INT J COMPUT COMMUN, ISSN 1841-9836 8(4):525-537, August, 2013. Parkinson’s Disease Prediction based on Multistate Markov Models O. Geman, H. Costin Oana Geman Faculty of Electrical Engineering and Computer Science, Stefan cel Mare University of Suceava Development and Human Health Department 13, University St., 720229 Suceava, Romania E-mail: geman@eed.usv.ro Hariton Costin 1. Faculty of Medical Bioengineering, Grigore T. Popa University of Medicine and Pharmacy, Iasi, Romania 2.Institute of Computer Science of Romanian Academy Iasi Branch, Romania E-mail: hncostin@mail.umfiasi.ro Abstract: In the real medical world, there are many symptoms or chronic diseases that cannot be characterized in a deterministic way, and which must be examined in a random way. In the study of these stochastic processes, Markov chains are used. There is a wide variety of phenomena that suggest a behavior in a Markov process manner such as: the probability that a patient’s health to improve, to get worse, to remain stable or to progress to death within a certain time slot, depending on what happened in the previous time window. Our goal is to show that the Markov chains can be applied to the patients with Parkinson’s disease in order to predict the evolution of the disease over time. So the doctor may decide a therapeutic solution that is adapted to the patient’s needs, and that can improve the quality of the patient’s life with Parkinson’s disease in terminal stage. Keywords: Parkinson’s disease, Markov chains, Multistate Markov Models, Predic- tion 1 Introduction Parkinson’s disease (PD) is a neurodegenerative disease that occurs due to loss of dopamine that is a neurotransmitter and due to slow and inexorable destruction of neurons. Brain area affected by progressive destruction of neurons is responsible for movements controlling [1]. For this reason, patients with Parkinson’s disease have rigid and uncontrollable gestures, postural instability, tremor, and speech disorders. Although Parkinson’s disease is considered specific old age, the average age is 50 years and can be confused with the normal aging process of the individual [2]. When first symptoms are manifested, it is believed that between 60% and 80% of the cells for the control of motor activity are destroyed [3]. Parkinson’s disease is a progressive disease, with signs and symptoms accumulated over time. Although this is potentially an invalidity disease, it progresses slowly so that most patients benefit from many years of active life after diagnosis. Moreover, unlike other serious neurological disorders, Parkinson’s disease is treatable. Treatment is surgical or based on drugs, but may also consist of an implanted device for brain stimulation [4]. Worldwide, the disease is diagnosed in 300,000 people each year [5]. Disease incidence and prevalence increase with age. Parkinson’s disease affects 1% of people aged over 65. Rarely, the disease occurs in childhood or adolescence. The incidence is 1.5 times higher among males than among women [6]. If Parkinson’s disease would be detected in an early stage, the physician may interfere with a proper treatment in order to slow the disease’s progression. Unfortunately, currently there is no screening test or biomarker that Copyright c⃝ 2006-2013 by CCC Publications 526 O. Geman, H. Costin can be highlighted in Parkinson’s disease. The three cardinal signs of Parkinson’s disease are resting tremor, rigidity and bradykinesia. Among them, two are essential for diagnosis. Postural instability is the fourth cardinal sign, but occurs late, usually after 8 years of disease evolution. In 70% of cases, uncontrollable rhythmic gestures of the hands, head and feet are the first symptoms and occur mainly at rest and during the stress’ periods (see [7]- [9]). Tremor is diminished during movements, disappears during sleep, and is exacerbated by stress and fatigue. Tremor becomes less evident as disease progression. This tremor, in the absence of other characteristic signs, indicates an early stage of disease or another diagnosis (see Table 1) [10]- [14]. Table 1: Neurological disorders characteristic signs Moment Speed Location Neurological Disorders Rest tremor 4-6 Hz arms, legs Parkinson’s disease Postural tremor 7-12 Hz hands Essential tremor Intention tremor 2-5 Hz