Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Original article    177 
ABSTRACT 
Athletes who are engaged in rowing are subject to a 
number of occupational health problems that require not 
only long-term rehabilitation but also long-term recovery. 
The purpose of the article is to highlight the rehabilitation 
of rowing athletes during the formation of respiratory 
rhythms in course of bone tissue prosthetics. The aspect of 
the restoration of disturbed breathing rhythms and the 
activity of the cardiovascular system is of particular 
relevance. Since disorders of the cardiovascular system 
usually take place when associated physical damage 
occurs, the issue of forming a model of medical 
intervention during operations and rehabilitation activities 
is of primary importance. The main research method was 
the experimental method. With its help, experimental 
measurements and analysis of the obtained results were 
carried out. The novelty of the study is determined by the 
fact that the formation of a system of rehabilitation 
measures should be determined by the requirements for the 
achievement of previously shown results. The authors 
determine the structure of the model, its practical 
application and the formation of a graphic model on its 
basis that will be used when replacing parts of the skeletal 
support. It was determined that this allows to form not only 
stable functioning support, but also the respiratory 
systems, and thus lay the foundation for the athlete’s 
further physical rehabilitation for practical activities and 
return to sports. The practical significance of the subject 
matter is determined by the need to form adaptive 
potential, which will allow athletes to maintain positive 
dynamics of performances in the future at the level of 
international competitions. 
Keywords: restoration, cardiovascular system, fracture, 
osteosynthesis, treatment 
1Department of the Physical Training, Beijing Sport 
University, Beijing, China 
2Department of the Olympic Sport History and Theory, 
National University of Ukraine on Physical Education 
and Sport, Kyiv, Ukraine 
3Department of the High Performance Sport, Beijing 
Sport University, Beijing, China 
 
IZVLEČEK 
Športniki, ki se ukvarjajo z veslanjem, so podvrženi 
številnim poklicnim zdravstvenim težavam, ki poleg 
dolgotrajne rehabilitacije zahtevajo tudi dolgotrajno 
okrevanje. Namen članka je osvetliti rehabilitacijo 
veslačev med oblikovanjem dihalnih ritmov pri protetiki 
kostnega tkiva. Vidik ponovne vzpostavitve motenega 
ritma dihanja in delovanja srčno-žilnega sistema je še 
posebej pomemben. Ker do motenj srčno-žilnega sistema 
običajno pride ob pojavu pridruženih telesnih poškodb, je 
vprašanje oblikovanja modela medicinskega posega med 
operacijami in rehabilitacijskimi aktivnostmi bistvenega 
pomena. Glavna raziskovalna metoda je bila 
eksperimentalna metoda. Z njegovo pomočjo so bile 
izvedene eksperimentalne meritve in analiza dobljenih 
rezultatov. Novost študije določa dejstvo, da je treba 
oblikovanje sistema rehabilitacijskih ukrepov določiti z 
zahtevami za doseganje predhodno prikazanih rezultatov. 
Avtorji določijo zgradbo modela, njegovo praktično 
uporabo in na njegovi osnovi oblikovanje grafičnega 
modela, ki bo uporabljen pri zamenjavi delov skeletnega 
nosilca. Ugotovljeno je bilo, da to omogoča oblikovanje ne 
le stabilne podpore za delovanje, temveč tudi dihalnih 
sistemov in s tem postavlja temelje za nadaljnjo fizično 
rehabilitacijo športnika za praktične aktivnosti in vrnitev v 
šport. Praktični pomen predmeta je določen s potrebo po 
oblikovanju adaptivnega potenciala, ki bo športnikom 
omogočil ohranjanje pozitivne dinamike uspešnosti v 
prihodnosti na ravni mednarodnih tekmovanj. 
Ključne besede: obnova, kardiovaskularni sistem, zlom, 
osteosinteza, zdravljenje 
Corresponding author*: Yan Geng,  
Department of the Olympic Sport History and Theory, 
National University of Ukraine on Physical Education and 
Sport, Kyiv, Ukraine 
E-mail: yageng350@gmail.com 
https://doi.org/10.52165/kinsi.29.1.177-202 
Yanbu Wang 1 
Yan Geng 2, * 
Wei Wang 2 
Chao Wang 3 
Qi Zhang 2 
 
REHABILITATION OF ROWING ATHLETES 
DURING THE FORMATION OF RESPIRATORY 
RHYTHMS IN COURSE OF BONE TISSUE 
PROSTHETICS  
REHABILITACIJA VESLAČEV MED 
NASTAJANJEM DIHALNIH RITMOV PRI 
PROTETIKI KOSTNEGA TKIVA 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   178 
INTRODUCTION 
Plate osteosynthesis holds a significant place in the surgical treatment of injuries and diseases 
of the musculoskeletal system to achieve high-quality results in rehabilitation after diseases and 
surgical interventions on the cardiovascular system (Gart and Wiedrich, 2017). An ideal fixator 
is one that, with minimal additional trauma to the soft and bone tissues, preserves the 
immobility of the fragments, and provides the function and supportability of the damaged limb 
throughout the entire treatment period (Malempati et al., 2015). Fixators should be produced 
from biologically inert materials (Bjerke et al., 2014). 
Fixator designs must be biomechanically justified (Hilliard et al., 2017). In the presence of 
alternating stresses, the destruction of material occurs at stresses significantly lower than the 
ultimate stress limit under a single static loading (Riley and Callahan, 2019). There are 
situations when the fixator does not withstand a long dynamic load (Bakrac, 2003). Stable plate 
osteosynthesis is an effective treatment for fractures (Hawson, 2011). A feature of the method 
is the creation of a large safety margin, which allows for early movements, thereby combining 
periods of consolidation and restoration of limb function (Hart et al., 1998). The scale of modern 
injuries has transformed this problem from medical to social (Arvinen-Barrow et al., 2014). 
Injuries in developed countries are the third leading cause of death and disability (Rudenko et 
al., 2015). Damage to the musculoskeletal system is detrimental to human health, which leads 
to significant material costs (Carlson, 2009). Annually, about 10% of the population receives 
bone fractures, and 0.5-16% of these patients experience non-union (Hagerman et al., 1995). 
An analysis of modern domestic and foreign literature indicates that there is no universal 
treatment method (Hagerman et al., 1995). Bone osteosynthesis is performed using structures 
such as wire, and metal rings (Arvinen-Barrow et al., 2015). This method has insufficiently 
strong fixation and does not find an independent application, but can be used together with 
other methods (Karpiel et al., 2021). 
The design of the plates provides for the introduction of screws in one area (Grindem et al., 
2018). Multi-fragmentary fractures are less sensitive to displacements; the absolute 
displacement separates into many insignificant relative displacements between individual 
fragments (Bejar et al., 2019). The situation with simple fractures is more critical (Wilk et al., 
2016). For simple diaphyseal fractures, bone osteosynthesis should be used, with anatomical 
repositioning and interfragmentary compression (Lu and Hsu, 2013). Today, there are many 
osteosynthesis systems, but all of them have certain disadvantages (Abassi et al., 2019). Given 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   179 
these circumstances, it is advisable to continue work on improving osteosynthesis systems 
(Wasik et al., 2015). The purpose of the article is to highlight the rehabilitation of rowing 
athletes during the formation of respiratory rhythms in course of bone tissue prosthetics. 
 
