Application of a mathematical model of a human lower limb for modeling shock-wave effects of contact explosion

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INTRODUCTION

At present, human body processes resulting from the effects of explosive and gunshot trauma, including the use of personal protective equipment, are primarily investigated through full-scale modeling. To achieve this, research of this kind employs various biological and synthetic materials. Biological materials include human cadavers and their constituent parts, as well as a range of experimental animals. Synthetic materials include bioimitators of human tissues and technical imitators of the human body or its parts. Nevertheless, conducting biomedical research using experimental animals presents considerable ethical challenges and poses significant difficulties in comparing the musculoskeletal systems of animals and humans owing to their inherent anatomical differences. The use of live humans or parts of human cadavers as test objects is even more problematic because it raises ethical concerns and presents practical difficulties [1, 2].

A potential solution to this problem is to incorporate various modeling techniques into the testing process. These techniques may include the use of biosimulators of living tissues, technical devices for parameter monitoring, and simulation models for comprehensive process modeling. At present, computer-aided engineering software systems are widely used in technical scientific studies. The main applied task of this methodology can be the selection of optimal parameters of complex materials and the creation of interaction models of these materials and their destruction [3, 4].

When developing numerical models of biomaterial behavior, even when solving similar problems, full correspondence between the results of modeling and in situ tests cannot be achieved because of the strong influence of geometric and physicomechanical parameters of each biological sample on the final result. In addition, calculating the expected biomaterial damage from the multifactorial effects of a close explosion is further complicated by the presence of protective structural elements to the computational model [5, 6].

Lower leg and foot injuries caused by contact explosion to the lower extremities protected by special anti-explosion footwear can be reasonably attributed to a separate type of mine explosion injury, called “barrier” mine explosion injury in individuals wearing anti-explosion footwear. If the protective effectiveness of the footwear is sufficient, characteristically, in most cases, most of the energy of a near explosion is spent on the destruction of the protective elements of the sole, and the remaining energy is transferred to the “shock shift” of the underlying foot structures. In this case, the casualty may sustain different closed injuries such as skin abrasions, soft tissue contusions, ligamentous injuries, and fractures of the bones of the foot and lower third of the tibia. In addition, if the protective structure is destroyed, injuries characteristic of a classic contact mine blast wound may be observed, with open injuries and even detachment of the foot [7, 8].

This study aimed to substantiate a finite-element model of the human lower extremity to simulate the destruction of musculoskeletal biomaterials under the shock-wave impact of a contact explosion, predict the nature and volume of damage to the human lower extremity, and solve applied problems in the design of special explosion-proof footwear.

MATERIALS AND METHODS

The finite-element technique was used to model shock-wave impact on the lower extremity and evaluate the behavior of the biological materials of which it is composed under the contact action of the blast. This modern computational mechanics is based on the decomposition of the studied structure into separate parts that are finite elements connected by nodes. The set of the interconnected finite elements attached to the base forms a design scheme called a finite-element (computational) model.

Using scientific data on the physical and mechanical properties of biological tissues of the human lower extremity and their behavior under shock-wave impact, an original full-scale finite-element model of the lower extremity of an adult male was developed with maximum consideration of all dimensional characteristics of its anatomical structures and physical and mechanical properties of the main biological tissues. With the labor-intensive and complicated process, the main stages of the model development are presented in the Results and Discussion.

The finite-element model was validated by comparing computational and experimental data (pressure and acceleration indices) obtained from detonating 75 g of explosives under a protective structure (a metal plate mimicking the protective composition of a sapper’s shoe) over which anatomical preparations of the human lower extremity were placed. This was conducted as part of the research project on the development of a sapper’s protective shoe, undertaken to advance the capabilities of the Russian engineering forces.

After research completion, radiographic signs of damage to the lower limb fragments, high-speed video recordings, and acceleration and pressure sensor data were analyzed. The schematic diagram of the full-scale experiment is shown in Figure 1.

 

Fig. 1. Presentation of the experiment and associated calculations: a — experimental scheme; b — calculation scheme

 

The results of the virtual (numerical) experiment simulating the impact of blast effects on the human lower extremity were analyzed in graphical format to determine the propagation of pressure and acceleration fields and predict the potential destruction of major anatomical structures.

The study was conducted in accordance with the project “Study of the creation of protective footwear for sappers” (project code “Foot”), which is part of the scientific work plan of the Armed Forces of the Russian Federation.

