An overview on platelet concentrates in tissue regeneration in periodontology

Kapil Bhangdiya; Anand Wankhede; Priyanka Paul Madhu; Amit Reche; Kapil Bhangdiya; Anand Wankhede; Priyanka Paul Madhu; Amit Reche

doi:10.3934/bioeng.2023005

AIMS Bioengineering

2023, Volume 10, Issue 1: 53-61. doi: 10.3934/bioeng.2023005

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Review Special Issues

An overview on platelet concentrates in tissue regeneration in periodontology

1.
Department Of Periodontology, Sharad Pawar Dental College and Hospital, Datta Meghe Institute Of Higher Education and Research (Deemed To Be University) Sawangi (Meghe) Wardha, Maharashtra 442001, India
2.
Department Of Public Health Dentistry, Sharad Pawar Dental College and Hospital, Datta Meghe Institute Of Higher Education and Research (Deemed To Be University) Sawangi (Meghe) Wardha, Maharashtra 442001, India

Received: 22 December 2022 Revised: 20 January 2023 Accepted: 06 February 2023 Published: 14 February 2023

Recent research on the use of platelet concentration in medicinal dentistry has enhanced the potential for tissue regeneration. The ability of platelets to enhance tissue regeneration by using different forms of platelet concentration (blood component) such as platelet-rich plasma (PRP), platelet-rich fibrin (PRF), standard platelet-rich fibrin, advanced platelet-rich fibrin, etc. PRP became widely used once its potential for tissue regeneration was discovered; however, the use of anticoagulants restricted the easy use of PRP as compared to PRF; however, after the discovery of PRF because of the easy formulation, it became widely used in medicinal dentistry. The purpose of this review is to elaborate on the importance of platelet concentration in periodontal regeneration.

Keywords:

Citation: Kapil Bhangdiya, Anand Wankhede, Priyanka Paul Madhu, Amit Reche. An overview on platelet concentrates in tissue regeneration in periodontology[J]. AIMS Bioengineering, 2023, 10(1): 53-61. doi: 10.3934/bioeng.2023005

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Abstract

1. Introduction

The microstructure of portable artificial kidneys is responsible for the blood filtration and urine formation (mechanical process of biofluid) which acts as a micromachine due to very small size but do the complex process of reabsorption, filtration, excretion and secretion in its different parts. Kidneys contain renal speck and renal tubule; renal speck is further split into glomerulus and Bowman's capsule where blood is filtered through the glomerulus circulates and filtered blood then comes to the renal tubule where reabsorption takes place. Within the nephron, glomerular purification comprises on reflexive movement of blood plasma from glomerulus capillaries to the Bowman's capsule that is easily permeable to H₂O, small solutes like urea, sodium ion, and glucose but not porous to hemoglobin and platelets. The next part of the renal tubule is the proximal convoluted tubule (PCT) where effective and reactive transport take place to reabsorb glucose, sodium chloride, and water from the glomerular filtrate. The current model describes biomechanics of plasma having variable viscosity depending upon the distance from centre to the proximal tube wall that is the main segment of kidneys where constant reabsorption and filtration take place.

Different studies of viscous flow through renal tubule having permeable wall property with constant, linear and nonlinear rate of reabsorption have been discussed. Burgen et. al. [1] introduced a study of viscous flow through kidney with glucose reabsorption and developed mathematical equations for solute reabsorption and used the results to check the high affinity of glucose. The dynamics of biofluid through renal tubule and flow behavior in the presence of constant reabsorption on the wall has been discussed by Macey et. al. [2]. He analysed the viscous flow through infinite cylinder using the Navier Stokes equations with constant and linear reabsorption on the walls of cylinder and noticed the Poiseuille flow for constant reabsorption and for the linear reabsorption the flow lines become flattens instead of curve lines. The dynamical properties of the viscous flow through a permeable tube had been examined by Marshall et. al. [3] and found the exact solution with the help of Navier-Stokes equation for velocity and pressure using the Darcy's law also, the approximate solution for the leakage flux has calculated for the proximal renal tubule. Berkstresser et. al. [4], Tewarson et. al. [5] and Gupta et. al. [6] introduced the numerical and analytic techniques for the solution of mathematical model of viscous flow through renal tubule. Ahmad et al. [7] discussed the exact solution of viscous flow through the renal tube and showed the periodic flow in radial direction of the tube. Muthu et al. [8] presented the finite difference method to find the viscous flow through the cross section of tube with permeable boundary, he found the axial velocity and mean pressure generated by the reabsorption on the wall and flux on the axial direction. Koplik et al. [9], Pozrikidis et al. [10], Kropinski et al. [11] and Haroon et al [12] obtained the results for volume flow rate, glucose reabsorption, velocity and shear stress for Newtonian flow through permeable tube and slit by different methods. They found the slip and no slip effect on the velocity, leakage flux, flow rate and used the resulting formulas to find the reabsorption rate and mean pressure against the fractional reabsorption.

