MATHEMATICAL MODELING AND OPTIMIZATION:

Demonstrating Changes in the Tone of those with Cerebral Palsy

James W. Fee, Jr., MS.

Applied Science and Engineering Laboratories Alfred I. duPont Institute/University of Delaware Wilmington, Delaware USA

ABSTRACT

This paper is a follow-up to one written for RESNA several years ago. The author proposed that mathe- matical modeling could offer insights into changes in muscle tone resulting from a form of vestibular stim- ulation. This paper presents progress the author has made in that direction. A model of a test to measure spasticity (the Leg Drop Pendulum Test) is outlined. The process by which the model is optimized to fit three subject's data is presented. Force inputs required for an optimized solution are compared in a set of triplet subjects (one triplet is without disabil- ity, the other two are diagnosed with spastic diplegic cerebral palsy). These forces demonstrate the higher excitability of muscle stretch receptors in cerebral palsy as well as a reduction in excitability after stim- ulation via whole body vertical oscillations.

BACKGROUND

Cerebral palsy is a nonspecific term used to describe a persistent qualitative motor disorder caused by nonprogressive damage to the central nervous sys- tem[5].

Mathematical modeling of neuromuscular systems is used to provide insight into how various levels of the nervous system might function. The purpose of this paper is to provide a follow-up to work the author advocated for in a previous RESNA paper[1]. In par- ticular, it was suggested that much can be gained from the application of systems modeling to various phenomena observed in the cerebral palsy clinic which are, at present, poorly understood. One phe- nomenon, which is the basis for the present work, is the effect of whole body vertical acceleration, a form of vestibular stimulation on those with cerebral palsy. Therapy programs often utilize a form of slow, rhythmical vestibular stimulation in order to inhibit abnormally high muscle tone[3].

If researchers and clinician alike are to make optimal use of vestibular stimulation for normalization of muscle tone, there must be a complete understanding of both the physiological mechanisms by which the desired change in tone is effected and the means by which the effect can be maximized.

As an illustration of the kinds of insights that can be gained from modeling efforts, work will be pre- sented on a model of the Leg Drop Pendulum Test[4]. This test is commonly used to assess spastic- ity. In the case presented here, the author uses the test to evaluate changes in spasticity following whole body vertical oscillations. The author and his co- workers have demonstrated that whole body vertical accelerations have a positive effect in decreasing some measures of spasticity in a small population with this disorder[2][6]. A set of triplets were among the subjects studied. Two of these three subjects had varying degrees of spasticity and one was without a disability. The triplets afforded an excellent opportu- nity to study the differences and similarities using modeling techniques.

This paper will demonstrate that by building a model of the limb of both the normal subject, and the subject with cerebral palsy, and by comparing model parameters from each, a greater understanding of the differences between each can be achieved. Furthermore by studying these parameters in both the before and after stimulation models of the subjects with cerebral palsy, insights into the changes in limb function can be better understood.

An approach will be adapted similar to Ramos and Stark[7] in that it will deal with the spinal level joint angle control system. For the present investigation all input to the muscles, either from higher centers of the nervous system or from feedback circuitry will be lumped into one or more force inputs. These force inputs represent final common pathway excitation of the muscle.

RESEARCH QUESTION

How might modern systems modeling be used to enhance the understanding of the effect of whole body vertical oscillations on the overall tone of those with spastic cerebral palsy?

METHOD

The model described in this paper will consist of a "passive plant" which represents the anatomy and physiology of the knee, acted on by a set of forces which represent the contractions of muscle groups about the knee. The Passive Plant, and the Active Forces can both be represented by an equation of motion.

This equation, stated simply, says that when a limb is dropped in a Leg Drop Pendulum Test, the accelera- tion, velocity and position changes it experiences are caused by a muscle force "F" and a gravity force "G" (see figure 1). For every pattern of motion that a limb might fall through, during the test, there is a unique set of values for all the parameters (symbols) in the equation below. Since the motion of the limb can be known (by recording limb motion with a video cam- era for example) a set of parameters can be found which makes the above equation represent that motion. This process, of finding parameters that makes the equation fit the real motion, is described as optimizing the equation on the real data.

FIGURE 1

The above equation of motion has been optimized on the real data of three subjects under two conditions. The three subjects were unique, they were triplets. This is special because it allows many of the parame- ters in the equation of motion to be the same for all three subjects. For example, since the height and weight of all three are the same, the force of gravity "G" is the same as is the inertia "J" of the lower leg. Since it is assumed that all three anatomies are the same, spring "K", "KP", damper "B" and coulomb friction "C" parameters can all be set to the same respective values for the three subjects. The only dif- ferences then, across the three models, and the two conditions (stimulated or not) are the muscle forces.

