PARKINSON S DISEASE: MODELING THE TREMOR AND OPTIMIZING THE TREATMENT Mohammad Haeri, Yashar Sarbaz and Shahriar Gharibzadeh Advaned Control System Lab, Eletrial Engineering Department, Sharif University of Tehnology, Tehran, IRAN Bioeletri Department, Faulty of Biomedial Engineering, Amirkabir University of Tehnology, Tehran, IRAN Abstrat: Parkinson s disease is a ommon neurologial disorder, the main symptom of whih is tremor. In this study, tremor is modelled using physiologial information and linial reordings. All bloks of the basal ganglia are onsidered exatly in this model and mathematial relations of eah blok are designed. Two main therapeuti approahes are implemented on the model. Finally, optimal dose and optimal time of presribing the drug, Levodopa, is offered. Our results indiate that this model an be used as a primary hoie for researh on Parkinson s disease. Copyright 005 IFAC Keywords: Medial, Optimization, Modelling, Osillation, Noise harateristis.. INTRODUCTION Parkinson s Disease (PD) is aused by destrution or malfuntioning of substantia nigra, a part of the Basal Ganglia (BG). The main funtion of BG, whih ontains several parts, is movement ontrol. The PD as the most ommon disease of the BG is aused by depletion of a neurotransmitter alled Dopamine. The main sign of the PD is tremor, an involuntary osillating movement with 4 6 Hz frequeny and high amplitude. The tremor is seen espeially in hands during both on voluntary movements and on the rest. Although several researhers have tried to model Parkinsonian tremor and analysed the reorded signals, none of them have inluded the preise physiologial information and neural strutures in their models (Titombe, et al., 00; Edwards, et al., 999; Asai, et al., 003). In addition, tremor has been supposed to be a simple sinusoidal signal in the previous researhes and the nonlinear behaviours of reorded tremors are not onsidered (Titombe et al., 00). In this study, we try to present a mathematial model ontaining physiologial information and linial data, as muh as possible. This model onsiders different parts of the brain involved in PD. Thus, tremor behaviour an be represented in a fairly omplete manner. On the other hand, nonlinear and noisy behaviours of reorded tremors are inluded, as well. In Setion we modelled the PD tremor in the BG struture with onsidering all available information about this part of brain. In Setion 3 two main treatments of the PD have been analysed. In Setion 4, problem of drug usage optimization is presented. Finally paper is onluded in Setion 5.. MODELING THE TREMOR Fig. illustrates a shemati diagram showing the different parts of the BG (Guyton, et al., 00).The main riteria for separating different bloks of the BG in physiology are apparent onfiguration, neuronal strutures, and input and output neurotransmitters. There is no preise information on internal funtioning of the bloks. Eah blok, onsisting a huge number of neurons, is assumed (at least in our study) to behave like a single neuron, having three passive eletrial elements: a membrane resistane, a membrane apaitane and a
longitudinal axonal resistane. Fig. shows the eletrial iruit equivalent representation of a single neuron (Guyton, et al., 00). Fig.. Different parts of the BG. G ( : SNo( t) = sgn( A( t)), 0 A( = g So s + 40 ( G ( : So ( = SNo( s + 30 0 So ( = SNo( s( s + 30) 0 G3 ( : GPo( = So ( + g s + 0 50 STNo ( g s + 0. G4 ( : STNo( = g GPo( s + 40 STNo ( = g GPo( s + 40 G5 ( : OUT ( = g s 00 + 0 STNo ( 00 g So ( s + 0 () () (3) (4) (5) Fig.. Eletrial iruit equivalent of a neuron. The relation between V o (membrane voltage) and I i (input urrent of membrane) is therefore given by a first order system. The behaviour of substantia nigra, the most important part involved in tremor, is supposed to be nonlinear. The model relations are as follow. G ( represents dynamis of the substantia nigra pars ompata. This part is taken as a first order system. At the ontinuation of this transfer funtion, a nonlinear element (sign funtion) is imposed. Input signal has inhibitory effet over the output signal, whih has been taken into