RESULTS AND DISCUSSION

The model of an elastic–viscous–plastic material with a fracture criterion based on the maximum value of effective plastic deformation was employed as the foundation for the developed mathematical model of materials comprising spongy and compact bone tissues (femur, fibula, and tibia) within the lower limb finite-element model. The effect of plasticity in this model must be considered because strain accumulation is a significant factor and, simultaneously, serves as an indicator of potential damage. The physical and mechanical characteristics of the materials comprising the patella, talus, cuboid, navicular, lateral cuneiform, intermediate and medial cuneiforms, metatarsals, and phalanges are analogous to those observed in tibial materials. Importantly, the calcaneal bone was subdivided into three distinct layers, namely. the compact substance, spongy substance, and marrow, in accordance with the expression of the “marrow component” (red bone marrow) [9, 10]. The physical and mechanical properties of the bones included in the numerical model are summarized in Table 1.

 

Table 1. Physical and mechanical properties of the bones of the lower limbs

Таблица 1. Физико-механические свойства костей нижней конечности

Indicators	Density (ρ), kg/m3	Young’s modulus (E), MPa	Poisson’s ratio (ν), rel. unit	Shear modulus (G), MPa	Bulk modulus of elasticity (K), MPa
Compact substance of the tubular bone	2 × 103	15 000	0.3	5769	12 500
Spongy substance of the tubular bone	1.1 × 103	445	0.3	–	–
Compact substance of cancellous bone	2 × 103	14 000	0.3	5385	11 667
Spongy substance of cancellous bone	1.1 × 103	292	0.3	–	–
Cancellous red marrow	2 × 103	15 000	0.3	5769	12 500
Compact substance of the tubular bone	6 × 103	445	0.45	–	–
Spongy substance of the tubular bone	9.75 × 103	2	0.167	–	–

 

A hyperelastic material model, which allows for the specification of viscous properties, was employed for the modeling of the lower limb muscles [11, 12]. The equations embedded in the material maps are described in detail in the user manual for LS-DYNA [13]. The physical and mechanical properties of muscle tissues are summarized in Table 2.

 

Table 2. Physical and mechanical properties of muscle tissues

Таблица 2. Физико-механические свойства мышечных тканей

Indicators	Value
Density, kg/m3	1.1 × 103
Poisson’s ratio, rel. unit	0.495
Shear modulus for frequency-independent damping, MPa	1
Stress limit for frequency-independent friction damping, MPa	0.001
Hyperelastic coefficients (C10), rel. unit	0.04
Additional shear relaxation modulus (GI_1), MPa	0.1
Attenuation parameter (BETAI_1), rel. unit	0.1
Fracture strain (MAT_ADD_EROSIONEFFEPS), rel. unit	1.1

 

Hyperelastic and viscoelastic material models are commonly used for human skin modeling. M. Ottenio et al. [14] presented a comparative analysis of three hyperelastic material models: Mooney – Rivlin, Ogden, and Neo-Hookean. These models were selected through an experimental process conducted using SIMULIA Abaqus. The Ogden model was chosen as a skin model given its accuracy in describing the mechanical properties of skin. The physical and mechanical properties of the skin are summarized in Table 3.

 

Table 3. Physical and mechanical properties of the skin

Таблица 3. Физико-механические свойства кожи

Indicators	Value
Density, kg/m3	1.1 × 103
Poisson’s ratio, rel. unit	0.495
Shear modulus, MPa	0.0096
Degree index, rel. unit	35.993
Shear relaxation modulus (GI_1), MPa	0.34
Attenuation constant (BETAI_1), rel. unit	0.593
Fracture strain (MAT_ADD_EROSIONEFFEPS), rel. unit	0.7

 

Discrete elements (springs) are usually used to model tendons. For single-degree-of-freedom discrete elements, the mathematical material model “MAT_SPRING_GENERAL_NONLINEAR” is used. This material models the properties of an elastic spring (compression or torsion) with variable stiffness. In addition, the strain rate effects can be considered using a velocity-dependent scaling factor. Therefore, the mathematical material model “S04_MAT_SPRING” with elastic mechanical behavior was used to model the tendons.

In the modeling of ligaments, the use of “flat” elements is an effective approach, as it allows for the consideration of shear deformations while reducing the computational cost compared with the volumetric element method. In scientific publications, two principal material models are presented: the elastic MAT_1 (MAT_ELASTIC), which reproduces an isotropic hypoelastic material, and the elastic–plastic MAT_19 (*MAT_STRAIN_RATE_DEPENDENT_PLASTICITY), which represents an isotropic elastic–plastic material for which the strain rate dependence can be specified [15, 16]. The physical and mechanical properties of ligaments are summarized in Table 4.