arms,legs Cerebellar lesions From the many symptoms or diseases that cannot be characterized in a deterministic way, but in a random way, PD is a prominent example. In our study, we used Markov chains as they characterize very well stochastic processes like diseases evolutions. So, our goal is to show that the Markov chains can be applied to the patients with PD in order both to predict the evolution of the disease over time, and illustrate the response to the specific treatment. In this way the doctor may decide upon a therapeutic solution that is adapted to the patient’s needs, and can improve the quality of the patient’s life in terminal stage. 2 Multistate Markov models In mathematics, a Markov process is a stochastic process having the property that, given its present state, the future states are independent of the past. This property is called the Markov property [15]- [19]. In a Markov process, the system can change or keep its state, according to a certain probability distribution. Changes of its state are called transitions. A random experiment that consists of a series of random sub-experiments is called a stochastic process. Such a special class of these processes is made by the Markov chains [20]- [26]. The evolution of a Markov process can be described by a transition matrix. We can consider the evolution of the health status of a patient as a Markov process that passes through the following states: Well, Suspicious, Ill (PD), or Dead, as is illustrated in Figure 1. For the Markov process illustrated in Figure 1, we can write the general matrix (1), where m = 4 (possible mutually exclusive results: E1 well, E2 suspicious, E3 ill/PD, E4 dead). P =   pww ... 0... ... ... 0 ... pdd   (1) As it can be seen, the transition matrix consists of pij elements, which represent the condi- tional probability that the system will change from the initial state (well) to next state j. The probability that the system remains in the same state after the experiment is given by pij with i = j, and the probability for the system to move from one state to another is given by pij with i̸=j. The transition matrix for the proposed system is a square matrix of order m = 4. The elements of the transition matrix must satisfy the following properties [19]: Parkinson’s Disease Prediction based on Multistate Markov Models 527 Figure 1: Four-state Markov Model for Parkinson’s disease stage (well, suspicious, ill/PD, dead) 1. 0 6 pij 6 1, i, j = 1, ..., m. 2. ∑m j=1 pij = 1, i = 1, 2, ..., m. The sum of the elements of each line must be 1 because E1, ..., Em is a complete system of events. 3. pdd = 1, for our application. Information about the transitions from one state to another in a Markov chain can be rep- resented by a transition matrix. It consists of elements pij - probability of crossing a step from state i to state j (i,j = 1,..., m, where m=4). We can talk about the transition probability of exactly k steps and a matrix formed by them. So, multistate Markov models in continuous time may be used to model the course of Parkinson’s diseases. Since Markov chains are stochastic processes, we cannot know exactly what it is happening on each state, so the system must be described in terms of probability. Definition 1. [19]: Consider a Markov chain with m states. A state vector for Markov chain is a probability vector X = [x1, x2, ..., xm]. The xi coordinates of the state vector X should be interpreted as the probability that the system be in the state i. The behavior of a Markov chain can be described by a sequence of state vectors. The initial state of the system can be described by a state vector noted X0. After a transition, the system can be described by a vector X1 and after k transitions the system is described by the state vector Xk. The relationship between these vectors can be summarized by the following theorem [19]: Consider a Markov process with the transition matrix P. If Xk and Xk+1 are vectors that describe a process state after k and k +1 transitions respectively, then Xk+1 = Xk ∗ P . We represent structural elements as a vector S = [s1t , ..., s i t, ..., s m t ], that for each t = 1,...,n and for each i = 1,...,m, sit varies between 0 and 1, and the sum of structural elements is 1 for any t. In order to model a Markov process, we must respect the following steps [19]: 1. First-order differences of the vector St will be calculated, thus ∆St/t−1 = St − St−1. 