aount. G ( represents the model of striatum. This omponent has two outputs and its input has an exitatory effet over them. The systemi blok diagram of the model is presented in Fig. 3. Parameter g of the model simulates Dopamine level. g = 0 represents the disease state and the resulted tremor is depited in Fig. 4. Setting g =, suppresses the tremor ompletely and therefore represents the healthy state. In transmission of signals between two neurons, several noise soures an be onsidered: ) Voltage gated hannels (pathways for movement of ions between inside and outside of the neuron, ) Amount of alium ion in the neurons (whih is a neessary stage for release of neurotransmitter), 3) Diffusion of neurotransmitter between two neurons and 4) Opening of ion hannels in response to released neurotransmitter in the destination neuron. Additional noise soures in the PD are: dereased uptake of Dopamine in synapses of striatum and a phenomenon alled up-regulation whih is the inrement of Dopamine reeptors in striatum, following a derease in Dopamine release level. Furthermore, beause of substantia nigra destrution, the Dopamine diffusion is higher in the PD. G 3 ( models the globus pallidus external segment. Here, there is an inhibitory input and an exitatory input with a single output. Transfer funtion of G 4 ( models the subthalami nuleus. This part has one inhibitory input with two outputs. A first order transfer funtion was onsidered for eah output. Finally, transfer funtion of G 5 ( models the globus pallidus internal segment and substantia nigra pars retiulata, whih produe the output of the BG. It has an exitatory input and an inhibitory input with only one output whih is the final output of the BG. Transfer funtion of eah input is onsidered a first order system. Fig. 3. Blok diagram of the model.
Ts ke G( = ( + st )( + st ) (8) Where k denotes amplifiation fator, T and T are time onstants in hours, s denotes the omplex frequeny and T is the delay time. Fig. 8 indiates plasma level of the drug and simulated results of the model (Eq. 8). Fig. 4. Response of the model for g = 0. Considering the above mentioned soures of noise in the model, modifies the primary model and makes it more realisti. Beause of the nature of noise soures, they are onsidered in the related sites, i.e. onnetions. Sine the noises show up with higher effets in the PD, we have onsidered the dependene of noise to parameter g and replaed it in the primary model by g ( + an( t)). n (t) is a white noise having mean of 0 and variane of. a is a free parameter to model the differenes among patients. Fig. 5. Power spetra of the non-noisy response. Power spetra are used to ompare linial and simulated data. Power spetra of non-noisy response are depited in Fig. 5. Noisy response is shown in Fig. 6. As it is apparent, the width of the spetra in noisy state is wider than the non-noisy state, just like the linial data. The power spetra of suh a linial data are depited in Fig. 7. Some linial reordings show frequeny ontents besides the peak frequeny. So, another noise soure must be onsidered just near the output of the model, whih simulates noises from soures other than the BG (as musles dynamis and the transmission pathway). A white noise is passed through a low pass filter whih orresponds to musles dynamis. The final response of the model is as follows. Fig. 6. Power spetra of the noisy response a = 0.. OUTo( = OUT ( + b g Glp ( n( (6) 50 G lp ( = s + 50 (7) 3. MODELING THE TREATMENT Fig. 7. Power spetra of the linial data. 3. Drug modeling Aording to linial findings, drug affets 0-5 minutes after the presription (Haisalihzade, et al., 989). Then the PD disorders start to derease and reah to its minimum state. Nearly after. 5 hours, the drug effets begin to redue. Finally after 4 hours, all drug effets are disappeared. We onsider behavior of the drug as a system with two parts: For the first part, dose of the drug is the input and plasma level of Levodopa is the output. This part an be modelled as a seond order system with the following transfer funtion (Haisalihzade, et al., 989). Fig. 8. Drug onentration.
