Diagnosis of Parkinson's Disease: A Limit Cycle Approach

Gaurav K. Singh, Vrutangkumar V. Shah, Harish J. Palanthandalam-Madapusi, SysIDEA Lab, IIT Gandhinagar

2013 IEEE International Conference on Control Applications (CCA) Part of 2013 IEEE Multi-Conference on Systems and Control Hyderabad, India, August 28-30, 2013

Abstract— Parkinson's disease is characterized by increased reaction times in both voluntary and involuntary motor responses and often results in unintended oscillatory motion of body parts, termed as Parkinsonian tremor. A simple and efficient method for diagnosing Parkinson's disease is still not available and furthermore, on the correct diagnosis of Parkinson's disease, estimating the severity of the disease is challenging. This is particularly important since the medications typically prescribed to address the symptoms of Parkinson's disease have adverse side effects and thus the dosages have to be finely optimized. In this paper, based on a systems theory perspective and a recently suggested control-system analogy [11], a simple method for the diagnosis of Parkinson's disease is explored. This paper also discusses the possibility of developing a low-cost diagnostic device and strategies for the same. Future work will focus on using data from real patients to validate the hypotheses and results in this paper.

I. INTRODUCTION

Parkinson's disease (PD) is an idiopathic and degenerative disorder of the central nervous system [1]. It is characterized by increased reaction times in both voluntary and involuntary motor responses [2] and often results in unintended oscillatory motion of body parts, especially in hands, termed as Parkinsonian tremor. These tremors are characterized by involuntary tremulous motion in parts not in action [3] [4]. Unfortunately, there is no lab test that will clearly identify the disease, but computer tomography (CT) and magnetic resonance imaging (MRI) brain scans, though usually appear normal with PD patients, are sometimes used to rule out disorders that could give rise to similar symptoms [5]. There are some well-reviewed therapies, such as deep brain stimulation (DBS), that suppress Parkinsonian tremor, but the fundamental mechanism behind Parkinsonian tremor and these therapies remains a mystery [6]. Moreover, DBS is at a relative technological standstill due to several factors: limited choice of stimulus waveforms, ability to stimulate at only a single location, and inefficient use of battery power.

G. K. Singh is an undergraduate student at NIT Tiruchirappalli, V. V. Shah is a Ph.D. scholar in Mechanical Engineering at Indian Institute of Technology (IIT) Gandhinagar, H. J. Palanthandalam-Madapusi is an assistant professor in Mechanical Engineering at IIT Gandhinagar (corresponding email: harish@iitgn.ac.in, Phone: +91 79 3245 9899). *This research was funded in part by IIT Gandhinagar.

For further understanding the mechanism behind Parkinsonian tremor, there have been two approaches to use model Parkinsonian tremor. The first approach is to experimental data and apply statistical tests to it to determine properties of the tremor such as frequency of oscillations, stochastic properties of the data, nonlinearity, etc. [7]. The objective of such an approach is to develop diagnostic tools that use statistical tests as diagnostic tools for Parkinson's disease. Such efforts have observed that the Parkinsonian tremors are nonlinear and deterministic, whereas a physiological tremor (arising in non-diseased subjects) is linear and stochastic. Although these statistical tests offer quick and simple diagnostic tools, they are not based on any known features or properties of the underlying disease and hence may not be applicable reliably over a wide range of cases and scenarios.

The second approach is to develop mathematical models that can simulate Parkinsonian tremor, for instance, a limit-cycle-exhibiting [8] system such as the Van-Der-Pol oscillator is fit to experimentally measured data. But such an approach lacks physical underpinnings and cannot explain some of the key features observed in experimental measurements such as why a patient trying to keep still would exhibit tremors (referred to as rest tremors) [9], whereas a patient involved in engrossing physical or mental activity may not exhibit tremors. These models are based on the hypothesis that Parkinsonian tremor may be a limit cycle [10] [6]. Recently [11] presented arguments based on a control-system analogy to support this hypothesis, and proposed a very simplistic model based on this analogy that, for the first time, established a logical connection between increased reaction times and limit-cycle-type tremors.