 

Table 4. Physical and mechanical properties of the ligaments

Таблица 4. Физико-механические свойства связок

Ligament/tendon	Behavioral mechanics	Density, kg/m3	Young’s modulus, MPa	Poisson’s ratio, rel. unit	Dependence on the strain rate
Tendon of the quadriceps femoris muscle	Elastic	1.1 × 103	800	0.49	–
Patellar ligament	Elastic	1.1 × 103	800	0.49	–
Lateral meniscus	Elastoplastic	103	12	0.33	+
Medial meniscus	Elastoplastic	103	12	0.33	+
Posterior ligament (meniscus)	Elastic	1.1 × 103	53	0.49	–
Transverse ligament (meniscus)	Elastic	1.1 × 103	53	0.49	–
Posterior cruciate ligament	Elastoplastic	1.1 × 103	543	0.49	+
Anterior cruciate ligament	Elastoplastic	1.1 × 103	543	0.49	+
External collateral ligament	Elastoplastic	1.1 × 103	543	0.49	+
Internal collateral ligament	Elastoplastic	1.1 × 103	543	0.49	+
Interosseous membrane	Elastic	1.1 × 103	53	0.49	–
Transverse ligaments (phalanges)	Elastic	500	1000	0.3	–
Ligaments (metatarsus — phalanges)	Elastic	103	50	0.3	–
Long plantar ligaments	Elastic	103	1000	0.3	–
Plantar calcaneonavicular ligament	Elastic	1.1 × 103	53	0.49	–
Deltoid ligament	Elastic	1.1 × 103	401	0.49	–
Anterior intertrochanteric ligament	Elastic	1.1 × 103	53	0.49	–
Posterior intertrochanteric ligament	Elastic	1.1 × 103	53	0.49	–
Medial ligament	Elastic	1.1 × 103	401	0.49	–
Posterior talofibular ligament	Elastic	1.1 × 103	401	0.49	–
Anterior talofibular ligament	Elastic	1.1 × 103	401	0.49	–
Posterior talofibular ligament	Elastic	1.1 × 103	53	0.49	–
Fibulocalcaneal ligament	Elastic	–	401	0.49	–
Talus ligament	Elastic	1.1 × 103	53	0.49	–
Cuboidal navicular cuneiform ligament	Elastic	103	100	0.3	–
Cuboidal navicular cuneiform ligament 2	Elastic	1.1 × 103	53	0.49	–

 

In the final stage of developing a numerical model of the human lower limb, a three-dimensional model was constructed using published data [17]. This enabled the creation of a comprehensive numerical model comprising a tetrahedral finite-element mesh, which incorporates specific geometric parameters (Fig. 2).

 

Fig. 2. Geometric dimensions of the calculated model of the human lower limb: a — dimensional parameters of the 3D model; b — dimensional parameters of the bone skeleton; c — parameters of the finite-element model

 

To test the model, four identical experiments were conducted. These included the recording of images and videos of the human lower limb fragments, which were then analyzed to determine the motion parameters (Fig. 3). The results demonstrated a high degree of correlation between the numerical model and the full-scale experiment.

 

Fig. 3. Modeling of the undermining of the lower limb through the protective structure: a — changing the contour of the lower limb in time; b — moving the “toe” of the lower limb when detonating explosives weighing 75 g

 

During the full-scale testing phase, two accelerometer sensors were installed on the studied object. These were positioned in the lower (H) and upper (B) thirds of the tibia. The sensor readings are presented as graphs in Figure 4.

 

Fig. 4. Accelerations of the lower limb during the detonation of explosives weighing 75 g obtained during field tests

 

In the numerical modeling phase, the calculated acceleration indices were obtained for sensor B with acceleration amplitude of 1450 g and duration of 0.39 ms, which correlates closely with the data obtained during the full-scale tests (Fig. 5). The results of the comparison of lower limb fractures are summarized in Table 5.