The parameters of the model are estimated using the least square approah and are given as (Haisalihzade, et al., 989). k = 48 T = 0.05473 T = 0.6073 T = 0.46 Fig. 9. Variation of g due to the drug presription. For the seond part, input is the drug onentration in the blood and output is the gain that represents Dopamine hanges ( g ). We divide the seond part into two segments: A linear dynami and a nonlinear relation. We assume that for drug onentration over 500, regardless of its exat value, the tremor is suppressed ompletely. So, the nonlinear part is a saturation element ats on 500. The linear part represents relation between the plasma level and the parameter g and is given as follow. 0. G ( = (9) s + 0. Fig. 9 shows the impulse response of the mediation on the tremor. As it is stated before there is a reverse relation between the gain parameter g and the level of Dopamine in the BG. We now exert the equivalent g (that represents the drug presription) on the main model of the tremor in Fig. 3. The tremor behaviour after drug presription is depited in Fig. 0. 3. Modeling of deep brain stimulation In general, the amplitude of stimulation voltage is nearly 3 v, pulse width is almost 90 µ s and the exerted frequeny is greater than 00 Hz. It seems that DBS auses hange in Dopamine ontent of brain, whih in turn affets the tremor amplitude diretly. We assume that the parameter g is ontrolled by DBS. It is supposed that eah DBS pulse releases a quantal amount δ of substane z whih is died with a time onstant t. τ is DBS period and we have f = / τ. Let s assume g 0 as the initial value of parameter g. For g 0 = 0, our model produes an osillating response like the tremor of PD. We relate the parameter g to amount of the substane z as follows. g( t) = g 0 z( t) (0) After some mathematial manipulation (Haeri, et al., 004), the following relation is obtained for g. Fig. 0. The tremor behaviour under drug mediation. a) Whole time sale, b) Short time sale. g t) = g ( 0 t t δ ( e ) t t e () It should be noted that frequeny of DBS, as well as its amplitude, an be hanged for regulation. We suppose that the amplitude hanges are represented by inrease or derease in δ (Limousin, et al., 998). Now we onsider effet of DBS on the main model of Fig. 3. δ is supposed to be. Frequeny of DBS is onsidered 5 Hz. In order to model different behaviors of patients, t is adjust in aordane. Considering the aeptable range of g ( to 0), in order to obtain g equal to, t is hosen 0.076. Larger t results in smaller steady state g. Sine the non-positive values of g are not defined in the model (and have no physial interpretation either) the final value of g should not be 0 or negative. In Fig., hanges of g due to DBS appliation are depited. Stimulations are exerted in seond 5 and stopped in seond 0. After stimulation, amount of g dereases rapidly and reahes in steady state. After stoppage of stimuli, g inreases again and approahes to its initial value of 0. Changes on g are applied to the PD model and the given response is depited in Fig.. As it is stated before t represents different patient behaviour, therefore effetiveness of DBS an be modelled by this parameter. For example, if t = 0.05, the patient will not benefit from this partiular DBS. In this ase, the tremor amplitude is
dereased, but not suppressed enough. Variation of g and the orresponding model response for t = 0. 05 are given in Figs. 3 and 4 respetively. 4. OPTIMIZATION OF LEVODOPA USAGE Use of Levodopa auses side effets in patients. Furthermore, hroni presription of drug redues its therapeuti advantages as time goes on and therefore requires inreasing amounts of the drug presription. Derease in amounts of the drug usage in the primary stages of the disease postpones onset of side effets. Usually the drug dose and usage time are determined by trial and error, whih obviously is not neessarily the optimal hoie. In next lines we explain how the drug dose and presription time an be optimized based on the mathematial model whih was developed in the pervious setions. Fig.. Variations of g due to DBS exertion, t = 0.076. 4. Defining a ost funtion for optimal drug usage Drug usage has two parameters: dose and time of presription. These two are onsidered as parameters of the ost funtion. Derement in the dose and inrement in the time interval between suessive presriptions are desired. The ost funtion is onsidered as: F ( δ, T ) = b δ + () T Fig.. Response of the model to DBS exertion, t = 0.076. Fig. 3. Variations of g due to DBS exertion, t = 0. 