Measurement of the dynamic axial load-share ratio in vivo could indicate sufficient callus healing in external fixators

Abstract

Background

Fracture healing is commonly evaluated through physical examination and radiographic results. However, these methods rely on the surgeons’ subjective experience, without including the objective biomechanical properties of the bony callus. This paper presents an innovative method for measuring the callus stiffness in vivo to evaluate fracture healing, further instructing surgeons to remove external fixator safely.

Methods

A novel dynamic axial load-share ratio (D-LS) index and its associated measuring system was introduced, including the system’s composition (hexapod and insole modules), theoretical model, and method for D-LS measurement. From Jan 2022 to May 2024, 36 patients with tibial shaft fracture treated by Taylor Spatial Frame were evaluated in this prospective study. Once the patient had reached clinical bone healing conditions, the in vivo D-LS measurement was conducted. The patients’ demographic data, clinical outcomes, particularly D-LS value and refracture rate were recorded.

Results

At a mean follow-up of 16.50 ± 5.79 months, a total of 36 patients completed the final follow-up. Fixators were removed with an average of 24.81 ± 4.51 weeks. The result of hexapod module’s precision examination were maximum errors of 3.72 N, 3.31 N and 2.68 N in x-, y- and z-axis, respectively. The measuring process took an average system installation time of 15.42 ± 4.88 min. Two patients (5.56%) reported fracture site pain. Each patient’s D-LS was determined after three rounds of measurement. The average D-LS value was 15.58 ± 2.77% (range, 9.60–20.52%). None of the 36 patients reported refracture at the last follow-up.

Conclusions

The novel D-LS measurement system can measure the dynamic forces of lower limb for patients with external fixator in vivo. An objective biomechanical indicator of the regenerate callus was provided by the D-LS. The D-LS measurement is a complement to standard radiological assessment only after radiologically confirmed bone union. Measuring the D-LS in vivo could indicate whether the callus healing is sufficient in external fixation, and 15.6% (average) was recommended as a reference D-LS value for safe fixator removal.

Introduction

External fixation is crucial in the treatment of bone defect, infected bone nonunion, complex extremity deformity, and high-energy fracture that are not suitable for internal fixation [1,2,3]. Deciding the most favorable timing for the removal of external fixator in fracture treatment is a considerable challenge. Refracture following the external fixator’s removal may occur if the bone has not fully healed [4, 5]. Therefore, effective and objective assessment of fracture healing is essential for the safe removal of external fixator.

Currently, there is no universal criteria for clinically determining the bone union. Fracture healing is commonly described as the restoration of bone’s biomechanical properties [6,7,8]. The generally applied criteria for assessing fracture healing in clinical and radiographic settings include painless weight bearing, absence of tenderness at the fracture site, and the presence of “bridging or callus formation across 3 of 4 cortices” on anteroposterior and lateral X-rays [9,10,11]. Nevertheless, the existing criteria are insufficient in providing information on the biomechanical characteristics of bone.

At present, the evaluation of bending or axial stiffness is the prevailing approach for assessing the biomechanical properties of bone [12, 13]. The process of evaluating bending stiffness involves applying a specified moment to the bone-fixator composite structure (BFCS) and then measuring the resultant angular displacement [14, 15]. However, the above process exerts a detrimental effect on callus formation. Moreover, researchers haven’t reached a consensus on the secure value of bending moment in testing. In comparison, axial stiffness is a more direct indicator of the limb’s weight-bearing capacity. Due to minor axial displacement of the BFCS under physiologic loading condition, effectively measuring the axial stiffness is difficult. Previous studies focused on changes in internal load of the fixator overtime to evaluate callus stiffness. When the axial load of external fixator gradually decreases and reaches to a stable point, it indicates that the bony callus achieves consolidation [16, 17]. This longitudinal method lacks comparability among individuals, so it cannot be served as a criterion to determine whether to remove the fixator.

Aarnes et al. reported a transverse method, measuring the axial load-share ratio (LS) of the external fixator, to evaluate callus axial stiffness [18]. The LS is calculated through dividing the load exerted on the fixator by the total load on the limb. What counts is the LS quantified the comparative relationship between the callus stiffness and the fixator stiffness. As the callus stiffness grows during the process of fracture healing, the LS declines and approaches a certain value. Several studies [5, 18, 19] have reached the conclusion that LS < 10% can be applied as a criterion for the safe removal of external fixator. Still, there are doubts regarding Aarnes’ approach, which utilizes a hyperstatic fixator structure with mounting stresses, and calculates the fixator load by simply summing the forces of each strut. In addition, the current methods for evaluating stiffness only obtain static data at a single moment, therefore failing to reflect the bone’s mechanical characteristics during locomotion. It is critical to accurately measure the dynamic forces on the lower limb in order to effectively evaluate the axial stiffness of callus.