The viscosity of plasma filtered in kidney is not a fixed quantity, but it changes according to the diameter of proximal tubule, also flow resistance in renal tube decreases strongly with reduced diameter of the tube which is less than 0.3mm as mentioned in the study of Gilmer et. al. [13]. The flow resistance of the fluid to be filtered in proximal tube is considerably the case of Poiseuille flow because of pressure gradient which describes that resistance in the flow is low at the centre of tube and high near the tube wall. Few studies have been presented the effect of viscosity on the blood flow through kidney (Lamport et. al. [14],[15], Gaylor et. al. [16] and Omori et. al. [17], Halfenstein et. al. [18], Mukhopadhyay et. al. [19],[20]) but in all these studies only theoretical and experimental data has been used. Recently, Mehboob et. al. [21] presented the effect of variable viscosity and constant reabsorption on the slow viscous flow through narrow straight tubule and found the formulas for the leakage flux, mean pressure drop and flow rate but this problem was formulated in cylindrical coordinate system. There is not a single study available in literature which describes the mathematical formulas of creeping flow having varying viscosity across the height of channel with uniform absorption on the permeable walls of the channel in rectangular coordinate system.

Keeping above studies in mind, it has been assumed that the viscosity is an exponential function of distance from the centre to the boundary and reabsorption on the wall is constant. In this paper to study the effect of exponential type of viscosity for the viscous flow through a kidney, authors have considered the uniform reabsorption on the wall and discussed the problem formulation in section 2, methodology of complex system is included in section 3, mathematical results are presented in fourth section, application of the problem is in section 5, discussion is in section 6 and conclusion is included in section 7.

2. Materials and methods

The present mathematical model suggests that a viscous fluid across a rectangular cross section is flowing under the effect of constant flux applied at the entrance of slit. The -axis locating at the middle of rectangular slit and -axis in the vertical direction of centre line as mentioned in Figure 1, due to symmetric behaviour, the upper half of a rectangular slit has considered to reduce the calculations.

Since the kidney contains the proximal renal tube and movement in this tube is of Poiseuille type Layton et al [22] which requires that axial velocity is highest at the middle of slit that is stated in Eq (7). Since boundaries of renal tubule are permeable and reabsorption takes place through these boundaries at a constant rate which is stated in Eq (8). These hypotheses support to assume that thickness is a decreasing function of distance from base to the boundary, therefore in the current research authors have assumed the Navier-Stokes equation with non-constant viscosity. The current problem of two-dimension, incompressible steady state creeping flow through slit become complicated due to non-constant viscosity.

Figure 1. Schematic diagram of the problem.

DownLoad: Full-Size Img PowerPoint

The suggested model recommends that movement of fluid through a slit satisfy the following flow field.

$V = [u (x, y), v (x, y), 0],$ (1)

where u and v represents horizontal and transverse velocity components.

The fluid flow through the nephron is incompressible i.e density is constant throughout the flow field and satisfy the following continuity equation

$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0.$ (2)

The creeping viscous flow across a slit suggests that in the absence of inertia term NSE's with variable viscosity satisfy the following set of equations.

