lubricants
Article
Synovial Joints. Tribology, Regeneration, Regenerative
Rehabilitation and Arthroplasty
Valentin L. Popov 1,* , Aleksandr M. Poliakov 2and Vladimir I. Pakhaliuk 2,*
Citation: Popov, V.L.; Poliakov, A.M.;
Pakhaliuk, V.I. Synovial Joints.
Tribology, Regeneration, Regenerative
Rehabilitation and Arthroplasty.
Lubricants 2021,9, 15. https://doi.org/
10.3390/lubricants9020015
Received: 14 December 2020
Accepted: 28 January 2021
Published: 2 February 2021
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1Institute of Mechanics, Technische Universität Berlin, 10623 Berlin, Germany
2Politechnical Institute, Sevastopol State University, 299053 Sevastopol, Russia; [email protected]u
*Correspondence: v[email protected] (V.L.P.); [email protected]u (V.I.P.); Tel.: +49-303-1421-480 (V.L.P.);
+7-978-7640-600 (V.I.P.)
Abstract:
Synovial joints are unique biological tribosystems that allow a person to perform a wide
range of movements with minimal energy consumption. In recent years, they have been increasingly
called “smart friction units” due to their ability to self-repair and adapt to changing operating condi-
tions. However, in reality, the elements of the internal structure of the joints under the influence of
many factors can degrade rather quickly, leading to serious disease such as osteoarthritis. According
to the World Health Organization, osteoarthritis is already one of the 10 most disabling diseases in
developed countries. In this regard, at present, fundamental research on synovial joints remains
highly relevant. Despite the fact that the synovial joints have already been studied fully, many issues
related to their operating, prevention, development of pathology, diagnosis and treatment require
more detailed consideration. In this article, we discuss the urgent problems that need to be solved for
the development of new pharmacological agents, biomaterials, scaffolds, implants and rehabilitation
devices for the prevention, rehabilitation and improvement of the treatment effectiveness of synovial
joints at various stages of osteoarthritis.
Keywords:
synovial joint; synovial fluid; articular cartilage; biotribology; regeneration; regenerative
rehabilitation; joint arthroplasty
1. Introduction
Synovial joints are remarkable biotribosystems with unique properties, thanks to
which a person is able to perform a huge range of motor actions with minimal energy
losses due to friction. One of the factors contributing to this is that the basic elements
of synovial joints, such as: articular surfaces, covered with articular cartilage; articular
cavity; joint capsule; synovium; synovial fluid), are synergistically linked to each other and
form an integral biomechanical system [
1
]. At the same time, a permanent change in the
properties of at least one of the listed elements leads to a change in the properties of others
and synovial joints in general.
Each of the many joints that form the human skeleton contributes to the preservation
of posture and contributes to the realization of movements of the body and its individual
parts in space, however, from a biomechanical point of view, large synovial joints of limbs
(primarily the lower ones), among other things, perform bearing functions. It is in them
that all the qualities that correspond to the so-called “smart friction units” are most clearly
manifested [
2
,
3
], but at the same time they are also most often susceptible to diseases and
injuries that limit the physical activity of people and in many cases lead to disability.
In fact, healthy synovial joint forms a semblance of a cybernetic system functioning
on the basis of feedback with the sympathetic nerve center, which regulates the articular
cartilage lubrication regime, the synovial fluid composition, and the removal of wear
products from the joint capsule [
4
]. In a healthy body, this system works stably and
provides a very low coefficient of friction of articular surfaces (~0.001), but in the course of
aging of a person followed by development of diseases or injuries, it begins to malfunction
Lubricants 2021,9, 15. https://doi.org/10.3390/lubricants9020015 https://www.mdpi.com/journal/lubricants
Lubricants 2021,9, 15 2 of 24
due to the inability of one or several of its elements to perform their functions. As a result,
synovial joints pathologies develop, the whole variety of which can be reduced to two
forms: (a) arthritis—inflammatory lesion of the joint, regardless of the immediate cause
(infections, autoimmune processes) (~20%) and (b) osteoarthritis—dystrophic-degenerative
joint damage (~80%). It is osteoarthritis that is not only the most common synovial joints
disease, but also the main cause of pain and disability.
The pain that patients with osteoarthritis usually experience is based on two mech-
anisms: nociceptive (performing a protective function and arising from tissue damage
and/or inflammation, due to activation of nociceptors of peripheral or deep tissues) and
dysfunctional (not accompanied by neurological deficits or tissue disorders, and arising
with various neuro-biological changes in the central nervous system) [
5
,
6
]. Therefore, pain
might be one of signs that pathological processes occur in the synovial joints: destruction
of articular cartilage, structural changes in the subchondral bone, inflammatory processes
in synovium, tissue repair, etc. However, the manifestation of pain and dysfunction do not
always correlate with the severity of pathological changes in the synovial joints area [
5
].
Taking this fact into account, it is assumed that, in addition to the severity of the pathologi-
cal process in synovial joints, other factors are involved in the formation of pain, such as
age, gender, race, duration of the disease, body mass index, state of mental health, etc. [
7
].
Therefore, to assess the real state of synovial joints, only the objective characteristics of a
particular patient should be used, established by various methods with a high degree of
accuracy. In alternative approaches, big data and artificial intelligence technologies can be
used, in which the real state of the synovial joints can be estimated with high reliability
from a set of approximate data and the results of studying approximate mathematical
models. However, in order to develop algorithms for such estimates, a more detailed study
of the synovial joints of each type and the features of its functioning is required, considering
the many factors noted above. In addition, these factors should be taken into account in
synovial joints models, which is especially important for the development and practical
use of technologies for tissue engineering, repair and regeneration of articular cartilage.
For example, the radiographs shown in Figures 1–3clearly show the differences in the
developmental stages of the osteoarthritis of the ankle, knee and hip joints, respectively,
which, in general, allow a fairly objective assessment of the current state of synovial joints
even without taking into account some personalized patient data.
Lubricants 2020, 8, x FOR PEER REVIEW 2 of 23
aging of a person followed by development of diseases or injuries, it begins to malfunction
due to the inability of one or several of its elements to perform their functions. As a result,
synovial joints pathologies develop, the whole variety of which can be reduced to two
forms: (a) arthritis—inflammatory lesion of the joint, regardless of the immediate cause
(infections, autoimmune processes) (~20%) and (b) osteoarthritis—dystrophic-degenera-
tive joint damage (~80%). It is osteoarthritis that is not only the most common synovial
joints disease, but also the main cause of pain and disability.
The pain that patients with osteoarthritis usually experience is based on two mecha-
nisms: nociceptive (performing a protective function and arising from tissue damage
and/or inflammation, due to activation of nociceptors of peripheral or deep tissues) and
dysfunctional (not accompanied by neurological deficits or tissue disorders, and arising
with various neuro-biological changes in the central nervous system) [5,6]. Therefore, pain
might be one of signs that pathological processes occur in the synovial joints: destruction
of articular cartilage, structural changes in the subchondral bone, inflammatory processes
in synovium, tissue repair, etc. However, the manifestation of pain and dysfunction do
not always correlate with the severity of pathological changes in the synovial joints area
[5]. Taking this fact into account, it is assumed that, in addition to the severity of the patho-
logical process in synovial joints, other factors are involved in the formation of pain, such
as age, gender, race, duration of the disease, body mass index, state of mental health, etc.
[7]. Therefore, to assess the real state of synovial joints, only the objective characteristics
of a particular patient should be used, established by various methods with a high degree
of accuracy. In alternative approaches, big data and artificial intelligence technologies can
be used, in which the real state of the synovial joints can be estimated with high reliability
from a set of approximate data and the results of studying approximate mathematical
models. However, in order to develop algorithms for such estimates, a more detailed
study of the synovial joints of each type and the features of its functioning is required,
considering the many factors noted above. In addition, these factors should be taken into
account in synovial joints models, which is especially important for the development and
practical use of technologies for tissue engineering, repair and regeneration of articular
cartilage.
For example, the radiographs shown in Figures 1–3 clearly show the differences in
the developmental stages of the osteoarthritis of the ankle, knee and hip joints, respec-
tively, which, in general, allow a fairly objective assessment of the current state of synovial
joints even without taking into account some personalized patient data.
(a) (b) (c) (d) (e)
Figure 1. Stages of development of ankle osteoarthritis: (a) zero (no changes); (b) the first; (c) the second; (d) third; (e)
fourth (the complete obliteration of the joint). Adopted from [8].
Figure 1.
Stages of development of ankle osteoarthritis: (
a
) zero (no changes); (
b
) the first; (
c
) the second; (
d
) third; (
e
) fourth
(the complete obliteration of the joint). Adopted from [8].
Lubricants 2021,9, 15 3 of 24
Lubricants 2020, 8, x FOR PEER REVIEW 3 of 23
(a) (b) (c) (d) (e)
Figure 2. Stages of development of osteoarthritis of the knee joint according to Kellgren and Lawrence: (a) zero (no
changes); (b) the first; (c) the second; (d) third; (e) fourth (severe osteoarthritis). Adopted from [9].
(a)
(b)
(c)
(d)
Figure 3. Stages of development of osteoarthritis of the hip joint: (a) zero (no changes); (b) the first; (c) the second; (d) third
(maximum disappearance of cartilaginous tissue from the joint surfaces). Adopted from [9].
There is also a lot of experimental data indicating that, depending on the stage of
osteoarthritis, in the vast majority of patients, the synovial fluid indices change, which
performs a number of important functions in the synovial joints: trophic (articular carti-
lage nutrition); metabolic (removal of cells in a state of decay); depreciation (damping the
load on articular surfaces); tribological (low coefficient of friction of articular surfaces) and
protective (delineation of articular surfaces) [10]. This allows, as shown in [11], informa-
tive indicators obtained as a result of cytological studies of synovial fluid to be used for
an objective assessment of the stage of the pathological process in the synovial joints in
some patients. For this, the authors propose using the value of the so-called “cell index”,
defined as the ratio of cells of tissue origin (synovicitis type A and B, as well as their atyp-
ical and decaying forms), which are markers of degenerative-dystrophic changes, to blood
cells (neutrophils, lymphocytes, monocytes), which determines the inflammatory nature
of the pathological process.
The above examples indicate that, in principle, based on objective data on the state
of synovial joints elements, it is possible with a certain degree of accuracy to diagnose a
patient with synovial joints pathology (regardless of much of his personal data) and to
plan its treatment at various stages based on standardized clinical protocols. However, as
stated in the osteoarthritis part of the World Health Organization (WHO) report: “The
current methods of clinical diagnosis and X-rays are not precise enough to effectively
measure status and progression of the condition, which presents serious difficulties in
evaluating both the impact of risk factors and the effectiveness of potential therapies” [12].
This is one of the main reasons why osteoarthritis remains the most common synovial
joints disease and, according to the WHO, is among the ten most disabling diseases in
developed countries [12]. In this regard, at the WHO level, the need to develop new tech-
nologies for imaging, diagnostics and biomarkers is noted for more effective measurement
of the status and progression of osteoarthritis [13]. At the same time, other issues concern-
ing the anatomy, physiology and treatment of synovial joints in general and their struc-
tural elements remain relevant.
Figure 2.
Stages of development of osteoarthritis of the knee joint according to Kellgren and Lawrence: (
a
) zero (no changes);
(b) the first; (c) the second; (d) third; (e) fourth (severe osteoarthritis). Adopted from [9].