The real-time loading of lower limbs during locomotion can be obtained by measuring the ground reaction force (GRF) through plantar pressure devices [20, 21]. These devices can be categorized into two types, the plantar platform system and the insole system [22,23,24,25]. Platform systems have the capability to measure the six-dimensional GRF precisely. However, their application is limited to laboratory due to the fixed nature of the platform. Besides, patients need to be trained for the adaptation of stepping positions to obtain valid data. Insole systems can be readily affixed to the planta and measure the simplified one-dimensional GRF [26, 27]. The insole system based on force-sensing resistors is of great cost-effectiveness to measure distributed pressure. Moreover, patients are familiar with using insoles, hence a more natural gait can be obtained compared with the platform system [28, 29]. Yet, there is no literature about measuring the dynamic LS in vivo to safely remove external fixators, while performing GRF measurements can effectively fill this gap.

This paper aims to introduce an innovative method measuring the dynamic LS in vivo to evaluate fracture healing, further instructing surgeons to remove external fixator safely.

Materials and methods

This paper presents a novel approach to measure the callus stiffness in vivo for the evaluation of fracture healing, thus provides instructions to surgeons on timely external fixator removal. The developed measurement system integrated a hexapod module and an insole module to measure the fixator force and plantar pressure in real-time, respectively. The study formulated six-dimensional mechanics model of the BFCS, then established a Back Propagation Neural Network model to measure GRF from plantar pressure, and eventually proposed an assessment method for callus stiffness using an equivalent index D-LS.

Development of D-LS measurement system

We developed a novel D-LS measurement system, which is composed of a hexapod module, an insole module, a data acquisition module and assistant computer software (Fig. 1A). The hexapod module measures the force in the fixator (FF) by exchanging the struts of patient’s original fixator. The insole module is adhered to the planta of the affected limb to measure GRF. The data acquisition module acquires signals from the sensors in hexapod and insole modules, and establishes wireless communication with the computer for portable operation.

Fig. 1

Overall appearance and detailed schematics of the novel D-LS measurement system. (A). Simulated picture of a patient applying the D-LS system to assess callus stiffness (the human model is adapted from “Chloe” by Q.SARDOR from the website https://sketchfab.com/qsardor57913, used under CC BY 4.0). (B). The hexapod module and connected data acquisition module. (C). Component composition of the force-measuring strut. (D). A pair of force-measuring insole and a connected signal box. (E). The assistant computer software of the measurement system

The hexapod module is a Gough-Stewart mechanism [30] encompassing six force-measuring struts and two fixator rings. Each strut has a same structure that contains two ball joints, a threaded telescopic shaft, and a one-dimensional force sensor. The force sensors (measuring range ± 400 N, DYMH-113, Dayang Sensing System Engineering, China) are connected to the acquisition module with flexible cables (Fig. 1B). The forces are gathered as 16-bit six-channel signals. A junction component was specifically designed to facilitate the rapid attachment and detachment of the strut and ring (Fig. 1C). After detaching an original strut, the exchanged force-measuring strut can be promptly installed in suitable locations on the ring. The telescopic shaft’s construction allows for a wide range of strut length adjustment. This feature makes it suitable for the majority of tibial fractures treated with an external fixator.

The insole module includes pairs of force-measuring insoles with several specifications (European shoe size of 38, 40, 42 and 44). Eight flexible force-sensing resistors (measuring range 0–100 N, ZNX-01, ChengTec Electronic Technology, China) with measuring consistency of ± 8% are embedded in the insole (Fig. 1D). The location of force-sensing resistors on the insole are determined according to the literature on foot morphology and plantar pressure distribution [31, 32]. The insole signal box, which is a component of the data acquisition module, collects pressure data from the connected force-sensing resistors as 16-bit eight-channel signals.

To achieve a balance between the system’s real-time capabilities and efficiency, the signal control of the D-LS measurement system is designed as a two-level architecture. The data acquisition module works as a lower unit, which runs real-time programs based on STM32 microcontroller. The FF and GRF are acquired synchronously with a sampling frequency of 50 Hz, then low-pass filtered and stored in the lower unit. The assistant computer software consisting of Windows-based time-sharing programs works as upper unit. The programs are written and compiled using MATLAB (R2021b, MathWorks Inc, USA). The assistant software has multi-tasking for data processing and visualization (Fig. 1E), which eases use for physicians and allows for future expansion of functionality. The upper and lower units communicate with each other via Bluetooth. The communication rate is 5 Hz when measuring D-LS, which satisfies the monitoring conditions and reduces the system performance loss. After the measurement, the entire dataset saved in the lower unit can be rapidly accessed to calculate the D-LS for callus stiffness assessment.

