TITLE OF THE MAP:
INVESTIGATING THE FORCE LENGTH
RELATIONS OF THE KNEE EXTENSORS
MUSCLES
Seminar: Diagnostics of the morphological, mechanical and neuronal
properties
Authors:
Cem Guezelarslan (621847; cem.guezelarslan@student.hu-berlin.de)
Ray Butendeich (621808; ray.butendeich@student.hu-berlin.de)
Jan Joppek (584322; jan.joppek@student.hu-berlin.de)
Surveyor: Univ.-Prof. Adamantios Arampatzis
T ABLE
OF
C ONTENTS
List of Figures
List of Tables
1
Introduction.
1.1
2
Force-length relation
1
1
Methods.
2
2.1
Subjects
2
2.2
Measurement method/setup
3
2.3
Knee extension moment measurements
5
3
Results.
7
4
Conclusion.
8
5
Discussion.
10
6
Limitations.
10
Literature
I
II List of Figures
Figure 1: Seating position during the measurement
Figure 2: Alignment of leg and force arm (using a laser)
Figure 3: Picture of the Matlap screen which subjects could see during the measurement
Figure 4: Maximal active isometric force of all four knee extensors at the different ankles of the participants
Figure 5: Theoretical strength curves of the knee extensor muscles for hip joint angles
of 90 ~ (thin line) and 180 ~ (thick line). Curves are the sums of force-length curves for
individual knee extensor muscles, at 90 ~ and 180 ~ Force is normalized to the maximal
force generated (Herzog et. Al. 1990)
III List of Tables
Table 1: Measured forces for the knee angles 20-100 degrees normalized and not normalized over body weight
III
1
I NTRODUCTION
The purpose of this study was to experimentally determine the force-length relationships of the knee extensor muscles, including the rectus femoris, vastus lateralis, vastus
intermedius, and vastus medialis. We aimed to derive a strength curve for knee extension based on these relationships. In the strength curve, knee extension angles and
measured isometric forces are combined to describe the optimal angles for force production.
1.1 F ORCE - LENGTH
RELATION
The force-length relationship describes the maximum isometric force that a muscle can
generate in relation to its length. Blix first reported these relationships in 1891, highlighting that muscles can exert greater force at certain lengths compared to others. To
calculate the maximal active isometric forces of the knee extensors, carefully executed
maximal effort contractions were performed, systematically altering the knee joint configurations. Determining force-length relationships is limited to selected multi-joint
muscles that cross at least one joint not crossed by any synergistic muscles acting at
another joint. In their study of knee extensor muscles from human cadavers (Herzog et.
al, 1990) showed that the two-joint muscles are not able to produce force throughout
their full anatomical range of motion, whereas the one-joint muscles can. The strength
curve, which was determined as the sum of the force-length relations of the individual
knee extensor muscles and showed good agreement with experimentally obtained knee
extensor strength curves. This characterization pertains solely to the isolated muscle and
does not account for the actual force production, which involves interactions between
muscles, tendons, and passive structures like ligaments.
In tests conducted on intact human skeletal muscles, maximal forces are associated with
maximal voluntary contractions. However, these contractions may be influenced by
factors such as the subject's emotional state and other uncontrollable variables, leading
to variations in the observed forces. Active forces result from contractions in the con-
1
tractile elements of a muscle and require metabolic energy. In contrast, passive forces in
a muscle do not require metabolic energy and arise from the forces generated in parallel
elastic elements. Isometric forces occur during contractions where the length of the
muscle fibers remains constant, which can be achieved by maintaining a fixed joint configuration throughout the contraction period.
The force-length properties have been established based on theoretical considerations
using the cross-bridge model of muscular contraction. According to this model, crossbridges extend from thick filaments to thin filaments within a sarcomere (Huxley,
1957). During contraction, these cross-bridges attach to the thin filaments, exert force
on them, and cause the sliding of thin filaments past the thick filaments. The crossbridges are arranged periodically on the thick filaments, functioning independently, and
possessing the same force-producing capability. The force generated by a sarcomere is
linearly related to the overlap between thick and thin filaments for sarcomere lengths
twice the length of the thin filaments or greater. Experimental evidence on single muscle fibers in animals (Gordon et al., 1966) has supported the cross-bridge model as the
mechanism responsible for maximal isometric force production in skeletal muscles.
The understanding of force-length characteristics in muscles has been applied in surgical procedures involving tendon transfers in the hand and foot, aiming to improve decision-making in such surgeries. This study seeks to enhance our comprehension of forcelength associations in intact human skeletal muscles, potentially providing valuable insights for informed decision-making in tendon transfer surgery
2
M ETHODS
2.1 S UBJECTS
Two healthy male subjects were recruited from the participants of our seminar. Prior to
participation, both subjects underwent screening to ensure they met the inclusion criteria. They were physically active individuals who had engaged in regular weight training.
Subjective clinical assessments were conducted in a small interview to rule out any
2
musculoskeletal injuries or orthopedic abnormalities. Subject A, aged 25, had a height
of 190 cm and weighed 88 kg. Subject B weighed 92 kg and stood at a height of 186
cm. The mean (± s.d.) values for the two subjects were calculated as follows: age 26 ± 1
years, height 188 ± 2 cm, and body mass 90 ± 2 kg.