 

Fig. 5. Accelerations of the lower limb during the detonation of explosives weighing 75 g in numerical simulations

 

Table 5. Verification of the destruction of the lower limb

Таблица 5. Верификация разрушений нижней конечности

Bone	Test result, %
	In situ	Virtual
Calcaneum	Fracture 100	Comminuted fracture
Talus	Fracture 75, crack 25	Fracture
Cuboid	Crack: probability 25	No
Navicular	No	No
Lateral	Defect: probability 25	No
Intermediate	No	No
Medial	No	No
Fibula	Fracture: probability 50	Fracture
Tibia	Fracture: probability 100	Fracture
Patella	No	No
Femur	No	No

 

In principle, all lower limb fractures can be divided into those with and without displacement of bone fragments. Displacement of bone fragments in a finite-element model is defined by the detachment of its elements, resulting in a defect. Therefore, the injuries shown in the calculation model should be interpreted as follows:

• Nondisplaced fractures with the removal of a few elements or an ordered series of elements (nondisplaced fractures).
• Displaced fractures/slip fractures/bone fractures (compound fractures) with disordered removal of multiple elements.

In general, the characteristics of injuries to the lower limb bones based on numerical modeling on the finite-element model coincided by 90% with that of injuries based on the field experiment.

CONCLUSIONS

Our analysis of available scientific data describing the physical and mechanical properties of biological materials and behavior of human lower limb tissues under shock-wave (explosive) impact allowed us to create an original finite-element model of the human lower limb with tuned interaction of its main constituent anatomical structures (skin, muscles, tendons, ligaments, and bone).

The proposed program for calculations and output of the main results, in which an algorithm for processing of the obtained graphical data in image format was implemented to obtain tolerance curves, allowed for a series of numerical tests simulating contact explosion of the developed lower limb model through the protective structure. The results of the calculations yielded the tolerance curves of pressures and accelerations. In addition, animations were created to illustrate the behavior of the anatomical structures of the lower limb under impact. The pressure field propagation within these structures was also visualized.

Virtual testing may be employed in the future to address applied issues. It can be used not only to assess the effectiveness of various design solutions for lower limb protection devices but also to investigate scientific problems related to forensic medical examination and military field surgery. This includes the study of features of blast trauma.

Therefore, the mathematical modeling of the destruction of biomaterials from explosive and shock impact will contribute to a reduction in the time and costs associated with the production of new samples of protective products developed for national defense.

ADDITIONAL INFORMATION

Authors’ contribution. Thereby, all authors made a substantial contribution to the conception of the study, acquisition, analysis, interpretation of data for the work, drafting and revising the article, final approval of the version to be published and agree to be accountable for all aspects of the study.

The contribution of each author. A.V. Denisov — development of the general concept, study design, literature review, data analysis, article writing; S.V. Matveykin — development of the general concept, study design, data analysis; S.V. Zaikin — statistical processing and analysis of data; A.V. Anisin — data analysis; S.N. Vasilyeva — literature review, article writing; E.A. Selivanov — collection and processing of materials, conducting experimental research.

Competing interests. The authors declare that they have no competing interests.

Funding source. This study was not supported by any external sources of funding.

About the authors

Alexey V. Denisov

Kirov Military Medical Academy

Author for correspondence.
Email: vmeda-nio@mil.ru
ORCID iD: 0000-0002-8846-973X
SPIN-code: 6969-0759

MD, Cand. Sci. (Med.)

Russian Federation, Saint Petersburg

Sergey V. Matveikin

Military Engineering Order of Kutuzov Academy named after Hero of the Soviet Union Lieutenant General of Engineering Troops D.M.Karbyshev

Email: sv-matv@bk.ru
ORCID iD: 0009-0002-9546-8425
SPIN-code: 6269-0498

MD, Dr. Sci. (Tech.)

Russian Federation, Krasnogorsk

Sergey V. Zaikin

Central Research Institute of Special Mechanical Engineering

Email: Sv.zaikin@mail.ru
ORCID iD: 0009-0002-9749-6665
SPIN-code: 7428-5580

MD, Dr. Sci. (Tech.)

Russian Federation, Khotkovo

Alexey V. Anisin

Kirov Military Medical Academy

Email: vmeda-nio@mil.ru
ORCID iD: 0000-0003-4555-953X
SPIN-code: 1213-3797

MD, Cand. Sci. (Med.)

Russian Federation, Saint Petersburg

Svetlana N. Vasilyeva

Kirov Military Medical Academy; Special Materials Corporation

Email: vmeda-nio@mil.ru
ORCID iD: 0009-0003-9731-6027
SPIN-code: 1276-3137

engineer

Russian Federation, Saint Petersburg; Saint Petersburg

Evgeny A. Selivanov

111th Main State Center for Forensic Medical and Forensic Examinations

Email: Selivanove@yandex.ru
ORCID iD: 0000-0001-8791-3707
SPIN-code: 4458-6793

forensic medical expert

Russian Federation, Saint Petersburg

References

Supplementary files