Principles of D-LS measurement

Mechanical modeling of BFCS

When the lower limb fracture is close to healing, full weight-bearing walking is generally adopted as a functional exercise. The ability to walk with full weight bearing is also an important indicator for clinical evaluation of bone healing. During the stance phase of the patient’s gait, the BFCS is subjected to compressive loading, and the schematic of its structure and forces is shown in Fig. 2A. Establish a coordinate frame $\:\left\{B\right\}$ at the callus site, with the z-axis along the bone’s mechanical axis and the y-axis pointing front of the body. Coordinate frame $\:\left\{A\right\}$ is set on the proximal ring of the fixator, with its axes oriented matching the ring’s geometry. The load exerting on the BFCS (CSF) mainly includes GRF and muscle force (MF), which is shared by the FF and the force in the callus (CF) within the structure. The equilibrium equation for these forces is established along the z-axis at the callus site $\:\left\{B\right\}$, writing as:

$$\:G+{F}_{M}={F}_{F}+{F}_{C}$$

(1)

.

Herein, $\:G$, $\:{F}_{M}$, $\:{F}_{F}$ and $\:{F}_{C}$ denote the GRF, MF, FF and CF along z-axis after coordinate transformation, respectively. Notice that these variables are scalar.

The FF can be described with a six-dimensional force $\:{\varvec{F}}_{A}$ acting at the origin of frame $\:\left\{A\right\}$. Based on the hexapod module we can measure the $\:{\varvec{F}}_{A}$ and transform it to frame $\:\left\{B\right\}$ to obtain the $\:{F}_{F}$. Denote the axial force in each strut of the hexapod module as $\:{f}_{Si}\:(i=\text{1,2},\cdots\:,6)$. Writing the spatial force balance equation of $\:{f}_{Si}$ and $\:{\varvec{F}}_{A}$ we have

$$\:{\varvec{F}}_{A}=\sum_{i=1}^{6}({f}_{Si}{\widehat{\varvec{s}}}_{i}+{f}_{Si}{\varvec{a}}_{i}\times\:{\widehat{\varvec{s}}}_{i})={\left(\begin{array}{cc}{\varvec{f}}_{A}^{T}&\:{\varvec{m}}_{A}^{T}\end{array}\right)}^{T}$$

(2)

.

Wherein, $\:{\widehat{\varvec{s}}}_{i}$ is the axis unit vector of the strut $\:i$, and $\:{\varvec{a}}_{i}$ is the center position vector of the hinge connecting the strut $\:i$ and the proximal ring. The vectors can be computed based on the mechanical dimensions of the fixator and the length of the struts. This involves solving the kinematics of the parallel mechanism, as previously detailed in our research [33]. The vectors $\:{\varvec{f}}_{A}$ and $\:{\varvec{m}}_{A}$ in Eq. (2) are the linear force and moment components of the $\:{\varvec{F}}_{A}$, respectively. Rewrite the equation into matrix form as

$$\begin{aligned}\left(\begin{array}{c}{\varvec{f}}_{A}\\\:{\varvec{m}}_{A}\end{array}\right)&=\left[\begin{array}{cccc}{\widehat{\varvec{s}}}_{1}&\:{\widehat{\varvec{s}}}_{2}&\:\cdots\:&\:{\widehat{\varvec{s}}}_{6}\\\:{\varvec{a}}_{1}\times\:{\widehat{\varvec{s}}}_{1}&\:{\varvec{a}}_{2}\times\:{\widehat{\varvec{s}}}_{2}&\:\cdots\:&\:{\varvec{a}}_{6}\times\:{\widehat{\varvec{s}}}_{6}\end{array}\right]\\&\quad{\left(\begin{array}{cccc}{f}_{S1}&\:{f}_{S2}&\:\cdots\:&\:{f}_{S6}\end{array}\right)}^{T}\end{aligned}$$

(3)

.

Fig. 2

(A). Schematic of the BFCS and its force state during the patient’s locomotion. (B, C). Force measurement accuracy examination of the hexapod module based on universal mechanical testing machine

Transform the reference frame of $\:{\varvec{F}}_{A}$ from frame $\:\left\{A\right\}$ to $\:\left\{B\right\}$ following the coordinate transformation of six-dimensional force, and take its projection along the z-axis to calculate $\:{F}_{F}$ as

$$\:{F}_{F}={\varvec{u}}_{z}^{T}\left[\begin{array}{cc}{\varvec{R}}_{BA}&\:0\\\:[{\varvec{p}}_{BA}{]}_{\times\:}{\varvec{R}}_{BA}&\:{\varvec{R}}_{BA}\end{array}\right]{\varvec{F}}_{A}$$

(4)

.