2. For each pair t/t − 1 of consecutive periods of time we will buid the partial transition matrices (MTP), as MTPt/t−1(m∗m) form. The elements of the MTPt/t−1(m∗m) matrix can be determined as follows: MTP ij t/t−1 = min(s i t−1, s j t)ifi = j (2) MTP ij t/t−1 = ∣∣∣∣∣∆sjt/t−1 ∗ ∆s j t/t−1∑m i=2(+∆s ij t/t−1) ∣∣∣∣∣ , (3) 528 O. Geman, H. Costin if i ̸= j and ∆si t/t−1 < 0 and ∆s j t/t−1 > 0. MTP ij t/t−1 = 0, (4) for the other elements, where i,j =1,...,m. In formula (3) the expression ∑m i=2(+∆s ij t/t−1) denotes the sum of positive values of the difference vector ∆st/t−1. 3. MTP(m ∗ m), total transition matrix is determined by summing the elements of partial transition matrixes. 4. MP(m ∗ m), transition probability matrix is calculated by ratio between each element of the total transition matrix and the sum of the line on which is located than item. 5. In the final stage of the algorithm, we obtain a forecast of the structural elements for future p periods by multiplying transposed of the matrix MP(m ∗ m) raised to the k power with the vector of structural elements for the last period. 3 Intelligent system for health status prediction using a Markov chain The architecture of the proposed system is shown in Figure 2. It consists from three modules. The first module will handle with the signal acquisition from patients suspected of Parkinson disease. In terms of software, this module is a software application that can acquire biomedical signals from WiiTM Remote device or other devices that can acquire signals generated by tremor. All data acquired from these devices are analyzed using the method presented in Section 2. Furthermore, the data are saved on a server. On this server, physicians can access data in order to establish a long history of patient evolution. The second module of this system is represented by the extracting knowledge from biomedical signals acquired from the patients. This module consists of a software application that runs on the server where there are kept biomedical signals acquired. The third module is the application that is executed in the doctor’s office. This application performs an interfacing of the doctor with the intelligent system, and presents the medical treatment and rehabilitation options. It must be said that bio-signals can be acquired in the doctor’s office but also at home if the patient has a PC and an internet connection. The design and development of this intelligent system used the newest technologies for distributed application development (WCF, SOAP), and the observations received from patients and specialists. 3.1 Database For the database we used the proposed methodology in previous papers [27], [28]. Database with affected patients has been provided by Suceava Emergency Hospital (Neurology Clinic). This dataset is composed of a range of biomedical tremor measurements from 88 people, 28 with Parkinson’s disease (PD), 30 "normal" tremor and 30 "suspicious" PD (undiagnosed). Each column in the table is a particular tremor measure, and each row corresponds one of 2500 tremor recordings from these individuals ("name" column). The main aim of the data is to discriminate healthy people from those with PD, according to "status" column which is set to 0 for healthy and 1 for PD or "Suspicious". All patients are suffering of moderate to severe postural tremor. This postural tremor cannot be differentiated on clinical features (frequency, Parkinson’s Disease Prediction based on Multistate Markov Models 529 Figure 2: Intelligent system for health status prediction of a patient using a Markov chain Table 2: Data: size, age, gender, and disease duration distribution of PD, SPD, and NT subjects PD SPD NT Number of patients 28 30 30 Mean age 64.54 63.24 64.52 (range in years) (40-90) (27-94) (24-86) Gender (male/female) 18/10 16/8 19/11 Mean disease duration 16,4 5,3 amplitude). Patients were kept under observation and investigation for 2 years, and data were acquired at 6 months, 1 year and 2 years (see Table 2). The mean disease duration (time for disease to install, in years), age and sex of PD patients were compared with the SPD or NT in Table 2. Notice in Table 2 that the mean age of PD, SPD and NT populations is similar, but the age ranges are different. This could be considered as an indicator that the PD starts years before actual diagnosis. 