In this paper, we leverage the models proposed in [11] to develop simple diagnostic tools. These diagnostic tools use the knowledge gained from the modeling efforts and take advantage of the existence and features of the limit cycle to not only diagnose Parkinson's disease but also its severity. This paper also discusses the possibility of developing a low-cost diagnostic device and strategies for the same.

II. MODELING OF PARKINSONIAN TREMOR

For the analysis in this paper, we employ the model structure proposed in [11]. The uncontrolled dynamics of the body part(e.g. human arm) is represented as a dynamical model whose transfer function or model equations can be obtained from first principles on making some simplifying assumptions such as a simple pendulum assumption. The total motor response can then be represented as a closed-

978-1-4799-1559-0/13/$31.00 ©2013 IEEE 252

loop feedback control system, in which the feedback path represents all sensory feedbacks, while the controller represents the neuro-system's logic that determines muscle actions. In such a setup, the reaction time is simply a transport delay in the closed-loop feedback system. The reaction time is known to be higher in patients suffering from Parkinson's disease by as much as 0.1 seconds [1] [2], than in disease-free individuals. Finally, approximating the physiological saturation of the transmission of control actions [12] as a saturation function in the forward path, we obtain a block diagram as shown in Figure 1. The intended initial velocity block denotes reference command, while the output is the angular velocity of arm.

For the numerical simulations in this paper, we use the simple pendulum model and choose length / of the pendulum to be 0.65m (average length of the typical human arm) and mass m of the pendulum to be 3.5 kg (average mass of the typical human arm), while the damping factor is 3.375Kg-m/s. Therefore, the state-space form for the above simple pendulum model is

X = Ax + Bu, y=Cx + Du,

where x = [ 0 0 ]T£ R2 is the state vector with 0 being the angle of the pendulum, 0 being the angular velocity of the pendulum, u £Ris the moment generated by the controller, ye R is the measured velocity of the tip of the pendulum, and

A-0

-15.07 1

..28 ; B =

"0"

1 ; C= [0 1] ; D= 0.

for the controller block, we use a proportional-derivative-integral (PID) controller with the proportional gain kv - 15, integral gain ki — 4 and derivative gain £D = 0.5.

III. Is SLOW REACTION TIME A CAUSE OF PARKINSONIAN TREMORS?

Following along the line of reasoning in [11], we first hypothesize that the increased reaction time in Parkinson's patients (~ 350 milliseconds) as compared to healthy individual (~ 250 milliseconds), modeled as a transport delay,is the key parameter that contributes to rest tremors in Parkinson's patients. Although increased reaction time in Parkinson's patients is a well-studied fact [13] [14], it has not been viewed as a primary contributor to tremors until it was first suggested in [11]. We perform a simple simulation study to explore the validity of the above hypothesis. We start with a simple pendulum model to represent a human arm and PID controller to represent the neuro-system's logic (motor control block) that determines the muscles action. The sluggish reaction is represented by the transport delay block and the physiological saturation block is represented by a simple saturation function. We choose the length and the mass of the human arm to be 0.65m and 3.5Kg,

respectively. Since Parkinson's tremor is classified as tremor at rest, we choose either zero intended velocity or very small intended velocity.

To explore the effect of reaction time, we start with zero intended velocity and small initial condition (initial angle of O.lrad) for four different time delays. From Figure 2, it can be seen that there are no oscillation for smaller time delays 0.05s and 0.1s, whereas for larger time delays of 0.15s and 0.2s we observe oscillations. To further explore this, we choose a small random intended velocity for the first 0.1s and zero intended velocity after that along with zero initial conditions. From Figure 3, again no oscillations are obtained for small delays while oscillations are observed for large delays. These tests suggest that the oscillations and thus tremors are independent of initial conditions and perturbations and are primarily dependent on the time delay in the feedback loop.

