Analysis on circumpelvic muscle force and spatial position and orientation of bone traction needle for pelvic fracture reduction

2.1. Skeletal muscle model and muscle force

Skeletal muscle is made up of two parts: muscle belly and tendon. Musculobelly is mainly composed of skeletal muscle fibers. The tendons are located at both ends of the musculobelly, connecting the bone to the musculobelly. Under the control of the nervous system, skeletal muscles can contract autonomously, generating muscle forces, which are transmitted by the tendons to the bones, and exerting the forces on the bones. When the pelvis is fractured under the action of external force, the broken end will be shortened, displaced, rotated and other deformities under the pulling action of muscle force [19].

Skeletal muscle can be divided into parallel muscle and pinnate muscle according to the arrangement pattern of muscle fibers. Musculoskeletal anatomy of the pelvis is shown in Fig. 1. Most of the skeletal muscle of human hip is the pinnacle muscle, which has shorter muscle fibers, larger physiological cross-sectional area and stronger muscle strength. The PA-MTM skeletal muscle model is adopted in this study, as shown in Fig. 2. CE is the contractile element, generating active muscle force; PEE is a parallel elastic unit, which produces elastic force when the muscle is pulled. SEE is a series elastic element with passive elasticity; SDE is a series damping unit to suppress high frequency oscillation. The pinnate angle $\alpha$ describes the angle between the orientation of the muscle fibers and the muscle force line

Fig. 1Musculoskeletal anatomy of the pelvis

Fig. 2Skeletal muscle model PA-MTM

The muscle contractility model can be expressed as:

where $F_{CE}$ is the force generated by the contraction unit, $F_{PEE}$ is the force generated by the parallel elastic unit, $\alpha \left(l_{CE}\right)$ is the angle between muscle fiber and tendon, $F_{SEE}$ is the force generated by the series elastic unit, and $F_{SDE}$ is the force generated by the series damping unit.

The $F_{CE}\left(l_{CE},{\dot{l}}_{CE},q\right)$, $F_{PEE}\left(l_{CE}\right)$, $\alpha \left(l_{CE}\right)$, $F_{SEE}\left(l_{CE},l_{MTM}\right)$, $F_{SDE}\left(l_{CE},{\dot{l}}_{CE},{\dot{l}}_{MTM},q\right)$ in Eq. (1) is determined by Eqs. (2)-(10), and the parameters related to the hip musculoskeletal model are shown in reference [20, 21]:

where, $F_{max}$ is the maximum equaling force when the length of contractile element $l_{CE}$ is the optimal length of muscle fiber $l_{CE,opt}$, $F_{isom}$ is a function of the length of contractile element $l_{CE}$, $A_{rel}$ is a function of the length of contractile element $l_{CE}$ and the degree of activation $q$, $B_{rel}$ is a function of the degree of activation $q$:

where $K_{SEE,l}=\frac{\mathrm{\Delta }{F}_{SEE,0}}{\mathrm{\Delta }{U}_{SEE,l}{l}_{SEE,0}}$, $l_{SEE,nll}=\left(1+\mathrm{\Delta }{U}_{SEE,nll}\right)l_{SEE,0}$, $l_{SEE,nll}=\left(1+\mathrm{\Delta }{U}_{SEE,nll}\right)l_{SEE,0}$, $K_{SEE,nl}=\frac{\mathrm{\Delta }{F}_{SEE,0}}{{\left(\mathrm{\Delta }{U}_{SEE,nll}{l}_{SEE,0}\right)}^{v_{SEE}}}$, and:

According to the force balance Eq. (1) composed of muscle fiber and tendon unit, the quadratic equation of one variable about the contraction speed of muscle fiber ${\dot{i}}_{CE}$ can be obtained: $0=C_2{{\dot{l}}_{CE}}^2+C_1{\dot{l}}_{CE}+C_0$:

Solve the quadratic equation of one variable Eq. (14) to get:

were $A_{rel,e}=-F_{e}q{F}_{isom}$, $B_{rel,e}=\frac{B_{rel}\left(1-F_e\right)}{S_e\left(1+\frac{A_{rel}}{q{F}_{isom}}\right)}$, $C_{1,e}$, $C_{2,e}$, $C_{0,e}$ are obtained by replacing $A_{rel}$, $B_{rel}$ with $A_{rel,e}$, $B_{rel,e}$.