3.2 Tremor recoding Yet, some researches have been made (including in Romania) in order to early diagnose the PD and its progress by means of the tremor or the gait analysis or other symptoms [29]- [34]. The tremor time series were acquired using an accelerometer sensor from a WiiTM console [35], connected via Bluetooth to a PC. The data were analyzed using an application implemented in Visual C 2010 Professional. The WiiTM Remote is the primary controller for Nintendo’s WiiTM i console. A main feature of the WiiTM Remote is its motion sensing capability, which allows the user to interact with and manipulate items on screen through the use of accelerometer and optical sensor technology [35]. Nintendo works on three axes: x - lateral, y - anteroposterior, and z - vertical. The device records both acceleration induced by hand movement and the component of gravitational force. If the controller is rotated, the gravity accelerometer affects the values on the x, y, and z axes (see Figure 3). This system using a WiiTM Remote is capable of analyzing frequency and estimated ampli- tude of tremor between 3 - 15 Hz (N tremor is between 5 - 12 Hz, and PD tremor is between 4-6 Hz). The WiiTM i Remote and PC are connected by Bluetooth - Human Interface Device Profile. The tremor analysis program was developed using Visual C 2010 Professional. The acceleration sampling period was set at 10 ms in the Nintendo device. Because the transmission rate through 530 O. Geman, H. Costin Figure 3: Interactive GUI using WiiTM Remote (tremor application) the Bluetooth device is limited, the sampling period of the tremor analysis was 40 ms. The ac- celerometer built into WiiTM Remote (Nintendo) measures gravitational and non-gravitational acceleration. The results of this paper suggest that Nintendo is useful for measurement and analysis of tremor using the methodologies described in [28], [29], [31]. We defined the following linguistic variables (for instance for X axis): • If x is between -0.10 mm and -1 mm then x is minimum Xmin; • If x is between -0.10 mm and 0.10 mm then x is medium Xmed; • If x is between 0.10 mm and 1 mm then x is maximum Xmax. We counted the number of spikes for each interval, and we used these values to describe the state vector. Next we proposed to predict the state of a patient using Markov chains. In this analysis the state vector is defined as: S = Xmin, Xmed, Xhigh, Ymin, Ymed, Yhigh, Zmin, Zmed, Zhigh (5) . Table 3 presents the number of spikes in each category for "normal" subjects, while Table 4 presents the values of the state vector for a subject with diagnosed Parkinson’s disease. For this paper we chose to exemplify the calculation of transition matrices from T0 to T1 and from T1 to T2 only for patients with Parkinson’s disease, by following the methodology presented in Section 2 (here notations Ti were used instead of ti and we illustrated the method only for PD patients and normal patients). In the first step we computed, according to the methodology, the deviations ∆ST1/T0 = ST1 − ST0 and ∆ST2/T1 = ST2 − ST1. We illustrate this in Tables 5 and 6 only with data acquired from a patient with PD. We computed next the transition matrices from T0 to T1 and from T1 to T2 (Tables 7 and 8, respectively). For example, the transition matrix from T0 to T1, MTPT1/T0(m∗m) is computed as follows: 1. the elements from the main diagonal are (SiT1, S i T0); Parkinson’s Disease Prediction based on Multistate Markov Models 531 Table 3: "Normal" subject vector, spikes number at T0, T1 and T2 for 60 seconds each record Features Vector T0 T1=6 months after T0 T2=12 months after T0 Total spikes Xmin 284 257 286 827 Xmed 1524 1458 1511 4511 Xmax 651 687 558 1896 Ymin 1289 1439 1435 4163 Ymed 1283 1247 1257 3787 Ymax 664 657 557 1878 Zmin 392 382 378 1152 Zmed 768 865 789 2422 Zmax 2031 1998 1875 5904 Table 4: Data: size, age, gender, and disease duration distribution of PD, SPD, and NT subjects Features Vector T0 T1=6 months after T0 T2=12 months after T0 Total spikes Xmin 382 358 379 1119 Xmed 785 758 688 2231 Xmax 897 857 912 2666 Ymin 578 547 524 1649 Ymed 457 479 487 1423 Ymax 