METHODS 
If the components of the displacement at the most dangerous point are known (the projections 
of the total vector on the coordinate axis (Fig. 1), then the condition will be written as: 
( ) ( ) ( )  
2 2 2
Λ Λ Λ Λ | Λi i imax x y z max= + + 
                    (1) 
Figure 1. Directions of coordinate axes and components of the main vector of cell 
displacements. 
 
The components of the displacements in equation (1) are determined mainly by the influence 
of such forces: 1
P
 – force acting along the longitudinal axis of the bone (for the lower 
extremities – this is mainly the strength of a person’s weight); 2
P
 – lateral force (muscle 
strength, limb weight in the supine position of the injured); 3
P
 – force that affects the foot when 
walking (the action of muscles during rotational movements of the body during walking, the 
weight of the foot in the supine position of the victim) and causes torsion deformities of the 
limb bones. 
Due to the action of these forces, bending moments and torsion moment appear: 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   180 
k k kM P e=       (2) 
where k
e
 – the distance from the line of action of the corresponding force to the place of 
fixation of the fracture; 1,2,3k =  – designations of forces and corresponding moments.  
Use of forces 1 2 3
, ,P P P
 and corresponding bending moments 1 2 3
, ,M M M
, acting in the 
fracture area. Presented: 1
e
 – distance from the line of action of force 1
P
 to the means of 
fixation; 2
e
 – distance from the line of action of force 2
P
 in the middle of the fracture; 3
e
 – 
distance from the line of action of force 3
P
 to the means of fixation. Thus, each component of 
the main linear displacement vector Λ
i
 depends on three concentrated forces and three 
moments of force. Let us denote: 
( )1 2 3Λ Λ ,Λ ,Λi i i ik
 – absolute displacement of point “i”, 
caused by forces k
P
; 
( )Λ Λ ,Λ ,Λi i i ij X Y Z
 – projections of the full displacement of a point “i” 
on axis “j”; 
Λijk  – projection on the axis “j” of absolute displacements Λ
i
k . 
It should be noted that most osteosynthesis systems are located on the lateral (external to the 
centre of the body) or on the medial (internal) side of the bone. The axis of the longitudinal 
force i
P
 passes through the frontal (the one that divides the body into the front and back) area 
of the bone. In the future will be considered the case when the line of action of the most 
dangerous force 2
P
 will also lie in the frontal plane. Therefore, displacements arising due to 
these forces and corresponding moments must be added algebraically. 
It should also be noted that the direction of action of external forces is unknown in advance. 
Therefore, in the future will be considered the most dangerous arrangement of forces 1 2 3
, ,P P P
, when the directions of the projections of displacement from all these forces coincide. In this 
regard, when constructing the criteria of functional reliability, it should be assumed that all 
components of the displacement vector have the same sign. Let us denote by   reduced 
displacements as the ratio of the absolute values of displacements to the acting forces, and by 
  displacements as the ratio of absolute values of displacements to current moments: 
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   181 
Λ Λ
;
i i
jk jki i
jk jk
k kP M
 = =
      (3) 
Λ Λ
;
i i
jk jki i
jk jk
k kP M
 = =
                                                            (4) 
The given displacements are determined experimentally by measuring displacements under the 
action of the corresponding forces and moments, for example, by digital recording using a 
camera. It should be noted that to determine the displacement 
i
jk  absolute displacement must 
be measured in the absence (or at least minimising) of the bending moment. This is done by 
applying a load to the system near the fixing means. When determining the given displacements 
i
jk  from general absolute displacements, it is necessary to remove displacements arising under 
the action of only concentrated forces.  
If the given displacements are known, then the components of the absolute displacements can 
be determined for any values of the components of the force vector k
P
 and corresponding 
components of the moment vector k
M
. Projection of general displacement Λ
i
 the entire X  
is defined as follows: 
1 1 2 2 3 3 1 1 2 2 3 3Λ λ λ
i i i i i i i
x x x x x x xP P P M M M   = + + + + +      (5) 
Projection of cell displacement Λ
i
 on axis Y  and Z  is defined similarly: 
( )
3
1
Λi i ij k jk jk k
k
P e 
=
= +
     (6) 
The condition of functional reliability using the maximum linear mutual displacement as a 
criterion is written in compressed form: 
( )  
1/2
2
3 3
1 1
Λ | Λi imax k jk jk k max
j k
P e 
= =
   