05 In whih δ is the dose of the drug in eah presription and T is the time interval between two suessive presriptions. Minimizing F will introdue the optimum state, beause it means dereasing of δ and inreasing of T. 4. Relation between drug dose and time of presription The relation between the drug presription dose and the plasma level of the drug is given by a seond order system with time delay (Eq. 8). As we know, the plasma level 500 or higher an effetively suppress the signs of the disease inluding the tremor. So, we have regulated the time of presriptions in a manner that just before reahing the onentration of 500 auses by the pervious presription, the effet of the next dose is initiated. If an impulse with amplitude of δ is implemented, the time response of the system will be:.6466( t 0.46) 8.75( t 0.46) y( t) = 40.89 δ ( e e ) (3) The main onstraint will be obtained by onsidering the ritial onentration of 500:.6466( T 0.46) 8.75( T 0.46) 500 = 40.89 δ ( e e ) (4) Where T is the time between onseutive drug presriptions and δ is the drug dose in eah presription. Another relation between these two parameters is imposed by the maximum daily allowable dose whih is 500 mg, so: Fig. 4. Response of the model to DBS exertion, t = 0.05. 4 δ 500 T (5)
4.3 Optimization We have simplified the formulas by determining following two variables: x = δ, x = (6) T 0.46 The ost funtion is rewritten as: F ( x = a x + b x (7), x ) And the onstraints are: (.6466 / x ) ( 8.75 / x ) 500 40.89 x ( e e ) = 0 (8) 500 4 x x + 65.5 x 0 (9) Sine the presription time and dose of the drug are both of similar important, we onsidered equal weights ( a = b ) for two parameters x and x. The optimization problem was solved using Lagrange method in MATLAB software. The solution is given as follows: x = 58.63, x =.7977 Returning to the main parameters, we have: motor oordination and omparison with Parkinson s disease data, Biosystems, Vol. 7, No. -, pp. -. Edwards, R., A. Beuter (999). Parkinsonian tremor and simplifiation in network dynamis, Bulletin of Mathematial Biology, Vol. 5, pp. 57-77. Guyton, A.C., J. Hall, 00 Text Book of Medial Physiology, 0 th ed., Philadelphia, Saunders. Haisalihzade, S.S., M. Mansour, C. Albani (989). Optimization of symptomati therapy in Parkinson s disease, IEEE Transations on Biomedial Engineering, Vol. 36, No. 3, pp. 363-7. Haeri, M., Sh. Gharibzadeh, and Y. Sarbaz (004). Modeling of the treatment on the Parkinsonian tremor, ICI-SCM 004, EC05-3, Cesme, Turkey. Limousin, P., P. Krak, P. Pollak, A. Benazzouz, C. Ardouin, D. Hofmann, A. L. Benabid (998). Eletrial stimulation of the subthalami nuleus in advaned Parkinson s disease, The New England Journal of Mediine, Vol. 339, No. 6, pp. 05. Titombe, M.S., L. Glass, D. Guehl, A. Beuter (00) Dynamis of Parkinsonian tremor during deep brain stimulation, Chaos, Vol., No. 4, pp. 766-773. δ = 58.63, T = 0.6035 These amounts will suppress the tremor ompletely. 5. DISCUSSION In this study, a omplete model was presented to emulate the tremor behaviour of the PD. In addition to a fairly aurate modelling of the effet of Dopamine as the main ause of the disease, the proposed model has simulated the behaviour of different parts in BG well enough. For eah part of the BG, the number of inputs and outputs are inluded exatly and in addition, the inhibitory and exitatory effets are onsidered and modelled as it is. Finally the differenes of strength of onnetion at health and illness states were onsidered as well. We also tried to inlude the noisy and nonlinear behaviour of the tremor observed in linial reordings. Comparing power spetra, we onluded that the model response and linial data are fairly similar. The effets of two main therapeuti approahes of the PD were also onsidered preisely. Finally, an optimization method was offered in order to derease side effets of the drug. The proposed optimal dose and optimal time of presription, an suppress the tremor exatly and it might be the best way of drug presription. It is worthy to examine these quantities experimentally. REFERENCES Asai, Y., T. Nomura, K. Abe, Y. Matsuo, Sh. Sato (003). Classifiation of dynamis of a model of