A novel fem framework for correlative study of mechanical properties of normal and osteoporotic human bone

Abstract

Background

Osteoporosis is a prevalent bone disease which results in increased bone porosity and decreased bone density, which in turn raises the risk of fractures. A conflict between bone formation (the process of creating new bone tissue) and bone resorption (the degradation and removal of old bone tissue) causes the disorder. This imbalance causes the process of bone remodelling to be disrupted, which weakens the bone structures.

Objective

Due to intrinsic anatomical differences, previous research on the prediction of bone failure has been imprecise. It requires improvement for load scenarios and validation for various demographics, ultimately leading to low accuracy.

Methods

To overcome these limitations, this study proposes a novel Finite Element analysis framework for predicting osteoporosis with the mechanical properties of human bone for stress, strain estimation.

Results

As a result of this proposed framework proves its significance with stress in healthy bones is 2.557541680090727e-04 and bones with osteoporosis is 1.814480251460656e-03, young's modulus of healthy bones and unhealthy bones are 7.019135266051970e + 10 and 0.6158529354739577e + 10, the von Mises stress for healthy bone is 2.4897e + 07, and for the unhealthy bone is 2.8638e + 07, finally, the maximum deflection in healthy bone is 1.0235e-03, for unhealthy bone is 2.1182e-03.

Conclusion

Thus the proposed model provides significant results in the presence or absence of osteoporosis disease.

Keywords

Osteoporosis Young's modulus bone density von Mises stress bone elasticity limit finite element analysis

Introduction

Osteoporosis is a popular bone metabolic disease that reduces bone mass and destroys tissue, increasing the risk of fracture. The T- and Z-score indices are used to quantify bone density, which is a significant global health concern. Osteoporosis is defined by the World Health Organisation (WHO) as a bone mineral density (BMD) with a 2.5 Standard deviation (SD).¹ Patients with densitometric osteoporosis are asymptomatic until the patient experiences a fragility fracture. The body parts that fracture most frequently are the spine, forearm, ribs, sternum, pelvis, proximal humerus, sacrum, clavicle, ankle, hip, hands, feet, and skull.² The amount of proximal humerus fractures (PHF) is rising as a result of life expectancy and osteoporosis. Osteoporosis is 5% of all fractures and is very common in older people. Although mechanical problems have decreased with locking plate systems, failure rates are still significant. Reducing failure rates can be aided by locking screws for inferomedial support and mechanical support in the medial region.³ Bone can be modelled and remodelled through the coordinated actions of osteocytes, osteoclasts, and osteoblasts. Mesenchymal stem cells give rise to osteoblasts, which differentiate into mechanical sensors called osteocytes, and osteoclasts, which become multi-nucleated bone-resorbing cells. When modelling, each operates separately.⁴

DXA, or dual-energy X-ray absorptiometry, is a clinical standard used to diagnose osteoporosis, often misclassifies patients with low-trauma fractures as osteopenia or normal bone density. This is due to the technique's lack of bone depth consideration, and its failure to capture cortical thickness, which is correlated with bone fragility. DXA is often performed at central sites, while fractures often occur at peripheral sites.⁵ DXA is the current technique for determining BMD, which can identify osteopenia, osteoporosis, or health. Osteoporosis is a disorder that requires screening for detection and treatment; several images can be utilised in order to diagnose it. External calibration phantoms are frequently employed, and ultrasonography testing has the potential to be used as a pre-screening tool. Because studies may expose patients to radiation, spinal radiographs are not routinely utilised in evaluation.⁶

Due to advanced computational approaches, multi-detector computed tomography (MDCT) is becoming a clinically meaningful metric for osteoporotic fracture risk identification, especially in research settings.⁷ Approaches such as deep learning and machine learning classifiers are done as an earlier work for detecting osteoporosis disease. Several existing classifications have been utilised for classifying osteoporosis diseases.⁸ In Quantitative Computed Tomography (QCT), high radiation doses are involved in identifying the bone quality compared to DXA.⁹ The cost is more expensive for this QCT process. However, compared to DXA and QCT, the ultrasound process provides less accuracy in diagnosing osteoporosis.¹⁰ The accuracy of diagnosing osteoporosis can be affected due to limited data on BMD norms.¹¹

The existing works provide several limitations such as lack of accuracy, reliability, sensitivity, geometric measurements, strength evaluation etc..¹² Current assessment methods like X-ray diagnostic radiography,¹³ photon absorptiometry,¹⁴ and magnetic resonance imaging¹⁵ face limitations like low bone mineral content sensitivity, high costs, trained personnel requirements, image standardisation, and radiation overexposure risks.¹⁶ To overcome these issues, this study proposed a novel framework which focuses on measuring bone mechanical strength. This method enhances accuracy and allows diagnosis and preventative actions. The contribution of this research is as follows:

This study introduces the Finite Element Analysis using the mechanical characteristics of human bone, both with and without osteoporosis.

The image of Femur bone is taken as the input image. Then, the presence of mineral content in the bone tissue is estimated using the Runge-Kutta order 4.

Furthermore, the young modulus and Seagull optimisation algorithm are combined to measure the bone stiffness.

The study's article is structured this way: Section 2 reviews the existing framework for predicting osteoporosis. The proposed algorithm's architecture is described in Section 3. Section 4 shows the materials and techniques. The outcome and analysis of the suggested work are presented in section 5. This research project is concluded in Section 6.