$\frac{\partial p}{\partial x} = \frac{\partial}{\partial x} (2 µ (y) \frac{\partial u}{\partial x}) + \frac{\partial}{\partial y} {µ (y) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})},$ (3)

$\frac{\partial p}{\partial y} = \frac{\partial}{\partial x} {µ (y) (\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})} + \frac{\partial}{\partial y} (2 µ (y) \frac{\partial v}{\partial y}) .$ (4)

2.1. Exponential viscosity

Case I: $µ (y) = e^{- β y}$

Since the viscosity is a decreasing function of radius of tube which is mentioned in Ref. (Gilmer et al [13]& Tripathi et al [23]) therefore, the variable viscosity in the following form has chosen

$µ (y) = e^{- β y}, β > 0.$ (5)

The assumptions of plasma flow through renal proximal tube and constant reabsorption on the permeable boundary satisfy the following boundary conditions

$\frac{\partial u}{\partial y} = 0, v = 0, a t y = 0,$ (6)

$u = 0, v = V_{0}, a t y = H,$ (7)

$Q_{0} = 2 W \int_{0}^{H} u (0, y) d y .$ (8)

Stream function and velocity profile are associated by the following expressions

$u = \frac{\partial ψ}{\partial y}, v = - \frac{\partial ψ}{\partial x} .$ (9)

The above expressions of stream function and Eqs (3,4) give the following form

$\frac{\partial p}{\partial x} = 2 e^{- β y} \frac{\partial^{3} ψ}{\partial x^{2} \partial y} + \frac{\partial}{\partial y} {e^{- β y} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}})},$ (10)

$\frac{\partial p}{\partial y} = e^{- β y} \frac{\partial}{\partial x} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}}) + 2 \frac{\partial}{\partial y} {e^{- β y} (- \frac{\partial^{2} ψ}{\partial x \partial y})} .$ (11)

Eliminating pressure gradient from Eqs (10,11), one can write the following expression.

$\nabla^{4} ψ - β^{2} E^{2} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (12)

where $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ a and $E^{2} = \frac{\partial^{2}}{\partial x^{2}} - \frac{\partial^{2}}{\partial y^{2}}$ .

Boundary conditions in terms of stream function are given as follows

$\frac{\partial^{2} ψ}{\partial y^{2}} = 0, \frac{\partial ψ}{\partial x} = 0, a t y = 0,$ (13)

$\frac{\partial ψ}{\partial y} = 0, \frac{\partial ψ}{\partial x} = - V_{0}, a t y = H,$ (14)

$ψ (0, 0) = 0, a n d ψ (0, H) = \frac{Q_{0}}{2 W} .$ (15)

The non-dimensional quantities are chosen from Ref. [12]

$\begin{array}{l} x^{*} = \frac{x}{L}, y^{*} = \frac{y}{L}, V_{0}^{*} = \frac{V_{0}}{U_{0}}, ψ^{*} = \frac{ψ}{U_{0} L}, β^{*} = β L, \\ Q_{0}^{*} = \frac{Q_{0}}{U_{0} W L}, τ_{w}^{*} = \frac{τ_{w}}{µ (y) U_{0} / L}, u^{*} = \frac{u}{U_{0}}, v^{*} = \frac{v}{U_{0}}, H^{*} = \frac{H}{L}, \end{array}$ (16)

where U₀ is reference velocity, L is the length of slit, H is the height and W is the width of slit. Nondimensional form of Eqs (12-15) after releasing “*” take the following form

$\nabla^{4} ψ - β^{2} E^{2} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (17)

$\frac{\partial^{2} ψ}{\partial y^{2}} = 0, \frac{\partial ψ}{\partial x} = 0, a t y = 0,$ (18)

$\frac{\partial ψ}{\partial y} = 0, \frac{\partial ψ}{\partial x} = - V_{0}, a t y = H,$ (19)

$ψ (0, 0) = 0, a n d ψ (0, H) = \frac{Q_{0}}{2 W} .$ (20)

2.1.1. Solution of the problem

To get the solution of problem given in Eqs (17-20), Inverse method [12] is used which requires the following function.

$ψ (x, y) = x R (y) + T (y),$ (21)

where R(y) and T(y) represent unknown functions.