Based on the theoretical analysis of the muscle force according to the PA-MTM skeletal muscle model, it can be known that the muscle force is related to the length of the muscle, the speed of muscle contraction, the degree of activation and other parameters.

2.2. Fracture reduction path

According to the CT data of the patient’s pelvic fracture, the reverse reconstruction is performed to obtain the 3D model of the pelvic fracture. CT data of pelvic fracture, as shown in Fig. 3. The model is imported into the GEOMAGIC software, the iliac bone on the affected side is registered with the mirror iliac bone on the healthy side, and the conversion matrix is derived to determine the fracture deviation. The 3D model of the registration of the iliac bone on the affected side and the mirror iliac bone on the healthy side of the pelvic fracture, as shown in Fig. 4, $O_B$ is the origin of the reference coordinate system $\left\{B\right\}$.

Fig. 3CT data of pelvic fracture

The displacement deviation and rotation deviation of pelvic fracture are obtained according to the transformation matrix of iliac bone registration on the affected side. In the pelvic fracture model analyzed in this paper, under the reference coordinate system $\left\{B\right\}$, the displacement deviations of the distal end of the fracture compared with the proximal end of the fracture along the $x$, $y$ and $z$ axes are 3.13 mm, –4.30 mm and –8.51 mm respectively, and the rotation deviations around the $x$, $y$ and $z$ axes are –4.90°, –6.36° and –4.83°, respectively.

According to the displacement deviation and rotation deviation of the pelvic fracture, the 3D A* algorithm is used to plan the reduction path. The result of planned reduction path is shown in Fig. 5. $O_1$ is the reduction start point and $O_2$ is the reduction end point.

2.3. Muscle force calculation of the circumpelvic main muscles

In the reduction of pelvic fractures, the main muscles that affect the reduction force of pelvic fractures are the gluteus maximus, the gluteus medius, the gluteus minimus, the m.obliquus externus abdominis, the m.obliquus internus abdominis and the iliacus. Since the gluteus maximus, gluteus medius, and gluteus minimus are muscles with wide attachment points, three points are used to mark the attachment points of the muscles to illustrate the respective functions of muscle fibers in different parts of the muscles. According to the position of hip muscle attachment points [22], the attachment points and action lines of each muscle are determined, as shown in Fig. 6.

Fig. 4Mirror-image healthy side registration model of pelvic fracture

Fig. 5Planned reduction path

Fig. 6The attachment points and action lines of the muscle

a) Anterior view

b) Posterior view

Firstly, assuming initial conditions ${\dot{l}}_{MTM}=0$, the initial length of the muscle$l_{MTM}$ is determined according to the coordinates of the starting and ending points of the muscle, relevant parameters of the muscle[23] and the degree of activation $q=0$ are determined, and then the initial muscle fiber length of the muscle $l_{CE}$ can be obtained by the force balance Eq. (1). Then, calculating the other unknowns:$F_{isom}$, $F_{PEE}$, $F_{SEE}$, $A_{{\mathrm{}}_{rel}}$, $B_{rel}$, $D_0$, $C_2$, $C_1$, $C_0$, ${\dot{i}}_{CE}$, $F_{SDE}$, $\alpha$ to get the initial muscle forces $F_{MTM}=F_{SDE}+F_{SEE}$.

According to the result of planned path, the length of each muscle in the reduction process is determined. Then, according to the Eq. (1-14) derived from the theoretical analysis of muscle force, muscle force in the reduction process is calculated. According to the skeletal muscle attachment points and the fracture types of this model, the main muscles that affect the reduction force of the pelvic fracture types in this paper are gluteus medius 2, gluteus medius 3, gluteus maximus 1, and m.obliquus externus abdominis. Muscle force generated by each muscle during reduction is shown in Fig. 7. It can be seen from the figure that: gluteus medius 3, gluteus maximus 1, and m.obliquus externus abdominis play a major role in the reduction process. The maximal force generated by gluteus medius 3, gluteus maximus 1, and m.obliquus externus abdominis during the reduction process are 207.32 N, 189.59 N, and 191.53 N, respectively, while gluteus medius 2 has the least force.