= +   
   
 
   (7) 
Denote (Fig.2 a): G – angle of rotation of the line that connects two opposite fracture points 
,M L ; 1 1,M L  – the position of two opposite fracture points to the application of the load; 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   182 
2 2,M L  – position of two opposite fracture points after application of load; L  – position of 
point L  after application of the load without taking into account the rotation of the fracture 
(parallel movement of the segment 1 1
,M L
); 1 1 2 2 2ML
S M L M L M L= = = 
 – distance between points ,M L  in different 
positions; 
Λ ,Λ ,ΛM M MX Y Z  – displacement projection of point M  on the coordinate axes; 
Λ ,Λ ,ΛL L LX Y Z  – displacement projection of point L  on the coordinate axes; 2L L

 – 
displacement of point L  only due to the mutual rotation of the parts of the fracture. 
 1 1 2 2 2ML
S M L M L M L= = = 
.                                     (8) 
Figure 2. Detailed (a) and simplified (b) diagram of movements of fracture points and rotation 
angles of fracture parts. 
 
  
a      b 
From the triangle 2 2
L L M
 (Fig. 2 b): 
 
( ) ( ) ( )
2 2 2
2 Λ Λ Λ Λ Λ Λ
L M L M L M
x x y y z zL L = − + − + −
                   (9) 
where, given the designations shown in Fig. 2: 
 2 2 2
;ML MLL M S L M S= =                                    (10) 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   183 
 
Explanations for formula (10) are given in Fig. 3, where for illustrative purposes are shown the 
projections of vectors 
1
1 2 1 2 2, ,M M L L L L  and projections of the corresponding displacements 
on the plane XoY . 
Figure 3. Projections of displacement vectors of fracture points on a plane XoY . 
 
 
In the final form, the angle of rotation of the parts of the fracture is defined as: 
𝐺 = 𝑎𝑟𝑐 𝑐𝑜𝑠 (1 −
|𝐿′𝐿2|
2
2𝑆𝑀𝐿
2 )                                                    (11) 
Functional reliability condition (2), using the maximum mutual angle as a criterion of rotation 
of the fracture parts (11), looks like this: 
𝐺𝑚𝑎𝑥 = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 ∑ [∑ 𝑃𝑘
3
𝑘=1 ((𝜆𝑗𝑘
𝐿 − 𝜆𝑗𝑘
𝑀 ) + (?̄?𝑗𝑘
𝐿 − ?̄?𝑗𝑘
𝑀 )𝑒𝑘)]
3
𝑗=1 } |𝑚𝑎𝑥 ≤ [𝐺] (12) 
 
RESULTS AND DISCUSSION 
The process of rehabilitation of the injured is accompanied by exposure of limbs to cyclically 
acting loads (walking, exercise). In this case, the mutual displacements of the opposite parts of 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   184 
the fracture can significantly increase, and the condition of functional reliability should take 
into account the deformations (displacements) that accumulate in the osteosynthesis system 
when cyclic loads are applied: 
( ) ( ) ( )( ) ( ) ( )( )( )  
1/2
2
3 3
0 0
1 1
Λ | Λi i i imax k k maxjk jk N jk jk N
j k
N P e   
= =
   
= + + +   
   
 
 (13) 
In (13), it was taken into account that the components of the point displacement consist of 
instantaneous displacement of cells ( )
0
Λi
jk
 and displacement of cells ( )
Λi
jk N
, arising during the 
term of N  load cycles (Figs. 3, 4): 
( ) ( )0
Λ Λ Λi i ijk jk jk N= +
      (14) 
Figure 4. Diagram of development of displacements of adjacent points and accepted notation. 
 
 
This can be expressed through reduced displacements arising due to the action of forces 
( )ijk
 
and moments 
( )ijk
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   185 
( ) ( )0
i i i
jk jk jk N
  = +
      (15) 
( ) ( )0
i i i
jk jk jk N
  = +
      (16) 
where, as in (13, 14), the index “0” means the instantly elastic component, the index N  – 
accumulated displacements: 
( )
( )
( )
( )0
0
Λ Λ
;
i i
jk jk Ni i
jk jk N
k kP P
 = =
     (17) 
( )
( )
( )
( )0
0
Λ Λ
;
i i
jk jk Ni i
jk jk N
k kM M
 = =
     (18) 
The most interesting from a practical point of view are the displacements of the fracture points, 
which accumulate under the action of a significant number of load cycles. In this case, in 
formula (13), displacements are given; it is advisable to replace the displacements that occurred 
on the N-th cycle with the limiting ones (under the influence of the maximum number of cycles): 
( ) ( )( ) ( ) ( )( )( )  
1/2
2
3 3
0 0
1 1
Λ | Λi i i imax k k maxjk jk gp jk jk gp
j k
P e   
= =
   
= + + +   
   
 
 (19) 
where indicated: 
( )
( )
( )
( )
Λ Λ
;
i i
jk gp jk gpi i
jk gp jk gp
k kP M
 = =
    (20) 
Given that: 
                        
( ) ( )( ) ( )0 0 0Λi i ik kjk jk jkP e + =
    (21) 
( ) ( )( ) ( )Λ Λ Λi i ik kjk gp jk gp jk gpP e+ =
    (22) 
  
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   186 
equation (22) can be written in this way: 
( )( )  
3 3
0
1 1
Λ | Λi i imax k jk jk k maxjk
j k
P k e 
= =
 
