rebounds, when present, have irregular structure with no evident periodicity. It appears that the arc length of the autocovariance function offers the best opportunity to numerically quantify this effect. We computed the arc length by normalizing the two axes to a same average increment, then summed the square roots of the sum of squares of the differenced time and autocovariance series; it is the usual formula for arc length. The rebound measure (or arc length of the autocovariance function) has a significant Spearman correlations with all the clinical variables listed in Table 3, and of about the same magnitude (plus or minus 10 percent). It has a Spearman correlation of 49.3 percent with the number of falls. Unlike the average residual, the rebound measure has the advantage of not being dependent on any smoothing, thus reflecting more objectively on the data.
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5. First crossing time—The waiting time until the autocovariance function a(t) hits zero for the first time is called the first crossing measure. It measures the time needed for the subject to lose postural control, in the following sense: if s denotes the tirst crossing time in seconds, then the position of the COP after s seconds is not related to its current position. Under Gaussian assumptions this means that the COP positions s seconds from now is independent from where it now is. This measure tends to take small values for parkinsonians and larger values for the normal controls, being consistent with the notion that parkinsonians are losing balance control sooner. The correlations with the UPDR scores are similar to those of the previously discussed measures. First crossing time is a measure that proves significant in the logistic model that we develop. Fourier transforms
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By computing the Fourier tansform of the x−coordinate of the COP series and integrating its absolute value over a range of clinical interest yields a Fourier measure. The idea is to capture differences between the groups at various frequency ranges. We selected to integrate up to 12.5 hertz for our specific selection as measure. The measure does not appear to be informative in discriminating parkinsonians from noemal, but it is very useful in detecting the fluency effect in both groups; see Section 4.
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Descriptive measures
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Several descriptive measure have been studied. Arc length of the COP trajectory proves useful and statistically significant in the logistic model. We also looked at the x-range and the y-range (sway amplitude of the COP projections. Though we realize the limitations of these last two measures and their sensitivity to ouliers, we found that the y-range is an important predictor of differences between groups, but not the x-range. Average velocities in the positive and negative directions along the coordinate axes were also studied. The forward velocity (along the y-axis) proves a useful predictor, but not the others.
3 The predictive models
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We develop a model that takes as input the platform measures and yields as output the probability that the subject in question belongs to a specific group. We investigated several predictive classes of models and found the logistic family to suit the task at hand best. There are several models that we developed, and we describe them below. The subjects are coded from 1 to 75, with the first 30 forming the parkinsonian group, and those with codes 31 to 75 being normal controls. Statistical analysis of all data taken on baseline conditions show that the y-axis projection of the COP trajectory data is a better predictor than the x-axis projection. The model we developed is a logistic regression that uses variables rebound, average residual, forward velocity, arc length, and crossing time as independent variables. Specifically, if p denotes the probability that a a (generic) subject is parkinsonian we express the (log-odds) ratio as a multiple regression of the independent variables
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where the β′s are unknown parameters subject to estimation, and the X′s represent the independent variables. In our case k = 5 and (X1,X2,X3,X4,X5) is the vector of the five variables in the order cited above. Maximum likelihood estimation yields estimates for the β ′s and exponentiation of the log-odds function then leads to the estimates of the individual probabilities.
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The model was developed on 80% of the available data and used to predict the remaining 20%. On the 80% of the data, under the hypothesis that the classification into groups is assigned randomly, we find a p-value less than .001, which reflects the fact that the model classifies correctly 18 out of the 24 parkinsonians selected in the 80% sample. The parameters associated to the variables were
Coefficient
Std. Error
t value
(Intercept)
2.906
1.938
1.499
ycrs.b
0.070
0.036
1.908
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Coefficient
Std. Error
t value
−0.378
0.129
−2.922
yavres.b
0.273
0.156
1.752
yarc.b
−0.405
0.182
−2.224
yvn.b
−6.805
3.044
−2.235
yreb.b
The 24 positions with lowest fitted values (low values are associated to the parkisonian group) are taken by subjects with codes written below 7 29 1 17 23 26 19 35 11 15 59 18 24 27 13 14 65 6 39 30 45 21 40 3
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among which we easily detect the 18 parkinsonians as the code numbers no greater than 30. Furthermore, of the 6 controls misclassified as parkinsonians 5 are family members of parkinsonians (those with codes between 31 and 60, inclusive). This allows the potential study of these subjects further since these subjects may be in a very early (preclinical or prodromal) stage of PD. Selective subjects identified to be at risk of developing PD may then undergo the more expensive brain PET imaging technique to confirm the diagnosis. Once diagnosed with PD, these subjects may be selected for novel preventive or symptomatic anti-PD treatment strategies. Prediction using this model on the 20% of the remaining data leads to correct classification of 4 out of the 6 remaining parkinsonians. On the entire data the model correctly predicts 22 out of the 30 parkinsonians. The model yields,
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Coefficient
Std. Error
t value
(Intercept)
1.959
1.652
1.185
ycrs.b
0.067
0.033
2.040
yreb.b
−0.405
0.118
−3.427
yavres.b
0.321
0.128
2.493
yarc.b
−0.326
0.135
−2.399
yvn.b
−6.255
2.563
−2.440
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To assess the significance of this prediction we note that, if subjects are randomly assigned into the two groups, a hypergeometric distribution results, yielding 12 parkinsonians correctly assigned with a standard error of 2.4; correct classification of 22 parkinsonians under such assumption carries a probability of less than 0.001. We thus conclude that the model is a helpful classification tool.