Invoking Eq (21) in Eqs (17-20), one can obtain the following set of equations

$\frac{d^{4} R}{d y^{4}} - 2 β \frac{d^{3} R}{d y^{3}} + β^{2} \frac{d^{2} R}{d y^{2}} = 0,$ (22)

$\frac{d^{4} T}{d y^{4}} - 2 β \frac{d^{3} T}{d y^{3}} + β^{2} \frac{d^{2} T}{d y^{2}} = 0,$ (23)

and boundary conditions are

$R (0) = 0, \frac{d^{2} R (0)}{d y^{2}} = 0,$ (24)

$R (H) = - V_{0}, \frac{d R (H)}{d y} = 0,$ (25)

$T (0) = 0, \frac{d^{2} T (0)}{d y^{2}} = 0,$ (26)

$T (H) = \frac{Q_{0}}{2}, \frac{d T (H)}{d y} = 0.$ (27)

Solving Eqs (22,23) under the boundary conditions (24-27), one can obtained the following expressions for R(y) and T(y)

$(y) = \frac{V_{0} (2 + e^{H β} y β (1 - H β) - e^{y β} (2 - y β))}{- 2 + e^{H β} (2 - H β (2 - H β))},$ (28)

and

$(y) = \frac{Q_{0} (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 4 + e^{H β} (4 - 2 H β (2 - H β))} .$ (29)

Following stream function can be found after using R(y) and T(y) in Eq. (21)

$ψ (x, y) = \frac{(Q_{0} - 2 V_{0} x) (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 4 + e^{H β} (4 - 2 H β (2 - H β))}$ (30)

2.1.2. Velocity distribution

With the aid of above stream function and Eq (9), one can get the next components of velocity

$u (x, y) = \frac{- (Q_{0} - 2 V_{0} x) β (1 - H β - e^{- (H - y) β} (1 - y β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (31)

and

$v (x, y) = \frac{V_{0} (- 2 - e^{H β} y β (1 - H β) + e^{y β} (2 - y β))}{- 2 + e^{H β} (2 - H β (2 - H β))} .$ (32)

Above equations show that horizontal velocity u(x, y) depends upon x and y, also horizontal velocity is maximum at y = 0 i.e.

$u_{m a x} = \frac{- (Q_{0} - 2 V_{0} x) β (1 - H β - e^{- H β})}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (33)

and v(x, y) is maximum at y = H.

$v_{m a x} = V_{0} .$ (34)

Flow rate across the slit can be calculated by the following formula

$Q (x) = 2 \int_{0}^{H} u (x, y) d y,$ (35)

$Q (x) = Q_{0} - 2 V_{0} x,$ (36)

which results from the inlet to the outlet of rectangular cross section.

2.1.3. Pressure

In biological movements pressure performs an important role therefore to find the pressure, Eqs. (9-10) with the help of Eq (31) and Eq (32) are reduced into the following form

$\frac{\partial p}{\partial x} = - β e^{- β y} (x \frac{d^{2} R}{d y^{2}} + \frac{d^{2} T}{d y^{2}}) + e^{- β y} (x \frac{d^{3} R}{d y^{3}} + \frac{d^{3} T}{d y^{3}}),$ (37)

$\frac{\partial p}{\partial y} = - e^{- β y} (\frac{d^{2} R}{d y^{2}} - 2 β \frac{d R}{d y}) .$ (38)

Using R(y) and T(y), Eqs (37,38) take the following form

$\frac{\partial p}{\partial x} = \frac{- (Q_{0} - 2 V_{0} x) β^{3}}{- 4 + e^{H β} (4 - 2 H β (2 - H β))},$ (39)

$\frac{\partial p}{\partial y} = \frac{V_{0} β^{2} (2 e^{- y β} (1 - H β) - e^{- H β} (2 - y β))}{2 - 2 e^{- H β} - H β (2 - H β)} .$ (40)

Integration of above expressions yield the following form

$p (x, y) = \frac{- V_{0} β (4 e^{- y β} (1 - H β) + e^{- H β} y β (4 - y β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)} + f (x),$ (41)

where f(x) i is given by as follows

$f (x) = \frac{- x (Q_{0} - V_{0} x) β^{3}}{- 4 + e^{H β} (4 - 2 H β (2 - H β))} + C,$ (42)

where C is an arbitrary factor. Inserting Eq (42) in Eq (41), we can compose the subsequent form of pressure

$p (x, y) - p (0, 0) = \frac{- β (4 (- 1 + e^{- y β}) V_{0} (1 - H β) + e^{- H β} β (4 V_{0} y + Q_{0} x β - V_{0} (x^{2} + y^{2}) β))}{4 - 4 e^{- H β} - 2 H β (2 - H β)},$ (43)

where p(0, 0) shows pressure at creek of rectangular cross section.