= +  
 
 
   (23) 
where: 
( )
( )0
Λ
1
Λ
i
jk gpi
jk i
jk
k = +
     (24) 
a ratio of cyclic creep, which shows an increase in the total displacement that occurs under 
cyclic loading, compared with the movement of the load at the single load.  
Under the action of cyclic loads, the angles of mutual rotation of the parts of the fracture can 
increase significantly. These changes compared to angles arising under the action of one-time 
loads can be taken into account by adding to instantly elastic displacements 
( ) ( )( )0 0Λ ,ΛL Mjk jk
 of 
viscoelastic origin (displacement  
ΛL
jk N
 and ( )
ΛM
jk N
, arising for N  load cycles) of points 
,L M  respectively. As in the case of linear displacements, this can be expressed in terms of 
displacements that arise due to the action of forces 
( ),L Mjk jk 
 and moments 
( ),L Mjk jk 
: 
 
𝐺𝑚𝑎𝑥(𝑁) = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 ∑
3
𝑗=1 [∑
3
𝑘=1 𝑃𝑘 (
(𝜆𝑗𝑘(0)
𝐿 + 𝜆𝑗𝑘(𝑁)
𝐿 − 𝜆𝑗𝑘(0)
𝑀 − 𝜆𝑗𝑘(𝑁)
𝑀 ) +
+(?̄?𝑗𝑘(0)
𝐿 + ?̄?𝑗𝑘(𝑁)
𝐿 − ?̄?𝑗𝑘(0)
𝑀 − ?̄?𝑗𝑘(𝑁)
𝑀 )𝑒𝑘
)]
2
|𝑚𝑎𝑥 ≤ [𝐺]}(25) 
 
In the equation (25), the expressions for the displacement components (9) – (11) are applied 
with the replacement of the notation of the point “i” by the points ,L M= ? = ? : 
( ) ( ) ( ) ( )0 0
Λ Λ Λ ;Λ Λ Λ ;L L L M M Mjk ikjk jk N jk jk N= + = +
                (26) 
( ) ( ) ( ) ( )0 0
λ λ λ ;λ λ λ ;L L L M M Mjk ikjk jk N jk jk N= + = +
    (27) 
( ) ( ) ( ) ( )0 0
λ λ λ ;λ λ λ ;L L L M M Mjk ikjk jk N jk jk N= + = +
    (28) 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   187 
Index 0= ? , as in (10) – (17), means the instantly elastic component, index N= ?  (26), 
(28) – accumulated displacements: 
( )
( )
( )
( )0 0
0 0
Λ Λ
;
L M
jk jkL M
jk jk
k kP P
 = =
     (29) 
 
( )
( )
( )
( )0 0
0 0
Λ Λ
;
L M
jk jkL M
jk jk
k kP P
 = =
.                                                (30) 
( )
( )
( )
( )
Λ Λ
;
L M
jk N jk NL M
jk N jk N
k kP P
 = =
     (31) 
 
( )
( )
( )
( )
Λ Λ
;
L M
jk N jk NL M
jk N jk N
k kP P
 = =
.                                                 (32) 
( )
( )
( )
( )0 0
0 0
Λ Λ
;
L M
jk jkL M
jk jk
k kM M
 = =
     (33) 
 
( )
( )
( )
( )0 0
0 0
Λ Λ
;
L M
jk jkL M
jk jk
k kM M
 = =
                                                (34) 
( )
( )
( )
( )
Λ Λ
;
L M
jk N jk NL M
jk N jk N
k kM M
 = =
     (35) 
 
( )
( )
( )
( )
Λ Λ
;
L M
jk N jk NL M
jk N jk N
k kM M
 = =
.                                               (36) 
As in condition (20) using linear displacement of fracture points, displacements arising on N
-th cycle replace the limit (under the action of a significant number of cycles) shows the 
displacement: 
𝐺𝑚𝑎𝑥 = 𝑎𝑟𝑘 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 ∑
3
𝑗=1 [∑
3
𝑘=1 𝑃𝑘 ((𝜆𝑗𝑘(0)
𝐿 + 𝜆𝑗𝑘(𝑔𝑝)
𝐿 − 𝜆𝑗𝑘(0)
𝑀 − 𝜆𝑗𝑘(𝑔𝑝)
𝑀 ) +
+(?̄?𝑗𝑘(0)
𝐿 + ?̄?𝑗𝑘(𝑔𝑝)
𝐿 − ?̄?𝑗𝑘(0)
𝑀 − ?̄?𝑗𝑘(𝑔𝑝)
𝑀 )𝑒𝑘]
2}|𝑚𝑎𝑥 ≤ [𝐺]
 (37) 
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   188 
where indicated: 
( )
( )
( )
( )
( )
( )
( )
( )
Λ Λ Λ Λ
; ; ;
L M L M
jk gp jk gp jk gp jk gpL M L M
jk gk jk gp jk gp jk gp
k k k kP P M M
   = = = =
  (38) 
Using the coefficient of influence of cyclic creep (24), condition (37) can be written in this 
form: 
𝐺𝑚𝑎𝑥 = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 ∑
3
𝑗=1 [∑
3
𝑘=1 𝑃𝑘 (𝑘𝑗𝑘
𝐿 (𝜆𝑗𝑘(0)
𝐿 + ?̄?𝑗𝑘(0)
𝐿 𝑒𝑘) − 𝑘𝑗𝑘
𝑀(𝜆𝑗𝑘(0)
𝑀 +
?̄?𝑗𝑘(0)
𝑀 𝑒𝑘))]
2
} ≤ [𝐺]          (39) 
A number of experimental studies have shown that the curve connecting the upper points of the 
cyclic creep diagram can be approximated with a sufficient degree of accuracy by the sum of 
exponential functions. As preliminary calculations indicated, the number of members of the 
sum is sufficient to be limited to three so that the approximation error does not exceed the 
instrumental error.  
This is actually the use of the generalised Kelvin-Voigt model by formally replacing the force 
action time with the number of load cycles. Represent the displacement in formula (24) in this 
way: 
 