Literature survey

From 3D models of an elderly patient based on computed tomography (CT) scans, B. Rhee et al.¹⁷ created a finite element model. First, linear finite element models that were analysed for peak von Mises stresses and element failure are used to investigate the impact of cannulated screws in the proximal femur in single-leg stance and lateral fall. Furthermore, in patients who had suffered lateral falls, prophylactic placement of cannulated screws greatly decreased osteoporotic proximal femur failure, with a 21% decrease in trabecular and 5% decrease in cortical failure. Furthermore, these results need to be considered before clinical use is required with biomechanical testing. However, this model provides inaccurate bone failure prediction because it does not provide the micro-mechanical properties of bone.

J.N. LaMonica et al.¹⁸ presented to determine whether the contralateral proximal femur can be strengthened and the risk of contralateral hip fractures reduced by inserting the Femoral Neck system into the bone. Next, using image processing software called ScanIP, a cadaver femur with the implant device is converted into three-dimensional models. Models were correlated and sent to Abaqus for finite element analysis (FEA) in order to evaluate the device's ability to lessen proximal femur stress. By significantly lowering peak stress in osteoporotic bone, the implant preserved 100% of cortical bone failure in the femoral neck and 5% of proximal femur trabecular bone failure. Because only one bone model was used, natural variations in human anatomy have an impact on the results.

The optimal screw design for FNFs was determined by F.M. Özkal et al.¹⁹ using the genetic algorithm (GA) and the finite element method (FEM). The most successful screw patterns are binary, triple, and quadruple, highlighting the importance of screw configuration in the femur treatment process. Healthcare disciplines like orthopaedics might find it easier to apply the recommended design approach, and cooperation between biomechanical and structural engineers is another significant outcome of this model. This model has to be improved for multi-load scenarios, as different forms of femoral neck fractures may occur in different implant configurations.

Park, H et al.²⁰ create deep learning scores that can be used to identify VF and osteoporosis in individuals 50 years of age and older using lateral spine radiography. There are 9276 patients from Severance Hospital in Korea are used in the study to assess two deep convolutional neural network scores such as VERTE-X osteo and VERTE-X pVF. The use of deep learning scores for the referral of high-risk people to bone-density testing resulted in a net reclassification improvement (NRI), according to the findings. The validation results have to be done for various populations to provide accurate results.

Widyaningrum, R et al.²¹ propose a machine learning and colour histogram-based automated trabecular bone segmentation approach for osteoporosis detection. In this model, 120 ROI images, 120 training datasets, and 42 testing datasets are used in the approach. The procedure consists of five steps: obtaining ROI images, converting them to grayscale, segmenting the images, extracting the distribution of pixel, and assessing the ML classifier performance. The best diagnostic outcomes were obtained when K-means segmentation and a multilayer perceptron classifier were used; the resulting accuracy, specificity, and sensitivity were 90.48%, 90.90%, and 90.00%, respectively. However, because there were only so many samples collected, the model had low accuracy requirements.

A machine learning-based method for osteoporosis grading using high-resolution computed tomography (HRCT) spine images was presented by Wang, J. et al..²² By precisely extracting the trabecular component of vertebral bodies, this model makes it possible to assess bone density with accuracy. Compared to conventional random sample methods, it demonstrated improved stability and decreased bias with 183 participants. This model did not accurately examine the prediction of fracture.

Cho, S et al..²³ In this study, two community-based cohorts were used to create the derivation set by stratified random sampling based on age and gender. Data from 380 individuals took place in the individual health assessment programme served as the first external test set. Body compositions from several abdomen CT scan levels were semi-automatically quantified to build a multi-layer perceptron (MLP) based multi-label classification model. The second external test set was constructed using data from patients who had abdominal CT scans at a tertiary-level facility to assess the predictive usefulness of the model output for mortality. This model is associated with long-term mortality. There are still concerns about radiation exposure and a lack of accessibility for CT images in low-resource environments.

In the earlier studies, bone failure prediction lacks micro-mechanical properties and is influenced by inherent variances in human anatomy, it is imprecise. It needs to be improved for several load scenarios and various forms of femur neck fractures. It is necessary to perform validation results for different demographics. Due to limited sample collection, worries about radiation exposure, and difficulties accessing CT images in low-resource locations, the model's accuracy is low.

Proposed methodology

In this research, a novel FEA framework is proposed for the mechanical properties of a human bone both with and without osteoporosis depends on the bone density and bone elasticity limit. Initially, the hip bone is taken as input for the proposed work to identify osteoporosis. Then, bone density is calculated how much amount of mineral content is present in the bone tissue via Runge-kutta order 4. Also, the bone elasticity limit is estimated by combining with the Seagull optimization algorithm (SOA) and Young's modulus for the elasticity limit process. Next, maximum deflection is utilised with different to identify whether the bone is displaced or not. Overall, using FEA, the bone displaces or not can be measured with accurate results by using the mechanical properties of human bone metrics von Mises stress. Figure 1 illustrates the proposed architecture of the quality of the femur bone to identify whether osteoporosis is present or not.

Figure 1.

Proposed architecture.

Finite element analysis

FEA is a computational method for analyzing the behaviour of physical products under loads and boundary conditions. It is one of the most popular approaches for solving partial differential equations (PDEs) that describe physical phenomena. Typical classes of engineering problems that can be solved using FEA are structural mechanics, heat transfer and vibration.

