A Validation Study for Estimating Vertical Stiffness and Leg Stiffness During Running in Children

Purpose : The purpose of this study was to validate for leg and vertical stiffness estimated by using Morinʼs method. Method : One hundred twenty seven children participated in this study. Each subject sprinted for 50 m. The motion through an interval from 30 m to 40 m was recorded with high-speed camera at 300 Hz from the sagittal plane. The running speed, contact time in stance phase and flight time were measured to estimate maximal force exerted on the foot (F max ), vertical stiffness (kʼ vert ) and leg stiffness (kʼ leg ) by Morinʼs method (2005). In gold standard method (MacMahon and Cheng, 1990), vertical excursion of center of mass and leg spring length variation was calculated from digitized landmarks and ratios of F max to vertical excursion of center of mass and to leg spring length variation were calculated as the vertical stiffness (k vert ) and leg stiffness (k leg ). Result : All values are represented in mean value ± standard deviation (SD). kʼ vert was 15.51 ± 5.97 kN/m, whereas k vert was 17.92±6.59 kN/m. The mean difference of the kʼ vert between k vert was -2.41±2.62 kN/m. ICC between kʼ vert and k vert was 0.851 (p<0.001). Mean kʼ leg and k leg was 5.21±1.91 kN/m and 6.81±2.57 kN/m. ICC between kʼ leg and k leg was 0.642 (p < 0.001), while kʼ leg was highly correlated with k leg (r = 0.839). Additionally kʼ leg was underestimated (-23.6%) to k leg . Conclusion : We could conclude that Morinʼs method might be able to estimate vertical stiffness and leg stiffness in children, although vertical and leg stiffness tended to be underestimated.


Introduction
In many sports, running ability is required to move quickly the body from one place to another. The time which a person spent time from start to goal in a given distance is generally used to assess the running ability. The time is determined by the distance of the event and by the personʼs average speed over the distance. The speed at which the person runs is equal to the product of two factors: the stride length and stride frequency. It is, however, a lack of information to assess particularly the running ability for children in development, because running ability for children in development is influenced mainly by the physical fitness as of then 1) .
During running, it is known that the leg supported the body behaves like a "spring" 2) 3) . MacMahon and Cheng (1990) justified spring-mass model consisting of a particle of the body mass and a leg supporting the particle as a linear spring, so called "leg spring" (Figure-1) 3) . A main parameter to describe lower limbs as the leg spring is leg stiffness which is defined as a ratio of peak ground reaction force to the maximal variation of leg spring length during stance phase. Additionally, vertical stiffness is used to describe the vertical motion of the COM during the foot is in contact with the ground. Spring-mass model has been widely used to describe storage and return of elastic energy in lower limbs 3)-5) . There is a possibility to be able to assess running ability for children in development by using leg and vertical stiffness. Morin et al (2005) developed a simple method for estimating vertical stiffness and leg stiffness 6) . By using this method, we were able to estimate leg and vertical stiffness from running speed, contact time, and flight time. These parameters could be measured by using video camera without using force platform and digitizing manually the anatomical landmarks of the body to calculate position of COM. Morinʼs method has, therefore, an advantage in the simplicity of the experimental system to measure parameters to estimate leg and vertical stiffness. In the fact, many previous studies employed Morinʼ s method to estimate leg and vertical stiffness 7)-10) .
Validity of Morinʼs method to estimate leg stiffness and vertical stiffness were tested to compare values of leg and vertical stiffness calculated from measured maximal ground reaction force and leg spring length 6) . Because the subjects in the validity test were ten young men, the tested validity is delimited in young adults. In our knowledge, it is not validated that vertical stiffness and leg stiffness in children estimated by using Morinʼs method. To apply Morinʼs method to children, it is necessary to investigate the validity of the leg and vertical stiffness estimated by using the method. The purpose of this study was to identify validity of Morinʼs method to estimate leg and vertical stiff-ness in children during running.

Subjects
A hundred twenty seven children (age 6-12 yrs, height 1.122-1.756 m, body mass 17.1-70.8 kg) participated in this study. The protocol of this study was approved by the Ethics Committee of Department of Sports Science at Juntendo University.

Data collection
Each subject sprinted for 50 m. The motion through an interval from 30 m to 40 m was recorded with high-speed camera at 300 Hz from the sagittal plane ( Figure-2). The running speed, contact time in stance phase and flight time was measured to estimate maximal value of normal force exerted on the foot (Fmax), vertical stiffness (kʼvert) and leg stiffness (kʼleg) by Morinʼs method (2005).

Estimation of maximal value of normal force Fmax
Maximal value of normal force Fmax was calculated as follows, where m, g, tf , tc represent the body mass, gravitational acceleration, flight time , contact time. The equation of Fmax, which based on a model used by Dalleau et al. (2004) 11) , was derived from inputting a half of tc into sin curve function of time F(t) 12) 13) .
F t=F sin  π t t The estimated Fmax was used to calculate leg stiffness and vertical stiffness in Morinʼ s method and reference method.