( ) ( ) ( )0
1
Λ Λ Λ 1
n
i i i
jk jk jk m i
m m
N
N exp
N=
  
= + − −  
  

\             (40) 
where ( )
0
Λi
jk
 – displacements that can be determined under static loads ( )
Λ ,i imjk m N
 – 
coefficients that can be determined by cyclic creep curves; N  – number of load cycles; m  – 
exponent number; n  – total number of exponents.  
In this case, there should be a condition: 
( ) ( )
1
Λ Λ
n
i i
jk m jk gp
m=
=
     (41) 
Dividing both sides (41) into k
P
, the reduced movement of the cell was got: 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   189 
( ) ( ) ( )0
1
1
n
i i i
jk jk jk m i
m m
N
N exp
N
  
=
  
= + − −  
  

   (42) 
Similarly, for reduced displacement 
( )ijk N : 
( ) ( ) ( )0
1
1
n
i i i
jk jk jk m i
m m
N
N exp
N
  
=
  
= + − −  
  

   (43) 
Substituting (43) into (10), the final form of the functional reliability condition using the 
criterion of maximum mutual displacements of fracture points under cyclic loads was obtained: 
 
( )
( ) ( ) ( )
( ) ( ) ( )
 
2
03 3
1
1 1
0
1
1 /
Λ | Λ
1 /
n
i i i
mjk jk m
m
max k max
n
j k i i i
m kjk jk m
m
exp N N
N P
exp N N e
 
 
=
= =
=
   
 + − − +    
   = 
   
   + + − −       

 

(44) 
Substituting (43) into (14), the final form of the criterion for the maximum angles of mutual 
rotation of the fracture parts under cyclic loads was obtained: 
𝐺𝑚𝑎𝑥(𝑁) = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 ∑ [∑ 𝑃𝑘 ((
𝜆𝑗𝑘(0)
𝐿 + ∑ 𝜆𝑗𝑘(𝑚)
𝐿 [1 − 𝑒𝑥𝑝(−𝑁/𝑁𝑚
𝐿 )]𝑛𝑚=1 −
−𝜆𝑗𝑘(0)
𝑀 − ∑ 𝜆𝑗𝑘(𝑚)
𝑀 [1 − 𝑒𝑥𝑝(−𝑁/𝑁𝑚
𝑀)]𝑛𝑚=1
) +3𝑘=1
3
𝑗=1
(
?̄?𝑗𝑘(0)
𝐿 + ∑ ?̄?𝑗𝑘(𝑚)
𝐿 [1 − 𝑒𝑥𝑝(−𝑁/?̄?𝑚
𝐿 )]𝑛𝑚=1 −
−?̄?𝑗𝑘(0)
𝑀 − ∑ ?̄?𝑗𝑘(𝑚)
𝑀 [1 − 𝑒𝑥𝑝(−𝑁/?̄?𝑚
𝑀)]𝑛𝑚=1
))]} |𝑚𝑎𝑥 ≤ [𝐺]     (45) 
As an example, it can be considered the application of the above criteria to assess the 
permissible loads of the system “tibia with a fracture-fixation plate”. The following fracture 
points were considered: point M (medial, point near the fixation of the plates, except for plate 
1L) and point L (lateral, point farthest from the plates). The external forces were applied in such 
a way that the directions of the projections of the displacement vectors of the point L, which 
arise due to individual forces and moments of forces, coincide (the most dangerous case). 
According to formula (3, 4), for an accurate assessment of the functional reliability of fracture 
fixation and determination of permissible loads, it is important to know 9 values of reduced 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   190 
displacements 
i
jk  and 9 values of
i
jk . Taking into account the displacements arising from the 
long-term impact of loads (18), the number of components of the given displacements increases 
to 36. Obviously, determining all the components, even using the digital method of recording 
displacements, is a rather complicated task. However, it turned out that the number of reduced 
displacements that affect the calculation results can be much smaller.  
When lowering the number of reduced displacements that affect the calculation results, use the 
example of osteosynthesis systems in which bone plates are used to fix fractures. The given 
displacements were determined by the results of experiments for osseous medial plates with 
angular stability (hereinafter 2M), medial blocked plates (3M) and X-shaped medial blocked 
plates for open corrective osteotomy (4M). The considered plates showed the best results when 
tested at single loads. 
The most dangerous fracture region is the one where under the action of loads the maximum 
mutual displacements of adjacent points occur. It is obvious (and this is confirmed 
experimentally) that in most cases the maximum displacements will occur at the point of 
fracture that is farthest from the fixation means (plate or rod apparatus of external fixation). 
Therefore, as the point “i” in the calculations of maximum displacements, it is necessary to take 
the point farthest from the latch. In the future, the movement would be denoted by points, unless 
it is necessary. The main contribution to the main displacement vector is made by the 
components 1 2 1 2 3
Λ ,Λ ,Λ ,Λ ,Λi i i i iX X Z Z Y . 
The first four components are longitudinal (along the X axis), and transverse (along the Z axis) 
displacement due to two forces 1 2
,P P
. The fifth component is displacement in a plane 
perpendicular to the longitudinal axis of the bone, due to the moment of torsion 3
M
. Table 1 
shows the results of experiments conducted with the tibia with simulated fracture-fixation plate 
systems under load 1 2 3
, ,P P P
. 
 
