**4. Analysis of knee joint motion using the simulator with waveforms measured**

In this section, we will deal with spastic patients as the subject of the analysis. Such patients have high phasic reflex.

### **4.1 Examples of waveforms measured**

The knee joint motion of a normal subject was measured in the pendulum test using the measurement system shown in Fig. 6 in section 3. Fig.16 shows examples of the waveforms measured. In the figure, (a) and (b) show the angle waveform and angular acceleration waveform, respectively. There is absolutely no restriction on motion of the lower leg, since only two accelerometers were attached to it. It is difficult to estimate the error in this measurement result quantitatively, but the measurement was probably made with about the same accuracy as obtained in the investigation in the preceding section. The collapse of the waveform that appears in the early stage of oscillation is noise produced by the state of contact between the hand of the investigator and the lower leg in the instant when the lifted leg was released. If this portion is eliminated, the angle waveform and angular acceleration waveform have typical damped oscillation with nearly the same periods, although the phases differ. These waveforms are the free oscillations mentioned in subsection 2.1. Both

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for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 35

contractile force in the quadriceps femoris muscle acts to extend the lower leg (see 2.2.3). The knee joint motion in this case is not free oscillation, but restricted one by the contractile force. Thus, as shown in the following equation, such a knee joint motion is given with an equation

02468

(a)

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(b)

The angle waveform and the angular acceleration waveform are accurately synchronized, because they are calculated directly from the outputs of the same linear accelerometers. Consequently, the results of measurement with the above-mentioned measurement system are suitable for the purpose of investigating behavior while rigorously referencing both

As seen from the above examples, according to the measurement system shown in Fig. 6 of section 3, the angle and angular acceleration of the knee joint motion in the pendulum test can be measured simultaneously with high accuracy. However, investigating in detail the phasic reflex generation mechanism of each subject or estimating the degree of increase in the reflex often requires values of physical quantities and/or waveforms for various

Fig. 17. Waveforms of knee-joint motion measured from a moderate spastic patient by

pendulum test. (a) Angle; (b) Angular acceleration.

(9)

time (s)

time (s)

that is obtained by adding this contractile force *Q*h to the right side of equation (8).

waveforms are described theoretically by the following differential equation derived by Vodovnik et al. (1984).

$$J\ddot{\theta} + B\dot{\theta} + K\theta + \frac{mgl}{2}\sin\theta = 0\tag{8}$$

Where *J*, *B*, *K*, *m*, , and *g* are the moment of inertia, viscosity, elasticity, mass, length of the lower leg and gravity acceleration.

Even if each coefficient value is taken as a constant, this is not simple to solve analytically. However, according to the result of numerical analysis, the waveforms have similar damped oscillations as in Fig. 16.

Fig. 16. Waveforms of knee-joint motion measured from a normal subject by pendulum test. (a) Angle; (b) Angular acceleration.

Next, we will look at the waveforms of spastic patients. Sample waveforms are shown in Fig. 17. The subject is a spastic patient with a moderately increased phasic reflex. Comparing them with the waveforms in Fig. 16, the first peak of the angular acceleration waveform is larger. This is because the muscle stretch velocity reaches a maximum in the vicinity where the acceleration first intersects the time axis and a phasic reflex is produced, and the resulting

waveforms are described theoretically by the following differential equation derived by

Where *J*, *B*, *K*, *m*, , and *g* are the moment of inertia, viscosity, elasticity, mass, length of the

Even if each coefficient value is taken as a constant, this is not simple to solve analytically. However, according to the result of numerical analysis, the waveforms have similar damped

02468

(a)

02468

(b) Fig. 16. Waveforms of knee-joint motion measured from a normal subject by pendulum test.

Next, we will look at the waveforms of spastic patients. Sample waveforms are shown in Fig. 17. The subject is a spastic patient with a moderately increased phasic reflex. Comparing them with the waveforms in Fig. 16, the first peak of the angular acceleration waveform is larger. This is because the muscle stretch velocity reaches a maximum in the vicinity where the acceleration first intersects the time axis and a phasic reflex is produced, and the resulting

(8)

time (s)

time (s)

Vodovnik et al. (1984).

oscillations as in Fig. 16.