The average pressure drop is defined as follows

$\overset{̅}{p} (x) = \frac{1}{H} \int_{0}^{H} (p (x, y) - p (0, 0)) d y .$ (44)

After inserting value of p(x, y) − p(0, 0) from Eq (43), the above equation takes the following form.

$\overset{̅}{p} (x) = \frac{12 V_{0} {(1 - H β)}^{2} + e^{- H β} (3 H Q_{0} x β^{3} + V_{0} (- 12 - H β (- 12 - 6 H β + (H^{2} + 3 x^{2}) β^{2})))}{12 e^{- H β} - 6 (2 - H β (2 - H β))} .$ (45)

Pressure drops over the length of rectangular cross section is given by the following formula

$Δ \bar{p} (L) = \bar{p} (0) - \bar{p} (L),$ (46)

which reduces to

$Δ \bar{p} (L) = \frac{- e^{- H β} L (- Q_{0} + L V_{0}) β^{3}}{4 - 4 e^{- H β} - 2 H β (2 - H β)} .$ (47)

2.1.4. Wall shear stress

Shear forces are very crucial in the viscous flow with variable viscosity and support to find the resistance on plane which are penned as follows

${τ_{w} |}_{y = H} = - µ (y) {(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x})}_{y = H},$ (48)

Invoking Eqs (31,32) in above equation, we attained

${τ_{w} |}_{y = H} = \frac{H (Q_{0} - 2 V_{0} x) β^{3}}{4 - 4 e^{- H β} - 2 H β (2 - H β)} .$ (49)

2.1.5. Fractional reabsorption and leakage flux

The ratio of reabsorption to filtration rate can be measured by the following formula

$F A = \frac{Q (0) - Q (L)}{Q (0)} = \frac{2 V_{0} L}{Q_{0}}$ (50)

Leakage flux is given by the following formula

$q (x) = - \frac{d Q (x)}{d x} = 2 V_{0} .$ (51)

2.2. Linear viscosity

Case II: $µ (y) = 1 - β y$

In this case we will take the following viscosity

$µ (y) = 1 - β y, β > 0,$ (52)

After injecting Eq (52) and Eq (9) in Eqs (2,3), we get

$\frac{\partial p}{\partial x} = 2 (1 - β y) \frac{\partial^{3} ψ}{\partial x^{2} \partial y} + \frac{\partial}{\partial y} {(1 - β y) (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}})},$ (53)

$\frac{\partial p}{\partial y} = (1 - β y) \frac{\partial}{\partial x} (\frac{\partial^{2} ψ}{\partial y^{2}} - \frac{\partial^{2} ψ}{\partial x^{2}}) + 2 \frac{\partial}{\partial y} {(1 - β y) (- \frac{\partial^{2} ψ}{\partial x \partial y})} .$ (54)

Reducing pressure gradient from Eqs (53,54), one can obtain the compatibility equation in terms of stream function

$(1 - β y) \nabla^{4} ψ - 2 β \frac{\partial}{\partial y} (\nabla^{2} ψ) = 0,$ (55)

where $\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}$ .

There will be no change in Eq (55) after using non-dimensional quantities given in Eq (16)

2.2.1. Solution of the problem

Substituting Eq (21) into the Eq (55) to get

$(1 - β y) \frac{d^{4} R}{d y^{4}} - 2 β \frac{d^{3} R}{d y^{3}} = 0,$ (56)

$(1 - β y) \frac{d^{4} T}{d y^{4}} - 2 β \frac{d^{3} T}{d y^{3}} = 0.$ (57)

On solving Eqs (56,57) under the boundary conditions (24-27), we get the solution

$R (y) = \frac{V_{0} (- 2 y β \ln (- β + H β^{2}) - 2 \ln (1 - y β) - y β (2 + 2 H β - y β - 2 \ln (- β + y β^{2})))}{- H β (- 2 - H β) + 2 \ln (1 - H β)},$ (58)

and

$T (y) = \frac{Q_{0} (2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{- 2 H β (- 2 - H β) + 4 \ln (1 - H β)},$ (59)