IV. DOES PARKINSONIAN TREMOR EXHIBIT LIMIT CYCLE BEHAVIOR?

In the field of dynamical systems, a limit-cycle is a closed trajectory or a closed loop in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity or as time approaches negative infinity [8] [15] [16]. If all the neighboring trajectories approach the limit-cycle as time approaches infinity, it is called a stable or attractive limit-cycle. If instead all neighboring trajectories approach it as time approaches negative infinity, it is an unstable or non-attractive limit-cycle. Some nonlinear systems exhibit such behavior, and further saturation in the forward path of a closed-loop unstable linear system often results in limit-cycle behavior [17].

To begin with, we consider phase plot of the model-generated tremor signal, that is, the plot of angular velocity v/s angular acceleration (or the signal v/s derivative). In this phase plot, as seen in Figure 4, we observe that the response converges to a closed loop thus indicating a stable limit-cycle behavior. We next try different initial conditions, some starting from inside the loop (Figure 4(a)) and some from outside the loop (Figure 4(b)) and observe that all of these trajectories with these different initial conditions converge to the same closed loop. Further, on observing the effect of small excitation such as a random (Figure 4(c)) forcing with zero initial conditions, we see similar close loops in phase plot. In all these cases, the closed loop appears to have flat top and bottom edges due to the presence of saturation.

intended initial

velocity

. \ J Motor 1 ,

- ] \ Control f

i

Transport

De-la\

Physiologies

Saturation _ J Body | _

| Dynamics | ^ Final output

Velocity

Figure 1. The closed-loop feedback system representing motor control in patients with Parkinson's disease.

253

Figure 2. Angular velocity for different time delays with zero intended velocity and initial angular position of 0.1rad.

Figure 3. Angular velocity for different time delays with a small random intended velocity for the first 0.1s and zero initial conditions. Although these observations strongly indicate a limit-cycle type behavior, these are not conclusive and hence cannot definitely show that this is limit cycle behavior. Further, this question can be answered definitely through a rigorous analysis of system dynamics, which is beyond the scope of current work. Nevertheless, that fact that a closed loop is obtained in the phase plot (irrespective of whether it is limit cycle or not) can be leveraged to develop diagnostic tests as discussed further in the following section. For simplicity however, we will refer to this closed loop as a limit cycle for the remainder of the paper.

Figure 4. The plot of angular velocity v/s angular acceleration. A limit-cycle type closed loop is obtained for all of the above cases, namely, (a) initial condition inside the loop (with zero input), (b) initial condition outside the loop(with zero input), (c) zero initial condition with random input of small variance.

V. LIMIT CYCLE FEATURES

Based on the above observations, we further hypothesize that the existence and certain features of the limit cycle obtained can be employed to come up with a possible diagnostic tool that may help in estimating the time delay, which has been established as the cause behind the tremor in the previous hypothesis, and thus may help optimize treatment strategies. In this regard, we notice that qualitatively, the shape of the limit cycle is similar for different saturation levels, transport delays, etc., however the size of the limit cycle is dependent on these factors. Hence we explore the idea of the area contained within the limit cycle being an indications of transport delay and thus possibly an indication of the presence of the disease and also its severity. To explore this conjecture further, we first have to establish that the area of the limit cycle is relatively (if not absolutely) insensitive to other factors that contribute to patient to patient differences such as length of the human arm, its mass, etc. On initial inspection, the size of the limit cycle measured through its area in phase space seems to be dependent on mass and length of the human arm and physiological saturation as well as the delay. Using the same model as in section IV, we observe that the effect of change in length and mass of the human arm has a smaller effect on the size of the limit cycle as compared to those of saturation and delay. This suggests using the area of the limit cycle as an indirect measure of the reaction time and saturation and thus may be useful as a diagnostic variable. Thus we compute the area of the limit cycle (based on the steady-state closed loop) using polyarea command of MATLAB.