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lower leg and gravity acceleration.

contractile force in the quadriceps femoris muscle acts to extend the lower leg (see 2.2.3). The knee joint motion in this case is not free oscillation, but restricted one by the contractile force. Thus, as shown in the following equation, such a knee joint motion is given with an equation that is obtained by adding this contractile force *Q*h to the right side of equation (8).

$$J\ddot{\theta} + B\dot{\theta} + K\theta + \frac{mgl}{2}\sin\theta = Q\_{\text{h}}\tag{9}$$

Fig. 17. Waveforms of knee-joint motion measured from a moderate spastic patient by pendulum test. (a) Angle; (b) Angular acceleration.

The angle waveform and the angular acceleration waveform are accurately synchronized, because they are calculated directly from the outputs of the same linear accelerometers. Consequently, the results of measurement with the above-mentioned measurement system are suitable for the purpose of investigating behavior while rigorously referencing both waveforms.

As seen from the above examples, according to the measurement system shown in Fig. 6 of section 3, the angle and angular acceleration of the knee joint motion in the pendulum test can be measured simultaneously with high accuracy. However, investigating in detail the phasic reflex generation mechanism of each subject or estimating the degree of increase in the reflex often requires values of physical quantities and/or waveforms for various

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**4.3 Applications of pendulum test simulator** 

level analysis of the stretch reflex using inverse simulation.

**4.3.1 Examples of waveforms obtained from simulation** 

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acceleration; (d) Muscle contraction.

as for muscle spindles and α-motoneurons can also be analyzed simply.

time (s)

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Fig. 18. Simulation result of a normal subject. (a) Angle; (b) Angular velocity; (c) Angular

angular velocity (rad/s)

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 37

Using the simulator described in the previous section, knee joint motion can be freely analyzed for arbitrary combination of input values of the simulator that cannot be directly measured from subjects. In addition, by slightly modifying the program as needed, the waveforms for arbitrary section of the model can be easily analyzed. Moreover, obtaining a model for each subject by inverse simulation using the simulator, it becomes possible to analyze the knee joint motion of that subject with the model only. The following 4.3.1, shows a simple analysis of knee joint motion using the simulator, and 4.3.2 shows a case of high-

Figures 18, 19, 20, and 21 are waveform examples of knee joint motion in a normal subject and patients with mild, moderate, and severe spasticity, respectively. In all of the figures, (a), (b), (c) and (d) show angle waveform, angular velocity waveform, angular acceleration waveform, and muscle contraction waveform, respectively. It is clearly understood from the figures that there is a relationship between knee joint motion and muscle contraction that changes as spasticity increases. Although not shown in the figure, output waveforms such

).We created this program in C language for execution on a personal computer.

), and angular acceleration waveform

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time (s)

time (s)

sections. Constructing a simulator for the pendulum test using the model described in the next subsection, the values of physical quantities and waveforms appropriate for an arbitrary purpose will be able to be calculated freely.

#### **4.2 Pendulum test simulator**

Since equation (9) described in 4.1 is a basic model expressing knee joint motion, a simulator can be constructed using it. However, to obtain a simulator with good accuracy, the nonlinearity of *B* and *K* needs to be considered together with a detailed mathematical formulation of *Q*h. The following briefly describes these implementation methods.

*Q*h is muscle contraction that occurs in the quadriceps femoris muscle during the stretch reflex, expressed by equations (10) and (11) (Jikuya et al., 2001).

$$Q\_{\rm h} = \exp(-T\_{\rm ds}\mathbf{s}) \frac{F\_{\alpha}}{(1 + T\_{\rm m}\mathbf{s})^2} \tag{10}$$

$$F\_{\alpha} = \frac{F\_{\gamma \text{d}} - k\_{\text{f}} \theta \text{s}}{F\_{\text{i}}} + F\_{\text{e}} \tag{11}$$

exp(-*T*ds) and 1/(1+*T*ms)2 are transfer functions that express the sum of the transmission delay times in Group Ia fiber and α-fiber (*T*d: time required for an impulse to pass through Group Ia fiber and α-fiber) and the characteristic of excitation contraction coupling (*T*m: twitch contraction time of muscle), respectively. s is a Laplace operator, and *k*f is a coefficient to convert knee joint angle to muscle length. *F* is the output of -motoneuron. *F<sup>d</sup>*, *Fi* and *Fe* are normalized command frequencies of *fγd*, *f*i and *f*<sup>e</sup> − *V*thα (*V*thα: α-motoneuron threshold), respectively. Equation (11) expresses that the phasic afferent information *F*γ<sup>d</sup> − *k*f*θ*s sent from muscle spindles, after being inhibited by presynaptic inhibition *F*i, is added to command *F*<sup>e</sup> from the brain and becomes the output of α-motoneuron.