Invoking R(y) and T(y) into Eq. (21) to get the following stream function

$ψ (x, y) = \frac{(2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{2 H β (2 + H β) + 4 \ln (1 - H β)} (Q_{0} - 2 V_{0} x) .$ (60)

2.2.2. Velocity field

With the aid of Eq (9) and Eq (60), flow field in axial and vertical direction are found as follows

$u (x, y) = \frac{- (Q_{0} - 2 V_{0} x) β (- (H - y) β - \ln (- β + H β^{2}) + \ln (- β + y β^{2}))}{H β (2 + H β) + 2 ln (1 - H β)},$ (61)

and

$v (x, y) = \frac{V_{0} (2 y β \ln (- β + H β^{2}) + 2 \ln (1 - y β) - y β (- 2 - 2 H β + y β + 2 \ln (- β + y β^{2})))}{H β (2 + H β) + 2 ln (1 - H β)} .$ (62)

Horizontal velocity is maximum at y = 0.

$u_{m a x} = \frac{- (Q_{0} - 2 V_{0} x) β (- H β + ln (- β) - \ln (- β + H β^{2}))}{H β (2 + H β) + 2 ln (1 - H β)},$ (63)

and v(x, y) is maximum at y = H.

$v_{m a x} = V_{0} .$ (64)

Flow rate within the four-sided slit is attained by using Eq (61) in Eq (35).

$Q (x) = Q_{0} - 2 V_{0} x .$ (65)

2.2.3. Pressure distribution

To find pressure, Eqs (53,54) can be composed in following manner

$\frac{\partial p}{\partial x} = - β (x \frac{d^{2} R}{d y^{2}} + \frac{d^{2} T}{d y^{2}}) + (1 - β y) (x \frac{d^{3} R}{d y^{3}} + \frac{d^{3} T}{d y^{3}}),$ (66)

$\frac{\partial p}{\partial y} = - (1 - β y) \frac{d^{2} R}{d y^{2}} + 2 β \frac{d R}{d y} .$ (67)

By using Eqs (58,59) in Eqs (66,67) to get

$\frac{\partial p}{\partial x} = \frac{(Q_{0} - 2 V_{0} x) β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)},$ (68)

$\frac{\partial p}{\partial y} = \frac{2 V_{0} β^{2} (- 2 H β + 3 y β - 2 \ln (- β + H β^{2}) + 2 \ln (- β + y β^{2}))}{H β (2 + H β) + 2 \ln (1 - H β)} .$ (69)

Integrating Eq (69) yields

$p (x, y) = \frac{(Q_{0} - V_{0} x) x β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)} + H (y),$ (70)

where H(y) is given by as follows

$H (y) = \frac{- V_{0} β (4 y β \ln (- β + H β^{2}) + 4 \ln (1 - y β) - y β (- 4 - 4 H β + 3 y β + 4 \ln (- β + y β^{2})))}{H β (2 + H β) + 2 \ln (1 - H β)} + C,$ (71)

where C is an arbitrary factor. Using Eq (71) in Eq (70), we can write the subsequent form of pressure

$p (x, y) - p (0, 0) = - β (- β (Q_{0} x β + V_{0} (- x^{2} β + y (- 4 - 4 H β + 3 y β))) + 4 V_{0} (\ln (1 - y β) - y β (- \ln (- β + H β^{2}) + \ln (- β + y β^{2})))) / H β (2 + H β) + 2 ln (1 - H β),$ (72)

where p(0, 0) shows the pressure at entrance of the slit.

Mean pressure drop can be find by substituting the value of p(x, y) − p(0, 0) in Eq (44) as follows

$\overset{̅}{p} (x) = V_{0} (\frac{1}{H} - 2 β) + \frac{(H^{2} V_{0} + x (Q_{0} - V_{0} x)) β^{3}}{H β (2 + H β) + 2 \ln (1 - H β)} .$ (73)

To get the pressure drop over the length L, invoke the values of p(0) and p(L) in Eq (47).