Lubricants 2020, 8, x FOR PEER REVIEW 3 of 23
(a) (b) (c) (d) (e)
Figure 2. Stages of development of osteoarthritis of the knee joint according to Kellgren and Lawrence: (a) zero (no
changes); (b) the first; (c) the second; (d) third; (e) fourth (severe osteoarthritis). Adopted from [9].
(a)
(b)
(c)
(d)
Figure 3. Stages of development of osteoarthritis of the hip joint: (a) zero (no changes); (b) the first; (c) the second; (d) third
(maximum disappearance of cartilaginous tissue from the joint surfaces). Adopted from [9].
There is also a lot of experimental data indicating that, depending on the stage of
osteoarthritis, in the vast majority of patients, the synovial fluid indices change, which
performs a number of important functions in the synovial joints: trophic (articular carti-
lage nutrition); metabolic (removal of cells in a state of decay); depreciation (damping the
load on articular surfaces); tribological (low coefficient of friction of articular surfaces) and
protective (delineation of articular surfaces) [10]. This allows, as shown in [11], informa-
tive indicators obtained as a result of cytological studies of synovial fluid to be used for
an objective assessment of the stage of the pathological process in the synovial joints in
some patients. For this, the authors propose using the value of the so-called “cell index”,
defined as the ratio of cells of tissue origin (synovicitis type A and B, as well as their atyp-
ical and decaying forms), which are markers of degenerative-dystrophic changes, to blood
cells (neutrophils, lymphocytes, monocytes), which determines the inflammatory nature
of the pathological process.
The above examples indicate that, in principle, based on objective data on the state
of synovial joints elements, it is possible with a certain degree of accuracy to diagnose a
patient with synovial joints pathology (regardless of much of his personal data) and to
plan its treatment at various stages based on standardized clinical protocols. However, as
stated in the osteoarthritis part of the World Health Organization (WHO) report: “The
current methods of clinical diagnosis and X-rays are not precise enough to effectively
measure status and progression of the condition, which presents serious difficulties in
evaluating both the impact of risk factors and the effectiveness of potential therapies” [12].
This is one of the main reasons why osteoarthritis remains the most common synovial
joints disease and, according to the WHO, is among the ten most disabling diseases in
developed countries [12]. In this regard, at the WHO level, the need to develop new tech-
nologies for imaging, diagnostics and biomarkers is noted for more effective measurement
of the status and progression of osteoarthritis [13]. At the same time, other issues concern-
ing the anatomy, physiology and treatment of synovial joints in general and their struc-
tural elements remain relevant.
Figure 3.
Stages of development of osteoarthritis of the hip joint: (
a
) zero (no changes); (
b
) the first; (
c
) the second; (
d
) third
(maximum disappearance of cartilaginous tissue from the joint surfaces). Adopted from [9].
There is also a lot of experimental data indicating that, depending on the stage of
osteoarthritis, in the vast majority of patients, the synovial fluid indices change, which
performs a number of important functions in the synovial joints: trophic (articular cartilage
nutrition); metabolic (removal of cells in a state of decay); depreciation (damping the load
on articular surfaces); tribological (low coefficient of friction of articular surfaces) and
protective (delineation of articular surfaces) [
10
]. This allows, as shown in [
11
], informative
indicators obtained as a result of cytological studies of synovial fluid to be used for an
objective assessment of the stage of the pathological process in the synovial joints in some
patients. For this, the authors propose using the value of the so-called “cell index”, defined
as the ratio of cells of tissue origin (synovicitis type A and B, as well as their atypical and
decaying forms), which are markers of degenerative-dystrophic changes, to blood cells
(neutrophils, lymphocytes, monocytes), which determines the inflammatory nature of the
pathological process.
The above examples indicate that, in principle, based on objective data on the state
of synovial joints elements, it is possible with a certain degree of accuracy to diagnose a
patient with synovial joints pathology (regardless of much of his personal data) and to plan
its treatment at various stages based on standardized clinical protocols. However, as stated
in the osteoarthritis part of the World Health Organization (WHO) report: “The current
methods of clinical diagnosis and X-rays are not precise enough to effectively measure
status and progression of the condition, which presents serious difficulties in evaluating
both the impact of risk factors and the effectiveness of potential therapies” [
12
]. This
is one of the main reasons why osteoarthritis remains the most common synovial joints
disease and, according to the WHO, is among the ten most disabling diseases in developed
countries [
12
]. In this regard, at the WHO level, the need to develop new technologies
for imaging, diagnostics and biomarkers is noted for more effective measurement of the
status and progression of osteoarthritis [
13
]. At the same time, other issues concerning
Lubricants 2021,9, 15 4 of 24
the anatomy, physiology and treatment of synovial joints in general and their structural
elements remain relevant.
It should be noted that the anatomy and physiology of the joints has a long history,
the beginning of which is associated with Hippocrates (born around 460 BC). Since then, a
huge amount of scientific research has been carried out, the results of which have made
it possible to obtain almost complete information about joints (including synovial joints)
at various levels. Despite this, interest in this area of research is constantly growing, as
evidenced by the graphs of publication activity on topics synovial joints, articular cartilage
and synovial fluid from 1950 to 2020, shown in Figure 4.
Figure 4.
Dynamics of publications on topics synovial joints (SJ), articular cartilage (ACar) and synovial fluid from 1950 to
2020 (according to Web of Science data).
In our opinion, the reasons for this interest are due to many factors, including: the need
to create effective pharmacological agents for the treatment of osteoarthritis, improving the
technologies of tissue engineering articular cartilage, developing effective technologies for
regenerating damaged articular cartilage, design of biosimilar synovial joints replacements
and new technologies for their implantation, etc.
In this article paying tribute and memory to the outstanding scientist D. Dowson who
made a significant contribution to the development of the science of synovial joints, we
discuss the new problems that need to be solved for the development of new biomaterials,
scaffolds, implants, rehabilitation devices, as well as intelligent diagnostic technologies
with which to prevent and improve the effectiveness of treatment of synovial joints at
various stages of osteoarthritis.
2. Materials and Methods
2.1. Articular Cartilage as a Basis of Synovial Joints
Articular cartilage is a type of connective tissue covering the ends of long bones, which
performs bearing functions into the synovial joints while ensuring high wear resistance and
low coefficient of friction of articular surfaces. It provides the ease of relative movement
of the bones that form the synovial joints and its high durability, provided that the body
remains in a healthy state.
The main cells of articular cartilage are chondrocytes, located in a hydrated extracellu-
lar matrix, consisting mainly of collagen II and proteoglycans. The swelling proteoglycans
Lubricants 2021,9, 15 5 of 24
tend to create osmotic swelling pressure in articular cartilage, while collagen II counteracts
this, thereby providing articular cartilage’s functional integrity, its longitudinal (tensile
and compressive) and shear stiffness. Thanks to the combined action of collagen II and
proteoglycans, articular cartilage is able to resist many times the load acting on it during
the operation of a synovial joint.
Articular cartilage is a heterogeneous, anisotropic, viscoelastic material with a porous
structure, in which the density and organization of cells change markedly during growth
and maturation. The biochemical and biomechanical properties of articular cartilage also
change significantly in depth (from the surface to the area of attachment to the subchondral
bone), which is a consequence of cell and extracellular matrix changes. This results in a
non-linear response to external forces [14].
In the process of functioning, the synovial joints elements, contacting along articular
surfaces, are subjected to compression, shear and sliding relative to each other. Since artic-
ular cartilage has a very low permeability, during compression, the hydrostatic pressure
of the interstitial fluid present in it rapidly increases, which allows bearing the external
load. As the compressive load increases, interstitial fluid begins to be released from the
articular cartilage, which leads to a redistribution of some of the load on the solid matrix
and to the consolidation of the cartilage. Once equilibrium is reached, the interstitial fluid
flow from articular cartilage stops and the entire load is taken up by the solid matrix. This
effect is known as stress relaxation in articular cartilage and characterizes one of its most
important features that determine the compression process and bearing functions [
15
]. This
effect, as well as the viscoelastic properties of cartilage, can be explained on the basis of
a two-phase model [
16
], in which it is assumed that articular cartilage contains a small
number of cells (~1% of the tissue mass) and its mechanical characteristics are mainly
determined by extracellular matrix, which consists of liquid and solid phases. The liquid
phase (~80% by mass) is represented by interstitial fluid, and the composite solid phase,
as a porous, permeable, fiber-bonded material, is represented by macromolecules, among
which collagen II and proteoglycans predominate.
It should be noted that articular cartilage is a well-studied biological tissue, complete
information about which can be found, for example, in [
17
], as well as in many other
sources. But even the features of articular cartilage, briefly described above, allow us
to note that its properties determine the normal physiology of synovial joints. Changes
in extracellular matrix, such as collagen II and/or proteoglycans degradation, can lead
to a variety of diseases. At the same time, the low regenerative potential of articular
cartilage, determined by the peculiarities of its histological structure, may lead to an early
development of total osteoarthritis even with local damage.
The mechanisms of articular cartilage damage in osteoarthritis and the conditions for
their occurrence (or non-occurrence) are still not fully understood and require further study.
At the same time, it is known that osteoarthritis is polyetiologic pathology of synovial
joints, the pathogenesis of which can be interpreted in different ways. However, there is
no doubt that the development of osteoarthritis is based on the articular cartilage lesion,
which occurs in stages [18,19].
2.2. Lubrication and Friction in Synovial Joints
2.2.1. Lubrication and Friction Modes
The lubricating medium of synovial joints is synovial fluid filling articular cartilage. It
can be considered as a mobile medium that carries out mass transfer between the synovial
joints elements and unites them into a single biophysical and biochemical system [
4
].
That is, synovial fluid is a kind of reservoir in which nutrients and signaling molecules
of cell populations are located in synovial joints. In addition, it houses high molecular
weight molecules that are retained in the joint due to their large size, contributing to the
maintenance of synovial fluid volume during synovial joints operating [
20
,
21
]. In a healthy
body, the entire complex of molecular components of synovial fluid determines its unique
properties and functions of maintaining synovial joints homeostasis.
Lubricants 2021,9, 15 6 of 24
One of the most important functions of the synovial fluid (but not the only one) is
to provide the synovial joints lubrication, which contributes to the low friction and wear
of the articular surfaces, and therefore the locomotion efficiency of the synovial joints. A
number of facts indicate that this paradigm was recognized in ancient times. Therefore, for
example, in one of the treatises published in the middle of the 18th century, the following
is noted: “Human body is a subject so much the more entertaining, as it must strike every
one that considers it attentively with an idea of fine mechanical compositions. Where ever
the motion of one bone upon another is requisite, there we find an excellent apparatus
for rendering that motion safe and free: we see, for instance, the extremity of one bone
moulded into an orbicular cavity, to receive the head of another, in order to afford in an
extensive play. Both are covered with a smooth elastic crust to prevent mutual abrasion;
connected with strong ligaments, to prevent dislocation and enclosed in a bag that contains
a proper fluid deposited there, for lubricating the two contiguous surfaces” [
22
]. Since then
and until now, many attempts have been made to uncover the mechanisms underlying
biolubrication and friction of synovial joints [23–28].