  
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   191 
Table 1. Geometrical characteristics, displacements, and rotation angles of the tibia with 
fracture systems – fixation plate 2 under external loads. 
Plate 
Impact force 1
P
 Fold force 2
P
 Torsion force 3
P
 
1e  1 1Λ /
L
X P  1 1Λ /
M
X P  1 1Λ /
M
Z P  2
e
 2 2
Λ /LX P  2 2Λ /
L
Z P  3
e
 ML
S
 3
M
 
𝐺𝑌𝑍(3) 
mm 103 N⋅mm mm 103 N⋅mm mm N⋅mm degree 
1L 25.3 19.36 5.40 1.25 18.3 9.90 33.0 63.2 29.6 632 1.346 
1M 15.0 5.94 2.12 0.87 21.6 5.62 6.60 46.1 39.5 461 0.668 
2M 11.0 8.07 3.03 1.16 21.5 9.90 4.40 59.5 34.8 595 0.688 
3M 17.5 12.25 4.75 2.36 20.6 8.80 3.42 56.9 31.6 569 1.252 
4M 21.1 4.88 4.88 0.50 35.8 6.78 3.62 51.5 38.1 515 0.247 
 
Table 2. Displacements of the fracture point L, most remote from the place of fixation by the 
plate. 
Plate 
1
L
x  1
L
x  1
L
z  2
L
z  2
L
x  2
L
z  3
L
y  
103 
N⋅mm 
103 mm/(N⋅mm) 
103 
N⋅mm 
103 mm/(N⋅mm) 
1L 5.40 0.552 0.049 32.1 0.541 0.049 1.100 
1M 2.12 0.255 0.058 5.35 0.260 0.058 0.999 
2M 3.03 0.458 0.105 2.13 0.460 0.105 0.702 
3M 4.75 0.429 0.134 0.64 0.427 0.134 1.213 
4M 0.80 0.193 0.023 2.77 0.189 0.023 0.319 
 
An analysis of the results of the action of these forces separately and methods for calculating 
the reduced displacements from the results of measurements of the absolute displacements of 
various fracture points were presented. Reduced point movements M  and L  in the direction 
Х under the force 1
P
. Displacement 1
ΛMX  is the result of longitudinal force 1
P
, which causes 
the displacement of part of the fracture along the axis X , regardless of bending moment 1 1 1M Pe=
. It is equal to displacement 1
ΛLX  of point L  without taking into account the moment 1M  
(parallel movement of the segment ML  without taking into account rotation in the plane XZ
). Therefore, 
 1 1 1
M Pe=
                                                           (46) 
1
1 1
1
ΛML M x
x x
P
 = =
     (47) 
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   192 
Reduced displacement of point L  in direction Х under the influence of the moment 1M . This 
displacement is equal 1 1
Λ ΛL MX X− , and the corresponding reduced displacement: 
1 1
1
1 1
Λ ΛL ML x x
x
Pe

−
=
     (48) 
Reduced displacement of points L  and M  in direction Z  under the influence of the moment 
1M . Because under the action of force 1P  lateral force in the direction Z  is absent, cell 
displacement 1
ΛLZ  of point L  in direction Z  caused only by the influence of the moment 1M
. At small angles of rotation 1 1
Λ ΛL MZ Z= , and the corresponding reduced displacement: 
1 1
1
1 1 1 1
Λ ΛL ML z z
z
Pe Pe
 = =
     (49) 
Displacement of point L  in direction Y  under the influence of force 1P . As the analysis of 
displacements of fracture points, displacements 1
ΛLy , 1Λ
L
z  due to force 1
P
 are dismissively 
small, which means that the quantities 1
ΛLy , 1Λ
L
z  are close to zero. No significant movement 
of fracture points in the direction Y  gives reason to neglect the given displacement 1
L
y . 
Displacement of point L  in direction Z  under the influence of the force 2P . Suppose that in 
the first state acts only force 1
P
. In this state, the transverse force in the direction Z  is absent 
and movement in this direction is caused only by a bending moment 1 1 1
M Pe=
: 
1 1 1 1 1 1Λ
L L L
z z zM Pe = =     (50) 
Suppose that in the second state acts only the force 2
P
. In this state, moving in the direction Z  
caused by the force 2
P
 and bending moment 2 2 2
M P e=
: 
2 2 2 2 2 2 2 2 2 2Λ
L L L L L
z z z x xP M P P e   = + = +     (51) 
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   193 
Displacements from bending moments do not depend on the method of application of these 
moments, that is: 1 2
L L
z z = . 
1
2
1 1
ΛLL z
z
Pe
 =
      (52) 
Substituting (52) into (53): 
2 1 2
2
2 1 1
Λ ΛL LL z z
z
e
P Pe
 = −
     (53) 
Values of relations 2 2
Λ /LZ P  and 1 1Λ /
L
Z P  are presented in Table. 1. 
Displacement of point L  in direction X under the force 2P . The force 2P  causes no shift in 
direction X  and 2Λ 0
M
X = . Displacement 2Λ
L
X  only caused by the moment 2 2 2
M P e=
. In this case, 
the displacement is shown: 
2
2
2 2
ΛLL x
x
P e
 =
      (54) 
Notice, that 1 2
L L
x x =   (Table 2). Deviations range from 0.44% (plate 2M) to 2.07% (plate 
4M), which corresponds to the measurement error.  
Displacement of point L  in direction Y  under the influence of force 3P . It is defined that 
 𝑡𝑔𝐺 = 𝛬𝑌3
𝐿 /𝑆𝑀𝐿.                                                (55) 
Thus, if: 
3
3
3 3
ΛLyL
y
Pe
 =
      (56) 
then in the final form: 
3
3 3
ΓL MLy
S
tg
Pe
 =
     (57) 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   194 
The results of calculations using formulas (52)-(53), (53)-(57) using the data from Table 1 are 
given in Table 2. Displacement of point M  in direction X  under the force 1P . Reduced 
displacement 1
M
x , which corresponds with 1
L
x , is given by the formula (47). Displacement of 
point M  in direction Z  under the effect of bending moments. Displacement of point M  due 
to the bending moment, it does not depend on the method of application of this moment (that 
is, on the direction of the load that causes this moment), but the movement of points L  and M  
in the transverse direction are equal. Therefore 1 2 1 2
M M L L
z z z z   = = = . 
Displacement of point M  in direction Z  under the force 2P . Calculation of reduced 
displacements 2
M
z  is carried out in the same way as 2
L
z : 
2 1 2
2
2 1 1
Λ ΛM MM z z
z
e
P Pe
 = −
     (58) 
Relations 2 2 2 2
Λ / Λ /M LZ ZP P=  and 1 1/
M
Z P  are given in the Table 1. Displacement of point 
M  under the influence of force 3P . The center of rotation of the fracture under the action of 
force 3
P
 coincides with the place of fixation of the fracture by the plate, therefore absolute and 
reduced moving points M due to the action of this force, it is extremely small, which is 
confirmed by torsion experiments. Reduced displacement of point M, calculated by formulas 
(47), (53), placed in the Table 3. 
Table 3. The reduced displacements of the fracture point M close to the place of fixation by the 
plate. 
Plate 1
M
x  1
M
z  2
M
z  2
M
z  
× 103, N⋅mm × 103, mm / (N⋅mm) × 103, N⋅mm × 103, mm / (N⋅mm) 
1L 5.40 0.049 32.1 0.049 
1M 2.12 0.058 5.35 0.058 
2M 3.03 0.105 2.13 0.105 
3M 4.75 0.134 0.64 0.134 
4M 0.80 0.023 2.77 0.023 
 