Here, $\:{\varvec{u}}_{z}$ stands for the unit vector of projection; the position vector $\:{\varvec{p}}_{BA}$ and rotation matrix $\:{\varvec{R}}_{BA}$ are the description of relative pose from frame $\:\left\{A\right\}$ to frame $\:\left\{B\right\}$, which are determined based on fluoroscopic images and clinical appearance measurements [34]. The matrix $\:[\cdot\:{]}_{\times\:}$ is a skew-symmetric matrix representation of the vector in it.

In the planta plane, establish a frame $\:\left\{P\right\}$ at the intersection point of the mechanical axis as illustrated in Fig. 2A. The GRF $\:{\varvec{f}}_{G}=(\begin{array}{ccc}0&\:0&\:{f}_{G}\end{array}{)}^{T}$ acting at the center of pressure (CoP) between the planta and ground is measured by the insole module, and the position of CoP is described with a vector $\:\varvec{c}$ in the frame $\:\left\{P\right\}$. The involved principle will be described in the following section. Within the stance phase, the MF $\:{\varvec{f}}_{M}$ form a moment equilibrium with the GRF $\:{\varvec{f}}_{G}$ about the center of the ankle joint [35, 36]. Therefore, the $\:{\varvec{f}}_{M}$ passing through the Achilles tendon can be calculated as

$$\:{\varvec{f}}_{M}=\left[\varvec{t}\right]_{\times\:}^{-1}\left[\varvec{c}-\varvec{e}\right]_{\times\:}{\varvec{f}}_{G}$$

(5)

,

in which $\:\varvec{t}$ and $\:\varvec{e}$ are the position vectors of the Achilles tendon junction and the ankle joint center, respectively. Applying similar transformations and projections to $\:{\varvec{f}}_{M}$ and $\:{\varvec{f}}_{G}$ as in Eq. (4) yield the $\:G$ and $\:{F}_{M}$ in Eq. (1).

The measurement accuracy of hexapod module was examined by a universal test machine (Model 5982, Instron Corp., USA). Quasi-static loading was performed in the x-axis, y-axis, and z-axis directions with forces of 500 N, 500 N, and 1800 N, respectively (Fig. 2B and C). Three rounds of examination in each direction were performed. The hexapod module’s measurement accuracy was analyzed by comparing the error to the standard values of the universal test machine. The examination results showed maximum errors of 3.72 N, 3.31 N and 2.68 N in x-, y- and z-axis, respectively. This accuracy was determined to be sufficient for the target application and providing the expected range of in vivo forces and the relative error.

GRF measurement based on the insole module

The primary cause of the axial load on the BFCS during locomotion is the vertical GRF, which could be measured using the insole module.

Firstly, each of the force-sensing resistors in the insole is individually calibrated to measure the channel pressure. As shown in Figs. 3A and 20 calibration nodes were taken at equal intervals between zero and 100 N (maximum range), and each resistor was loaded and the response signal was acquired. The signal mapping function for each channel was obtained based on the Newton interpolation method. For the channel signal $\:{D}_{j}$ of the j-th resistor, use the interpolation function $\:{g}_{j}\left({D}_{j}\right)$ resulted from calibration yields the channel pressure

$$\:{F}_{Dj}={g}_{j}\left({D}_{j}\right)$$

(6)

.

A loading test was applied to check the calibration accuracy of each resistor. The results showed that the maximum error for all channels is 11.3 N, aligning with the resistor’s inherent measurement reliability. This accuracy was determined to be sufficient for measuring GRF and providing the expected range of in vivo forces and the relative error.

Fig. 3

(A). Calibration of each force-sensing resistor. (B). A volunteer with insole module and force plate measurements for Back Propagation Neural Network training. (C). The measured and reference GRF curves across 15 volunteers

Secondly, the CoP is calculated based on the channel pressure. Determine the positions of the force-sensing resistor $\:{\varvec{r}}_{j}=\begin{array}{ccc}({x}_{j}&\:{y}_{j}&\:0\end{array}{)}^{T}$ in the frame $\:\left\{P\right\}$ based on the insole’s geometry, and the position vector of CoP is

$$\:\varvec{c}=(\sum_{j=1}^{8}{F}_{Dj}{\varvec{r}}_{j})/(\sum_{j=1}^{8}{F}_{Dj})$$

(7)

.