For further image processing, the femur bone is transferred into the medical image analysis software. Before being imported into the 3-Matic software programme to create a finite element mesh utilising a linear tetrahedral element, the bone image is transformed into a 3D geometric model. Following meshing, the density is used to determine the femur's material qualities. Bone geometry is based on the femur bones and is calculated using FEM. FEMs are numerical methods for solving PDEs or functional minimisation problems. The typical FEA workflow is depicted in Figure 2.

Figure 2.

Work flow of finite element analysis.

Bone geometry construction

Tetrahedral components are arranged to form the structure of bones. The adult human femur bone has dimensions of 7 cm in width and 20 cm in length, which is half of its initial length. Femur bone is formed in these two dimensions. There are 1580 components and 144 points in the bone geometry. Every element in the experiment has a Tetrahedral shape with a width of 1 cm and a length of 0.2 cm, as determined by the results of the element size test. The smaller element is, the better simulation result. This is a result of the small size of each bone cell. The femur bones’ shape is created by constructing the bone geometry using FEM. The accuracy of FEA is highly dependent on the quality of the mesh used for discretizing the geometry. In this study, the mesh consists of 6 × 1115 tetrahedral elements and 2 × 2436 nodes, ensuring sufficient granularity for capturing the complex mechanical behaviour of both healthy and osteoporotic bones. Tetrahedral elements are particularly effective for modelling irregular geometries, such as bone structures, as they allow for precise approximation of the stress-strain distribution. This refined mesh provides a robust framework for analyzing the mechanical properties, ensuring accurate results for stress, strain, deflection, and Young's modulus in both healthy and osteoporotic bone models. Table 1 illustrates the summary of FFA for applied load, stress, and strain.

Table 1.

Summary of finite element analysis for applied load, stress, and strain by category.

Performance	Load (kg)	Category
Performance	Load (kg)	Osteopenia	Osteoporosis	Healthy	Male
Maximal Tip Deflection (m)	25	6.2480e-04	5.1509e-04	2.5524e-04	1.1949e-04
	35	8.6279e-04	8.0719e-04	3.3586e-04	1.6213e-04
	69	1.6975e-03	1.3964e-03	6.8385e-04	3.2145e-04
	150	3.8751e-03	3.4997e-03	1.4563e-03	7.2554e-04
	250	5.7532e-03	5.5142e-03	2.5514e-03	1.1944e-03
	414	1.0634e-02	8.8187e-03	3.9760e-03	1.9706e-03
Von Mises Stress (N/m² or Pascals)	25	1.0381e + 07	7.0189e + 06	6.1031e + 06	3.1230e + 06
	35	1.4643e + 07	9.8205e + 06	8.5456e + 06	4.3728e + 06
	69	2.8724e + 07	1.9360e + 07	1.6840e + 07	8.6220e + 06
	150	6.2443e + 07	4.2127e + 07	3.6629e + 07	1.8744e + 07
	250	1.0407e + 08	7.0190e + 07	6.1022e + 07	3.1235e + 07
	414	1.7191e + 08	1.1623e + 08	1.0105e + 08	5.1724e + 07
Output Strain Value	25	5.2713e-04	6.2269e-04	7.6355e-05	8.2570e-05
	35	6.4714e-04	2.6667e-04	9.2833e-05	N/A
	69	1.1905e-03	1.7817e-03	2.1572e-04	2.1270e-04
	150	3.4156e-03	8.7219e-04	4.1263e-04	4.9813e-04
	250	7.7350e-03	5.0785e-03	7.9555e-04	1.7350e-03
	414	1.0804e-02	9.3892e-03	1.1900e-03	1.3458e-03

In this study, the elements of these components are represented in various forms such as the element Tetrahedral given below:

Element tetrahedral

To facilitate calculations, femur bone is treated as an isotropic material in this study, which means that it behaves the same way when treated in different ways. The Tetrahedral part is stretched and displays stress represented by ${σ} = [D] {ε}$ . The matrix material characteristics are transformed for the bone quality ${σ} = {σ_{x} σ_{y} τ_{x y}}^{T} {ε} = {ε_{x} ε_{y} ε_{x y}}^{T} [D]$ is determined in equation (1):

[D] = \frac{E}{1 - v^{2}} [\begin{matrix} 1 & v & 0 \\ v & 1 & 0 \\ 0 & 0 & \frac{1 - v}{2} \end{matrix}]

(1)

The weighted Residual technique for material elasticity gives the femur bone's elasticity a finite element representation is in equation (2).

\int_{Ω} {\begin{matrix} \frac{\partial ω_{1}}{\partial x} & 0 & \frac{\partial ω_{1}}{\partial y} \\ 0 & \frac{\partial ω_{2}}{\partial y} & \frac{\partial ω_{2}}{\partial x} \end{matrix}} [D] {\begin{matrix} \frac{\partial u}{\partial x} \\ \frac{\partial v}{\partial y} \\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \end{matrix}} d Ω = \int_{Ω} {\begin{matrix} ω_{1} f_{x} \\ ω_{2} f_{y} \end{matrix}} d Ω + \int_{τ_{n}} {\begin{matrix} ω_{1} {\overset{}{\bar{Φ}}}_{x} \\ ω_{2} {\overset{}{\bar{Φ}}}_{y} \end{matrix}} d r

(2)

The algorithm of the proposed FFA is illustrated in Algorithm 1 as follows:

Algorithm 1.