A. Saturation limits Vs. Size of the limit cycle

During our analysis, we find that when we increase the saturation, say n times, then the coordinates, both abscissa and the ordinates, of the corner points approximately increase n fold. Therefore, we should expect an n-squared (n times n) fold increase in area of the limit cycle, and that is indeed observed as seen from figure 5. In a similar fashion, the area of the limit cycle decreases nsquared times with n times reduction of the saturation limits. Here we keep the saturation function symmetric, that is, the lower saturation limit is simply the negative of the upper saturation limit.

254

B. Time delay Vs. Size of the limit cycle

Along similar lines, an increase in area of limit cycle is observed with increase in time delay as seen from figure 6. However, instead of an n-squared relationship, in this case, an exponential relationship seems to exist between the increase in delay and the increase in area of limit cycle.

C. Model Parameters Vs. Size of the limit cycle

Now, when we vary the mass of the arm within the range 2.5Kg - 4.5Kg, a significant difference in the shape and sizeof the limit cycle is not observed (variation of about 5%) and hence can be neglected. However, the length of the human arm appears to have a larger impact (about 12%) on the area of the limit cycle albeit a smaller effect than those of saturation and delay. Nonetheless, since it is not difficult

Figure 5. Limit cycles obtained for two saturation values.

Figure 6. Limit cycles obtained for two delay values.

Figure 7. Finding the area of the limit cycle from actual approach and alternative (approximate) approach

Figure 8. 3-D plot of area of the limit cycle as a function of delay and saturation. Delay values are from 0.12 s to 0.4 s and saturation values are from 0.5 N to 2.5 N. (a) Approximate area of the limit cycle and (b) actual area of the limit cycle at different delay and saturation. Here we have assumed equal saturation in both sides. The values are calculated for an arm of mass 3.5 Kg, length 0.65 m and spring constant 3.375 Kg/m. to obtain a this data can be used as an input for the diagnosis.

A simplified approach To further simplify the approach, we explore using an approximate indicator of the area of the limit cycle by calculating the area of the rectangle constructed from by the maximum and minimum of the angular velocity (horizontal axis) and angular acceleration (vertical axis) as shown in figure 7. This approach on account of being simpler is also computationally cheaper and hence easier to implement on a microcontroller. Figure 8(a) shows the approximated area and 8(b) shows an actual area of the limit cycle. As seen from these figures, the simplified approach also shows an almost identical trend as the more accurate approach. Hence, an alternative approach can be adopted as an indicator of the actual area of the limit cycle.

VI. DIAGNOSIS

The envisioned diagnostic device would first check the existence of limit cycles and then use the size and area of the limit cycle to determine the delay and thus indirectly the

In addition to using the feature of the limit cycle, since it is an established fact that the frequency of Parkinson's tremor is 4Hz-6Hz, a spectral analysis of the tremor signal can also be used as an additional check.

disease using this approach is outlined in Figure 9. The diagnosis starts with obtaining the angular velocity data

analysis, the phase portrait, in which derivative of the signal is obtained and plotted against the original signal, is observed. If a repeating closed trajectory in the phase space is obtained, then it may be an indication of a limit cycle and

. This can be concluded by observing the phase plane plot of the oscillations (excluding the first few possibly transient cycles) and determining whether the cycles are repeating to within a small tolerance.

255

Next, to obtain an estimate of the delay, we perform area analysis on the steady-state closed loop. Based on the steady-state closed loop, we compute the area contained within it. From this area, using a look-up table of area values for various saturations and delay computed ahead of time using simulations, a rough range of saturation and delay values are arrived at. At this point however, one major challenge is that multiple combinations of saturation and delay can yield the same area of the closed loop as seen in Figure 8. On further exploring this aspect through simulations, we find that although the same area is obtained for multiple combinations, the actual shape of the limit cycle is different for these different combinations of delay and saturation. Also, the point worth mentioning is that the area computation in the table is made assuming that the saturation function is symmetric, that is, both positive and negative saturation limits are same in magnitude. Since the delay, and not the physiological saturation, has been established as the cause behind the tremor in the first hypothesis, we need to estimate the delay separately. Further simulation analysis reveals that the saturation limits can be estimated by looking at the 4 corner points of the limit cycles (as shown in the Figure 10). The ordinates of the coordinates can be subjected to some simple mathematical operations to yield a relatively small range of average saturation. Now, the area of the limit cycle is analyzed for the estimated small range of saturation and thus we obtain a corresponding delay within a small range. Note that the delay may be viewed as an indication of the severity of the disease and thus may help to optimize the treatment for PD. Furthermore, by repeating the same diagnostic test at regular interval (e.g. every three months) one can track the progress of the disease and judge the effectiveness of any treatment strategies. Further, this approach ensures that even if the value of the delay is not estimated correctly due to other uncertainties, the trends are captured correctly and thus help in optimizing treatment strategies. These ideas are further illustrated through the following example.