$Δ \bar{p} (L) = - \frac{L (Q_{0} - L V_{0}) β^{3}}{H β (2 + H β) + 2 ln (1 - H β)} .$ (74)

2.2.4. Wall shear stress

Invoking Eqs (61,62) in Eq (48) to attain wall shear stress

${τ_{w} |}_{y = H} = - \frac{H (Q_{0} - 2 V_{0} x) β^{3}}{(1 - H β) (H β (2 + H β) + 2 \ln (1 - H β))} .$ (75)

3. Constant viscosity

Case III: If we apply the limit β → 0 on Eq 31, 32 and 43 and solving the resulting expressions by the L' Hospital rule, one can attain the results of constant viscosity µ which are same as mentioned in Ref. [12].

$ψ (x, y) = \frac{1}{2} (\frac{Q_{0}}{2} - V_{0} x) \frac{y}{H} (3 - {(\frac{y}{H})}^{2}),$ (76)

$u (x, y) = \frac{3 (Q_{0} - 2 V_{0} x)}{4 H} (1 - {(\frac{y}{H})}^{2}),$ (77)

$v (x, y) = \frac{V_{0}}{2} \frac{y}{H} (3 - {(\frac{y}{H})}^{2}),$ (78)

$p (x, y) - p (0, 0) = - \frac{3 µ (Q_{0} - V_{0} x) x}{2 H^{3}} - \frac{3 µ V_{0} y^{2}}{2 H^{3}} .$ (79)

4. Application to kidneys flow

We have utilized biological data of mammalian kidneys to find the values of V₀ and Δp in artificial renal tubule which is assumed to be the permeable channel. In this study length of artificial renal tubule (slit) is chosen to be L = 0.12cm and height of slit has considered as H = 0.0005cm, average viscosity µ = 6.9×10⁻³dynsec/cm², Q₀ = 1.25×10⁻⁴cm³/sec, β = 2.5×10⁻³/cm and W = 0.1 cm (as $H ≪ W$ ) from Zhang et al [24]. Table 1 shows the calculated values of mean pressure and reabsorption rate for the constant and variable viscosity. As cited in Refs. (Mehboob et. al. [21], Layton et al [22], Korkmaz et al [25], Herranz et al [26] and Tortora et al [27] in usual conditions glomerulus pressure is assumed to be about $45 m m H g (59995.07433675 d y n / c m^{2})$ which is above the average pressure than that observed in tubes elsewhere in the body. It has stated in Table 1 the reducing values of reabsorption rate causes to lower the fractional reabsorption, but the average pressure drop in the renal tubule enhances with the falling values of reabsorption rate, also the assumption of variable viscosity indicate that average pressure drop is elevated as matched with the case of constant viscosity.

The results of this study are beneficial to measure the change in pressure compared to the fractional reabsorption because the viscosity of the liquid decay near the boundary which is the consequence of lubricating nature of permeable wall. The numerical values given in table 1 indicate that the various rate of reabsorption requires the different mean pressure which may be supportive for the patients of nephrotic disorder. The present data from table 1 also support the person that how much intake of fluid is needed to generate the mean pressure next to the filtration rate which has examined by the blood CP and urine test.

Table 1. Fractional reabsorption, reabsorption rate of fluid and pressure variation for persistent (constant) viscosity and variable viscosity in a renal tubule.

Viscosity	FA	90%	80%	70%	60%
	V₀cm/sec	4.7×10⁻³	4.2×10⁻³	3.6×10⁻³	3.1×10⁻³
Constant	Δp(0.12)dyn/cm²	1212.19	2404.51	3835.3	5027.62
Variable	Δp(0.12)dyn/cm²	6460.62	7025.69	7703.78	8268.84

| Show Table

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5. Results

Figure 2. Effect of V₀ at entrance region (x = 0.1).

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Figure 3. Effect of β at entrance region (x = 0.1).

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Figure 4. Effect of V₀ at middle region (x = 0.5).

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Figure 5. Effect of β at middle region (x = 0.5).

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Figure 6. Effect of V₀ at exit region (x = 0.9).

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Figure 7. Effect of β at exit region (x = 0.9).

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Figure 8. Effect of V₀ in the tranverse direction of flow.

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Figure 9. Effect of β in the transverse direction of flow.

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Figure 10. Effect of V₀ on pressure difference.