Finite element analysis (FFA)

Input: Geometry description matrix (gd), structural properties (Poisson's ratio, Young's modulus, density), boundary loads, and boundary conditions.

Output: Displacement, Von Mises stress, and strain distribution.

Step 1: Initialise the geometry description (gd), set formula (sf), and name space matrix (ns).

Step 2: Decompose the constructive 2D solid geometry into non-overlapping minimal regions.

Step 3: Create the structural model using static plane stress formulation.

Step 4: Assign material properties

Specify material properties for each region such as, Poisson's ratio (ν), Young's modulus (E), and density (ρ).

Step 5: Generate 2D geometry from the decomposed matrix

Step 6: Reconstruct the geometry and visualise it by plotting.

Step 7: Assign the structural properties of materials to the structural model.

Step 8: Specify the boundary loads for the structural model.

Step 9: Specify the boundary conditions for the structural model.

Step 10: Generate a Tetrahedral mesh for the geometry. Ensure appropriate mesh density to balance accuracy and computational cost.

Step 11: Solve the system Solve the system using FEA.

Step 12: Analyze the results, including displacement, Von Mises stress, and strain.

Step 13: Visualise the results through plots.

End

Material and methodology

Simulation of the distribution of stress, force, and strain rate

By applying a compressive force (F) to point 144 (Figure 3), which represents the connection of the pelvis and femur, the pressure simulation is performed. The point at which the force flows on the vertical axis connects the femur and pelvis is 144. The boundary condition of the simulation is a zero limiting at the lowest point of the bone stem. To account for standing and walking, a change in force is made. While walking requires a force of (2.7 × body weight) N, standing requires a force of (2 × body weight) N.

Figure 3.

Edge detection in the femur bone.

The nodes i, j, k, and m indicate the rectangle element with attributes, as shown in Figure 4. The ij side, km side, and noodle side of a rectangle with four vertices that constitute an element are parallel to the x-axis and the y-axis. For the rectangular elements’ shape functions to be specified as a global coordinate system (x, y).

Figure 4.

Rectangular elements in global coordinates.

Using the interpolation equation (3), shape functions for rectangle elements can be determined.

u = a_{1} + a_{2} x + a_{3} y + a_{4} x y

(3)

By demonstrating the link among strain and stress ${σ} = [D] {ε}$ . The stress is formed by the equation ${σ} = {1 σ_{x} σ_{y} τ_{x y}}^{T}$ . Using this relationship strain is formed by the equation. ${ε} = {1 ε_{x} ε_{y} ε_{x y}}^{T} [D]$ .

[D] = \frac{E}{(1 + v) (1 - 2 v)} [\begin{matrix} 1 - v & v & v & 0 \\ v & 1 - v & v & 0 \\ v & v & 1 - v & 0 \\ 0 & 0 & 0 & \frac{1 - 2 v}{2} \end{matrix}]

(4)

The matrix format of the relationship among strain and stress is denoted in the equation (4).

s t r e s s = f o r c e / a r e a

(5)

Equation (5) expressed the linear stress calculation. This assumes the force is a product of stiffness and displacement, where the force calculation is expressed in equation (6).

F o r c e = s t i f f n e s s \times D i s p l a c e m e n t

(6)

Hence, the response of a component to stress is influenced by its stiffness, geometry, and the material's inherent properties. Strain is a critical metric in FEA, quantifies the deformation of a material and the calculation is illustrated in equation (7):

S t r a i n = c h a n g e i n l e n g t h / o r i g i n a l l e n g t h

(7)

For materials with a linear stress-strain relationship, the behaviour is characterised by Young's modulus (E), defined as:

M o d u l u s o f E l a s t i c i t y = L o n g i t u d i n a l s t r e s s / s t r a i n

(8)

Young's modulus determines the slope of the stress-strain curve, which remains linear until failure occurs either through fracture or plastic deformation. At this point, the material no longer follows the linear model, and Young's modulus cases to apply. In materials modelled with linear FEA, such as healthy human bone, the relationship between stress and strain remains linear up to the ultimate tensile strength (UTS). However, in bones affected by osteoporosis, this relationship deviates due to reduced density and elasticity, resulting in increased stress and strain under similar loads. For nonlinear materials like rubber, the stress-strain relationship becomes more complex and cannot be described using Young's modulus beyond small strain values. Hyperelastic materials exhibit significant deformation before failure, unlike bone tissues, which exhibit lower strain tolerance when diseased. To address the limitations of previous research in bone failure prediction, this study introduces a novel FEA framework tailored to osteoporosis assessment. By incorporating mechanical properties such as bone density and elasticity, the model delivers significant insights into stress, strain, and deformation differences between healthy and osteoporotic bones.

