Introduction

The force-length relation (FLR) and force-velocity relation (FVR) are two frequently studied relationships in skeletal muscle physiology that have practical significance. The FLR describes the isometric contractile force of a muscle as a function of muscle length. The curve is characterized by a positive slope at shorter muscle lengths, a zero slope at medium lengths, and a negative slope at longer lengths (if only active tension is being measured) (1). The FVR describes the contractile force of a muscle as a function of velocity and is characterized by a double-hyperbolic pattern (1). In general, peak torque decreases as concentric velocity increases. The practical applications of the FLR and FVR range from sports training to rehabilitation to biomedical engineering.

Given the popularity and practical significance of these curves, it is important for those who are working within the domain of skeletal muscle physiology and biomechanics to have a theoretical understanding of these relationships using applied examples. Therefore, the purpose of this study is to assess the FLR and FVR and to qualitatively and quantitatively compare the results with past literature

Methods

Five isometric maximum voluntary contractions (MVCs) of the knee extensors were performed by one male at 45, 75, 90, 105, and 120 degrees; four MVCs were performed by one female at 45, 75, 90, and 105 degrees. Angles were measured as the anatomical position of the knee. During each MVC, attachment angle, attachment angular velocity, and torque was sampled at 100 Hz using a Biodex System 4 dynamometer. Data was also collected during one trial where the knee was moved through its range of motion by the Biodex while the subject was relaxed. Data was post-processed with a 5-point moving average filter for both the passive and MVC trials. The peak torque value from each of the MVC trials was subtracted from the corresponding length during the passive trial and then plotted.

Eight MVCs of the knee extensors were then performed by the same male at the following angular velocities: -240, -180, -120, -90, 90, 120, 180, and 300 . During each MVC, attachment angle, attachment angular velocity, and torque were sampled at 100 Hz using the same Biodex. A knee angle of 75 degrees was used for analysis. The torque at 75 degrees during the passive trial was subtracted from each of the MVC trials and then plotted.

Data Analysis

library(ggplot2)
library(dplyr)
library(purrr)
# Passive import
setwd("../data/Part_1")
files <- list.files(pattern = "Passive")
comb_passive <- map(files, read.delim, skip = 5, sep = "")
names(comb_passive) <- files
comb_passive <- map2(comb_passive, c("Male", "Female"), function(x, y) {
  x$sex <- y
  return(x)
})
comb_passive <- do.call(rbind, comb_passive)
# Active import
setwd("../data/Part_1")
files <- list.files(pattern = "isom")
comb_active <- map(files, read.delim, skip = 5, sep = "")
names(comb_active) <- files
setwd("../data/Part_2")
data_passive <- read.delim("BradyPassive.txt", header = TRUE, skip = 5, sep = "")
data_0 <- read.delim("Brady75isom.txt", header = TRUE, skip = 5, sep = "")
data_90_90 <- read.delim("Brady90con_90ecc.txt", header = TRUE, skip = 5, sep = "")
data_120_120 <- read.delim("Brady120con_120ecc.txt", header = TRUE, skip = 5, sep = "")
data_180_180 <- read.delim("Brady180con_180ecc.txt", header = TRUE, skip = 5, sep = "")
data_300_300 <- read.delim("Brady300con_240ecc.txt", header = TRUE, skip = 5, sep = "")
# Passive Cleaning --------------------------------------------------------
names(comb_passive) <- c("time", "torque", "biodex_angle", "knee_angle", 
                      "biodex_velocity", "sex")

comb_passive$index <- c(0, rep(1:(nrow(comb_passive)-1)%/%5))

comb_passive_avg <- 
    comb_passive %>%
    group_by(index) %>%
    summarise(time = mean(time),
              torque = mean(torque),
              biodex_angle = mean(biodex_angle),
              knee_angle = mean(knee_angle),
              biodex_velocity = mean(biodex_velocity),
              sex = sex[1])
# Active Cleaning ---------------------------------------------------------
Degrees.1 <- as.numeric(stringr::str_extract(names(comb_active), "[0-9]{2,3}"))
sex <- c(rep("Female", 4), rep("Male", 5))