Fig. 7The muscle force during reduction process

3.1. Reduction force analysis

Skeletal muscles that affect the reduction of Tile C pelvic fractures mainly include: gluteus medius 2, gluteus medius 3, gluteus maximus 1, and m.obliquus externus abdominis. From the anatomical diagram of human pelvis muscles, we can know the working points and directions of each muscle. In this paper, Tile C type pelvic fracture is taken as an example. Three groups of muscles that have great influence on reduction are selected and represented by four forces. Force analysis of the injured pelvis is conducted, as shown in Fig. 8.

In the process of pelvic reduction, the pelvis moves at a constant speed and is in a balanced state, and the resultant force and moment are zero, then:

where, ${\mathbf{F}}_{M1}$ is the muscle force generated by the gluteus medius muscle 2, ${\mathbf{F}}_{M2}$ is the muscle force generated by the gluteus medius muscle 3, ${\mathbf{F}}_{M3}$ is the muscle force generated by the gluteus maximus 1, ${\mathbf{F}}_{M4}$ is the muscle force generated by the m.obliquus externus abdominis. ${\mathbf{F}}_1$ is the force that the robot acts on the bone traction needle 1, ${\mathbf{F}}_{{\mathrm{}}_2}$ is the force that the robot acts on the bone traction needle 2, and $\mathbf{G}$ is the gravity of the pelvis; ${\mathbf{r}}_1$ is the ${\mathbf{F}}_{M1}$ action point vector, ${\mathbf{r}}_2$ is the ${\mathbf{F}}_{M2}$ action point vector, ${\mathbf{r}}_3$ is the ${\mathbf{F}}_{M3}$ action point vector, ${\mathbf{r}}_4$ is the ${\mathbf{F}}_{M4}$ action point vector, ${\mathbf{r}}_{F_1}$ is the ${\mathbf{F}}_1$ action point vector, ${\mathbf{r}}_{F_2}$ is the ${\mathbf{F}}_2$ action point vector, ${\mathbf{r}}_G$ is the $\mathbf{G}$ action point vector.

Fig. 8Force analysis of injured pelvis

a) Anterior view

b) Posterior view

3.2. Reduction force calculation

According to the muscle force (Eqs. (1-14)) of each muscle and the reduction force (Eq. (15)) during the reduction process, total reduction force and component reduction force along the $x$, $y$, $z$ directions, as shown in Fig. 9.

Fig. 9Total reduction force and component reduction force during reduction process

It can be seen that in the process of pelvic fracture reduction, the reduction force along the transverse direction of the pelvis (i.e. the $x$ direction) is the largest, the maximum value is 238.82 N, and the maximum fracture reduction force is 301.67 N. In the process of pelvic fracture reduction, the reduction force changes with the change of muscle length. The longer the muscle length is, the greater the muscle force and reduction force are.

4.1. Finite element model of pelvic musculoskeletal tissue

Using the pelvic CT scan data, the inverse modeling technique is used to reconstruct the 3D digital model of the damaged Tile C type pelvic musculoskeletal tissue in the Mimics software. The reconstruction process of musculoskeletal model is shown in Fig. 10. The established 3D digital model of pelvic musculoskeletal tissue is shown in Fig. 11.

The pelvis musculoskeletal assembly model is imported into the finite element software ANSYS20.0 in the format of x_t, and the material attributes of each pelvic tissue are assigned, as shown in Table 1. The tetrahedral elements are used to divide the grid, and the number of elements and nodes of the pelvis musculoskeletal model is 292514 and 436917.

Fig. 10Musculoskeletal model modeling process

Fig. 113D digital model of pelvic musculoskeletal tissue

4.2. Different spatial position and orientation of bone traction needles

When a pelvic fracture is reduced, two bone traction needles are usually inserted into the affected side to pull the bone. According to doctors' clinical experience in the reduction of pelvic fractures [4], the bone traction needle is inserted into the pelvic bone 70 mm at three entry points A, B and C, respectively, and the spatial position and orientation of two bone traction needles are inserted into pelvic, as shown in Table 2 and Fig. 12.