Thirdly, an Artificial Neural Network model is established to measure the value of vertical GRF $\:{f}_{G}$ using channel pressure (the specific methodology can be found in our previous research [29]). The model is a Back Propagation Neural Network, which has ten variables in its input layer. Eight of the variables are the channel pressures $\:({F}_{D1},{F}_{D2},\cdots\:{F}_{D8})$, and the other two are the x- and y-coordinate of CoP. The output layer of the Back Propagation Neural Network has one variable $\:{f}_{G}$. The Back Propagation Neural Network has two hidden layers, one with 10 and one with 5 neurons, and are respectively connected to the input and output nodes. The modeling process was based on 15 healthy volunteers within our research team (12 men and 3 women, age 27.6 ± 2.41 years, height 172.63 ± 4.58 cm, weight 73.26 ± 10.90 kg). Subjects walked in a natural gait wearing the insole module, and the vertical GRF measured by a high-precision force plate (FP4060-10, Bertec, USA) was used as a reference for the Back Propagation Neural Network’s training (Fig. 3B). Perform network training with 1000 iterations, and terminate the training if the gradient does not drop for six consecutive iterations.

Figure 3C illustrates a comparison of the measured and reference GRF across all 15 participants. The root-mean-square error of the GRF derived from the offered Back Propagation Neural Network model was 0.113 ± 0.042 Body Weight. This value was determined to be sufficient for the target application and providing the expected range of in vivo forces and the relative error.

D-LS index establishment

The callus and the fixator bear the CSF in parallel and the structure undergoes a small amount of deformation (submillimeter level). The second order term of the derivative of the displacement can be neglected, thus the relationship between FF and CF can be equivalently described by the Hooke’s law as

$$\:\varDelta\:=\frac{{F}_{C}}{{k}_{C}}=\frac{{F}_{F}}{{k}_{F}}\:\text{a}\text{n}\text{d}\:{F}_{F}=\frac{{k}_{F}}{{k}_{C}}{F}_{C},$$

(8)

where $\:\varDelta\:$ is the total elastic deformation. As the fracture heals, the stiffness of callus $\:{k}_{C}$ gradually increases while the stiffness of the external fixation apparatus (including the pin and the fixator) $\:{k}_{F}$ remains nearly constant. Thus, apply a certain CSF and measure the $\:{F}_{F}$ can indicate the stiffness of callus. It is on this basis that the LS index was initially defined as [18]

$$\:\text{LS}=\frac{{F}_{F}}{G}=\frac{{F}_{F}}{{F}_{F}+{F}_{C}}=\frac{{k}_{F}\varDelta\:}{{k}_{F}\varDelta\:+{k}_{C}\varDelta\:}=\frac{{k}_{F}}{{k}_{F}+{k}_{C}}$$

(9)

.

The patient’s LS value decreases with bone healing, until a threshold indicating safe removal of external fixation is reached. Notice should be taken that the LS is measured in a static body pose. Regarding locomotion, the loading state of CSF can be represented by Eq. (1). Based on the principles mentioned above, $\:G$, $\:{F}_{M}$ and $\:{F}_{F}$ are obtained from the developed D-LS measurement system. Hence, by computing the time integrals of FF and CSF and examining their ratio, we may obtain the D-LS index, which indicates the stiffness of the callus when subjected to dynamic loads:

$$\:{\eta}_{\tau\:}=\sum_{\tau\:}\frac{{F}_{F}}{G+{F}_{M}}\delta\:t\:\text{a}\text{n}\text{d}\:{\eta}_{a}=\frac{1}{N}\sum_{\tau\:=1}^{N}{\eta}_{\tau\:}.$$

(10)

Here, the first equation represents the D-LS value $\:{\eta}_{\tau\:}$ corresponding to each gait cycle, which is obtained by summing over all data within the gait cycle $\:\tau\:$, and $\:\delta\:t$ is the sampling period of the system. The second equation defines the average D-LS $\:{\eta}_{a}$ of $\:N$ gait cycles, representing the callus stiffness during the patient’s complete locomotion.

D-LS measurement process

Diagnosis and patient selection

This study was approved by the Ethics Committee of Tianjin Hospital before the clical trial started and complied with the Declaration of Helsinki. All participants provided informed written consent prior to participating. The methods were carried out in accordance with the relevant guidelines and regulations. A prospective analysis was conducted on patients treated by the Taylor Spatial Frame (TSF) at Tianjin Hospital from Jan 2022 to May 2024, including 26 males and 10 females with a mean age of 43 years (range, 20–64). Inclusion criteria: (1) patients with tibial shaft fractures; (2) open fractures or closed fractures with poor surrounding soft tissues; (3) D-LS measurement was conducted when the fracture achieved bone union under clinical and radiographic examinations; (4) a minimum follow-up period of 6 months after the fixator removal. Exclusion criteria: (1) patients with bilateral tibial shaft fractures; (2) severe medical conditions; (3) pediatric fracture; (4) age exceeding 65 years. The surgery and D-LS measurement were performed by the same team.