Bone density estimation

Bone density is the measure of how much amount of mineral content is present in the femur bone. The MATLAB 2021b uses Runge-Kutta order 4 for this estimate technique. The simulation method involving stress distribution, strain, and strain rate yields the strain rate variable. The kinematic equations for mono-nucleoid cells, old bone, osteoblast, osteocyte, and young bone is shown in the following Table 2 and the equation (9) to equation (15) is given as follows:

\begin{matrix} \frac{d n_{M C E L L}}{d τ} = - δ_{1} (β_{1} + n_{M C E L L}) n_{M C E L L} + J_{3} + J_{N e w_{-} B} - D_{1} \end{matrix}

(9)

\frac{d n_{O l d_{-} B}}{d τ} = - (β_{3} - n_{M C E L L} + n_{O l d_{-} B}) n_{O l d_{-} B} - D_{2} + J_{N e w_{-} B}

(10)

\begin{matrix} \frac{d n_{O B}}{d τ} = δ_{3} (β_{6} - n_{O l d_{B}} - (n_{O B} + n_{O S t e o i d} + n_{N e w_{B}})) \\ (β_{8} - (n_{O B} + n_{Osteoid} + n_{N_{N -} B})) \dots \\ - δ_{4} (β_{11} - (n_{Osteoid} + n_{N e w_{-} B}) n_{O B}) + D_{3} - D_{4} \end{matrix}

(11)

\begin{aligned} \begin{matrix} \frac{d n_{Osteoid}}{d τ} = δ_{4} (β_{11} - (n_{Osteoid} + n_{N e w_{-} B})) n_{O B} \\ - δ_{5} (β_{14} - n_{N e w_{-} B}) n_{Osteoid} + D_{4} - D_{5} \end{matrix} \end{aligned}

(12)

\frac{d n_{N e w_{-} B}}{d τ} = δ_{5} (β_{14} - n_{N_{N e w_{-} B}}) n_{Osteoid} - J_{N e w_{-} B} + D_{5}

(13)

D_{α} = \frac{l_{α v} d_{(1)}}{k_{+ 2} {[B o]}^{2}}

(14)

ρ (I) = ρ_{0} (N_{O l d_{B}} (I) + N_{N e w_{B}} (I))

(15)

Table 2.

Mathematical equations for thermodynamic variables of bone remodelling.

Parameter	Description
$n_{M C E L L}$	Changes in the cell concentration of mono-nucleoid
$n_{O l d_{-} B}$	Changes in the concentration of old bone
$n_{O B}$	Changes in the osteoblast concentration
$n_{Osteoid}$	Changes in the osteoid concentration
$n_{N e w_{-} B}$	Changes in the new bone concentration
$β_{i}$	Total normalisation of the starting substance of the material
$D_{α}$	Dynamic stimulus influence on the rate of chemical to-equation $α$
$J_{i}$	Flux to substance i

Equation (16) yields the finite element domain integral form for the bone measuring density based on the weighted function's result.

\int_{Ω} [B]^{T} [D] [B] d Ω {d}

(16)

Calculation of bone elasticity limits

Furthermore, the bone elasticity limit estimation is obtained by combining Young's modulus with the SOA. Since the young modulus cannot assess bone elasticity directly, so SOA is used in conjunction with it to perform the elasticity limit procedure. After that, maximal deflection is applied differently with the mechanical properties of human metrics von Mises stress to determine whether the bone is displaced or not. The SOA composed of two phases such as Exploitation and Exploration.

i) Migration (Exploration)

Avoidance of collisions: It is necessary find position of the new search agent denoted by equation (17), a variable $A_{m}$ is used.

C_{s} = A_{m} \times P_{s} (y)

(17)

Where, $C_{s}$ indicates position of the search agent that avoids colliding with other search representatives in equation (13). $P_{s}$ represent the current position of the search agent, y denotes the current iteration of the process. $A_{m}$ is a representation of the search agent's movement patterns in a certain search space given in equation (18).

A_{m} = f_{e} - [y \times (\frac{f_{e}}{m a x_{i t e r a t i o n}})]

(18)

Where: $y = 0, 1, 2, \dots m a x_{i t e r a t i o n} ., f_{e}$ represent to regulate how often variables $A_{m}$ are used, which are reduced linearly $f_{e}$ to 0. Here, set 2 as $f_{e}$ value.

Moving in the direction of the best neighbour: After stopping a collision between neighbours, the search agents move in the direction of the best neighbour.

M_{s} = B \times [P_{b s} (y) - P_{s} (y)]

(19)

The positions of search agents $P_{s}$ with respect to the best-fit search agent $P_{b s}$ (i.e., the fittest seagull) are represented by $M_{s}$ in equation (19). B denotes randomised behaviour is in charge of maintaining the right balance between exploitation and exploration. B is computed as follows in equation (20):

B = 2 \times A_{m}^{2} \times r a n d

(20)

Where $r a n d$ stands for random number ranges in $[0, 1]$ .

Staying near the top search agent: Finally, the search agent might change where it stands in relation to the leading search agent.

D_{s} = | C_{s} + M_{s} |

(21)

Where, $D_{s}$ is the distance between the best-fit search agent and the search agent (i.e., the fitness value is less to find the best seagull) is determined in equation (21).

ii) Attacking (Exploitation)

The spiral movement behaviour takes place in the air during the attack on the prey. This behaviour can be explained as follows in the x, y, and z planes.

\vec{x} = r \times \cos (k)

(22)

\vec{y} = r \times \sin (k)

(23)

\vec{z} = r \times k

(24)

r = c \times w^{k d}

(25)

Where, $k$ is a random number in the interval $[0 \leq k \leq 2 π]$ , and r is the spiral's radius for each turn. The spiral shape is represented by the parameters c and $d$ , while the natural logarithm's base is $w$ . Equations (22) through (25) are used to compute the updated position of the search agent.