Next is a description of the modeling of *B* and *K* with large nonlinearity. Each extrafusal muscle fiber that make up muscle contain many actin and myosin molecules. It is known that, in a resting state, the vast majority of these molecules are in a gel state, but when the muscle starts to flex or extend movement, these molecules solate in accordance with the velocity of the flexion or extension (Lakie et al., 1984). Using the properties of these actin and myosin molecules, the temporal changes in *B* and *K* values are described by the following differential equations (Jikuya et al., 1995).

$$B = a(B\_\text{M} - B) - b(B - B\_\text{m}) \mid \theta \mid \tag{12}$$

$$K = c(K\_M - K) - d(K - K\_m) \mid \theta \mid \tag{13}$$

Here, *B*M(*K*M) and *B*m(*K*m) are the maximum and minimum values, respectively, of *B*(*K*), and *a*, *b*, *c*, and *d* are all constants. These equations express equality of the differential of viscosity and elasticity coefficients to the value when the proportion that is solated (*b*(*B*-*B*m)| |) is subtracted from the portion of actin and myosin molecules that is gelated (*a*(*B*M-*B*)).

The pendulum test simulator is a program specifically for analyzing knee joint motion, prepared according to the mathematical models in equations (9)–(13) above. The inputs to the program are values of the constants that appear in these equations, and the output is angle waveform (*θ*), angular velocity waveform ( ), and angular acceleration waveform ( ).We created this program in C language for execution on a personal computer.

### **4.3 Applications of pendulum test simulator**

36 Advanced Topics in Measurements

sections. Constructing a simulator for the pendulum test using the model described in the next subsection, the values of physical quantities and waveforms appropriate for an

Since equation (9) described in 4.1 is a basic model expressing knee joint motion, a simulator can be constructed using it. However, to obtain a simulator with good accuracy, the nonlinearity of *B* and *K* needs to be considered together with a detailed mathematical

*Q*h is muscle contraction that occurs in the quadriceps femoris muscle during the stretch

exp(-*T*ds) and 1/(1+*T*ms)2 are transfer functions that express the sum of the transmission delay times in Group Ia fiber and α-fiber (*T*d: time required for an impulse to pass through Group Ia fiber and α-fiber) and the characteristic of excitation contraction coupling (*T*m: twitch contraction time of muscle), respectively. s is a Laplace operator, and *k*f is a coefficient

are normalized command frequencies of *fγd*, *f*i and *f*<sup>e</sup> − *V*thα (*V*thα: α-motoneuron threshold), respectively. Equation (11) expresses that the phasic afferent information *F*γ<sup>d</sup> − *k*f*θ*s sent from muscle spindles, after being inhibited by presynaptic inhibition *F*i, is added to command *F*<sup>e</sup>

Next is a description of the modeling of *B* and *K* with large nonlinearity. Each extrafusal muscle fiber that make up muscle contain many actin and myosin molecules. It is known that, in a resting state, the vast majority of these molecules are in a gel state, but when the muscle starts to flex or extend movement, these molecules solate in accordance with the velocity of the flexion or extension (Lakie et al., 1984). Using the properties of these actin and myosin molecules, the temporal changes in *B* and *K* values are described by the

Here, *B*M(*K*M) and *B*m(*K*m) are the maximum and minimum values, respectively, of *B*(*K*), and *a*, *b*, *c*, and *d* are all constants. These equations express equality of the differential of viscosity

The pendulum test simulator is a program specifically for analyzing knee joint motion, prepared according to the mathematical models in equations (9)–(13) above. The inputs to the program are values of the constants that appear in these equations, and the output is

and elasticity coefficients to the value when the proportion that is solated (*b*(*B*-*B*m)|

subtracted from the portion of actin and myosin molecules that is gelated (*a*(*B*M-*B*)).