Operating method

After surgeons confirmed the tibial shaft fracture had healed according to clinical and radiographic examination, the D-LS measurement is conducted for the patient. A typical case is shown in Figs. 4 and 5, and this patient’s measurement results are further displayed in Fig. 6. The detailed D-LS measurement process is as follows.

Fig. 4

A patient who suffered tibial shaft fracture in right leg and treated by the TSF external fixator. (A, B). The last follow-up images before removing the TSF. (C, D). AP and lateral views of X-rays 4 weeks after TSF removal

Fig. 5

Images of the D-LS measurement procedure using the novel system of the same patient shown in Fig. 4. (A). Clinical view of affected limb before removing the TSF. (B). Install the hexapod module by exchanging the struts. (C). Clinical view of installed hexapod module and corresponding screenshot of software. (D). Patient performed a level, straight walking trial with the D-LS system

Firstly, the struts’ lengths and installation positions were recorded. Then, the hinges connecting the struts and rings were loosened. The patient was guided to participate in weight-bearing exercises like normal walking and soft stair climbing for roughly 30 min. If somewhat pain occurred at the fracture site, surgeons restored the original state of fixator. The next LS measurement would be conducted after a 4-week interval. If not, surgeons proceeded to the subsequent steps. In this study, 2 patients experienced pain at the fracture and completed LS measurement 4 weeks later.

The original struts were exchanged with force-measuring struts, constituting the hexapod module. During the process, each rod was replaced one by one, with particular attention paid to gentle movements to prevent any discomfort. The affected limb was horizontally lifted, and then the struts’ lengths were adjusted to create a minimal compression force ranging from 0 to 5 N. The software automatically performed theoretical calculations according to installation information of force-measuring struts. Surgeons verified the congruity between the software’s graphical simulation and the mechanism’s actual configuration.

Prior to walking, the patient sat on a chair and raised the lower legs wearing the insole module. The insole’s pressure data was recorded during unloading state, which was subsequently subtracted in the measurement. Patients performed a level, straight walking trial with the measurement system at a comfortable speed for 8 m without assistive device protection. The FF and GRF during walking were recorded continuously. Once the testing was completed, the original rod lengths were restored to minimize any impact on the healing callus.

Following each measurement round, the software interface automatically presented the FF and GRF curves and calculated the related MF and CSF. The staged D-LS $\:{\eta}_{\tau\:}$ corresponding to each gait cycle was calculated. Finally, the D-LS result $\:{\eta}_{a}$ was determined by calculating the average of all staged D-LS across three rounds of measurement.

Clinical treatments after measurement

All patients were encouraged to resume soft daily activities on the second day after fixator removal. Within two weeks after fixator removal, patients can exercise quadriceps muscle, actively flex and extend the knee and ankle joints. Within the subsequent two weeks, patients can begin walking but should avoid running and engaging in physical exercise. Clinical visits and radiographs were routinely taken 4 weeks later.

Statistical analysis

The following parameters of patients were recorded: general data, frame-wearing duration, system installation time, walking time, gait cycle, velocity, D-LS value and refracture rate. The statistical study was conducted using SPSS software (22.0, IBM Corp, USA). The count variables were expressed in form of number and percentage. The continuous variables were presented as the mean, standard deviation, and range of the observations.

Results

Demographic data

The demographic data is shown in Table 1.

Table 1 Demographic data

Clinical outcomes

The clinical outcomes are depicted in Table 2.

Table 2 Clinical outcomes

Figure 6A displays the frequency distribution of the D-LS value $\:{\eta}_{a}$ among patients including trends in central tendency. The histogram bar and the red curve indicate that the D-LS approximate a normal distribution, with values ranging from 9 to 21%. The distribution is centered around a mean of 15.58% and has a standard deviation of 2.77%. The red curve peaks in the central region, indicating that the data is concentrated between 14% and 16%. The blue curve further reveals the accumulative characteristics of the D-LS, with the d₅₀ value of 14.82% and the d₉₀ value of 18.50%.

The mechanical data measured in a typical case (the same patient of Fig. 4) during the trial is shown in Fig. 6B and C. The gait time of the patient is normalized according to the stance phase, and the mean values of GRF, MF, FF and CF in terms of Body Weight for each stance phase is illustrated in Fig. 6B. The total load exerting on the BFCS (CSF, unimodal curve) peaks at 3.8 Body Weight, which is the resultant force of the calf muscle force (MF, unimodal curve) and the ground reaction force on the planta (GRF, bimodal curve). It can be seen that the influence of MF on CSF exceeds that of GRF, making the curve shape of CSF close to MF and the peak appears at about 75% of stance phase. The FF is relatively small in comparison, and its peak value of about 0.48 Body Weight appears in the middle of the stance phase. This is consistent with the fact that the fixator bears only a small amount of CSF when the callus has healed.