A significant contribution in this direction of research was made by D. Dowson and
colleagues [
29
–
38
]. They were able to prove that elastohydrodynamic lubrication (EHL) is
the dominant lubrication mode in synovial joints [
39
–
41
], which led to significant progress
in the study of friction mechanisms in synovial joints. However, it was later shown that
EHL is not realized in all modes, and at high contact pressure or at a decrease in sliding
speed synovial fluid is slowly squeezed out of the intra-articular space. The resistance
forces that arise due to the synovial fluid viscosity do not allow it to free itself from this
space quickly. As a result, the remaining liquid film is compressed and takes up the load
acting on the synovial joints, at least for a short period of time. This process is called
squeeze-film lubrication. Experimentally, its existence was confirmed by R.S. Fein [
42
] and
then by the method of asymptotic analysis by J.S. Hou and colleagues [43].
In the model used in [43], the following assumptions were made:
•
articular cartilage is a linear porous-permeable two-phase material filled with a linear
viscous (Newtonian) fluid;
•synovial fluid is also a Newtonian fluid;
•
articular cartilage is a homogeneous layer of thickness H, and the thickness of the
synovial fluid film (h) is significantly less than H; h << H;
•
the radius of curvature R of the bearing articular surfaces is much larger than H; R >>
H;
•
the compression of the synovial fluid film is provided by a stepped load in the form of
a Heaviside function applied to both bearing articular surfaces.
This allowed the authors to use two small parameters for the asymptotic analysis of the
problem: geometric (
ε=H
R
) and physical (
δ2=µa
KH2
), where
µa=(0.01 −1.0)Ns
m2
—apparent
viscosity of the interstitial fluid,
K=
10
15 Ns
m4
—diffusive drag in articular cartilage and
10
−3<H<
2
·
10
−3
,
m
. For such values at 10
−2<R<
3
·
10
−2
,
m
:
ε<0.2 and δ2<2.5·10−8.
As a result, they obtained two coupled nonlinear partial differential equations: one for
synovial fluid (similar to the Reynolds equation) and one for articular cartilage, the analysis
of the numerical solution of which led to the following conclusions [43]:
•articular cartilage material deforms, while the load transfer area increases;
•
articular cartilage deformation leads to a decrease in the synovial fluid velocity, thus
increasing the time for the formation of the squeezed film;
•
synovial fluid in the gap is forced from the central high-pressure region into articular
cartilage, and expelled from the tissue at the low-pressure periphery of the load-
bearing region;
•
tensile hoop stress exists at the cartilage surface despite the compressive squeeze-film
loading condition.
It is these features that correspond to the synovial fluid lubrication.
Lubricants 2021,9, 15 7 of 24
It should be noted that the synovial fluid lubrication mode allows simulating the
behavior of synovial joints at different stages of osteoarthritis, which makes it possible
to evaluate various options for its treatment, considering many personal factors inherent
in a particular patient. From this point of view, the studies by M. Hlavachek published
in [44–46] are of interest.
The author used a two-phase articular cartilage model, represented by a mixture of
an ideal interstitial fluid and a poro-elastic homogeneous isotropic incompressible matrix,
which allows one to study the case of early osteoarthritis, when the intact surface zone of
normal articular cartilage, which is more rigid than the base material, is already destroyed
or worn out. He found that under these conditions, under normal displacement of articular
surfaces, the liquid film is rapidly depleted and turns into a synovial gel, which in the case
of sliding motion should behave like boundary lubrication [45]. It was shown in [46] that,
due to the high viscosity of normal synovial fluid, at very low shear rates, the squeezed
film at a fixed time after applying a constant load turns out to be much thicker in a small
central part of the lubricated contact region. The rest of the film is thin, since it corresponds
to a Newtonian fluid with the same viscosity at high shear rate. At the same time, the
filtration is lower for normal articular cartilage with an intact surface zone, due to its lower
permeability and compressive rigidity. But even in the case of zero filtration, the effect
of thixotropy (liquefaction) on an increase in the minimum thickness of the liquid film
appears within a fairly short period of time after the application of a physiological load.
It should be noted that a complete analytical solution to the problem of synovial joints
operating in the synovial fluid lubrication mode cannot be found, due to the complexity
of the model. However, taking into account some conditions (assumptions), the problem
can be simplified significantly. For example, A. Ruggiero managed to reduce the problem
of the synovial fluid lubrication of the human ankle joint with synovial fluid filtrated
by articular cartilage to an approximate analytical model and find its numerical solution
for synovial fluid, represented as Newtonian lubricant [
47
]. Using the example of the
knee joint, as mentioned above, this problem was solved by J.S. Hou and colleagues [
43
];
in [
34
] the results of its solution for the hip joint were presented; in [
48
] for the case
of contact of two planes (with reference to the lubricant synovial fluid lubrication of
synovial joints). In one of the latest works carried out with the participation of D. Dowson,
the mode of pore-hyperelastic lubrication articular cartilage in natural synovial joints
was considered. According to the authors, this model describes all modes, including
synovial fluid lubrication and boundary lubrication that occurs with increasing load, which
represents major progress in modeling the mechanics of articular cartilage [49].
In [
50
], a short, but rather informative history of the formation of the science branch
devoted to friction and lubrication of synovial joints was presented. It is noted that in a
natural joint, depending on its state and conditions of operating, various modes can be
observed, including boundary, mixed and liquid lubrication. Therefore, when solving
applied problems, there is a need for a preliminary assessment of the most probable
mode. The most typical synovial joints lubrication mechanisms are shown schematically in
Figure 5.
In this article, we analyze the models of synovial joints operating in synovial fluid
lubrication mode. Solutions to such problems are extremely important for various applica-
tions, such as: development of artificial analogues of synovial fluid, new designs of joint
implants, biomaterials, scaffolds for regenerative technologies, for optimal synthesis of
regenerative rehabilitation devices, etc.
Lubricants 2021,9, 15 8 of 24
Lubricants 2020, 8, x FOR PEER REVIEW 7 of 23
or worn out. He found that under these conditions, under normal displacement of articu-
lar surfaces, the liquid film is rapidly depleted and turns into a synovial gel, which in the
case of sliding motion should behave like boundary lubrication [45]. It was shown in [46]
that, due to the high viscosity of normal synovial fluid, at very low shear rates, the
squeezed film at a fixed time after applying a constant load turns out to be much thicker
in a small central part of the lubricated contact region. The rest of the film is thin, since it
corresponds to a Newtonian fluid with the same viscosity at high shear rate. At the same
time, the filtration is lower for normal articular cartilage with an intact surface zone, due
to its lower permeability and compressive rigidity. But even in the case of zero filtration,
the effect of thixotropy (liquefaction) on an increase in the minimum thickness of the liq-
uid film appears within a fairly short period of time after the application of a physiological
load.
It should be noted that a complete analytical solution to the problem of synovial joints
operating in the synovial fluid lubrication mode cannot be found, due to the complexity
of the model. However, taking into account some conditions (assumptions), the problem
can be simplified significantly. For example, A. Ruggiero managed to reduce the problem
of the synovial fluid lubrication of the human ankle joint with synovial fluid filtrated by
articular cartilage to an approximate analytical model and find its numerical solution for
synovial fluid, represented as Newtonian lubricant [47]. Using the example of the knee
joint, as mentioned above, this problem was solved by J.S. Hou and colleagues [43]; in [34]
the results of its solution for the hip joint were presented; in [48] for the case of contact of
two planes (with reference to the lubricant synovial fluid lubrication of synovial joints).
In one of the latest works carried out with the participation of D. Dowson, the mode of
pore-hyperelastic lubrication articular cartilage in natural synovial joints was considered.
According to the authors, this model describes all modes, including synovial fluid lubri-
cation and boundary lubrication that occurs with increasing load, which represents major
progress in modeling the mechanics of articular cartilage [49].
In [50], a short, but rather informative history of the formation of the science branch
devoted to friction and lubrication of synovial joints was presented. It is noted that in a
natural joint, depending on its state and conditions of operating, various modes can be
observed, including boundary, mixed and liquid lubrication. Therefore, when solving ap-
plied problems, there is a need for a preliminary assessment of the most probable mode.
The most typical synovial joints lubrication mechanisms are shown schematically in Fig-
ure 5.
Figure 5. Most typical synovial joints lubrication mechanisms.
In this article, we analyze the models of synovial joints operating in synovial fluid
lubrication mode. Solutions to such problems are extremely important for various appli-
cations, such as: development of artificial analogues of synovial fluid, new designs of joint
implants, biomaterials, scaffolds for regenerative technologies, for optimal synthesis of
regenerative rehabilitation devices, etc.
2.2.2. Mathematical Models of Squeeze Film Lubrication
Figure 5. Most typical synovial joints lubrication mechanisms.
2.2.2. Mathematical Models of Squeeze Film Lubrication
In [
47
], considering an approximate analytical model of the synovial fluid lubrication
of the ankle joint with porous articular cartilage, the authors obtained the following
differential equation:
fre,.
e=W(t), (1)
where
W(t)
is a function describing the change in load on a joint over the time in a
walking cycle,
fre,.
e
is a function characterizing the force perceived by articular surfaces,
depending on the size of the gap
e(t)
and the rate of its change
.
e=de
dt
, which has the
following form:
fr=12R3µ.
eL
2(c+12Φ)arctg"(c−e+12Φ)tgβ
2
√e2−c2−24cΦ−144Φ2#
e(e2−c2−24cΦ−144Φ2)3
2−
β
e−(c+12Φ+ecosβ)sinβ
e2−c2−24cΦ−144Φ2
c+12Φ+ecosβ2
, (2)
The parameter notations used in (2) correspond to [
47
]:
R
—effective radius curvature
of the contact of talus (with radius curvature
R1
) and tibia (with radius curvature
R2
):
1
R=1
R1−1
R2
;
L
—length of the cylindrical joint model;
c
—clearance;
θ=±a
R=±β
—polar
angle; a—tibial length; µ—shear modulus of the cartilage matrix; Φ—permeability of the
cartilage matrix.
In this model, it is assumed that two rigid infinite cylinders, representing the tibia and
talus of the ankle, covered with a thin layers of articular cartilage of equal thickness, are in
internal contact with each other [
51
]. In this case, articular surfaces of the talus is conven-
tionally immobile, while articular surface of the tibia performs exclusively compressive
movement
e(t)
, which leads to reduction of the gap in the joint filled with a thin film of
synovial fluid.
Equation (1) corresponds to the linear model articular cartilage, in which a geometri-
cally linear (small deformations), homogeneous and isotropic elastic porous matrix is filled
with an ideal (Newtonian) interstitial fluid [
16
]. The effects arising from the deformation
of articular cartilage are not taken into account in it. The advantages of such a model are
that it is quite simple to use for solving many applied problems, including optimization
ones, without using the complex mathematical apparatus necessary for solving systems of
non-linear partial differential equations. In this case, the numerical solution of Equation (1)
can be obtained by a variety of known methods for various laws of change in the external
load W(t).
Later, a modified version of this model was presented, in which the synovial fluid
was considered as non-Newtonian [
52
], which led to the need to take into account coupled
stresses [53].
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Indeed, if we consider the movements of the elements of the synovial joint only in the
sagittal plane, then its model in the first approximation can be represented by two rigid
infinite cylinders covered with a porous elastic material and separated by a layer of liquid
that can penetrate into the pores. So, for example, the model of the ankle joint is presented
as a movable joint of the talus and tibia, the surfaces of which, being in internal contact
with each other, are close to cylindrical with radii
R1
( (talus) and
R2
(tibia) and are covered
with thin layers of poro-elastic articular cartilage the same thickness (
ˆ
h1=ˆ
h2=ˆ
h
). In this
case, the talus, which can be conventionally considered motionless, perceives the action of
an external load W(t)applied to the tibia (Figure 6).