P_{s} (y) = (D_{s} \times \vec{x} \times \vec{y} \times \vec{z}) + P_{b s} (y)

(26)

Where $P_{s} (y)$ modifies the other search agents’ positions and records the optimal response from equation (26). $A_{m}$ randomly chosen population is used to begin SOA. Throughout the iteration process, the search agents might adjust the positions concerning the top search agent. From $f_{e}$ to 0, it $A_{m}$ decreases linearly. Variable B is in charge of ensuring a seamless transition between exploration and exploitation.

Where Ω represents the domain items. The constituents of the elasticity stiffness matrix of the femur bone can thus be produced in equation (27)

[K^{e}] = \int_{Ω} [B]^{T} [D] [B] d Ω

(27)

Thus, the following finite element equation (28) describes the material's elasticity.

E (I) = E_{0} C + P_{s} (y) (\frac{ρ (I)}{ρ_{o}})^{3} with C = c o n s t a n t

(28)

Result and discussion

In MATLAB, geometry for FEA is constructed using built-in tools and libraries. Providing the geometry parameters, creating the node coordinates, determining element connection, creating the mesh, and presenting the geometry as visualised in the mesh are typical steps in the process. The geometry's measurements and characteristics, including form and size, are described. Next, the nodes that will comprise the coordinates of the mesh are listed using the MATLAB tools such as mesh grid, Ingrid and linspace can be used to produce the coordinates or these can be manually entered. Then, Delaunay Tetrahedral is often utilised to establish how nodes are connected to generate elements. Also, the mesh is created by using the node coordinates and element connectivity. The MATLAB plotting function trisurf for tetrahedral meshes and patch for quadrilateral meshes is used to display the mesh to verify the correctness of the image.

MATLAB offers extra tools and libraries for more complicated meshes, such as the PDE Toolbox and DistMesh. Meshes for FEA can be created and modified using the PDE Toolbox's tools. It comes with both 2D and 3D meshing tools. A versatile MATLAB library called DistMesh can be used to create unstructured tetrahedral and tetrahedral meshes. To characterise the geometry, a signed distance function is needed.

To confirm and display the accuracy of the mesh in MATLAB, one must use plotting tools like trisurf, trimesh, and plot. These functions assist you in examining the nodes and elements to make sure the descriptions and connections are accurate. Here, trisurf is utilised to visualise 3D tetrahedral meshes. It plots a Tetrahedral-defined surface in three dimensions. Trimesh is perfect for generating a wireframe view of the mesh, which facilitates structural and connection inspection without surface filling. Then, the plot can used to plot nodes and edges independently for an in-depth assessment. Figures 5 and 6 depict the simulated geometry and meshing produced by MATLAB.

Figure 5.

Geometry of femur bone.

Figure 6.

The meshing of the designed model.

Von Mises stress

In the field of solid mechanics, the von Mises stress referred to as the von Mises equivalent stress or the von Mises-Hencky stress is a scalar value that is obtained from the stress tensor. When evaluating yield and failure criteria in materials that are subjected to complicate loading circumstances, it is especially crucial. Engineers can ascertain whether a particular material will undergo permanent deformation under a complex state of stress by the von Mises stress criterion.

The von Mises stress $σ_{v}$ is calculated as follows in equation (29):

σ_{v} = \sqrt{\frac{1}{2} [(σ_{1} {- σ_{2})}^{2} + (σ_{2} {- σ_{3})}^{2} + (σ_{3} {- σ_{1})}^{2}]}

(29)

Where, $σ_{1,} σ_{2}, σ_{3}$ denoted as the principal stress. Alternatively, the von Mises stress can be represented in equation (30) in terms of the stress components in Cartesian coordinates:

σ_{v} = \sqrt{\frac{1}{2} [(σ_{x x} {- σ_{y y})}^{2} + (σ_{y y} {- σ_{z z})}^{2} + 6 (τ_{x y}^{2} + τ_{y z}^{2} + τ_{z x}^{2})]}

(30)

Where, $σ_{x x}, σ_{y y}, σ_{z z}$ are normal stresses, $τ_{x y}^{2}, τ_{y z}^{2}, τ_{z x}^{2}$ are shear stresses. The von Mises stress is a scalar quantity that represents an identical stress that would generate the same amount of yield as the actual condition of stress if applied equally in all directions. It offers a method for contrasting the uniaxial yield stress derived from basic tensile testing with complicated, multi-axial stress states. It is a useful tool when working with ductile materials since it provides an accurate assessment of the material's yield and failure. By evaluating the von Mises stress to the material's yield strength or ultimate strength, engineers can ascertain if a material is likely to fail under a given stress scenario.

In solid mechanics, the von Mises stress is a crucial instrument for assessing the robustness as well as safety of an object under intricate stress conditions. It makes determining multi-axial stress assessments easier and ensures that components and structures are built to sustain operating loads without yielding. Engineers may forecast and prevent material failure by contrasting the von Mises stress to the yield material strength. This ensures the dependability and longevity of engineering projects. Figure 7 represents the von Mises stress of femur bone.

Figure 7.

Von Mises stress.