2.2 M EASUREMENT
METHOD / SETUP
To investigate the force-length relationship of the knee extensor muscles, a series of
measurements of maximal voluntary isometric contractions (MVC) at different knee
angles was planned. The objective was to determine whether there is a universal forcelength relationship shared among individuals or if there are significant individual differences. Two subjects were selected for comparison to identify any significant correlations. To standardize the measurements, the moments were normalized based on the
subjects' bodyweight. Nine measurements were taken for each participant, resulting in a
strength curve that represented the measured moments at specific knee angles. The
measurements were performed using an isokinetic dynamometer (Biodex 3) in a randomized order at knee angles of 20°, 30°, 40°, 50°, 60°, 70°, 80°, 90°, and 100°. A custom MATLAB script and GUI in MATLAB were used for data recording.
3
Figure.1 Seating position during the measurement
During the measurements, the Biodex backrest was fixed at 45 degrees to maintain a
consistent hip angle across all trials. This was crucial as hip angle influences the knee
extensor moment as shown by (Herzog, 1990), and any changes in hip angle during the
measurements could impact the results. The subject's hip was secured with a belt to prevent any pelvic movement during contractions. The tested leg was fastened to the force
arm just above the ankle using Velcro straps, while the other leg remained hanging. The
knee and ankle were aligned with the axis of the force arm using a laser, ensuring proper alignment as depicted in figure 2. The subjects maintained their arms crossed in front
of their chest throughout the trials. The final position, as shown in picture 2, was replicated after each change in knee angle. Additionally, the force arm was reset before each
trial to eliminate any passive forces on the sensor in the starting position.
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Figure. 2 Alignment of leg and force arm (using a laser)
2.3 K NEE
EXTENSION MOMENT MEASUREMENTS
The participants were instructed to perform static contractions using their knee extensors with maximal effort at nine specific knee angles. Particular emphasis was placed on
minimizing strain on unprepared muscles to reduce the risk of injury. To ensure adequate recovery and minimize the effects of previous maximal contractions, appropriate
rest periods of approximately 30-45 seconds were provided between each trial. During
each contraction, participants were encouraged to maintain the effort for approximately
3-5 seconds. It was essential to achieve a plateau in the maximal contraction to accurately calculate the maximum force. To provide real-time feedback, a computer screen
displayed the target force and the force produced by the participants. This visual feedback, resembling a game-like scenario where participants aimed to elevate the force line
further with each attempt, may have positively influenced their performance. Furthermore, external motivation was provided through supportive cheering from the rest of the
class during each trial, maintaining a consistent approach across all participants and
5
trials. Prior to initiating the measurements, participants performed two to three warm-up
trials to prepare their muscles for the subsequent testing.
We are well aware that a real worthwhile break would have to be a lot longer.
Figure. 3 Picture of the Matlap screen which subjects could see during the measurement
In the graph, the x-axis corresponds to time in seconds, while the y-axis represents the
moment in Nm (Newton-meters). The blue line represents the moment measurements
captured by the Biodex device during the trial. The measurement procedure commenced
when the Start button was clicked at the 0-second mark and concluded approximately at
8 seconds when the Stop button was pressed. The red vertical line indicates the specific
moment of interest recorded for the projects. During the sustained contraction period of
3-5 seconds, there may be fluctuations in the produced moment. To analyze the trial, we
established an analysis region, visually delineated by two horizontal green lines. This
region was manually determined and served as the designated area of interest for evaluation. By clicking the Evaluate button, the maximum measured moment within this defined region is determined. Additionally, the graphical user interface (GUI) draws a red
vertical line to mark this moment on the graph. To facilitate comparison between the
two subjects, we employed normalization techniques by dividing the moments by their
respective body weights. This normalization process yielded the normalized values,
which were subsequently utilized to construct strength curves for knee extension. The
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resulting curves are displayed in table 1 and are further elaborated upon in the subsequent chapter.
Angle
A
A (normalised)
20°:
106,6 Nm
1,21 Nm/kg
112,03 Nm
1,21 Nm/kg
30°:
177,68 Nm
2,01 Nm/kg
155,12 Nm
1,68 Nm/kg
40°:
197,61 Nm
2,24 Nm/kg
188,22 Nm
2,04 Nm/kg
50°:
221,44 Nm
2,51 Nm/kg
239,28 Nm
2,6 Nm/kg
60°:
302,8 Nm
3,44 Nm/kg
244,33 Nm
2,65 Nm/kg
70°:
296,34 Nm
240,85 Nm
2,61 Nm/kg
80°:
296,84 Nm
3,37 Nm/kg
203,43 Nm
2,21 Nm/kg
90°:
228,54 Nm
2,59 Nm/kg
175,18 Nm
1,9 Nm/kg
100°:
227,43 Nm
2,58 Nm/kg
178,38 Nm
1,93 Nm/kg
3,36 Nm/kg
B
B (normalised)
Tab.1 Measured forces for the knee angles 20-100 degrees normalized and not normalized over body
weight
3
R ESULTS
After comparing the normalized values from both subjects, we found that the optimal
angle for maximal voluntary isometric knee extension was identical. Both subjects were
able to exert the highest force at 60° knee angle. The minimal force they produced was
at 20-degree knee angle for both subjects. Overall, the extracted strength curves shown
in figure 4 are quite similar showing just slight deviations. The minimal difference in
both participants could have many reasons. For example, different from Herzog et. al.
we did not measure the exact seating position, which was shown to have a significant
effect on maximal knee extension. we didn't measure the body constitution from the
participants as well, like if they were active in their life or if they had some injuries in
the past. In addition to that, subjects didn't have a warmup before the measurement, so
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the data could be more meaningful if the mentioned parameters had been considered.