Figure 6C depicts the CSF and FF of several gait cycles in a round of D-LS measurement. Although both curves are unimodal in each cycle, their peak time are different. Taking each peak moment of CSF, the ratio of FF to CSF is calculated according to the traditional LS method, and the result is 8.7–14.8%. On the contrast, the D-LS of each gait cycle $\:{\eta}_{\tau\:}$ ranges 14.3–16.7%. It shows that the D-LS has better stability as an evaluation index of callus stiffness.

Fig. 6

Data measured in clinical trials. (A). The frequency distribution of patients in different D-LS intervals, shown with histogram and overlaid cumulative frequency plot. (B). The mean values of GRF, MF, FF and CF for each stance phase of the patient shown in Fig. 4. (C). The values of CSF and FF from several gait cycles during a single walking test of the same patient

Discussion

Timing fixator removal has always been a great challenge for surgeons. Prolonged wearing of a fixator in patients can lead to considerable discomfort and require continuous nursing care. Conversely, if the mechanical stiffness of the callus is insufficient, removing the fixator early can lead to refracture, which is a highly serious complication. The refracture rate after fixator removal was reported in several studies, 3% in De Bastiani, 6% in Krettek, 7.7% in Liu and 9.4% in Simpson [4, 5, 37,38,39]. It is essential to assess the condition of fracture healing and determine the ideal timing for fixator removal.

The conventional approaches to assess the fracture healing rely on the clinical and radiographic results. Evidently, the final determination made by surgeons is subjective, which means the reliability of the fixator removal judgment needs to be improved. Corrales et al. [40] reviewed 123 studies on the fracture healing assessment through radiographic methods, and found that 11 different criteria for fracture union were involved. However, the reliability of the radiographic assessment was only reported in two of those studies. Anand et al. [41] examined the consistency of intraobserver and interobserver evaluations in evaluating fracture healing using radiographs of patients treated with external fixator. They discovered that the agreement among the surgeons engaged was below 50%.

Measuring the bone’s biomechanical properties in vivo is of great interest in fracture healing research. The assessment of fracture healing can be accomplished by applying compressive stresses on the BFCS and measuring the related load distribution [42, 43]. The current studies on the LS only analyze static conditions and fail to reflect the bone’s properties under dynamic loads. Those methods employ Aarnes’s design [44], which incorporate a force-measuring device consisting of three parallel struts that are equipped with force sensors. The drawbacks include: The two rings should be parallel to each other and perpendicular to the axis of the diaphysis, which necessitates complicated modification of the components when facing spatial or unilateral fixators. The device’s hyperstatic structure is easy to produce internal stresses, resulting in additional callus loads and measurement errors. By comparison, the D-LS system presented in this paper benefits of fast installation and precise measurement. The spatial fixators (like Taylor Spatial Frame and TrueLok Hexapod System) or unilateral fixators can be quickly converted to the system’s hexapod module based on its Gough-Stewart mechanism and junction components. In our research, the average system installation time was 15.42 min, which was significantly shorter than the 81.3 min in Liu’s research [19], applying Arne’s approach in LS measurement.

In order to assess the dynamic stiffness property of the bony callus during patients’ locomotion, the load on the BFCS and its distribution between the fixator and the callus need to be measured and analyzed in real time. The hexapod module and the insole module of D-LS system measure FF and GRF synchronously during the gait cycle, respectively. The MF can be calculated based on the GRF, and further determine the total load exerted on the BFCS. Consequently, we defined the D-LS index based on the time integral ratio of the FF to the BFCS’s total load. The D-LS reflects the stiffness of callus relative to the fixator, and it illustrates the cumulative effect of load by analyzing force-time integral in each gait cycle. Furthermore, it helps to reduce the impact of patient gait differences on measurement results. Ma et al. [29] discovered that there is not always a simultaneous occurrence of muscle contraction and plantar touch, which means that there is a discrepancy in the occurrence time of the maximum GRF and internal tibial force. Using the data from a case depicted in Fig. 6C, calculate the conventional LS at the time of the peak CSF for each gait cycle. The LS values vary from 8.7 to 14.8%. In comparison, the D-LS $\:{\eta}_{\tau\:}$ that used time integral showed a variation of 14.9–16.3% for each gait cycle, and an average $\:{\eta}_{a}$ of 15.6% for the entire walk. The D-LS index described in this research offers a more stable and comprehensive evaluation of callus stiffness during locomotion.