Deformation plane

The deformation plane in FEA refers to the particular plane or surface inside a structure that is being studied or evaluated with deformation or strain. This is especially important for 2D analysis, but the idea also applies to 3D investigations where a particular plane or cross-section inside the 3D structure is of relevance. Engineers frequently concentrate on particular planes or cross-sections across the structure during 3D analysis to comprehend localised deformation behaviour. The following factors determine the deformation plane selection are areas predicted to undergo significant deformation or stress through Area of Interest. Also, Locations where structural failure is most likely to occur are known as critical cross-sections. The deformation or strain through the chosen deformation plane is then investigated. This comprises the vector field that depicts the nodes’ displacements inside the plane is called the displacement field. Therefore, strain and stress distribution are derived from the displacements, these metrics aid in evaluating the performance and structural integrity.

Engineers can plot and examine deformation, stress, and strain inside the deformation plane using visualisation capabilities in FEA software. Usually, these visualisations consist of plots with contours that display the amount of strain, stress, or deformation. Then, vector plots show the nodal displacements’ direction and amplitude. Furthermore, a deformed shape shows how a structure or component changes from its initial shape when under load.

In FEA, the notion of the “deformation plane” is essential for comprehending stress and localised deformation in a structure. Engineers can obtain a profound understanding of the behaviour of the material under different loading situations by splitting its structure into finite elements, applying loads and boundary conditions, solving the governing equations, and visualising the outcomes. This method guarantees that the structures can bear the applied loads without experiencing severe deformation or failure, which aids in the construction of safer and more effective structures. Figure 8 denotes the femur bone deformation.

Figure 8.

Bone deformation.

Comparative analysis of healthy and unhealthy bone

This section analyses the comparison of the FEA results of healthy and sick bones and sheds light on how illnesses impact bone mechanics. Researchers and engineers can better understand the consequences of bone illnesses and enhance clinical outcomes through improved treatment and implant designs by methodically analysing stress, strain, and displacement. This method aids in designing preventive measures to preserve bone health as well as in estimating fracture risks.

Stress

A key idea in the study of material mechanics, the internal forces that develop in a material as a consequence of external loads or deformations are referred to as stress. The intensity of these internal forces per unit area within the material is measured by stress. To predict failure, ensure structural integrity, and analyse how materials react to external forces, it is essential to understand stress. Shear stress and normal stress are two different kind of stress. Tensile stress and compressive stress are two types of normal stress. Tensile stress is the result of pulling a material apart. The stress functions for the cross-sectional area. The mathematical expression for the stress is formulated in equation (31).

σ = \frac{F}{A}

(31)

Where, the force $(F)$ perpendicular to that area $(A)$ . Compressing two pieces of material together causes compressive stress. In addition, it functions in the opposite path of tensile stress and is perpendicular to its cross-sectional area. When forces are applied parallel to a material's surface, shear stress occurs. It interacts with the cross-sectional area indirectly.

The comparison study clearly shows how osteoporosis negatively affects bone strength. When opposed to osteoporotic bones, healthy bones show significantly lower stress values under the same circumstances. This emphasises how crucial it is to treat osteoporosis as soon as possible to preserve bone health and avoid fractures. The information emphasises the need for measures that can help people who are at risk of osteoporosis or who already have the disease maintain or increase their bone density and structural integrity. The stress on healthy and diseased bones is shown in the Figure 9 and Table 3.

Figure 9.

Analysis of output stress for healthy and osteoporosis.

Table 3.

Performance analysis of output stress.

S.N	Category	Sample	Output Stress
1	Healthy	Healthy bone (H100)	2.557541680090727e-04
		Healthy bone (H101)	3.093277168916645e-04
		Healthy bone (H104)	3.259084085641746e-04
		Healthy bone (H102)	3.191637353667540e-04
2	Osteoporosis	Osteoporosis (P100)	2.589519455445487e-03
		Osteoporosis (P101)	1.814480251460656e-03
		Osteoporosis (P111)	2.200243899876646e-03
		Osteoporosis (P381)	1.814480251460656e-03

Young's modulus

Young's modulus, is a measurement of the stiffness or rigidity of a material, is sometimes known as the elastic modulus or the modulus of elasticity. Within a material's elastic limit the point at which the load has been eliminated and the material recovers to its initial shape it describes the link among strain (deformation) and stress (force per unit area). The ratio of tensile stress $(σ)$ to tensile strain $(ϵ)$ along with the linear elastic area of the stress-strain curve is defined as Young's modulus $(E)$ . The mathematical expression for Young's modulus E is formulated in equation (32).

E = \frac{σ}{ϵ}

(32)

Where $σ$ is a tensile stress, $ϵ$ is the tensile strain. The tensile strain can be formulated in equation (33)

ϵ = \frac{Δ L}{L_{0}}

(33)

Where $Δ L$ denotes the change in material length, $L_{0}$ represent the original material length. A crucial factor in determining and measuring the stiffness of materials is Young's modulus. It is essential to material science and engineering because it affects the choice of materials, design choices, and the forecasting of material behaviour under diverse loading scenarios.

The information supplied shows the notable variation in Young's modulus between osteoporotic and healthy bones. Osteoporotic bones show noticeably reduced stiffness, making them more prone to deformation and fractures, while healthy bones demonstrate substantially higher stiffness, showing strong resistance to deformation. Clinical evaluations, treatment planning, and the creation of biomedical implants all depend on this data. The Young's modulus on healthy and diseased bones is shown in Figure 10 and Table 4.

Figure 10.

Analysis of Young's modulus output for healthy and osteoporosis.

Table 4.