354 349 357 1060 Zmin 257 282 253 792 Zmed 578 549 754 1881 Zmax 1300 1329 1348 3977 Table 5: The deviations ∆ST1/T0 = ST1 − ST0 (PD patient) T1 vs. T0, for the state vector S Time Xmin Xmed Xmax Ymin Ymed Ymax Zmin Zmed Zmax SUM T1 358 758 857 547 479 349 282 549 1329 T0 382 785 897 578 457 354 257 578 1300 Deviation -24 -27 -40 -31 22 -5 25 -29 29 Deviation+ 22 25 549 29 76 Table 6: The deviations ∆ST1/T0 = ST1 − ST0 (PD patient) T2 vs. T1, for the state vector S Time Xmin Xmed Xmax Ymin Ymed Ymax Zmin Zmed Zmax SUM T2 379 688 912 524 487 357 253 754 1348 T1 358 758 857 547 479 349 282 549 1329 Deviation 21 -70 55 -23 8 8 -29 205 19 Deviation+ 21 55 8 8 205 19 316 532 O. Geman, H. Costin 2. if i ̸= j, ∆Si T1/To < 0 and ∆Sj T1/To > 0, so the matrix equals the absolute value of ∆Si T1/T0 ∗ ∆S j T 1/T 0∑ ∆S ij T 1/T 0 >0 ; 3. the rest of elements equals 0. Table 7: The transition matrix from T0 to T1 Features Vector Xmin Xmed Xmax Ymin Ymed Ymax Zmin Zmed Zmax Xmin 358 0 0 0 0 0 0 0 0 Xmed 2.548 758 1.625 10.244 0 0 4.345 0 0 Xmax 0 0 857 0 0 0 0 0 0 Ymin 0 0 4.548 547 0 0 0 0 0 Ymed 7.413 0 1.021 12.547 479 0 0 12.457 0 Ymax 1.124 0 1.245 6.333 0 349 0 1.125 0 Zmin 1.354 0 0 6.687 0 0 257 2.548 0 Zmed 0 0 0 0 0 0 0 549 0 Zmax 4.211 0 0 24.442 0 0 0 8.457 1.300 Table 8: The transition matrix from T1 to T2 Features Vector Xmin Xmed Xmax Ymin Ymed Ymax Zmin Zmed Zmax Xmin 358 0 2.387 0 0 0 0 0 0 Xmed 0 688 0 6.257 0 0 4.345 6.211 0 Xmax 1.250 0 857 0 4.587 0 0 0 5.244 Ymin 0 0 1.287 524 0 0 0 0 0 Ymed 0 5.687 1.021 8.985 479 0 0 6.258 0 Ymax 1.124 0 1.245 0 0 349 0 1.125 0 Zmin 1.354 0 0 6.154 5.698 2.542 257 2.548 0 Zmed 0 0 0 0 0 0 0 549 2.241 Zmax 4.211 0 0 3.587 2.587 2.325 0 9.237 1.329 In the third step we calculated the total transition matrix (Table 9), which is the sum of partial transition matrices computed in the previous stage. In the fourth stage we computed the probability transition matrix by the ratio of each element of the total transition matrix to the sum of the line where the element is located. In the final stage of the algorithm we obtained the forecast of the structural elements for next year by multiplying the transposed matrix of transition probabilities with the vector of the structural elements for T2, i.e. the vector corresponding to T2 = 12 months. We get the following transition probabilities between the 9 elements of the features vector Xmin, ..., Zmax. The values of the main diagonal are the probabilities that the patient progress to state that is described by the features vector (which corresponds to a stage of the disease). The forecast of the Xmin...Zmax for the next year is obtained by multiplying the two matrices (transposed and Parkinson’s Disease Prediction based on Multistate Markov Models 533 Table 9: The total transition matrix (in %). Features Vector Xmin Xmed Xmax Ymin Ymed Ymax Zmin Zmed Zmax Total % Xmin 100 0 2.387 0 0 0 0 0 0 100 Xmed 9.59 90 0.0005 0.0051 0 0 0 0.0015 0 100 Xmax 8.54 0 91.46 0 0 0 0 0 0 100 Ymin 1.07 0 0 98.93 0 0 0 0 0 100 Ymed 12.91 0 0.17 1.69 84.71 0 0 0.5 0 100 Ymax 7.96 0 0.0004 0.47 0 91.37 0 0.14 0 100 Zmin 8.94 0 0.0009 0.90 0 0 89.79 0.26 0 100 Zmed 8.1 0 0 0 0 0 0 91.9 0 100 Zmax 15.19 0 0.0008 0.85 0 0 0 0.25 83.6 100 elements for T2). Thus we obtain the patient’s evolution for next year, for "normal" and "PD" (Table 10 and Table 11). Table 10: The "normal" subject’s evolution for the next year (no. of spikes) Features T0 T1=6 months T2=12 months T3=24 months sfter T0 T4=24 months after Vector after T0 after T0 (with Markov chain) T0(recorded) Xmin 244 257 286 295 299 Xmed 1442 1458 1511 1657 1656 Xmax 651 687 688 689 694 Ymin 1412 1439 1442 1420 1421 Ymed 1233 1247 1257 1243 1240 Ymax 614 627 665 688 686 Zmin 392 399 410 412 414 Zmed 768 788 789 786 785 Zmax 1992 1998 1999 1995 1994 From the last two tables one can see, by using Markov chains, the tremor symptom evolution of certain patients. Also we may note the very good prediction power of this method, as the features vector elements for the predicted tremor signal after 24 months from the first recording are very similar with the same vector elements, but acquired and measured by means of WiiTM Remote and the appropriate software. The maximum error between prediction and measured values was 1.33%. Similar judgement was used and corresponding good results concerning the prediction of disease evolution were obtained in the case of "suspicious PD" patients, for whom some early signs were found (insomnia, constipation, loss of smell, equilibrium and postural impairment, tremor symptom or speech difficulties) and they became to be attentively monitored. Also, another remark may be made related to the similarity between features vectors measured for "suspicious PD" patients and "diagnosed PD" patients, when using the same Markov chains for status prediction. 