4 Identification of the verbal fluency effect To study the verbal fluency cognitive-motor dual task effect we perform paired comparisons, that is, fluency - baseline across parkinsonian subjects, then across normals. Since the data Int J Appl Math (Sofia). Author manuscript; available in PMC 2017 September 25.
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consists of measures that are averages, t-tests are appropriate to test the hypothesis that there is no difference between fluency and baseline. Several measures capture significant differences. In the case of the parkinsonians we perceive significant differences in the measures listed in Table 3. We conclude that the hypothesis of a null fluency effect is untenable and should be rejected in favor of a significant non-zero effect for each of the measures listed in Table 3. The associated p-values are for one-sided departures from the null hypothesis. Paired comparisons of the fluency effect across the normal subjects yield the results in Table 4. Data informs that the fluency effect is significantly different from zero in normal subjects as well.
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Measures that perceive differences in both groups of subjects are: the x- and y-Fourier measures, the arc length of the two-dimensional trajectory, and a mediolateral velocity. It is interesting that in parkinsonians all velocities are significantly affected by the fluency test but not in normals. On the other hand, the stiffness measure detects the fluency effect in normals but not in parkinsonians. The main conclusion is that a verbal fluency effect is clearly differentially present in the PD patient and control subject populations. Performance of a verbal fluency task in the PD patients alters the velocity of the COP motion whereas this dual-task performance selectively affects measures of rigidity in the control subjects. Therefore, performance on the verbal fluency task may have diagnostic value in the diagnosis of subjects suspected of having PD.
5 Discussion and Conclusions Author Manuscript Author Manuscript
Langston and Koller (1991) noted that PD can be viewed as a disease having two phases. The first phase is a preclinical period that covers the period from disease inception to the time when the disease becomes symptomatic. The second phase represents the symptomatic period where the classical symptoms of PD such as bradykinesia, tremor and rigidity occur. Before the appearance of the classic parkinsonian signs and symptoms, there may be a prodromal stage where signs and symptoms occur but do not specifically indicate PD (Horstink and Morrish, 1999). The clinical diagnosis of PD is most difficult early in the disease when the signs and symptoms are subtle (Koller and Montgomery, 1997). Montgomery et al. (1999), using a test battery evaluating motor function, found that 22.5 percent of first degree relatives of patients with PD had abnormal test scores compared to 9 percent of normal control subjects of the general population. None of the subjects who had an abnormal test score had neurological signs or symptoms of PD. The increased frequency of abnormal test results in first degree relatives of patients with PD may represent a genetic or shared environmental factor in the pathogenesis of PD (or a combination of both). The causes of PD are being unravelled and neuroprotective therapy is close to reality. Neuroprotective therapies are likely more effective when given early in the course of the disease rather than late. Therefore, identification of very early disease has become crucial to select subjects for treatments that may have the potential of secondary prevention of PD. A biomarker, to be useful in screening large populations to identify preclinical disease, should
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be inexpensive, easily administered, and sufficiently sensitive and specific to avoid unacceptable false negatives or positives (Tetrud, 1991). Measures of dopamine nerve terminal integrity with positron emission tomography (PET) have allowed preclinical disease to be detected in relatives of patients with PD (Bernheimer et al., 1973, Brooks, 2000). However, PET imaging is very expensive and not suitable for mass screening of at-risk subjects. In this paper we focused on the development of measures of postural stiffness and motor dysfunction by making use of a Kistler electronic platform. There is preliminary evidence that some of these measures are significantly correlated with descriptive clinical assessments, such as those used in the Unified Parkinson’s Disease Rating Scale, particularly the Motor Sub-scale (Lauk et al., 1999). The platform measures are more objective measures than those offered by the UPDRS. The predictive model that we developed here has the potential to identify subjects with or who are at risk of developing PD. Although PET imaging can be used for the diagnosis of PD, the high costs prohibit the use of PET imaging for screening purposes of large numbers of subjects. Therefore, the above described platform measures can be used as an inexpensive and easy-to-use screening tool in a medical doctor’s office. Subjects with abnormal screening results can then undergo the more expensive PET imaging technique to confirm the diagnosis of PD and then be selected for novel preventive treatment strategies. We are now in the process of collecting covariate data on combined platform and PET imaging testing in first degree relatives of patients with PD in an on-going study at the University of Pittsburgh Medical Center.