More about these limitations and others in the last chapter.
Figure 4: Maximal active isometric force of all four knee extensors at the different ankles of the
participants
4
C ONCLUSION
After reviewing our data and comparing them to the results of important studies about
the force-length relation of the four knee extensors we found similar effects. The discovery from the study of Herzog et. Al. 1990, that there is an optimal angle in the knee
joint, where all four knee extensors can produce the maximal active isometric force, can
also be supported by our own results. Herzog et. al. collected a similar survey, but they
considered additional factors, such as additional muscles and cross bridge theories. The
following graph shows their results in relation to our survey. The other results do not
provide a good basis for comparison because Herzog included additional analysis tools
and other factors.
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Figure 5: Theoretical strength curves of the knee extensor muscles for hip joint angles of 90 ~ (thin
line) and 180 ~ (thick line). Curves are the sums of force-length curves for individual knee extensor
muscles, at 90 ~ and 180 ~ Force is normalized to the maximal force generated (Herzog et. Al. 1990)
The optimal angle of the knee is at 60°, to have the maximal active isometric force of
the four knee extensors. Unlike us, Herzog and his colleagues also examined the angle
of the hip to discuss the meaning of the rectus femoris as a two-joint muscle. Their results displayed in figure 5 show a thinner and a thicker line, because of the two hip angles they measured. Behind the thinner line, they positioned the participants with a 90°
hip angle. The thicker line shows the strength curve with a hip angle of 180°. Herzog et.
Al. find out that, with the hip angle of 90° the knee is more flexed, that's why the maximal active isometric forces could occur with a 10°-20° smaller angle (Herzog 1990).
Since the effects of the hip angle on the maximal knee extension are substantial these
deviations could explain our results that are shown in the table and the figure 4.
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5
D ISCUSSION
In the previous chapter, we compared our result with the results from a similar study of
Herzog. To underline the differences, within our measurement we only use the parameter of the knee angle for the maximal active isometric force of all four knee extensors,
as well as Herzog with his colleagues. As the Results are shown the optimal ankle for
all four knee extensors is at 60° but we can't apply statements about the muscle length at
this angle. To measure the exact length of the muscle we need more data, i. e. the muscle sonography. With the system of the muscle sonography, it is possible to make visual
images of muscles for an exact length determination. The muscle sonography must be
used to make a relative force-length relation. Otherwise, an EMG could also be used, to
record the muscle activity of minimum three of the four knee knee extensors. This could
show more about the activity of the muscle in the different lengths. With those two instruments we could apply better statements about the muscle activity in relation to the
angle and the muscle length, not only about the angle of the knee.
6
L IMITATIONS
To sum up the Limitations that are already mentioned in the other chapters, the whole
measurement could have been more professional and standardized. The way we did it in
the seminar was due to the limited time we had available for each measurement.
Started with the participants' preparation, like the warm-up and the measurement of the
same break time between the sets. This could help the participants to start every measurement with the same energy level to not be exhausted. In addition to that, to have a
valid result, there would have had to be many more participants than only two. With
only two participants we cannot generalize the results to the population or estimate the
parameters in the population with the desired precision therefore we needed the comparison with other studies, like Herzog. Also, the hip angle parameter, that Herzog and his
colleagues integrated, should also be a parameter in the measurement. To come to an
end like in the discussion already mentioned. To hypothesize the relative force-length
relation there had to be an exact determination of muscle length in the different knee
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angles. Summing up these are some of the main limitations of our measurement affecting our data, but as the comparison to other studies has shown that our result, although
having slight deviations, were in line with their findings.
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IV Literature
Abrahamse SK, Herzog W, and ter Keurs HEDJ (1988) Considerations regarding forcelength relations of human rectus femoris muscle. Proceedings of the Canadian Society
for Biomechanics Ottawa, pp 30-31
Blix M (1891) Die Länge und die Spannung des Muskels. Scand Arch Physiol 3:295318
Gordon AM, Huxley AF, Julian FJ (1966) The variation in isometric tension with sarcomere length in vertebrate muscle fibers. J Physiol (London) 184:170-192
Herzog, W., Abrahamse, S. K., & ter Keurs, H. E. (1990). Theoretical determination of
force-length relations. European Journal of Physiology, 113-119.
HUXLEY A. F. (1957). Muscle structure and theories of contraction. Progress in biophysics and biophysical chemistry, 7, 255–318.
IV