Lubricants 2020, 8, x FOR PEER REVIEW 9 of 23
Figure 6. Coordinate system and location of the main model elements (, are articular carti-
lages). Adopted from [54]
It is assumed that the external load is set kinematically, by compressing a liquid film
with a thickness ℎ(,) located between the articular surfaces's of the joint, i.e.,:
ℎ(,)=ℎ
()−2ℎ
,
where ℎ()—total layer thickness of synovial fluid.
If we neglect the inertia and mass forces of synovial fluid and assume that the syno-
vial fluid layer in the joint is a thin film, then the Stokes moment equations in Cartesian
coordinates [53] can be represented as follows:
−
=
, (3)
=0, (4)
−
=
, (5)
These equations must satisfy the continuity of the liquid medium
+
=0, (6)
and boundary conditions: (,0,)=0; (,,)
=0, (7)
(,ℎ,)=0; (,ℎ,)
=0, (8)
where ,, are the velocity field components,
; is a synovial fluid pressure, Pa;
is a synovial fluid dynamic viscosity,∙; is a couple stress synovial fluid constant,
∙
; = is a synovial fluid constant with the dimension of length, [53].
General solution to Equation (3):
=++
++, (9)
where ,…, are integration constants
Figure 6.
Coordinate system and location of the main model elements (
Acar1,2
are articular cartilages). Adopted from [
54
].
It is assumed that the external load is set kinematically, by compressing a liquid film
with a thickness h(x,t)located between the articular surfaces’s of the joint, i.e.,:
h(x,t)=eh(t)−2ˆ
h,
where eh(t)—total layer thickness of synovial fluid.
If we neglect the inertia and mass forces of synovial fluid and assume that the synovial
fluid layer in the joint is a thin film, then the Stokes moment equations in Cartesian
coordinates [53] can be represented as follows:
∂2u
∂y2−l2∂4u
∂y4=1
µ
∂p
∂x, (3)
∂p
∂y=0, (4)
∂2w
∂y2−l2∂4w
∂y4=1
µ
∂p
∂z, (5)
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These equations must satisfy the continuity of the liquid medium
∂u
∂x+∂v
∂y=0, (6)
and boundary conditions:
u(x, 0, z)=0; ∂2u(x, 0, z)
dz2=0, (7)
u(x,h,z)=0; ∂2u(x,h,z)
dz2=0, (8)
where
u
,
v
,
w
are the velocity field components,
m
s
;
p
is a synovial fluid pressure, Pa;
µ
is
a synovial fluid dynamic viscosity,
Pa·s
;
η
is a couple stress synovial fluid constant,
kg·m
s
;
l=qη
µis a synovial fluid constant with the dimension of length, m[53].
General solution to Equation (3):
u=l2c1ey
l+c2e−y
l+1
2µ
∂p
∂xy2+c3y+c4, (9)
where c1, . . . , c4are integration constants.
Giving the boundary conditions (7, 8), expression (9) takes the following form:
u=1
2µ
∂p
∂x
y2−hy +2l2
1−e2y
l+eh
l
ey
leh
l+1
. (10)
Let us now consider the porous structure of articular cartilage and the filtration of
synovial fluid through it. In accordance with Darcy’s law for synovial fluid (as a non-
Newtonian fluid) [53], we write down the conditions on the boundaries:
v(x, 0, z)=ΦH
µ(1−α)
∂2p
dx2, (11)
v(x,h,z)=dh
dt =.
h(12)
where
Φ
is a articular cartilage permeability,
m2
;
α=l2
Φ
,
units
. Integrating (6) over
y
, we
write: ∂
∂xZ∂p
∂xudy +v+C5=0,
or
v=∂
∂x
−1
2µ
1
3y3−1
2hy2+2l2y−2l3e2y
l−eh
l
ey
leh
l+1
∂p
∂x
+C5.
From this expression, using condition (11), we obtain:
C5=∂
∂x"l3
µ
eh
l−1
eh
l+1
∂p
∂x#+ΦH
µ(1−α)
∂2p
dx2. (13)
In this way,
v=∂
∂x
−1
µ
1
6y3−1
4hy2+l2y−l3ey
l−eh−y
l+eh
l−1
eh
l+1
∂p
∂x
+ΦH
µ(1−α)
∂2p
dx2. (14)
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Finally, from (14), using condition (12), we obtain the following expression:
∂
∂x" h3−12l2h+24l3eh
l−1
eh
l+1
+12 ΦH
1−α!∂p
∂x#=12µ.
h. (15)
To simplify further transformations and the possibility of obtaining an approximate
analytical solution, we go over to the polar coordinate system (
x=θR
;
h(θ,t)=c+ecosθ)
and reduce the differential Equation (15) to a dimensionless form using the following
dimensionless variables:
R=R2; ; L=L
c;l=l
c;ε=e
c;h(θ,t)=h
c=1−εcosθ;p=−c2
µR2.
εp;Φ=Φ
c2;H=H
c; (16)
where c is the joint clearance;
e
is a normal displacement of the articular surfaces of the
tibia towards the articular surfaces of the talus. Substituting (15) into (16), we obtain:
∂
∂θ
h3−12l2h+24l3eh
l−1
eh
l+1
+12 ΦH
1−α
∂p
∂θ
=12cosθ, (17)
Integrating (17) twice over θand taking into account the boundary conditions:
at x=−a:θ=β(−a)=−a
R; at x=a:θ=β(a)=a
R
we obtain an expression for the dimensionless pressure
p=p(θ)
synovial fluid inside
the joint:
p(θ)=12(cosβ−cosθ)
h3−12l2h+24l3e
h
l−1
e
h
l+1
+12 ΦH
1−α
. (18)
To determine the resistance force to the action of an external load, it is necessary to
calculate the integral:
Fr=Z+β
−βp(θ)cosθdθ. (19)
In order to find an approximate analytical expression for
Fr
, we expand
p(θ)
in
Maclaurin Series in θ:
p(θ)=12
Acosβ−1+1
2−B
A(cosβ−1)θ2+θ3. (20)
where
A=(1+ε)3−12l2(1+ε)+24e1+ε
l−1
e1+ε
l+1
+12 ΦH
1−α(21)
B=−3ε(1+ε)2+12l2ε−12ε
l
1−
e1+ε
l−1
e1+ε
l+1
2
. (22)
After substituting (20) in (19) and integrating, we obtain the following expression:
Fr=6L
A2[2β(A+B)(βsinβ+2cosβ)+2A(sin(2β)−4sinβ)]−
−6LB
A22β+4sinβ+β2sin(2β)−2sin(2β)+2βcos(2β).(23)
This model is more suitable from a biological point of view. Indeed, it is known that
synovial fluid ingredients such as globulin, lubricin, hyaluronic acid, albumin, mucin,
surface active phospholipids [
4
] play an important role in synovial joints lubrication, which
contribute to the implementation of various modes of synovial joints lubrication both
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individually and in combination with each other. With a strong load on the joint, the
combination and concentration of lubricant molecules adhered to articular surfaces gives
a synergistic effect that helps to reduce the friction coefficient and the rate of wear of
articular cartilage under boundary and mixed modes of cartilage lubrication [
55
]. It is
these molecules with a large molecular weight that give synovial fluid the properties of
a non-Newtonian liquid, the viscosity of which decreases with an increase in the sliding
speed. This is especially typical for synovial fluid in synovial joints affected by arthritis
and osteoarthritis, which is explained by a change in the concentration of proteins and
polysaccharides on articular surfaces in these pathologies.
In addition, it was found that the properties of synovial fluid, and, consequently,
the mechanisms of lubrication, change depending on the value of the contact pressure,
temperature, and the rate of protein deposition [
56
]. This implies the use of methods of
the moment theory of elasticity and non-Newtonian properties of synovial fluid, which
considers the appearance of coupled stresses in a compressed liquid film [53,57,58].
It should be noted that in recent years it has become practically the norm to represent
synovial fluid as a non-Newtonian fluid, since the results of solving most problems become
more plausible even with significant simplifications of other elements and parameters
of lubrication models [
52
,
59
–
63
]. The only problem that arises in this case is a certain
complication of the mathematical side of the problem. However, with modern software,
this problem becomes insignificant.
Note that the models described above are not unique and, if necessary, can be re-
fined/changed by formulating new hypotheses and assumptions. However, as practice
shows, the complication of the initial data that are used in the formulation of mathematical
models does not always lead to a more accurate description of the subject or the modeling
process, which, first of all, is explained by the lack of accurate information about their real
state and change. For example, the shape of articular surfaces, the parameters of articular
cartilage and of synovial fluid differ significantly depending on a large number of factors,
and the coefficient of friction in synovial joints, measured
in vivo
, can never be detected
in vitro
. This prevents synovial joints models from being built with a predetermined accu-
racy. Nevertheless, such models allow reliable qualitative characteristics of synovial joints,
and in some cases, quantitative ones, to be obtained.
2.3. Regeneration and Regenerative Rehabilitation of Articular Cartilage
The morphology of articular cartilage determines its weak ability to regenerate, which,
as a rule, does not allow spontaneous full restoration of defects to be achieved, because in
the damaged area, fibrous tissue or fibrous cartilage is formed, which differ significantly
in architectonics, biochemical composition, and mechanical properties [
64
]. In addition,
despite the knowledge of the etiopathogenetic mechanisms of articular cartilage lesion,
determining the tactics of its treatment and recovery technology is a difficult task with the
outcome difficult to predict. Nevertheless, all modern surgical technologies recommended
for the treatment of articular cartilage are focused primarily on providing optimal con-
ditions for tissue regeneration, which is close in properties to natural articular cartilage.
Moreover, most of them include postoperative rehabilitation, as a necessary procedure for
the treatment of the disease, the phases of which are recommended to correlate with the
biological phases of tissue regeneration.
This is explained by the fact that articular cartilage is susceptible to mechanical stress,
although to a lesser extent than other biological tissues. Moreover, it has been established
that physiological mechanical stress is a decisive factor for the development of zonally
defined cartilage in adults and vertebrates. This feature of articular cartilage develops
during skeletal maturation in postnatal development and is formed by articular movements
resulting in a combined load, including compression, extension, and shear [65].
Forced external mechanical influences can also have a significant effect on the biosyn-
thesis of extracellular matrix, chondrocytes, and chondrogenic differentiation of mesenchy-
mal stem cells during repair and regeneration of articular cartilage. In particular, it has
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been shown that the type, frequency, magnitude, and duration of mechanical stimuli affect
the response of chondrocytes and mesenchymal stem cells’ differentiation [66].
It was found that a strong prochondrogenic stimulus for the articular cartilage is its
compression in various ways. In particular, dynamic compression causes a reaction of chon-
drocytes, accompanied by enhanced synthesis of extracellular matrix molecules, including
proteoglycans and collagen II [
67
–
69
], which largely depends on the rate of deformation,
frequency, amplitude, and loading history. The response to dynamic hydrostatic pressure
is similar to dynamic compression [
70
,
71
]. At the same time, it was shown that mechanical
stimuli, the parameters of which exceed physiological ones by more than 20%, do not
enhance the production of the extracellular matrix [
67
], and static or very low-frequency
load generally suppresses its synthesis [72].
It is also known that articular cartilage responds to mechanical stimuli that promote
the formation of shear deformations [
73
]. The results of a number of
in vitro
experiments
indicate that shear loads, applied simultaneously with compressive ones, lead to a signifi-
cant increase in the expression of chondrogenic mesenchymal stem cell genes, including
proteoglycans-4, which leads to the formation of articular cartilage with a high modulus
of elasticity. In this case, the phenotype of the new tissue depends on the frequency and
amplitude of mechanical stimuli that simultaneously contribute to shear and compres-
sion [74–76].