comb_active <- pmap(list(comb_active, Degrees.1, sex), function(x, y, z) {
  x$Degrees.1 <- y
  x$sex <- z
  return(x)
})

comb_active <- do.call(rbind, comb_active)

colnames(comb_active) <- c("time", "torque", "biodex_angle", "knee_angle", 
                           "biodex_velocity", "sex")

comb_active <- 
    comb_active %>% 
    group_by(knee_angle, sex) %>% 
    filter(torque == max(torque))

comb_active <- comb_active[-2,]

# Subtracting Passive from Active -----------------------------------------
peak_passvie <- 
    comb_passive %>%
    select(knee_angle, torque, sex) %>% 
    group_by(knee_angle, sex) %>% 
    filter(torque == max(torque), knee_angle %in% c(45, 75, 90, 105, 120))

comb_active$passive <- c(-11.1, -9.7, -2.2, -1, 2.2, 3.7, 7.3, 9.1, 15.2)

comb_active <- 
    comb_active %>% 
    mutate(active_component = torque - passive)

# ANALYSIS PART 2 ---------------------------------------------------------
# Cleaning ----------------------------------------------------------------
# Passive
ecc_passive <- max(subset(data_passive, Degrees.1 == 75 & DEG.SEC < 0)$FT.LBS)
con_passive <- max(subset(data_passive, Degrees.1 == 75 & DEG.SEC > 0)$FT.LBS)

# Active
data <- 
rbind(
mean(subset(data_0, Degrees.1 == 74 & DEG.SEC == 0)$FT.LBS),
max(subset(data_90_90, Degrees.1 == 75 & DEG.SEC < 0)$FT.LBS),
max(subset(data_90_90, Degrees.1 == 75 & DEG.SEC > 0)$FT.LBS),
max(subset(data_120_120, Degrees.1 == 75 & DEG.SEC < 0)$FT.LBS),
max(subset(data_120_120, Degrees.1 == 75 & DEG.SEC > 0)$FT.LBS),
max(subset(data_180_180, Degrees.1 == 75 & DEG.SEC < 0)$FT.LBS),
max(subset(data_180_180, Degrees.1 == 75 & DEG.SEC > 0)$FT.LBS),
max(subset(data_300_300, Degrees.1 == 76 & DEG.SEC < 0)$FT.LBS),
max(subset(data_300_300, Degrees.1 == 75 & DEG.SEC > 0)$FT.LBS))

velocity <- c(0, -90, 90, -120, 120, -180, 180, -300, 300)

data <- as.data.frame(cbind(data, velocity))

colnames(data) <- c("FT.LBS", "velocity")
comb_passive_avg %>% 
    ggplot(mapping = aes(x = knee_angle, y = torque, linetype= sex)) +
    geom_smooth(se = FALSE, color = "black") +
    labs(title = "Biodex Passive Trial", linetype = "Sex") +
    xlab("Knee Angle (degrees)") + 
    ylab ("Torque (Nm)") +
    theme_bw() +
    theme(panel.grid.minor.x = element_blank(),
          text = element_text(family = "Arial", size = 12),
          plot.title = element_text(size = 16),
          panel.grid.minor.y = element_blank()) +
    guides(linetype = guide_legend(reverse = TRUE))

comb_active %>% 
    ggplot(mapping = aes(x = knee_angle, y = active_component, linetype = sex)) + 
    geom_smooth(se = FALSE, color = "black") + 
    geom_point(shape = 15, size = 2) +
    labs(title = "Biodex MVC trials: Torque and Knee Angle") + 
    ylab("Torque (Nm)") + 
    xlab("Knee Angle (degrees)") + 
    labs(linetype = "Sex") +
    xlim(30,130) +
    theme_bw() +
    theme(panel.grid.minor.x = element_blank(),
          text = element_text(family = "Arial", size = 12),
          plot.title = element_text(size = 16),
          panel.grid.minor.y = element_blank()) +
    guides(linetype = guide_legend(reverse = TRUE))