VII. AN EXAMPLE

3.5Kg and length 0.65m. We assume spring constant to be 3.375N/m. We put delay as 0.3s and symmetrical saturation limits as 2N and -2N with an average saturation limit of 2N. The limit cycle obtained (steady state closed loop) is shown in the Figure 10. Now, the objective is to estimate the delay (the reaction time) of the patient from the size and shape of this limit cycle. First, to estimate the saturation, consider the four corner points of the limit cycle. The four points are (-0.29, 3.23), (0.43, 0.93), (0.25, -3.37) and (-0.44, -1.09). The ordinates are directly related to the saturation values, and a rough relationship is obtained by heuristics. Using these heuristic relationships, we compute the average saturation to be between 1.66N and 2.09N. Saturations in either direction

are obtained as 1.60N to 2.02N in positive direction and -1.72N to -2.16N in negative direction. Next, we compute the area of the limit cycle using polyarea function in MATLAB to be 4.41units. Assuming an error of

10% in area computation due to inaccuracies in measurements and external disturbances, we use the area range of 4.85 to 3.97 units to lookup the possible values of delay from area data (displayed in Figure 8). For looking up the delay values, we use the saturation range mentioned in the previous paragraph and an arm length of 0.65m. From this analysis, we obtain delay values to be between 0.28 seconds to 0.32 seconds from the look up table.

VIII. DIAGNOSTIC DEVICE

Based on the methodology discussed in the previous section, an inexpensive diagnostic device can be envisioned. This diagnostic device could either be a separate inexpensive pocket device or can be implemented in a smartphone as a smartphone application. A simple pocket device, which contains an accelerometer or gyro sensor, microcontroller and a display, can be used as an inexpensive diagnostic discussed in sections VI and VII. This pocket device is first held by a patient suffering from

sensor is used to detect acceleration and/or angular velocity. The angular velocity measured by the gyros will be processed in the microcontroller as per the flowchart discussed in Figure 9. As per the sequence of steps discussed in Section VI, the presence of limit cycle is first checked and then the delay is estimated from the area of the limit cycle. Based on these computations, the appropriate values are sent to the display and a simple diagnosis based on the frequency, limit cycle, and delay can be performed. Although at this point, this diagnosis process is not yet finely tuned and robustness and accuracy are still open questions, a big advantage of such a device is its simplicity, cost, ease of use (can be used by untrained people and non-professionals). As mentioned earlier, the same test for the same individual can be repeated at different time instances to get an idea of the progress of the disease. The idea here is that for the same individual all other parameters are likely to remain constant and the differences between limit cycles at different time instances will purely be a function of the time delay/reaction time which indirectly is likely a measure of the severity of the disease. Note that this idea may also be implemented as a smartphone app since smartphone have in-built sensors and processing power.

IX. DISCUSSION

The above simulation studies show that such simplistic

feasible, but clearly a much more rigorous study with extensive analysis of data from real patients is needed to flesh out these ideas. These in-depth studies will not only help validate some of the hypotheses, but also

256

help characterize the feasibility, robustness, and accuracy of such a methodology and device.

Figure 9. Al

Figure 10. The limit cycle obtained after putting saturation values 2N and -2N and delay value 0.3s for an arm of mass 3.5 Kg and length 0.65 m. Coordinates of the four corner points are shown. Although the above results use simplistic models, simulations with more complex dynamical models and controllers were tested and they revealed that qualitatively, the conclusions drawn in these studies are retained even for more complex models. Nonetheless, these studies are not complete until they are validated by real data.