Morinʼs method to estimate leg stiffness kʼleg and vertical stiffness kʼvert
Maximal displacement of COM Δyʼ during stance phase was required to estimate leg stiffness kʼleg and vertical stiffness kʼvert. Therefore, Δyʼ was estimated from Fmax, tc, and m, as follows.

Δy=
F t  mπ  +g t  8 Maximal displacement of leg spring length Δlʼ was calculated as follows.
Δlʼ=Δyʼ+l− l  −vt/2  The l0 represents the leg spring length at landing to the ground calculated from body weight×0.53 14) . The υ represents the running speed calculated from the 30-40 m interval time.
From the above, follow equations to calculate kʼleg and kʼvert were derived.

Reference method to calculate leg stiffness and vertical stiffness
In gold standard method 2) , ratio of Fmax to vertical excursion of COM and to leg spring length variation was calculated as the vertical stiffnes (kvert) and leg stiffness (kleg). Locations of COM in stance phase were calculated from digitized coordinates and body segment parameters 15) . And, leg spring tilt angle θ at landing was calculated from directional vector r=r, r form ankle joint to hip joint as follows, where rx and ry represented horizontal and vertical components of the directional vector. This θ is an angle of the directional vector to the vertical axis; therefore the angle means leg spring tilt angle. Maximal displacement of leg spring length in reference method Δl was calculated from following equation.

Δl=Δy+l1−cos θ
From the above, follow equations to calculate kleg and kvert were derived.

Data analysis
To estimate a bias of leg stiffness and vertical stiffness in Morinʼ s method, kʼleg and kʼvert were compared to kleg and kvert in reference method. Intra-class correlation coefficients (ICCs) of vertical stiffness and leg stiffness were calculated to assess a validity of the variables estimated by Morinʼs method relative to gold standard method. The statically significant level was set in 0.05. All values are represented in mean value ± standard deviation (SD).

Discussion
In the present study, leg stiffness and vertical stiffness during running were calculated using Morinʼ s method and using reference method. We found that kʼleg was highly correlated with kleg ( Figure-4). Additionally kʼleg was underestimated (-23.6%) to kleg. Where mean kʼvert demonstrated slightly lower value (-13.4%) relative to mean kvert, kʼvert was also highly correlated with kvert. The tendency of the underestimation for leg stiffness was general agreement with the results reported by Morin et al (2005) 6) . However, the biases for kʼleg and kʼvert were found high in this study. This result represents that the bias of kʼleg affected to accuracy of the individual values; however the tendency in the group of children could be validated because of the high relationship between variables in Morinʼs estimation and gold standard method.
A significance of this study was to test a validity of Morinʼ s method to estimate leg and vertical stiffness for children. Leg and vertical stiffness during running over ground for one hundred twenty seven children showed small value relative to values of leg and vertical stiffness for ten young adults in previous study 6) . 95% continence intervals of kʼleg and kʼvert in one hundred twenty seven children were ranged from 4.87 to 5.53 kN/m, and from 14.53 to 16.61 kN/m, whereas kʼleg and kʼvert for ten young adults were ranged from 11.63 to 14.27 kN/m, and from 37.57 to 62.85 kN/m. The leg stiffness and vertical stiffness for children represented about 20-40% to the values for young adults. Morin et al. (2005) reported that the values estimated for leg and vertical stiffness highly correlated (leg stiffness: r = 0.94; vertical stiffness: r = 0.99) with reference values 6) . These results indicated that accuracy of Morinʼs method for children was lower than for young adults.
The mean values of main parameters to calculate kleg, kʼleg, kvert, kʼvert were in Table-   the differences between kleg and kʼleg were a) the difference between Δyʼ and Δy, and b) the difference between l1−cos θ and l− l  −vt/2  . The difference between Δyʼ and Δy determined 13.2% of the difference between kleg and kʼleg, whereas the difference between l1−cos θ and l− l  −vt/2  determined 86.8% of the difference between kleg and kʼleg. The main factor of the difference between kleg and kʼleg, therefore, was the difference between l1−cos θ and l− l  −vt/2  . Morin et al. (2005) analyzed the effects of variations of the different mechanical parameters consisting of the Morinʼ s model on the estimated leg and vertical stiffness 6) . They concluded that contact time was most sensitive parameter in the mechanical parameter.
The main factor of the difference between kleg and kʼleg in the result of this study included the contact time; therefore it might be affected from sensitivity of contact time.

Conclusion
We could conclude that Morinʼs method might be able to estimate vertical stiffness and leg stiffness in children, although vertical and leg stiffness tended to be underestimated. The evidence of this study should be taken care of to interpret correctly the values.