The force plate system is commonly employed for measuring GRF because of its high accuracy. However, the force plate system is mainly used in laboratory due to its large size, high cost, and complicated device. Many researchers prefer adopting the pressure insole to facilitate more convenient measurement during locomotion. The force-sensing resistor often serves as the sensor for pressure insole thanks to its wide sensing range, simple measurement circuit, and good electromagnetic interference immunity. To overcome the nonlinear performance of the force-sensing resistor and the discrete sensor distribution, we first conducted data calibration for each sensor and then developed a Back Propagation Neural Network model for estimating GRF. The force signal and CoP of force-sensing resistor are used as inputs of the Back Propagation Neural Network. The Back Propagation Neural Network incorporates information regarding limb kinematics to enhance its capacity to forecast GRF throughout various gait phases. As shown in Fig. 3C, the approach presented in this paper achieves a good accuracy of GRF measurement with a root-mean-square error of 0.113 ± 0.042 Body Weight. The results obtained in this study, utilizing cost-effective insole sensors, are equivalent to those achieved through the utilization of standard commercial devices [22, 26, 45].

The curves of measured forces versus time during the normalized stance phase of all patients are depicted in Fig. 6A. Their overall trends exhibit similarities, with notable rates of change during the initial and ending stance phases. The patients’ GRF shows bimodal characteristics with insignificant fluctuations in the midst of the stance, which is comparable to the GRF of slow walking healthy individuals [29, 46, 47]. During the loading response period, the foot makes contact with the ground and slows down the body, creating a rapidly increasing load that reaches its first peak. In the mid-stance and terminal stance periods, the foot propels the body forward and the center of gravity experiences slight fluctuations, resulting in small changes between the first and second peaks. In the pre-swing period, the base of the foot lifts and shifts the body’s weight to the opposite foot, causing a fast-dropping load. We observed that a significant trough did not occur in the middle of the stance phase for most of the patients. The average gait cycle duration was 1.15 s, and the average velocity was 1.31 m/s, both falling within the normal range [48, 49]. Patients became generally accustomed to the fixator frame in the later post-operative stage, which leads to gait performance resembling that of healthy individuals.

Given the challenges of directly detecting tibia force in vivo during gait, current researches mostly concentrate on calculating muscle force using inverse dynamic musculoskeletal models. Wehner et al. firstly reported the internal forces along the human tibial axis during normal gait [50], which are obtained by simulation-based analysis of healthy volunteers’ motion capture data. The results revealed the highest internal load in the tibial shaft being the axial force with up to 4.7 Body Weight. Subsequent studies [29, 51,52,53] showed that the average axial force exerted on the tibia in the stance phase was approximately 2.9–3.5 times greater than the GRF. The results obtained in this study measuring GRF and calculating CSF are in accordance with those reported in these studies. The CSF in patients is biomechanically equivalent to the tibial axial force in healthy individuals. The CSF is slightly larger than the latter due to the patients wearing a fixator with a specific weight. In this study, the average D-LS value $\:{\eta}_{a}$ of 36 cases was 15.58 ± 2.77%. According to Eq. (8), The proportion of FF sharing the CSF depends on the stiffness of the callus relative to the fixator, which means that during the specific healing period of the patient, the proportion is approximately unchanged in the walking or stationary state. Upon the patient’s bone healing, the ratio of FF to the CSF in the stance phase resembled LS in the static state. The experimental results of the D-LS in our research are comparable to the LS value of 8–10% reported in earlier studies [5, 19]. Furthermore, the developed D-LS method enables the evaluation of the dynamic load-bearing capacity of callus, a capability that was not achievable using the conventional LS approach. The approach presented in this paper also frees surgeon form time-consuming musculoskeletal model simulation analysis.

This research has several study limitations: Firstly, a conservative attitude should be adopted regarding the interpretations of our results due to a relatively small sample size. Secondly, the study lacked a control group for patients whose fixator removal was only determined by clinical criteria. Thirdly, the accuracy and convenience of the new system should be validated through a comparative analysis with traditional Aarnes’ approach. There are certain system deficiencies that require improvement: For greater convenience, the data acquisition module is presently connected to the computer wirelessly, while the fixator and insoles are wired. In the future, there will be enhanced integration of electronic components to create seamless and wireless communication among different modules. Additionally, only four sizes of insoles can be chosen for patients. The range of insole types should be expanded to cater to patients of varying sizes. Moreover, the development of an integrated D-LS measurement system that patients can utilize post-surgery to measure D-LS in real-time is the primary focus of ongoing research efforts.

Conclusions

The novel system in this study has portable hardware and user-friendly software to in vivo measure the dynamic forces of lower limb. This method serves as an effective supplement to the biomechanics properties associated with bone healing. When combined with radiologically confirmed union and pain-free application, it offers a more comprehensive indication of sufficient callus healing in external fixators. Our study provides for the first time the D-LS, expressed as a biomechanical index of tibial callus. The final mean D-LS value, roughly 15.6% (average), could be served as a reliable guide to safely removing external fixators.