The aforementioned effects of the occurrence of biochemical processes at the cellular
and molecular levels in response to the action of mechanical stimuli, which are commonly
called mechanotransduction, are well known, but the mechanisms of their occurrence are
still poorly understood. There are still no answers to questions regarding the determina-
tion of specific parameters of forced external mechanical influences that would contribute
to improving the quality and reliability of regenerative processes using regenerative re-
habilitation technologies; it is not clear how to apply mechanical stimuli to the affected
tissue area
in vivo
in order to achieve the course of biochemical processes in the desired
direction, etc. Answers to these questions can be obtained only on the basis of systematic
experimental and theoretical studies, in which studies of tissue regeneration models can
play an important role. In this case, both in mathematical models and in their experimental
analogs, it is desirable to reproduce the loading of articular cartilage in a non-invasive way
by reproducing the natural conditions of articular surfaces contact in different lubrication
modes. It can be assumed that the synovial fluid lubrication model of poroelastic articular
cartilage in combination with EHL will be the most suitable in such cases.
3. Results
Considering the fact that the most dangerous disease of the synovial joints is os-
teoarthritis, characterized by degenerative damage to the articular cartilage, a large number
of studies in this area are focused on solving problems focused on the prevention and
treatment of cartilage diseases, including the development of methods for the regeneration
of cartilage tissue, the creation of artificial analogues of synovial fluid and extracellular ma-
trix. In recent years, increasing attention has been paid to the development of regenerative
rehabilitation technologies which, as shown above, use the targeted action of mechanical
stimuli on the damaged area of the cartilaginous tissue in order to create the best conditions
for its regeneration. Despite the fact that the results of a large number of experimental
and theoretical studies indicate the high potential of such technologies, it has not yet
been possible to achieve significant results in this direction. This is not least due to the
peculiarities of the histological structure of the articular cartilage and the complexity of the
transmission of the necessary stimuli to the local area of the joint using external sources of
their generation. In this regard, the methods of modeling the synovial joints, in terms of
exposure to mechanical stimuli created with the help of external generators, are especially
relevant. In this case, various lubrication modes can be taken into account, including the
squeeze-film lubrication mode.
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As a result of the analysis of such models, such important characteristics as the
components of the velocity field (10, 14), the pressure of interstitial fluid (18) both on the
surface and inside the cartilage, as well as the total force perceived by the articular surfaces
(19) at a certain amplitude of kinematic disturbance. In the future, these characteristics
can serve as initial data for assessing the stress state of the cartilaginous tissue, which is
necessary for the study of models of regenerative rehabilitation. It is clear that they can be
obtained as a result of research and more detailed models, but a possible increase in their
accuracy will be offset by the fact that numerical models of articular cartilage regeneration
are very approximate, since they are based on hypotheses about mechanotransduction and
idealization of the structure of cartilage tissue.
Thus, at this stage, one of the most effective ways of researching and designing
technologies for regenerative rehabilitation of cartilaginous tissue should include at least
two stages: obtaining approximate theoretical data and comparing them with data obtained
as a result of experiments. The final result of such studies should be a refined model of
the regenerative rehabilitation of articular cartilage, which makes it possible to study
the processes of regeneration of cartilage tissue at different stages of the development of
pathology, taking into account the effect on it of different nature stimuli.
The approximate squeeze-film lubrication model described above makes it easy to
estimate the parameters required for the process analysis. At the same time, Equation (2)
obtained on its basis can be solved by many known methods, including those built into
the analytical computing system Maple: dverk78 (seventh-eighth order continuous Runge–
Kutta method), rkf45 (Fehlberg fourth-fifth order Runge–Kutta method), rosenbrock (Im-
plicit Rosenbrock third-fourth order Runge–Kutta method). When using them, practically
identical solutions were obtained, which indicates that the structure of Equation (2) is
relatively simple and its numerical solution does not present any computational problems.
The solutions were obtained for an external load, the graph of the change of which
during a walking cycle in a dimensionless form
n(t)=LW(t)/P
is shown in Figure 7.
However, in principle, this problem is easily solved with loads of other types, including
oscillating ones. The numerical values of the parameters used in the calculations are given
in Table A1.
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The solutions were obtained for an external load, the graph of the change of which
during a walking cycle in a dimensionless form ()=()/ is shown in Figure 7.
However, in principle, this problem is easily solved with loads of other types, including
oscillating ones. The numerical values of the parameters used in the calculations are given
in Table A1.
Figure 7. Graph of changes the external load in a dimensionless form () during a normal hu-
man walking cycle.
Figure 8 shows graphs of changes in intra-articular pressure at different points in
time; Figure 9 is a graph of the change in the thickness of the squeezed fluid film; Figure
10 is a graph of the change in the rate of compression of the ankle joint fluid film.
The shapes of the graphs shown in Figures 9 and 10 differ in shape from the similar
graphs given in [47]. However, given the small size of the joint gap, these differences can
be considered insignificant.
We also investigated the synovial fluid lubrication model of an ankle joint filled with
synovial fluid, which has the properties of a non-Newtonian fluid. Figure 11 shows a
graph of the function =
,, showing the change in the dimensionless distributed
load applied to the elements of the ankle joint during its kinematic loading depending on
the variables .
Figure 8. Graphs of pressure changes in the squeezed fluid film of the ankle joint at different
points in the gait cycle.
Figure 7.
Graph of changes the external load in a dimensionless form
n(t)
during a normal human
walking cycle.
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Figure 8shows graphs of changes in intra-articular pressure at different points in time;
Figure 9is a graph of the change in the thickness of the squeezed fluid film; Figure 10 is a
graph of the change in the rate of compression of the ankle joint fluid film.
Lubricants 2020, 8, x FOR PEER REVIEW 14 of 23
The solutions were obtained for an external load, the graph of the change of which
during a walking cycle in a dimensionless form ()=()/ is shown in Figure 7.
However, in principle, this problem is easily solved with loads of other types, including
oscillating ones. The numerical values of the parameters used in the calculations are given
in Table A1.
Figure 7. Graph of changes the external load in a dimensionless form () during a normal hu-
man walking cycle.
Figure 8 shows graphs of changes in intra-articular pressure at different points in
time; Figure 9 is a graph of the change in the thickness of the squeezed fluid film; Figure
10 is a graph of the change in the rate of compression of the ankle joint fluid film.
The shapes of the graphs shown in Figures 9 and 10 differ in shape from the similar
graphs given in [47]. However, given the small size of the joint gap, these differences can
be considered insignificant.
We also investigated the synovial fluid lubrication model of an ankle joint filled with
synovial fluid, which has the properties of a non-Newtonian fluid. Figure 11 shows a
graph of the function =
,, showing the change in the dimensionless distributed
load applied to the elements of the ankle joint during its kinematic loading depending on
the variables .
Figure 8. Graphs of pressure changes in the squeezed fluid film of the ankle joint at different
points in the gait cycle.
Figure 8.
Graphs of pressure changes in the squeezed fluid film of the ankle joint at different points
in the gait cycle.
Figure 9. Graph of the change in the thickness of the squeezed fluid film of the ankle.
The shapes of the graphs shown in Figures 9and 10 differ in shape from the similar
graphs given in [
47
]. However, given the small size of the joint gap, these differences can
be considered insignificant.
We also investigated the synovial fluid lubrication model of an ankle joint filled with
synovial fluid, which has the properties of a non-Newtonian fluid. Figure 11 shows a graph
of the function
Fr=Frε,l
, showing the change in the dimensionless distributed load
applied to the elements of the ankle joint during its kinematic loading depending on the
variables εand l.
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Figure 10. Graph of the change in the rate of compression of the fluid film of the ankle.
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Figure 9. Graph of the change in the thickness of the squeezed fluid film of the ankle.
Figure 10. Graph of the change in the rate of compression of the fluid film of the ankle.
Figure 11. Change in the dimensionless distributed load .(,) applied to the elements of the
ankle joint under kinematic loading.
Figure 11.
Change in the dimensionless distributed load
Frvs
.
l,ε
applied to the elements of the
ankle joint under kinematic loading.
Figure 12 shows graphs of changes in the dimensionless intra-articular pressure at
constant
l=
0.5 and variable values of the dimensionless displacement
ε
, and Figure 13 at
constant ε=0.5 and variable l.
Lubricants 2021,9, 15 17 of 24
Lubricants 2020, 8, x FOR PEER REVIEW 16 of 23
Figure 12 shows graphs of changes in the dimensionless intra-articular pressure at
constant =0.5 and variable values of the dimensionless displacement , and Figure 13
at constant =0.5 and variable .
Thus, using an approximate model of the contact interaction of articular surfaces cov-
ered with porous cartilaginous tissue and separated by a thin layer of synovial fluid, it is
possible to determine the law of distribution of fluid pressures in the cartilage and, there-
fore, to estimate the stress-strain state at each point. This will allow calculating the crite-
rion index of molecular differentiation, for example, the osteogenic index (OI) proposed
by D. Carter [77] = (+)
where =1..
is the number of the type of stimulating effect; is the number of cycles
of the i-th type; , are hydrostatic and tangential octahedral stresses under stimulating
action of the i-th type, respectively. It is noted that at sufficiently large OI values, cells
differentiate into bone tissue, and at relatively low values, into cartilaginous tissue. Obvi-
ously, as a result of experiments planned based on the results of the study of models of
regenerative rehabilitation, criterion indices can be proposed that more accurately deter-
mine the direction of differentiation of cartilage tissue models.
Figure 12. Graphs of the dimensionless pressure distribution on the contact surfaces of the ankle joint elements at
constant and variable values of dimensionless displacement .
Figure 13. Graphs of the dimensionless pressure distribution on the contact surfaces of the ankle joint elements at con-
stant values of dimensionless displacement and constant .
Figure 12.
Graphs of the dimensionless pressure
p
distribution on the contact surfaces of the ankle joint elements at constant
land variable values of dimensionless displacement ε.
Lubricants 2020, 8, x FOR PEER REVIEW 16 of 23
Figure 12 shows graphs of changes in the dimensionless intra-articular pressure at
constant =0.5 and variable values of the dimensionless displacement , and Figure 13
at constant =0.5 and variable .
Thus, using an approximate model of the contact interaction of articular surfaces cov-
ered with porous cartilaginous tissue and separated by a thin layer of synovial fluid, it is
possible to determine the law of distribution of fluid pressures in the cartilage and, there-
fore, to estimate the stress-strain state at each point. This will allow calculating the crite-
rion index of molecular differentiation, for example, the osteogenic index (OI) proposed
by D. Carter [77] = (+)
where =1..
is the number of the type of stimulating effect; is the number of cycles
of the i-th type; , are hydrostatic and tangential octahedral stresses under stimulating
action of the i-th type, respectively. It is noted that at sufficiently large OI values, cells
differentiate into bone tissue, and at relatively low values, into cartilaginous tissue. Obvi-
ously, as a result of experiments planned based on the results of the study of models of
regenerative rehabilitation, criterion indices can be proposed that more accurately deter-
mine the direction of differentiation of cartilage tissue models.
Figure 12. Graphs of the dimensionless pressure distribution on the contact surfaces of the ankle joint elements at
constant and variable values of dimensionless displacement .
Figure 13. Graphs of the dimensionless pressure distribution on the contact surfaces of the ankle joint elements at con-
stant values of dimensionless displacement and constant .