X. CONCLUSION

In this paper, based on a control-system analogy, the possibility of using the presence of a limit cycle and its features for the diagnosis of Parkinson's disease was explored. First, the hypothesis that the Parkinson's tremor exhibits a limit-cycle-type behavior was leveraged to distinguish it from other tremors like essential tremors. A simple model of a closed loop feedback system involving

components like motor control, transport delay, physiological saturation and body dynamics, is constructed and used for testing this hypothesis. Further, ideas to exploit the existence and features of the limit cycle (area in particular) to come up with possible diagnostic tools that may help optimize treatment strategies were explored. Based on this idea a methodology and an algorithm to estimate the transport delay in PD patients were proposed. This estimate in turn would be an indication of the severity of the disease. An illustrative example is included to test the effectiveness of algorithm. This paper also discussed the possibility of developing a low-cost diagnostic device and strategies for the same. Although these ideas show potential, a number of challenges and future tests that are needed to develop this idea further were also discussed.

REFERENCES

1. M.Hallett, R.Jasper, J. R.Daube, F.Mauguire, Movement

Disorders, Elsevier Health Sciences, 2003. 2. S. A. K.Wilson, The Croonian Lectures on Some Disorders of

Motility and of Muscle Tone, with special reference to the corpus striatum, TheLancet, vol. 206, no. 5319, pp. 268-276, 1925.

3. J. Parkinson, An Essay on Shaking Palsy, Sherwood, Neely, and Jones, London, 1817.

4. A. Guyton and J. Hall, Textbook of medical physiology, 11th edition, International print-O Pac, 2006, 711

5. D. J.Brooks, April 2010, Imaging approaches to Parkinson disease, The Journal of Nuclear Medicine, vol.51, no.4, pp.596-609, 2010.

6. J. Modolo, A. Beuter, Linking brain dynamics, neural

An integrated perspective, Medical Engineering and Physics, vol. 31, no. 6, pp. 615-623, 2009.

7. A. Beuter, J. Bélair, C.Labrie, Feedback and delays in neurological diseases: A modeling study using dynamical systems, Bulletin of Mathematical Biology, vol. 55, no. 3, pp. 525-541, 1993.

8. S. H.Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering, Westview Press, 2001.

9. J. Jankovic, Parkinsons disease: clinical features and diagnosis,J NeurolNerosurg Psychiatry, vol. 79, no. 4, pp. 368-376, 2008.

10. G. Austin, C. Tsai, A physiological basis and development of a model for parkinsonian tremor, ConfiniaNeurologica, vol. 22, no. 3-5, 1962.

11. H.Palanthandalam-Madapusi, S.Goyal,Is Parkinsonian tremor a limit cycle ?, Journal of Mechanics in Medicine and Biology,vol.11, no.5, pp. 1017-102, 2011.

12. R. Edwards, A. Beuter, L. Glass, Parkinsonian tremor and simplication in network dynamics, Bulletin of Mathematical Biology, vol. 61, pp. 157 177, 1999.

13. E. Scholl, G. Hiller, P. Hovel, Dahlem MA, Time-delayed Feedback in neuro-systems, Philos Transact A Math PhysEngScivol. 367, pp. 1079 1096, 2009

14. G. Orosz, J. Moehlis, RM. Murray, Controlling biological networks by time-delayed signals, Philos Transact A Math PhysEngScivol. 368, pp. 439, 2010

15. M. Vidyasagar, Nonlinear Systems Analysis, second edition, Prentice Hall, Englewood Cliffs, New Jersey 07632, 2002.

16. P. Hartman, Ordinary Differential Equation, Society for Industrial and Applied Mathematics, 2002

17. G. F.Franklin, Powell D. J., Emami-Naeini A, Feedback Control of Dynamical Systems, Prentice Hall, Fifth Edition, 2006.

257