Performance analysis of Young's modulus.

S.N	Category	Sample	Young's modulus
1	Healthy	Healthy bone (H100)	7.019135266051970e + 10
		Healthy bone (H101)	7.868503932886549e + 10
		Healthy bone (H104)	7.353038130696133e + 10
		Healthy bone (H102)	7.529637394120473e + 10
2	Osteoporosis	Osteoporosis (P100)	0.6158529354739577e + 10
		Osteoporosis (P101)	0.8524114257721721e + 10
		Osteoporosis (P111)	0.7998386478881680e + 10
		Osteoporosis (P381)	0.8524114257721721e + 10

Von Mises stress

The three principal stresses (σ₁, σ₂, and σ₃) that are applied to a material at any given point are used to calculate the von Mises stress. The material experiences maximum and minimum normal stresses in three mutually perpendicular directions, which are represented by these principal stresses.

The information provided in Table 5 is divided into two categories: osteoporosis and healthy. There are several samples in each category, along with the appropriate Von Mises stress values. The information supplied demonstrates the variation in Von Mises stress between osteoporosis-affected and healthy bones. Although both show very low-stress levels, cumulative stresses tend to be slightly higher in osteoporotic bones. The Von Mises Stress on healthy and diseased bones is shown in Figure 11 and Table 5.

Figure 11.

Analysis of von Mises stress for healthy and osteoporosis.

Table 5.

Performance analysis of von Mises stress.

S.N	Category	Sample	Von Mises Stress
1	Healthy	Healthy bone (H100)	2.4897e + 07
		Healthy bone (H101)	2.4886e + 07
		Healthy bone (H104)	2.4894e + 07
		Healthy bone (H102)	2.4897e + 07
2	Osteoporosis	Osteoporosis (P100)	2.8638e + 07
		Osteoporosis (P101)	2.8629e + 07
		Osteoporosis (P111)	2.8594e + 07
		Osteoporosis (P381)	2.8629e + 07

Maximum deflection

Materials like bones have a property called elastic deformation that develops to external forces. This temporary bending or deformation of bone tissue is characterised by its reversibility upon removal of stress. Collagen fibres are essential to provide bone strength, structural stability and flexibility. The direction and force of the applied force influence how much elastic deformation occurs in bones. The size, shape, and structural characteristics affect a bone which responds to external forces; then the ability to deform is impacted more by thicker, more inflexible bone tissue. Long bones with stiffer shafts, like the femur, are better suited to distribute and absorb stresses.

The presented data in Table 6 emphasises how healthy bones and bones impacted by osteoporosis differ in the maximal deflection. When compared to healthy bones, osteoporotic bones bend or flex noticeably more, a symptom of reduced structural rigidity and integrity. This data can be useful for improving the comprehension of bone biomechanics, tracking the efficacy of treatments, and estimating fracture risk. The maximal deflection on healthy and diseased bones is shown in Figure 12 and Table 7.

Figure 12.

Analysis of maximum deformation for healthy and osteoporosis.

Table 6.

Performance analysis of Maximal deflection.

S.N	Category	Sample	Maximal deflection
1	Healthy	Healthy bone (H100)	1.0235e-03
		Healthy bone (H101)	0.98155e-03
		Healthy bone (H104)	1.0067e-03
		Healthy bone (H102)	0.96019e-03
2	Osteoporosis	Osteoporosis (P100)	2.1182e-03
		Osteoporosis (P101)	2.2154e-03
		Osteoporosis (P111)	2.2270e-03
		Osteoporosis (P381)	2.2154e-03

Table 7.

Comparison of result.

Loads	Existing method (Mpa) ²⁴	Proposed Method (Mpa)
25kg	10.70	6.10
35kg	14.75	8.54
150kg	43.20	36.62
250kg	63.3	61.02
414kg	93.73	101.05

The table compares the stress values (in MPa) under varying load conditions using an existing method²⁴ and the proposed method. For lighter loads, such as 25 kg and 35 kg, the proposed method demonstrates significantly lower stress values (6.10 MPa and 8.54 MPa) compared to the existing method (10.70 MPa and 14.75 MPa), highlighting improved efficiency in stress prediction for smaller loads. As the load increases to 150 kg and 250 kg, the stress values for both methods converge, with the proposed method showing slightly better alignment to real-world bone mechanics. However, for the heaviest load of 414 kg, the proposed method predicts a higher stress value (101.05 MPa) compared to the existing method (93.73 MPa), suggesting that it more accurately captures the nonlinear behaviour of osteoporotic bone under extreme conditions. These results highlight the superior performance of the proposed method in predicting stress values across a wide range of loading conditions.

Conclusion

The study effectively builds and applies a unique FEA framework by examining the mechanical characteristics of human bone, including maximum deflection, Young's modulus, Von Mises stress, bone density, and bone elasticity limit to anticipate osteoporosis. Through the evaluation of bone mechanical properties, this research offers a precise and dependable tool for osteoporosis prediction. This method improves our understanding of bone mechanics under various stress circumstances and corrects earlier errors in bone failure predictions. The simulation result shows that the von Mises stress of healthy bone is generally higher than that of osteoporotic bone. The potential risk of osteoporosis decreases with increasing bone flexibility. The results may result in more effective preventive interventions and patient-specific treatment plans, which could have a significant impact on osteoporosis diagnosis and therapy.

Footnotes

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

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