Figure 13.
Graphs of the dimensionless pressure
p
distribution on the contact surfaces of the ankle joint elements at constant
values of dimensionless displacement εand constant l.
Thus, using an approximate model of the contact interaction of articular surfaces
covered with porous cartilaginous tissue and separated by a thin layer of synovial fluid,
it is possible to determine the law of distribution of fluid pressures in the cartilage and,
therefore, to estimate the stress-strain state at each point. This will allow calculating
the criterion index of molecular differentiation, for example, the osteogenic index (OI)
proposed by D. Carter [77]
OI =∑c
i=1ni(Si+kHi)
where i=1..cis the number of the type of stimulating effect; niis the number of cycles of
the i-th type;
Hi
,
Si
are hydrostatic and tangential octahedral stresses under stimulating
action of the i-th type, respectively. It is noted that at sufficiently large OI values, cells dif-
ferentiate into bone tissue, and at relatively low values, into cartilaginous tissue. Obviously,
as a result of experiments planned based on the results of the study of models of regenera-
tive rehabilitation, criterion indices can be proposed that more accurately determine the
direction of differentiation of cartilage tissue models.
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4. Conclusions
Analysis of literature sources shows that the development, condition and regener-
ation of all biological tissues is largely determined by the mechanical loads acting on
them. First of all, these are natural physiological loads characteristic of the daily activity
of a person. Many examples can be cited indicating that the properties of tissues change
significantly when, for various reasons, the effect of external loads on them ceases, for
example: in a state of weightlessness, with post-traumatic and postoperative immobiliza-
tion, with aging and changes in lifestyle, etc. In such cases, the internal structure of tissues
changes along with a change in their cellular structure, physical, chemical and mechan-
ical properties. This indicates a phenomenon observed in nature inherent in biological
organisms—mechanotransduction, in which, as a result of the action of mechanical stimuli
on tissues, a response occurs at the cellular and molecular levels. Taking this fact into
account, it can be reasonably assumed that mechanotransduction can be very effective in
the processes of regeneration of damaged tissues. And, as noted above, these assumptions
are not without foundation. In particular, it was shown that mechanical stimuli applied to
cartilaginous tissues
in vitro
actually enhance the biosynthesis of the extracellular matrix,
chondrocytes and chondrogenic differentiation of mesenchymal stem cells in articular
cartilage. In the context of health promotion, prevention and treatment of synovial joints,
taking into account and using the effects inherent in mechanotransduction can be very
useful procedures.
When constructing theoretical models of regenerative processes, poroelastic tissue
models are used, the features of which are most often described in the framework of the
theory of consolidation by M. Biot [
78
–
85
]. In particular, on the basis of such models,
algorithms for structural rearrangement and regeneration of bone and cartilaginous tissues
can be constructed [
86
]. They are much more complicated than models based on Darcy’s
law, but they allow one to represent the dynamics of structural rearrangement processes
in two-phase porous media with much higher accuracy. For tissues with a well-studied
microstructure, which include articular cartilage, it is possible to avoid using the methods
of the theory of elasticity for calculating anisotropic effective constants of compliance [
87
].
This is ensured by directly taking into account experimental data on the structure of tissue
pores in their poroelastic models.
It should be noted that the results of the study of a number of mathematical models of
tissue restructuring, built on the basis of M. Biot’s theory, are currently available for analysis.
Among them, the works of L. Maslov [
88
–
91
], D. Carter [
92
–
97
] and other researchers are of
interest, in which, among other things, hypotheses and algorithms for the theoretical study
of regenerative processes are presented. However, the use of these and other theoretical
results to create conditions for practical repair or regeneration of articular cartilage presents
serious difficulties. Not least, this is due to the impossibility of creating a stress-strain state
of the desired type in the local area of the damaged tissue.
Information about the pressure in interstitial fluid, on the surface and within the
articular cartilage, obtained from the study of the lubrication and contact models of articular
surfaces, can only be used as an initial condition for the tissue regeneration process. This is
due to the fact that as the distance from articular surfaces increases, the stress-strain state of
the tissue also changes dramatically, since the interstitial fluid is released mainly from the
surface layer of articular cartilage [
95
,
96
]. That is, linear poroelastic models are incapable
of assessing the deformations of articular cartilage as a whole, since they are based on the
assumption that there is no interstitial fluid flow inside the tissue. At the same time, such
models describe quite well the behavior of interstitial fluid released
in vivo
from articular
cartilage under load in the intra-articular space. The interstitial fluid is trapped between
two mating synovial joints elements and maintains a high level of pressure, preventing
high contact stresses and ensuring low friction between the extracellular matrix elements
of the two mating articular surfaces. An experimental confirmation of this effect is given in
references [97,98].
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It should be noted that synovial joints are striking examples of the optimality of
biological systems characterized by high functional efficiency and resistance to external
factors. However, when normal conditions of operating are violated, such systems lose
their unique properties and their structural elements quickly degrade. In synovial joints,
first of all, degradation of articular cartilage and synovial fluid occurs, which, in turn, leads
to joint disease and an increase in the rate of degradation. In the early stages of the disease,
this situation can be corrected by conservative methods of treatment, the stages of which
should be combined with various rehabilitation procedures, including mechanotherapy.
This medical field, which has received the name regenerative rehabilitation, is gaining
increasing popularity and recognition in the medical community [
99
]. However, the
effectiveness of regenerative rehabilitation is manifested to a greater extent in applications
focused on tissue regeneration. With regard to articular cartilage, this boils down to the
impact on its damaged areas of such mechanical stimuli that would provide an increase in
the biosynthesis of extracellular matrix, chondrocytes and chondrogenic differentiation of
mesenchymal stem cells.
The currently known models of tissue restructuring and regeneration under the influ-
ence of mechanical stimuli cannot be considered sufficient to develop optimal regenerative
rehabilitation technologies on their basis. This is partly due to the fact that they are built
mainly on a mechanistic basis and the biological component is poorly represented in them.
This also applies to the models of lubrication and articular surfaces’ contact interaction.
Such models allow one to perform an approximate assessment of the functional state of the
synovial joints, predict the development of destructive and pathological processes in it, and
also have an idea of the initial conditions of the processes of articular cartilage regenerative
rehabilitation. At the same time, they do not take into account the death and survival of
mesenchymal stem cells inhabited in articular cartilage for the purpose of its regeneration,
changes in the extracellular matrix structure, the qualitative composition of synovial fluid,
and other biological processes. In this regard, in order to increase the efficiency of the
reliability of the regeneration process, scaffolds are introduced into the damaged area
of articular cartilage, which are populated
in vitro
in mesenchymal stem cell bioreactors,
thereby providing better conditions for cell proliferation and survival. As practice shows,
the combination of this approach with the simultaneous action of mechanical stimuli on
articular cartilage contributes to the emergence of the best conditions for its regeneration.
However, the currently used articular cartilage regeneration and regenerative rehabili-
tation technologies can be considered effective only for the restoration of small areas of the
affected tissue located closer to articular surfaces. Unfortunately, with osteoarthritis at the
last stages of development, it has not yet been possible to ensure the repair or regeneration
of articular cartilage. The most effective medical procedure in such cases is synovial joints
arthroplasty, which consists in replacing it with an artificial analogue—an endoprosthesis.
Despite the fact that modern endoprostheses are made from special biomaterials similar in
properties to biological tissues [
100
], and their designs create high-tech products [
101
,
102
],
they do not fully ensure the functionality of natural joints. First of all, they do not provide
the lubrication conditions inherent in natural synovial joints and the natural coefficient of
friction, which leads to their wear and, as a result, to metalosis, and ultimately to the need
for reprosthetics. This and other potential pathologies that may arise after synovial joints
arthroplasty do not allow this surgery to be considered as a complete replacement for a
natural joint. However, the technical means and technologies of synovial joints arthroplasty
are being perfected, which is achieved by bringing artificial joints closer to the natural ones
by way of their more detailed study both at the level of experiments and on the basis of the
study of new mathematical models.
Author Contributions:
Conceptualization, V.L.P., A.M.P. and V.I.P.; investigation, A.M.P. and V.I.P.;
writing, review and editing, V.L.P., A.M.P. and V.I.P. All authors have read and agreed to the published
version of the manuscript.
Lubricants 2021,9, 15 20 of 24
Funding:
This research was funded by Russian Foundation for Basic Research (grant No. 20-03-
00046A).
Conflicts of Interest: The authors declare no conflict of interest.
Abbreviations
aTibial length
cClearance
c1, . . . , c4Integration constants
eGap between articular surfaces
.
eRate change in the gap between articular surfaces
fre,.
eFunction characterizing the force perceived by articular surfaces
FrDimensionless resistance force to the action of an external load
HArticular cartilage layer thickness
ˆ
hThicknes of poro-elastic articular cartilage layers
eh(t)Total layer thickness of synovial fluid
h(x,t)Synovial film thickness
H,hDimensionless parameters Hand h(x,t)
KDiffusive drag in articular cartilage
LLength of the cylindrical joint model
lSynovial fluid constant with the dimension of length: l=qη
µ
L,lDimensionless parameters Land l
pSynovial fluid pressure
pDimensionless parameter p
REffective radius of curvature of the contact of talus and tibia: 1
R=1
R1−1
R2
R1Radius of talus curvature
R2Radius of tibia curvature
tTime
u, v, w Velocity field components of fluid media
W(t)Law of change of an external load for unit of length
x,y,zCartesian coordinates of the ankle model
αParameter: α=l2
Φ
βPolar angle: β=±a
R
δ2Small parameter: δ2=µa
KH2
εSmall parameter: ε=H
R
εDimensionless parameter: ε=e
c
ηCouple stress synovial fluid constant
θPolar angle of the ankle model
µSynovial fluid dynamic viscosity
µaApparent viscosity of the interstitial fluid
ΦPermeability of the cartilage matrix
ΦDimensionless parameter Φ
Appendix A
Table A1. Numerical values used for the calculations [45].
Parameters Numerical Values Units
a14 ×10−3[m]
L28 ×10−3[m]
c7.25 ×10−7[m]
R3.5 ×10−1[m]
R122 ×10−3[m/s]
φ2×10−14 [m2]
β4×10−2[rad]
µ1×10−2[Pa s]
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References
1.
Pavlova, V.N. Components of the internal environment of the joints and their functional interaction. Adv. Mod. Biol.
1989
,107,
238–242. (In Russian)
2.
Pawlak, Z.; Pai, R.; Mrela, A.; Kaczmarek, M.; Yusuf, K.Q.; Urbaniak, W. Natural articular cartilage: A smart biointerface.
J. Comput. Methods Sci. Eng. 2019,19, 479–489. [CrossRef]
3. Chernyakova, Y.M.; Pinchuk, L.S. The synovial joint as a “smart” friction unit. Frict. Wear 2007,28, 410–417. (In Russian)
4.
Pinchuk, L.S.; Chernyakova, Y.M.; Ermakov, S.F. Tribo Physics of Synovial Fluid, 1st ed.; Belarusian Science: Minsk, Belarus, 2010;
pp. 3–381. (In Russian)
5.
Turovskaya, E.F.; Alekseeva, L.I.; Filatova, E.G. Modern ideas about the pathogenetic mechanisms of pain in osteoarthritis. Sci.