(10)

(11)

(12)

(13)


*<sup>d</sup>*, *Fi* and *Fe*

is the output of -motoneuron. *F*

formulation of *Q*h. The following briefly describes these implementation methods.

arbitrary purpose will be able to be calculated freely.

reflex, expressed by equations (10) and (11) (Jikuya et al., 2001).

to convert knee joint angle to muscle length. *F*

from the brain and becomes the output of α-motoneuron.

following differential equations (Jikuya et al., 1995).

**4.2 Pendulum test simulator** 

Using the simulator described in the previous section, knee joint motion can be freely analyzed for arbitrary combination of input values of the simulator that cannot be directly measured from subjects. In addition, by slightly modifying the program as needed, the waveforms for arbitrary section of the model can be easily analyzed. Moreover, obtaining a model for each subject by inverse simulation using the simulator, it becomes possible to analyze the knee joint motion of that subject with the model only. The following 4.3.1, shows a simple analysis of knee joint motion using the simulator, and 4.3.2 shows a case of highlevel analysis of the stretch reflex using inverse simulation.

#### **4.3.1 Examples of waveforms obtained from simulation**

Figures 18, 19, 20, and 21 are waveform examples of knee joint motion in a normal subject and patients with mild, moderate, and severe spasticity, respectively. In all of the figures, (a), (b), (c) and (d) show angle waveform, angular velocity waveform, angular acceleration waveform, and muscle contraction waveform, respectively. It is clearly understood from the figures that there is a relationship between knee joint motion and muscle contraction that changes as spasticity increases. Although not shown in the figure, output waveforms such as for muscle spindles and α-motoneurons can also be analyzed simply.

Fig. 18. Simulation result of a normal subject. (a) Angle; (b) Angular velocity; (c) Angular acceleration; (d) Muscle contraction.

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that subject can be gained.

application of inverse simulation.

for Knee Joint Motion During the Pendulum Test Using Two Linear Accelerometers 39

(a) (b)

muscle contraction (Nm)

(c) (d)

Determination of the input values for the simulator, so that the waveforms obtained with the pendulum test simulator agree as closely as possible with the waveforms for actual knee joint motion measurements, is called inverse simulation. If inverse simulation is conducted for knee joint motion measured with the pendulum test, the waveforms generated in the simulator, as already mentioned, are nearly the same as the measured waveforms for that subject. The constant and command frequency values at this time are values that characterize the individual subject. Therefore, if simulation is conducted based on these constant and command frequency values and waveforms and information for each part of the reflex arc are analyzed, a detailed understanding of the enhancement of the reflex for

The results of inverse simulation for one patient with spasticity are shown in Fig. 22. The solid line shows the result of actual measurement, and the broken line shows the result of simulation. There is extremely close agreement between the two results. Considering the sufficient accuracy of the knee joint motion measurement system and this kind of good agreement between the two with this method, the simulator is also assumed to have sufficient accuracy. At the current stage, however, some problems remain in aspects such as the accuracy of constant and command frequency values obtained from the inverse

If these problems are solved, it is expected that the following issues can be resolved with

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Fig. 21. Simulation result of a severe spastic patient. (a) Angle; (b) Angular velocity; (c)

**4.3.2 High-level analysis of the stretch reflex by inverse simulation** 

simulation and the time required to implement inverse simulation.

Fig. 19. Simulation result of a mild spastic patient. (a) Angle; (b) Angular velocity; (c) Angular acceleration; (d) Muscle contraction.

Fig. 20. Simulation result of a moderate spastic patient. (a) Angle; (b) Angular velocity; (c) Angular acceleration; (d) Muscle contraction.

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Fig. 19. Simulation result of a mild spastic patient. (a) Angle; (b) Angular velocity; (c)

time (s)

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Fig. 20. Simulation result of a moderate spastic patient. (a) Angle; (b) Angular velocity; (c)

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Fig. 21. Simulation result of a severe spastic patient. (a) Angle; (b) Angular velocity; (c) Angular acceleration; (d) Muscle contraction.