ggplot(data, mapping = aes(x = velocity, y = FT.LBS)) + 
    geom_line(color = "black", se = F) +
    geom_point(shape = 15, size = 2) +
    labs(title = "Biodex MVC trials: Torque and Velocity")+ 
    ylab("Torque (Nm)") + 
    xlab("Knee Angular Velocity (deg/sec)") +
    theme_bw() +
    theme(text = element_text(family = "Arial", size = 12),
          plot.title = element_text(size = 16),
          panel.grid.minor.y = element_blank(),
          panel.grid.minor.x = element_blank())

Discussion

Qualitative comparisons
Qualitatively speaking, the observed data moderately reflected the typical FLR and FVR. The observed FLR data (Fig. 1) for both males and females showed a roughly-linear positive slope followed by a roughly-linear negative slope, which reflects the typical FLR curve. The FVR data (Fig. 2) had an overall positive slope for the negative velocity, which does not reflect a typical FVR curve, and a relatively steep, negative slope for the positive velocity. Fig. 3 depicts torque during the passive trial, and the observed data is very similar to the theoretical relationship; that is, a linear relationship between torque and knee angle.

Although the data in general reflect the theoretical FLR, there are several apparent differences worth noting. 1. The negative slope for the male participant appears significantly less steep than the positive slope, whereas the theoretical curve depicts the slopes as roughly symmetrical. 2. A small plateau was not observed for the female participant. However, if torque were measured at more knee angles between 75 and 90 degrees, a plateau may have been observed. 3. The male participant appears to have a wider FLR curve compared to the female; at 120 degrees, the male achieved a higher peak torque than what he achieved at 45 degrees, indicating that a larger knee angle would have been needed to equate with the torque observed at 45 degrees. In contrast, the female participant already had a lower peak torque at 105 degrees than 45 degrees. (4) The male participant had a much larger torque at all knee angles.

Similarly, there are several differences between the observed data and the theoretical FVR worth noting. 1. The concentric side (positive velocity) of the curve does have an expected negative and steep slope, but there is a clear outlier at 120 degrees. This could be explained by a change in subject effort from one trial to the next. 2. As stated earlier, the eccentric (negative velocity) values had an overall positive slope as they approached zero, which does not reflect the usual FVR curve. It is difficult to interpret this section of the data because there is a combination of positive and negative slopes with no clear trend.

Quantitative Comparisons
Several aspects of the FLR and FVR from the obtained data were quantitatively compared with studies from the literature. A study by Borges (2) found that the isometric torque of the knee extensors in women varied between 54% and 62% of that in men. In the present study, we found it to be between 44% and 63% of that in men. Even though the sample size of the present study was n=1, the results are consistent with past literature. Borges also had his subjects perform isokinetic trials of the knee extensors at 12, 90, and 150 and found the average for young men to be 155, 122, and 96 Nm, respectively. In the present study, the most similar corresponding data points were at 0, 90, and 120 , and the average values for the young male participant were 192, 77, and 97 Nm, respectively. Interestingly, our data does not follow the same decreasing slope that is theoretically expected and observed in the literature. Potential explanations for the observed data will be discussed in the following section. Thorstensson and colleagues (3) performed isokinetic contractions of the knee extensor that consisted of positive velocities ranging from 0-180 . He found that the knee torque produced was higher during the static trial than any of the dynamic trials, which is consistent with the theoretical curve as well as our observed data. Similarly, one study showed that torque was highest during the static trial, and also found a linear decrease with increasing angular velocities (4). In contrast, a linear decrease was not observed in the present study.

Sources of error
Torque, knee angle and knee angular velocity were measured to assess the FLR and FVR, as opposed to directly measuring force, muscle length and contractile velocity. Although these measurements can serve as a good approximation, it is important to keep in mind that they are not the same thing. For example, the exact moment arms and center of mass of the shank is unknown, so we cannot obtain precise force measurements. Similarly, the exact change in sarcomere length is unknown, so the change in knee angle is used as an approximation. Additionally, when measuring the torque, the knee must be perfectly aligned with the dynamometer or else there will be translational forces that are unaccounted for. In the present study, it is highly unlikely that all (or any) of the trials were performed with the knee perfectly in line.