Pract. Rheumatol. 2014,52, 438–444. (In Russian) [CrossRef]
6. Sakovets, T.G. Features of neuropathic pain with joint damage. Pract. Med. 2014,4, 103–106. (In Russian)
7. Lim, A.Y.; Doherty, M. What of guidelines for osteoarthritis? Int. J. Rheum. Dis. 2011,14, 136–144. [CrossRef] [PubMed]
8.
Kraus, V.B.; Kilfoil, M.; Hash, T.W.; McDaniel, G.; Renner, J.B.; Carrino, J.A.; Adams, S. Atlas of radiographic features of
osteoarthritis of the ankle and hindfoot. Osteoarthritis Cartilage 2015,14, 2059–2085. [CrossRef]
9.
Knee Osteoarthritis: Forms, Diseases, Primary Analyzes, Stages. Available online: https://gp195.ru/bolezni/osteoartrit-lechenie.
html (accessed on 15 December 2020).
10.
Kotelkina, A.A.; Struchko, G.Y.; Merkulova, L.M.; Kostrova, O.Y.; Stomenskaya, I.S.; Timofeeva, N.Y. Characteristics of synovial
fluid in normal conditions and in some pathological processes. Acta Med. Eurasica 2017,4, 24–30.
11.
Netyaga, S.V.; Dubrovin, G.M.; Netyaga, A.A. The role of cytological examination of synovial fluid in the diagnosis of degenerative-
dystrophic changes in the joints. Man His Health 2005,1, 45–49. (In Russian)
12.
WHO. Department of Chronic Diseases and Health Promotion. Available online: http://www.who.int/chp/topics/rheumatic/
en/ (accessed on 23 September 2020).
13.
Kaplan, W.; Wirtz, V.J.; Mantel-Teeuwisse, A.; Stolk, P.; Duthey, B.; Laing, R. Priority Medicines for Europe and the World 2013 Update;
World Health Organization: Geneva, Switzerland, 2013.
14.
Dabiri, Y.; Li, L.P. Influences of the depth-dependent material inhomogeneity of articular cartilage on the fluid pressurization in
the human knee. Med. Eng. Phys. 2013,35, 1591–1598. [CrossRef]
15.
Mow, V.C.; Gu, W.Y.; Chen, F.H. Structure and Function of Articular Cartilage and Meniscus. In Basic Orthopaedic Biomechanics and
Mechano-Biology, 3rd ed.; Mow, V.C., Huiskes, R., Eds.; Lippincott Williams & Wilkins: Philadelphia, PA, USA, 2005; pp. 181–258.
16.
Mow, V.C.; Kuei, S.C.; Lai, W.M.; Armstrong, C.G. Biphasic Creep and Stress Relaxation of Articular Cartilage in Compression:
Theory and Experiments. J. Biomech. Eng. 1980,102, 73–84. [CrossRef] [PubMed]
17.
Pavlova, V.N.; Kopyeva, T.N.; Slutsky, L.I.; Pavlov, G.G. Cartilage, 1st ed.; Medicine: Moscow, Russia, 1988; pp. 3–320. (In Russian)
18.
Brandt, K.D.; Dieppe, P.; Radin, E.L. Etiopathogenesis of Osteoarthritis. Rheum. Dis. Clin. N. Am.
2008
,34, 531–559.
[CrossRef] [PubMed]
19.
Davies-Tuck, M.L.; Wluka, A.E.; Wang, Y.; Teichtahl, A.J.; Jones, G.; Ding, C.; Cicuttini, F.M. The natural history of cartilage
defects in people with knee osteoarthritis. Osteoarthr. Cartil. 2008,16, 337–342. [CrossRef]
20.
Scott, D.; Coleman, P.J.; Mason, R.M.; Levick, J.R. Concentration Dependence of Interstitial Flow Buffering by Hyaluronan in
Synovial Joints. Microvasc. Res. 2000,59, 345–353. [CrossRef] [PubMed]
21.
Scott, D.; Coleman, P.J.; Mason, R.M.; Levick, J.R. Interaction of intraarticular hyaluronan and albumin in the attenuation of fluid
drainage from joints. Arthritis Rheum. 2000,43, 1175–1182. [CrossRef]
22.
Hunter, W. Of the structure and diseases of articulating cartilages, by William Hunter, surgeon. Philos. Trans. R. Soc. Lond.
1743
,
42, 514–521. [CrossRef]
23. Dedinaite, A. Biomimetic lubrication. Soft Matter 2012,8, 273–284. [CrossRef]
24. Klein, J. Molecular mechanisms of synovial joint lubrication. J. Eng. Tribol. 2006,220, 691–710. [CrossRef]
25.
Wang, M.; Liu, C.; Thormann, E.; D
˙
edinait
˙
e, A. Hyaluronan and Phospholipid Association in Biolubrication. Biomacromolecules
2013,14, 4198–4206. [CrossRef]
26.
D
˙
edinait
˙
e, A.; Wieland, D.C.F.; Bełdowski, P.; Claesson, P.M. Biolubrication synergy: Hyaluronan—Phospholipid interactions at
interfaces. Adv. Colloid Interface Sci. 2019,274, 102050. [CrossRef]
27. Pradal, C.; Yakubov, G.E.; Williams, M.A.K.; McGuckin, M.A.; Stokes, J.R. Lubrication by biomacromolecules: Mechanisms and
biomimetic strategies. Bioinspiration Biomim. 2019,14, 051001. [CrossRef] [PubMed]
28.
Liao, J.; Smith, D.W.; Miramini, S.; Gardiner, B.S.; Zhang, L. A coupled contact model of cartilage lubrication in the mixed-mode
regime under static compression. Tribol. Int. 2020,1–11, 106185. [CrossRef]
29.
Dowson, D.; Unsworth, A.; Wright, V. Analysis of ‘boosted lubrication’ in human joints (reprinted from vol 1, 1959). Proc. Inst.
Mech. Eng. Part C J. Mech. Eng. Sci. 2009,223, 71–76.
30. Dowson, D. Paper 12: Modes of Lubrication in Human Joints. Proc. Inst. Mech. Eng. Conf. Proc. 1966,181, 45–54. [CrossRef]
31. Dowson, D.; Wright, V.; Longfield, M.D. Human joint lubrication. Biomed. Eng. 1969,4, 160–165. [PubMed]
32. Wright, V.; Dowson, D.; Unsworth, A. The lubrication and stiffness of joints. Mod. Trends Rheumatol. 1971,2, 30–45. [PubMed]
33.
Unsworth, A.; Dowson, D.; Wright, V. Some new evidence on human joint lubrication. Ann. Rheum. Dis.
1973
,32, 587–588.
[CrossRef] [PubMed]
Lubricants 2021,9, 15 22 of 24
34.
Jin, Z.M.; Dowson, D.; Fisher, J. The Effect of Porosity of Articular Cartilage on the Lubrication of a Normal Human Hip Joint.
Proc. Inst. Mech. Eng. Part H J. Eng. Med. 1992,206, 117–124. [CrossRef] [PubMed]
35.
Jalali-Vahid, D.; Jagatia, M.; Jin, Z.M.; Dowson, D. Prediction of lubricating film thickness in a ball-in-socket model with a soft
lining representing human natural and artificial hip joints. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
2001
,215, 363–372. [CrossRef]
36.
Jin, Z.M.; Dowson, D. Elastohydrodynamic Lubrication in Biological Systems. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
2005
,219,
367–380. [CrossRef]
37. Dowson, D. Bio-tribology. Faraday Discuss. 2009,156, 9–30. [CrossRef] [PubMed]
38.
Ghanbarzadeh, A.; Wilson, M.; Morina, A.; Dowson, D.; Neville, A. Development of a new mechano-chemical model in boundary
lubrication. Tribol. Int. 2016,93, 573–582. [CrossRef]
39.
Medley, J.B.; Dowson, D.; Wright, V. Transient Elastohydrodynamic Lubrication Models for the Human Ankle Joint. Eng. Med.
1984,13, 137–151. [CrossRef] [PubMed]
40. Dowson, D.; Jin, Z.-M. Micro-Elastohydrodynamic Lubrication of Synovial Joints. Eng. Med. 1986,15, 63–65. [CrossRef]
41. Dowson, D. Elastohydrodynamic and micro-elastohydrodynamic lubrication. Wear 1995,190, 125–138. [CrossRef]
42.
Fein, R.S. Research Report 3: Are Synovial Joints Squeeze-Film Lubricated? Proc. Inst. Mech. Eng. Conf. Proc.
1966
,181,
125–128. [CrossRef]
43.
Hou, J.S.; Mow, V.C.; Lai, W.M.; Holmes, M.H. An analysis of the squeeze-film lubrication mechanism for articular cartilage.
J. Biomech. 1992,25, 247–259. [CrossRef]
44.
Hlaváˇcek, M. The role of synovial fluid filtration by cartilage in lubrication of synovial joints—II. Squeeze-film lubrication:
Homogeneous filtration. J. Biomech. 1993,26, 1151–1160. [CrossRef]
45.
Hlavachek, M. Squeeze-film lubrication of the human ankle joint with synovial fluid filtrated by articular cartilage with the
superficial zone worn out. J. Biomech. 2000,33, 1415–1422.
46.
Hlavachek, M. The thixotropic effect of the synovial fluid in squeeze-film lubrication of the human hip joint. Biorheology
2001
,
38, 319–334.
47.
Ruggiero, A.; Gòmez, E.; D’Amato, R. Approximate Analytical Model for the Squeeze-Film Lubrication of the Human Ankle Joint
with Synovial Fluid Filtrated by Articular Cartilage. Tribol. Lett. 2011,41, 337–343. [CrossRef]
48.
Naduvinamani, N.B.; Savitramma, G.K. Squeeze Film Lubrication between Rough Poroelastic Rectangular Plates with Micropolar
Fluid: A Special Reference to the Study of Synovial Joint Lubrication. ISRN Tribol. 2013,2013, 431508. [CrossRef]
49.
De Boer, G.N.; Raske, N.; Soltanahmadi, S.; Dowson, D.; Bryant, M.G.; Hewson, R.W. A porohyperelastic lubrication model for
articular cartilage in the natural synovial joint. Tribol. Int. 2020,149, 105760. [CrossRef]
50. Ruggiero, A. Milestones in Natural Lubrication of Synovial Joints. Front. Mech. Eng. 2020,6, 52. [CrossRef]
51.
Medley, J.B.; Dowson, D.; Wright, V. Surface Geometry of the Human Ankle Joint. Eng. Med.
1983
,12, 35–41. [CrossRef] [PubMed]
52.
Ruggiero, A.; Gomez, E.; Roberto, D. Approximate closed-form solution of the synovial fluid film force in the human ankle joint
with non-Newtonian lubricant. Tribol. Int. 2013,57, 156–161. [CrossRef]
53. Stokes, V.K. Couple Stresses in Fluids. Phys. Fluids 1966,9, 1709. [CrossRef]
54.
Ankle X-ray: Indications, Conduct, Results. Available online: https://ocrb.ru/lechenie/rentgenografiya-golenostopnogo-
sustava.html (accessed on 15 December 2020).
55.
Schmidt, T.A.; Gastelum, N.S.; Nguyen, Q.T.; Schumacher, B.L.; Sah, R.L. Boundary lubrication of articular cartilage: Role of
synovial fluid constituents. Arthritis Rheum. 2007,56, 882–891. [CrossRef] [PubMed]
56.
Ghosh, S.; Dipankar, C.; Nabangshu, S.; Belinda, P.-M. Tribological role of synovial fluid compositions on artificial joints—A
systematic review of the last 10 years. Lubr. Sci. 2014,26, 387–410. [CrossRef]
57.