Fig. 12Position of bone traction needle inserted into the pelvic

4.3. Simulation analysis of the application point of reduction force

The possible position of the bone traction needles inserted into the pelvis is point A, point B or point C. The pelvis with two bone traction needles inserted in different positions was divided into group AB, AC and BC. Then, the finite element analysis of the pelvic musculoskeletal model with bone traction needles is compared.

The loading and boundary conditions are as follows: three positions $S_1$, $S_2$ and $S_3$ are selected on the bone traction needle in group AB, respectively, and the maximum reduction force is applied. The force points are shown in Fig. 13. In order to approach the anatomical state of pelvic fracture reduction, the proximal femoral section and the sacroiliac joint on the healthy side are fixed and restrained, limiting 6 DOF in the direction, and then the static analysis of pelvic fracture reduction with bone traction needle is performed. The force applied at $S_1$ position of group AB is set as group AB1, and the finite element analysis results of group AB1 are shown in Fig. 14. The force applied at the $S_2$ position of group AB is set as group AB2, and the finite element analysis results of group AB2 are shown in Fig. 15. The force applied at $S_3$ position of Group AB is set as Group AB3, and the finite element analysis results of Group AB3 are shown in Fig. 16.

Fig. 13Different force points Si

Fig. 14Finite element analysis results of AB1 group

a) Stress diagram

b) Displacement diagram

Fig. 15Finite element analysis results of AB2 group

c) Stress diagram

b) Displacement diagram

Fig. 16Finite element analysis results of AB3 group

d) Stress diagram

b) Displacement diagram

The simulation results of different force applying point of the bone traction needle are shown in Fig. 17. The maximum stresses at the needlebone interface in AB1, AB2 and AB3 groups are 210.48 MPa, 419.72 MPa and 595.75 MPa, respectively; The maximum displacements of AB1, AB2 and AB3 groups are 6.9612 mm, 9.4036 mm and 11.85 mm respectively. In terms of strength and stiffness, the pelvic stability is AB1 group > AB2 group > AB3 group. When pelvic fracture is reduced, the force point of reduction force on bone traction needle is $S_1$ point.

Fig. 17Comparison of maximum stress and maximum displacement of different force points

4.4. Simulation analysis of different spatial position and orientation of bone traction needle

According to the determined force point $S_1$, the spatial position and orientation of the bone traction needle inserted into the pelvis during the pelvic fracture reduction is compared and analyzed.

The loading and boundary conditions are as follows: the maximum reduction force of 301.67 N is loaded at the application point $S_1$ of bone traction needle in AC group and BC group, respectively. The fixation constraints are carried out at the proximal femur section and the sacroiliac joint on the healthy side, limiting 6 degrees of freedom. Then, the finite element simulation of the pelvic musculoskeletal tissue with two bone traction needle is performed. Finite element analysis results of AC1 and BC1 groups are shown in Fig. 18 and Fig. 19.

Fig. 18Finite element analysis results of AC1 group

e) Stress diagram

b) Displacement diagram

Fig. 19Finite element analysis results of BC1 group

f) Stress diagram

b) Displacement diagram

The simulation analysis results of the position and orientation of the bone traction needle are shown in Fig. 20. The maximum stresses of AB1 group, AC1 group and BC1 group are located at the interface of needle bone and are 210.48 MPa, 206.4 MPa and 192.77 MPa, respectively. The maximum displacements of AB1, AC1 and BC1 are 6.9612 mm, 4.943 mm and 5.4761 mm, respectively. The maximum stress values around two bone traction needles in AB1 group are 210.48 MPa and 120.89 MPa, the maximum stress values around two bone traction needles in AC1 group are 206.40 MPa and 165.12 MPa, the maximum stress values around the two bone traction needles in BC1 group are 192.77 MPa and 192.77 MPa. So the maximum stresses difference between bone needles and their surrounding of AB1, AC1 and BC1 group are 80.59 MPa, 41.28 MPa and 0 MPa. From three aspects of strength, stiffness and stress distribution uniformity, the stress state of the pelvis with bone traction needle is the best in the BC group under the action of reduction force. When the pelvic fracture is reduced, the best positions of the bone traction needle are point B and point C.

Fig. 20Comparison of maximum stress and maximum displacement of bone traction needle in different position and orientation