The matrix representation of the reduced displacements of the points L and M. The reduced 
displacements are conveniently represented in the form of matrices (59) – (62). The negligibly 
small quantities in these matrices are indicated by zeros. 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   195 
 
 
1 1 1 1 1
2 2 2 2 2
3
0 0 0 0 0 0 0
0 0 ; 0 ; 0 0 ; 0 0
0 0 0 0 0 0 0 0 0 0 0
L L L M M
x x z x z
L L L M M
z x z z z
L
y
    
    

       
       
       
       
                                                       (59) 
 
1 1 1 1 1
2 2 2 2 2
3
0 0 0 0 0 0 0
0 0 ; 0 ; 0 0 ; 0 0
0 0 0 0 0 0 0 0 0 0 0
L L L M M
x x z x z
L L L M M
z x z z z
L
y
    
    

       
       
       
       
                                                         (60) 
 
1 1 1 1 1
2 2 2 2 2
3
0 0 0 0 0 0 0
0 0 ; 0 ; 0 0 ; 0 0
0 0 0 0 0 0 0 0 0 0 0
L L L M M
x x z x z
L L L M M
z x z z z
L
y
    
    

       
       
       
       
                                                       (61) 
 
1 1 1 1 1
2 2 2 2 2
3
0 0 0 0 0 0 0
0 0 ; 0 ; 0 0 ; 0 0
0 0 0 0 0 0 0 0 0 0 0
L L L M M
x x z x z
L L L M M
z x z z z
L
y
    
    

       
       
       
       
                                                    (62) 
The rigidity conditions of the systems “fracture – plate” at one-time loads. Given only those 
given displacements 
,i ijk jk  , which are different from zero in matrices (59) – (62), the 
conditions of functional reliability (3, 4), (12) at single loads are significantly simplified: 
( ) ( ) ( ) ( )  
2 2 2
1 1 1 1 2 2 2 2 3 3 1 1 1 2 2 2 2Λ | Λ
L L L L L L L
max x x x y z z zP e P e P e P e P e           = + + + + + +      
 (63) 
𝐺𝑚𝑎𝑥 = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 [(𝑃1?̄?𝑥1
𝐿 𝑒1 + 𝑃2?̄?𝑥2
𝐿 𝑒2)
2
+ (𝑃3?̄?𝑦3
𝐿 𝑒3)
2
]} |𝑚𝑎𝑥 ≤ (𝐺) 
  (64) 
The stiffness conditions for the systems “fracture – plate” under cyclic loads (23), (37) are 
reduced to the form (16), (37) without taking into account small reduced displacements in 
matrices (59) – (62). Note that the displacements of these points in the corresponding directions 
due to creep are also close to zero (they are at the level of instrumental error). To take into 
account the influence of cyclic loads, let`s assume that the ratio between the projections ( )
ΛL
jk gp
 
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   196 
and ( )
0
ΛL
jk
 in coefficient (24), depend little on the direction of the coordinate axis and depend 
only on the direction of the reduced force. In this case, complete displacements arising due to 
force k
P
, can be calculated by introducing the same coefficients in (24): 
( )
( )0
Λ
1
Λ
i
k gpL L L L
xk yk zk k i
k
k k k k= = = = +
    (65) 
The error that may arise due to this assumption will not be significant, since the values of the 
coefficients 
i
kk  are in rather narrow ranges: 1.28 ... 1.31 for a 2M plate; 1.21 ... 1625 for the 
3M plate and 1.29 ... 1.32 for the 4M plate (Table 4) It should be noted that a large value 
i
kk  
for a 4M plate does not indicate a high level of creep of this system; in this system, the cyclic 
creep strains are the smallest. Given these assumptions, the rigidity condition (23), a 
generalisation of condition (63) for the case of cyclic creep for a system with plates, will have 
the form: 
( )
( )
 
2 2
1 1 1 1 1 2 2 2 2 3 3 3 3
2
1 1 1 1 2 2 2 2 2
Λ | Λ
L L L L L L L
x x x y
max max
L L L L L
z z z
Pk e P k e P k e
Pk e P k e
   
  
   + + + +  
= 
 + + +
   (66) 
Similarly, given that 1 1 1 2 2 2 2 2
; ; ; .L M L M L M L Mx x z z z z z x       = = = = , condition (37) and 
generalised condition (64) for the case of cyclic creep for a system with plates will have the 
form: 
𝐺𝑚𝑎𝑥 = 𝑎𝑟𝑐 𝑐𝑜𝑠 {1 −
1
2𝑆𝑀𝐿
2 [(𝑃1𝑘1
𝐿?̄?𝑥1
𝐿 𝑒1 + 𝑃2𝑘2
𝐿?̄?𝑥2
𝐿 𝑒2)
2
+ (𝑃3𝑘3
𝐿?̄?𝑦3
𝐿 𝑒3)
2
]} ≤ [𝐺]               (67) 
 