The optimization problem is now much simpler. The Passive Plant parameters ("K", "KP", "B", "C" and "G") need be found only once and used for all three subjects. In doing so the data collected from the subject without disability was used. This was done because it was expected that his/her force input would be a minimum. Instructions to the subjects before the test were to relax the limb completely; the subject without disability found it easiest to comply.

Once the anatomically dependent Passive Plant parameters have been identified the remaining task is to identify the force input required to match model data to real data. By examining these forces one can see the differences in patterns of contractions for the three subjects both before and after stimulation.

RESULTS

Figure 2. below illustrates forces which are gener-

FIGURE 2

ated in a normal limb during a Leg Drop Pendulum Test. It can be assumed that there is some activity in the muscle due to the stretch reflex, however on the whole, the muscle is at rest. This figure should be compared to the next which illustrates forces opti-

FIGURE 3

mized for the model of one of the triplets with cere- bral palsy. The differences in forces needed to generate this model data are obvious, almost 14 times as much force is generated in this model than the last. This illustrates a highly sensitive reaction to the muscle stretch which accompanies the leg drop. This result is certainly consistent with present knowledge of the stretch reflex, which would be expected to be more sensitive in those with cerebral palsy.

Finally, Figure 4 below illustrates forces generated for a model of the same subject's position data after whole body stimulation. Differences here are seen in the amplitude and directions of the forces. The dras- tic reduction in force by four fold, 28.0 to 7.0 Nwtns. clearly suggests that the excitability of this muscle is reduced. Changes in direction of force indicate that agonist as opposed to antagonist muscles must be activated in order to recreate the real data using the model.

     TABLE 1 : Force Amplitudes and Durations

FIGURE 4

DISCUSSION

Table 1 above presents a summary of all the forces needed to optimize each of the five modeled subjects and conditions. Trends seen in the data suggest that muscle reactions to the Leg Drop Pendulum Test may be more normalized after vertical whole body oscillations. On a more fundamental level the values illustrate what can be learned from some simple modeling techniques. These numbers are clearly consistent with clinical observation and in fact rein- force known physiological phenomenon.

The changes in forces illustrated in these models are interesting, yet the author believes they beg a more fundamental question, what are their nature and form what mechanisms do they arise? Further modeling of the system will shed light on these questions as well. Replacing the square wave input forces with a good model of the muscle spindle should allow insight into the role these signals play in generating spastic- ity in the muscles of those with cerebral palsy.

REFERENCES

[1] Fee, J.W., Jr., "Mathematical Modeling: A Tool for Cerebral Palsy Investigations", Proceedings of the 15th Annual RESNA Conference, June 1992.

[2] Samworth, K.T., Fee, J.W., Jr., "Measuring Leg Motion Changes Following Vertical Vestibular Stimulation: A Case Study", Proceedings of the `94 Annual RESNA Conference, June 1994.

[3] Trombly C.A, Occupational Therapy for Phys- ical Dysfuction, Williams & Wilkins, Balti- more, 1987, p 86

[4] Bajd, T., Vodovnik, L, "Pendulum testing of Spasticity", J. Biomed Engr. 6:9-16, 1984.

[5] Miller, F., Bachrach, S.J., "Cerebral Palsy A Complete Guide for Caregiving", John Hopkins University Press, Baltimore, 1995, Chap. 1.

[6] Fee, J.W., Jr., Samworth, K.T."Passive Leg Motion Changes in Cerebral Palsy Children Af- ter Whole Body Vertical Accelerations, IEEE Trans. Rehab. Engr. 3(2):228-232 June 1995

[7] Ramos, C.F., Stark, L.W. "Postural Mainte- nance During Fast Forward Bending: A Model Simulation Experiment Determines the `Re- duced Trajectory'", Experimental Brain Re- search, 82:651-657, 1990.

James W. Fee, Jr. Applied Science and Engineering Laboratories Alfred I. duPont Institute P. O. Box 269 Wilmington, DE 19899 Phone: (302)651-6830

ACKNOWLEDGEMENT

Funding for this research was provided by the Nemours Foundation. The author wishes to thank Katherine Samworth for her help with this research.

Figure 1 Equation of Motion

Figure 2 Timing and Amplitude of active Forces- Normal Model

Figure 3 Timing and Amplitude of Active Forces Spastic Model--Before Stimulation

Figure 4 Timing and Amplitude of Active Forces Spastic Model--After Stimulation