534 O. Geman, H. Costin Table 11: The PD patient’s evolution for the next year (no. of spikes) Features T0 T1=6 months T2=12 months T3=24 months sfter T0 T4=24 months after Vector (with Markov chain) T0(recorded) Xmin 382 385 396 398 399 Xmed 785 792 784 796 795 Xmax 837 857 912 944 946 Ymin 518 537 544 586 585 Ymed 457 459 467 489 488 Ymax 354 359 373 382 380 Zmin 257 282 278 310 308 Zmed 528 549 558 568 566 Zmax 1300 1329 1348 1399 1398 4 Conclusions In this paper we describe a general purpose model of PD prognosis based on Markov process and show how this simple mathematical tool may be used to generate detailed and accurate assessments of Parkinson’s disease stage and therefore may be applicable in medical screening for PD. Markov models consider a patient to be in one of a finite number of discrete states of health. All clinically important events are modeled as transitions from one state to another. Thus, the use of Markov models has the potential to allow the development of decision models that more faithfully represent clinical problems. Our study used a database where there are subjects who are considered normal, but with some tremor symptoms, and subjects considered "suspects", for whom we can apply the above methodology and can see if certain subjects move to the "normal" state or the first symptoms of Parkinson’s disease will appear. Thus, medical staff can intervene with specific medication for Parkinson’s disease. Using Markov chain is an efficient way to find the features vector for an individual patient at a given time, and this state vector may be used to predict and identify a stage in Parkinson’ disease. So, the physician can choose a treatment, based on this forecast with an appropriate level of medication. The system was validated for 88 patients under observation: 28 with PD tremors, 30 with SPD ("Suspicious" PD tremor), and 30 with NT (Normal tremor), and we plan to expand the study to more patients with PD. Already results interpretation and discussions with involved neurologists are directed to the validation of the study. The next step will be the creation of an expert or decision-support system based on fuzzy logic for Parkinson’s disease screening, which will help a physician to diagnose PD in its early stages, especially of individuals in the class "Suspicious" of PD. So, future research approaches will include the testing and validation of a screening test, in order to detect Parkinson’s disease or other neurological disorders in their early stages. Acknowledgment The authors would like to thank Radu Vasilcu, MD, and Prof. Mihai Covasa, PhD, MD, for their help in providing data for PD patients and for clinical evaluation of results during experiments. Parkinson’s Disease Prediction based on Multistate Markov Models 535 Important notice: the experiments were not prejudicial in any way to the health of human subjects investigated and they were not subject to any invasive maneuvers. All the subjects were free to decide whether or not they wish to participate in this study. They did not lose any benefits to which they are entitled, if they did not accept the participation. The duration of this study was 3 years. All personal information was and will be kept confidential. Medical infor- mation may be made available to the institution that houses the research, Ethics Commission, or other persons/institutions where the law requires. The benefits will be strictly medical. The information obtained in this study may help physicians to find a method of early diagnosis for those suffering from PD and to identify the best options for their treatment. It was no financial compensation during the study for the participants. 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