Acknowledgments This work was supported by grants from the Scaife Family Foundation - Pittsburgh, the NIH, and the Department of Veterans Affairs.
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References
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Bernheimer H, Birkmayer W, Hornykiewicz O, Jellinger K, Seitelberger F. Brain dopamine and the syndromes of Parkinson and Huntington. Clinical, morphological and neurochemical correlations. J Neurol Sci. 1973; 20:415–55. [PubMed: 4272516] Bloem BR, Beckley DJ, van Hilten BJ, Roos RA. Clinimetrics of postural instability in Parkinson’s disease. J Neurol. 1998; 245:669–73. [PubMed: 9776467] Brockwell, PJ., Davis, RA. Time Series: Theory and Methods. New York: Springer–Verlag; 1991. Brooks DJ. Morphological and functional imaging studies on the diagnosis and progression of Parkinson’s disease. J. Neurol. 2000; 247(Suppl 2):11–18. Chow CC, Collins JJ. Pinned polymer model of posture control. Physical Review E. 1995; 52:907– 912. Constantine GM, Bohnen NI, Chow CC. Electronic platform measures of balance impairment in parkinsonians and first degree relatives, I. Journ of Pure and Applied Mathematics. 2004; 13:259– 273. Dietz V. Human neuronal control of automatic functional movements: interactions between central programs and afferent input. Physiological Reviews. 1992; 72:33–69. [PubMed: 1731372] Horstink, MWIM., Morrish, PK. Preclinical diagnosis of Parkinson’s disease. In: Stern, GM., editor. Parkinson’s disease: Advances in Neurology. Vol. 80. Philadelphia: Lippincott Williams Wilkins; 1999. p. 327-33. Koller W, Montgomery E. Issues in the early diagnosis of Parkinson’s disease. Neurology. 1997; 49(Suppl 1):S10–S25. Langston JW, Koller WC. The next frontier in Parkinson’s disease: presymptomatic detection. Neurology. 1991; 41(Suppl 2):8–13. [PubMed: 2041599]
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Lauk M, Chow CC, Lipsitz LA, Mitchell SL, Collins JJ. Assessing muscle stiffness from quiet stance: applicability to Parkinson’s disease. Muscle & Nerve. 1999; 22:635–639. [PubMed: 10331364] Montgomery EBJ, Baker KB, Lyons K, Koller WC. Abnormal performance on the PD test battery by asymptomatic first-degree relatives. Neurology. 1999; 52:757–62. [PubMed: 10078723] Press, W., Flannery, B., Saul, S., Vetterling, W. Numerical Recipes. 2. Cambridge: Cambridge University Press; 1992. Tetrud JW. Preclinical Parkinson’ disease: Detection of motor and non-motor manifestations. Neurology. 1991; 41(Suppl 2):69–72.
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TABLE 1
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Average squared residual findings Number of Subjects
Average Squared Residual (ASR)
Standard Deviation of ASR
Parkinsonians
30
14.27
1.05
Nonparkinsonians
45
3.78
0.22
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TABLE 2
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Spearman correlations between UPDRS measures and the average squared residual Clinical measure
Spearman correlation coefficient
Rigidity
0.58 (P < 0.0009)
Bradykinesia
0.55 (P < 0.0010)
Posture
0.63 (P < 0.0001)
Foot agility (right)
0.52 (P < 0.0002)
Foot agility (left)
0.68 (P < 0.0000)
UPDRS motor score
0.58 (P < 0.0006)
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2.4517
0.0102
t-value
p-value
xf
0.0044
−2.8105
xrate
0.0191
2.1713
arc
0.0013
3.2998
yf
0.0262
2.0227
yreb
0.0071
2.6111
xp
0.0108
2.4267
xvp
0.0081
−2.5553
xvn
0.0030
2.9591
yvp
0.0052
−2.7362
yvn
Fluency effects for parkinsonians: the t-values and associated p-values for each Measure
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3.3206
0.0009
t-value
p-value
xf
0.0097
−2.4260
xarea
0.0162
2.2084
xaveres
0.0026
2.9474
arc
0.0134
2.2909
yf
0.0013
−3.1964
yarea
0.0162
2.2084
yaveres
0.0080
2.5068
xvb
Fluency effects for normal subjects: the t-values and associated p-values for each measure
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