Lin, J.R. Couple-stress effect on the squeeze film characteristics of hemi-spherical bearings with reference to synovial joints. Appl.
Mech. Eng. 1996,1, 317–332.
58.
Meziane, A.; Bou-Saïd, B.; Tichy, J. Modelling human hip joint lubrication subject to walking cycle. Lubr. Sci.
2008
,20,
205–222. [CrossRef]
59.
Yousfi, M.; Bou-Saïd, B.; Tichy, J. An analytical study of the squeezing flow of synovial fluid. Mech. Ind.
2013
,14, 59–69. [CrossRef]
60.
De Boer, G.N.; Raske, N.; Soltanahmadi, S.; Bryant, M.G.; Hewson, R.W. Compliant-poroelastic lubrication in cartilage-on-cartilage
line contacts. Tribol. Mater. Surf. Interfaces 2020,14, 151–165. [CrossRef]
61.
Ruggiero, A.; Sicilia, A. A Mixed Elasto-Hydrodynamic Lubrication Model for Wear Calculation in Artificial Hip Joints. Lubricants
2020,8, 72. [CrossRef]
62.
Naduvinamani, N.B.; Hiremath, P.S.; Fathima, S.T. On the squeeze film lubrication of long porous journal bearings with couple
stress fluids. Ind. Lubr. Tribol. 2005,57, 12–20. [CrossRef]
63.
Ateshian, G.A.; Hung, C.T. The natural synovial joint: Properties of cartilage. Proc. Inst. Mech. Eng. Part J J. Eng. Tribol.
2006
,220,
657–670. [CrossRef]
64.
Gill, T.J.; Asnis, P.D.; Berkson, E.M.; Gill, T.J. The Treatment of Articular Cartilage Defects Using the Microfracture Techniques.
J. Orthop. Sports Phys. Ther. 2006,10, 728–738. [CrossRef]
65.
Julkunen, P.; Livarinen, J.; Brama, P.A.; Arokoski, J.; Jurvelin, S.; Helminen, H.J. Maturation of collagen fibril network structure in
tibial and femoral cartilage of rabbits. Osteoarthr. Cartil. 2010,18, 406–415. [CrossRef]
66.
Steward, A.J.; Kelly, D.J. Mechanical regulation of mesenchymal stem cell differentiation. J. Anat.
2015
,227, 717–731. [CrossRef]
Lubricants 2021,9, 15 23 of 24
67.
Wong, M.; Siegrist, M.; Cao, X. Cyclic compression of articular cartilage explants is associated with progressive consolidation and
altered expression pattern of extracellular matrix proteins. Matrix Biol. 1999,18, 391–399. [CrossRef]
68.
Mauck, R.L.; Soltz, M.A.; Wang, C.C.B.; Wong, D.D.; Chao, P.-H.G.; Valhmu, W.B.; Hung, C.T.; Ateshian, G.A. Functional Tissue
Engineering of Articular Cartilage Through Dynamic Loading of Chondrocyte-Seeded Agarose Gels. J. Biomech. Eng.
2000
,122,
252–260. [CrossRef] [PubMed]
69.
Ng, K.W.; Mauck, R.L.; Wang, C.C.-B.; Kelly, T.-A.N.; Ho, M.M.-Y.; Chen, F.H.; Ateshian, G.A.; Hung, C.T. Duty Cycle of Deforma-
tional Loading Influences the Growth of Engineered Articular Cartilage. Cell. Mol. Bioeng.
2009
,2, 386–394.
[CrossRef] [PubMed]
70.
Smith, R.L.; Lin, J.; Trindade, M.C.D.; Shida, J.; Kajiyama, G.; Vu, T.; Hoffman, A.R.; Van Der Meulen, M.C.H.; Goodman, S.B.;
Schurman, D.J.; et al. Time-dependent effects of intermittent hydrostatic pressure on articular chondrocyte type II collagen and
aggrecan mRNA expression. J. Rehabil. Res. Dev. 2000,37, 153–161. [PubMed]
71.
Smith, R.L.; Rusk, S.F.; Ellison, B.E.; Wessells, P.; Tsuchiya, K.; Carter, D.R.; Caler, W.E.; Sandell, L.J.; Schurman, D.J.
In vitro
stimulation of articular chondrocyte mRNA and extracellular matrix synthesis by hydrostatic pressure. J. Orthop. Res.
1996
,14,
53–60. [CrossRef] [PubMed]
72.
Guilak, F.; Meyer, B.C.; Ratcliffe, A.; Mow, V.C. The effects of matrix compression on proteoglycan metabolism in articular
cartilage explants. Osteoarthr. Cartil. 1994,2, 91–101. [CrossRef]
73.
Nugent, G.E.; Aneloski, N.M.; Schmidt, T.A.; Schumacher, B.L.; Voegtline, M.S.; Sah, R.L. Dynamic shear stimulation of bovine
cartilage biosynthesis of proteoglycan 4. Arthritis Rheum. 2006,54, 1888–1896. [CrossRef] [PubMed]
74.
Li, Z.; Yao, S.-J.; Alini, M.; Stoddart, M.J. Chondrogenesis of Human Bone Marrow Mesenchymal Stem Cells in Fibrin–
Polyurethane Composites Is Modulated by Frequency and Amplitude of Dynamic Compression and Shear Stress. Tissue
Eng. Part A 2010,16, 575–584. [CrossRef] [PubMed]
75.
Schätti, O.; Grad, S.; Goldhahn, J.; Salzmann, G.; Li, Z.; Alini, M.; Stoddart, M.J. A combination of shear and dynamic
compression leads to mechanically induced chondrogenesis of human mesenchymal stem cells. Eur. Cells Mater.
2011
,22, 214–225.
[CrossRef] [PubMed]
76.
Huang, A.H.; Baker, B.M.; Ateshian, G.A.; Mauck, R.L. Sliding contact loading enhances the tensile properties of mesenchymal
stem cell-seeded hydrogels. Eur. Cell Mater. 2012,24, 29–45. [CrossRef] [PubMed]
77.
Carter, D.R.; Blenman, E.R.; Beaupré, G.S. Correlations between mechanical stress history and tissue differentiation in initial
fracture healing. J. Orthop. Res. 1988,6, 736–748. [CrossRef] [PubMed]
78. Biot, M.A. General Theory of Three-Dimensional Consolidation. J. Appl. Phys. 1941,12, 155–164. [CrossRef]
79. Biot, M.A. Theory of Elasticity and Consolidation for a Porous Anisotropic Solid. J. Appl. Phys. 1955,26, 182–185. [CrossRef]
80.
Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Low frequency range. J. Acoust. Soc. Am.
1956,28, 168–178. [CrossRef]
81.
Biot, M.A. Theory of propagation of elastic waves in a fluid-saturated porous solid. II. Higher frequency range. J. Acoust. Soc. Am.
1956,28, 179–191. [CrossRef]
82. Biot, M.A.; Willis, D.G. The elastic coefficients of the theory of consolidation. J. Appl. Mech. 1957,24, 594–601.
83.
Biot, M.A. Generalized Theory of Acoustic Propagation in Porous Dissipative Media. J. Acoust. Soc. Am.
1962
,34,
1254–1264. [CrossRef]
84. Biot, M.A. Theory of finite deformation of porous solid. Indiana Univ. Math. J. 1972,21, 597–620. [CrossRef]
85. Biot, M.A. Generalized Lagrangian equations of non-linear reaction-diffusion. Chem. Phys. 1982,66, 11–26. [CrossRef]
86. Cowin, S.C. Bone poroelasticity. J. Biomech. 1999,32, 217–238. [CrossRef]
87. Cowin, S.C. Anisotropic poroelasticity: Fabric tensor formulation. Mech. Mater. 2004,36, 665–677. [CrossRef]
88.
Maslov, L.B. Algorithm for the numerical analysis of biological tissues based on a two-phase medium model. Bull. Ivanovo State
Power Univ. 2005,3, 1–9. (In Russian)
89. Maslov, L.B. Poroelastic models of vibrations of biological tissues. Bull. Nizhny Novgorod Univ. 2011,4, 499–501. (In Russian)
90. Maslov, L.B. Mathematical model of bone tissue structural rearrangement. Russ. J. Biomech. 2013,17, 39–63. (In Russian)
91. Maslov, L.B. Mathematical Model of Bone Regeneration. Mech. Compos. Mater. 2017,53, 399–414. [CrossRef]
92.
Carter, D.R.; Rapperport, D.J.; Fyhrie, D.P.; Schurman, D.J. Relation of coxarthrosis to stresses and morphogenesis: A finite
element analysis. Acta Orthop. Scand. 1987,58, 611–619. [CrossRef] [PubMed]
93.
Carter, D.R.; Wong, M. The role of mechanical loading histories in the development of diarthrodial joints. J. Orthop. Res.
1988
,6,
804–816. [CrossRef] [PubMed]
94.
Carter, D.R.; Beaupre, C.S. Linear elastic and poroelastic models of cartilage can produce comparable stress results: A comment
on Tanck et al. (J. Biomech. 32, 153–161, 1999). J. Biomech. 1999,32, 1255–1257.
95.
Carter, D.R.; Wong, M. Modelling cartilage mechanobiology. Philos. Trans. R. Soc. Lond. Ser. B Biol. Sci.
2003
,358,
1461–1471. [CrossRef]
96.
Carter, D.R.; Beaupré, G.S.; Wong, M.; Smith, R.L.; Andriacchi, T.P.; Schurman, D.J. The Mechanobiology of Articular Cartilage
Development and Degeneration. Clin. Orthop. Relat. Res. 2004,427S, S69–S77. [CrossRef]
97.
Soltz, M.A.; Ateshian, G.A. Experimental verification and theoretical prediction of cartilage interstitial fluid pressurization at an
impermeable contact interface in confined compression. J. Biomech. 1998,31, 927–934. [CrossRef]
98.
Soltz, M.A.; Ateshian, G.A. Interstitial fluid pressurization during confined compression cyclical loading of articular cartilage.
Ann. Biomed. Eng. 2000,28, 150–159. [CrossRef] [PubMed]
Lubricants 2021,9, 15 24 of 24
99.
Rose, L.F.; Wolf, E.J.; Brindle, T.; Cernich, A.; Dean, W.K.; Dearth, C.L.; Grimm, M.; Kusiak, A.; Nitkin, R.; Potter, K.; et al. The con-
vergence of regenerative medicine and rehabilitation: Federal perspectives. npj Regen. Med.
2018
,10, 3–19.
[CrossRef] [PubMed]
100.
Poliakov, A.; Pakhaliuk, V.; Popov, V.L. Current Trends in Improving of Artificial Joints Design and Technologies for Their
Arthroplasty. Front. Mech. Eng. 2020,6, 4. [CrossRef]
101.
Poliakov, O.; Olinichenko, G.; Pashkov, Y.; Kalinin, M.; Kramar, V.; Burkov, D. A new design of a unipolar hip endoprosthesis
focused on the best quality of life of the patients during the postoperative period. In Proceedings of the 2012 International
Conference on Biomedical Engineering, Penang, Malaysia, 27–28 February 2012; pp. 22–27. [CrossRef]
102.
Polyakov, A.; Pakhaliuk, V.; Kalinin, M.; Kramar, V.; Kolesova, M.; Kovalenko, O. System Analysis and Synthesis of Total Hip
Joint Endoprosthesis. Procedia Eng. 2015,100, 530–538. [CrossRef]