Another potential source of error is subject performance. During the isometric trials, it is possible that subjects were more fatigued during the last trial than the first trial, which may have altered the results. The same male subject was then used for the isokinetic trials, which again may have altered the results due to fatigue. Or, similarly, variation in subject performance may have caused errors. The slope of the concentric FVR was a particularly unusual result, and this may have been caused by variation in subject performance.

Other factors affecting force production
The data in the present study did not perfectly replicate the FLR or FVR, and that is to be expected. Not only are there multiple sources of error contributing to the observed data points, but we also had a sample size of n=1, and there is always some amount of variance, even without measurement error. Not only this, but other factors can also affect force production in addition to the F-L and F-V relations. Pennation angle can affect muscle force production. Most muscles have fascicles that are not perfectly parallel to the line of pull but instead fan out; when the fascicles fan out, more fibers are able to fit into a given volume, which leads to a greater cross-sectional area. Because there is a linear relationship between cross-sectional area of force production (5), pennate muscles are able to produce more force compared to non-pennated muscles. A study by Ikegawa et al. demonstrated this, by showing a positive correlation (r = 0.832) between cross-sectional area and pennation angle for bodybuilders (r = 0.832) and weightlifters (r = 0.682) (6).

Musculoskeletal geometry does not affect force production, but it does affect muscular torque. As previously stated, it is important to note that muscle torque was measured, not muscle force. Given a certain amount of force being produced by a muscle, the observed muscular torque can vary depending on the moment arm of the muscle. It can also vary depending on the length of the moment arm of the force that is being overcome (e.g. dumbbell).

A more obvious factor that can affect force production is physical activity. Depending on the type of physical activity being performed, motor unit fatigability can be reduced, and/or contraction speed and motor unit force can be increased (7). The latter is generally accomplished through strength training. By taking a muscle biopsy, Keijo et al. (7) found an increase in the proportion of type IIa fibers following a 10-week progressive strength training program aimed at increasing muscle mass, maximal peak torque, and explosive strength. They also found an increase in the cross-sectional area of the quadriceps femoris by an average of 12.2% for young men following the program. Increasing the number and proportion of type IIa fibers and increasing the cross-sectional area of the muscle via strength training lead to an increase in force production.

Force production can be influenced by motor unit recruitment. The size principle is well-known in skeletal muscle physiology and it states that motor units are recruited according to their size as more force is required. An individual’s ability to effectively recruit and coordinate larger motor units will influence maximal force production. This principle also has implications for the FL and FV curves as a whole; it suggests that differences in motor unit recruitment could be one explanatory variable for deviations from the typical FLR and FVR. Even if a subject has well-developed motor recruitment of a muscle, the motor units can become fatigued and alter force production. Contessa et al. (8) has shown increases in motor unit firing rate variability during fatigue, which led to increased variability of force production.

Conclusion
The purpose of this study was to generate FLR and FVR curves and qualitatively and quantitatively compare our findings with the literature. In the present study, the two curves moderately reflect the theoretical curves and the curves found in the past literature, but there were several differences. One of the most notable differences was the positive velocity slope for the FVR curve, which did not show a steady, negative slope as one would expect. It is also important to note that there are multiple potential sources of error for the present study, including an imperfect knee alignment with the knee dynamometer, fatigue, variation in subject performance, and the fact that muscular torque, knee angle, and knee angular velocity were used as measurements as opposed to in vivo force production and sarcomere length. And lastly, length and velocity are not the only two factors contributing to force production. Muscular structure, musculoskeletal geometry, physical activity and motor unit recruitment can all play a role.

References

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  6. Ikegawa SF, et. al. J of Strength and Cond Research,22(1), 128-131, 2008.
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