  
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   197 
Table 4. Coefficient values 
, , m
i j j
jk m m
k N N
 of functional reliability conditions (66) and (67). 
Characteristic 
Plates 
2M 3M 4M 
Stress 1
i
jk  
1.29 1.21 1.31 
Bending 2
i
jk  
1.28 1.25 1.29 
Torsion 3
i
jk  
1.31 1.21 1.32 
Coefficients 1 1
,N N
 
7 7 12 
Coefficients 2 2
,N N
 
28 28 28 
Coefficients 3 3
,N N
 
70 70 70 
 
Using the criterion of maximum linear displacement recorded in the form (3), and the value of 
reduced displacements given in Tables 2 and 3, were calculated the permissible loads 
( ) ( ) ( )1 , 2 , 3dop dop dop
P P P
 with their simultaneous action. The calculation results are given in Table 
5. For the permissible mutual movement of the fracture points was taken 
 Λ 1mm=
. Using 
the criterion for the maximum mutual angle of rotation of the fracture parts recorded in the 
formula (8), and the values of the reduced displacements given in Table 2 and 3, permissible 
loads calculated ( ) ( ) ( )
1 , 2 , 3dop dop dop
P P P
 with their simultaneous action. Permissible loads 
calculated according to the specified condition when [𝐺] = 3∘ had significantly larger values 
than those shown in Table 5. Because the permissible loads were chosen, calculated according 
to the permissible linear displacements under the condition (3, 4). 
 
  
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   198 
Table 5. Permissible loads ( ) ( ) ( )
1 , 2 , 3dop dop dop
P P P
, not leading to mutual displacement of fracture 
points greater than 1 mm when the fracture is fixed by various types of plates. 
Plate 2M Plate 3M Plate 4M 
 
( )1 ,dop
P
 
 
( )2 ,dop
P
 
 
( )3 dop
P
 
 
( )1 ,dop
P
 
 
( )2 ,dop
P
 
 
( )3 dop
P
 
 
( )1 ,dop
P
 
 
( )2 ,dop
P
 
 
( )3 dop
P
 
0 0 9.50 0 0 5.49 0 0 20.89 
40.0 8.56 40.0 5.09 40.0 190.90 
80.0 4.75 80.0 3.61 100.0 13.44 
92.4 0 105.9 0 130.0 0 
60 0 8.28 40 0 4.76 60 0 19.98 
20.0 6.79 20.0 4.00 30.0 17.94 
40.0 3.88 40.0 2.68 60.0 14.07 
48.1 0 53.5 0 94.2 0 
80 0 7.20 60 0 3.64 100 0 18.23 
10.0 6.21 10.0 2.97 20.0 16.16 
20.0 4.81 20.0 1.95 40.0 13.03 
33.0 0 27.0 0 69.1 0 
122.6 0 0 80.1 0 0 203.7 0 0 
 
For clarity, the data given in Table 5 is presented in the form of three-dimensional diagrams 
(limiting areas) that limit the areas of permissible loads (Fig. 5). Any point located under each 
of the surfaces corresponds to three values of the loads P1, P2 and P3, the combined action of 
which does not lead to the occurrence of displacements and rotation angles of the fracture parts 
that are larger than permissible: Λ 1mm  and [𝐺] < 3∘. If the point belongs to the surface, 
then this combination 1 2 3
, ,P P P
 causes the cell displacement Λ 1mm= . In the case when the 
point is outside the boundary surface, such combination of loads is dangerous, while moving 
the cell fracture point distant from the plate Λ>1mm . 
  
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   199 
Figure 5. Area of permissible loads ( ) ( ) ( )
1 , 2 , 3dop dop dop
P P P
, not leading to physiologically 
dangerous displacements larger than Λ 1mm= . 
 
2M 
 
3M 
 
4M 
y = -0.0732x + 7.3366
R² = 0.4209
0
1
2
3
4
5
6
7
8
9
10
0 20 40 60 80 100
y = -0.0312x + 3.8801
R² = 0.2742
0
1
2
3
4
5
6
0 20 40 60 80 100 120
y = -0.2253x + 38.007
R² = 0.0344
0
50
100
150
200
250
0 20 40 60 80 100 120 140
Kinesiologia Slovenica, 29, 1, 177-202 (2023), ISSN 1318-2269  Rehabilitation of Athletes   200 
CONCLUSION 
The value of permissible loads calculated using the given deformations only corresponds to the 
initial state of bone regeneration. The process of rehabilitation of the injured is accompanied 
by exposure of limbs to cyclically acting loads (walking, exercise). To determine the 
permissible loads for the later stages of fracture fusion, it is necessary to take into account the 
kinetics of changes in the deformation properties of bone tissue in the fracture region during its 
regeneration. Calculation of permissible loads provided that the force  acts at a distance 
 from the longitudinal axis of the system (lateral load on the distal region of the 
1st metatarsal bone). It should be noted that the given deformations (Tables 2 and 3) were 
determined experimentally on full-scale bone specimens with simulated fractures with diastases 
(distances between the edges of the fracture) not filled with bone regenerate.  
This fractured state corresponds only to the initial stages of fusion when the regeneration is 
absent or its mechanical properties do not affect the deformation (displacement) of the fracture 
point under the influence of physiological loads. The prospects were determined by the need to 
form adaptive potential, which will allow athletes to maintain positive dynamics of 
performances in the future at the level of international competitions. 
Declaration of Competing Interest 
The authors declared no potential conflicts of interest with respect to the research, authorship, 
and/or publication of this article. 
Funding 
This research did not receive any specific grant from funding agencies in the public, 
commercial, or not-for-profit sectors 
 
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