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Figure 11. Effect of β on pressure difference.

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Figure 12. Effect of V₀ on wall shear stress.

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Figure 13. Effect of β on wall shear stress.

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Figure 14. (a) V₀ = 0.4, (b) V₀ = 1.2, (c) V₀ = 1.5.

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Figure 15. (a) β = 0.1, (b) β = 0.2, (c) β = 0.65.

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6. Discussion

In this research the mathematical results of flow speed, flow pattern, pressure and shearing force are presented by the graphs which are plotted in software Mathematica under the effect of different values of reabsorption velocity V₀ and viscosity parameter β at x = 0.1, x = 0.5, x = 0.9, which are entry, central and leaving points of the rectangular cross section respectively.

Figures (2), (4) and (6) show the decaying effect of reabsorption rate V₀ on axial velocity at the points x = 0.1, 0.5, 0.9 on the slit and observe that decay is swifter near the center region (x = 0.5) but near the departure region (x = 0.9) back flow has been examined. It has been noted that near the centre of slit drift is highest due to change in pressure and near the borders of slit (y = h) the fluid movement become motionless due to deposition of the particles (reabsorption V₀).

Figures (3), (5) and (7) explain the decelerating impact of viscosity parameter β on axial rate near the center of the slit at the points x = 0.1, 0.5, 0.9 of the slit. The axial flow rises away from the centre (y = h) and decays near the centre line (y = 0) due to the growing values of viscosity parameter β, because resistance at the boundary wall (y = h) makes the motionless flow at the boundary but the resistive forces become stronger near the centre of slit (y = 0) due to pressure gradient $\frac{d p}{d x}$ which helps to make the decreasing effect on axial velocity.

Figure 8 illustrates the variation of reabsorption rate V₀ at the vertical direction of flow field, which displays the symmetry about the centre line (y = 0). The presence of reabsorption V₀ indicates the growing effect on the transverse flow near the permeable membrane of renal tube. Decaying effect of viscosity parameter β on the vertical flow is displayed through the Figure 9, it is observed that the resistive force retards the movement of the fluid molecules in vertical direction.

The pressure p plays a vital role for the flow of viscous fluid, therefore graphical results for pressure p are displayed in Figure 10 and Figure 11. The decreasing effect of reabsorption velocity V₀ on pressure difference is displayed in Figure 10 which shows that presence of reabsorption V₀ on the membrane require the low change in pressure for the viscous flow through permeable channel and the pressure at the entrance x = 0.1 is high as compared with the other region x = 0.9 of the slit. Figure 11 indicates that rising values of β help to reduce the pressure difference from entry point x = 0.1 to the centre line y = 0, which is evidenced that decreasing viscosity β help to reduce the pressure for the viscous fluid flow.

When the fluid is in contact with the surface then wall shear stress is an important phenomenon on the boundary surface. The variation in wall shear stress against V₀ is described in Figure 12 which shows the reduction in the shear stress on the wall with the mounting values of reabsorption V₀. The constant reabsorption V₀ on the wall helps to reduce the shearing forces on the permeable wall, which indicates that flow across the tube move easily without the large load. The rising values of parameter β results to improve the shearing forces on the wall because the growing values of β increases the resistance near the boundary wall which requires the high load for the fluid flow near the boundary.

The pattern of the flow field can be observed by the streamlines which are plotted in Figure 14 (a-c) and Figure 15 (a-c). It is examined that addition in reabsorption velocity V₀ triggers the number of lines which indicates that the fluid is becoming thin in the presence of reabsorption V₀, but the number of streamlines reduces with the mounting values of viscosity parameter β which indicates that fluid is going to be thin, and it contains more resistance for the fluid flow.

7. Conclusions

This research presents the mathematical modelling of plasma flow through the renal tube, the two-dimensional slit in rectangular coordinate is considered in the analogy of proximal renal tube and viscous fluid is assumed as a plasma with variable viscosity from centre line to the boundary surface due to reabsorption on the permeable wall. The complex mathematical model is solved by the inverse method and exact expression for flow speed, load, shear force and filtration are calculated. The application of the present study is presented with the numerical data present in the literature. The special cases for the linear and constant reabsorption are also obtained and observed that results of constant reabsorptions match with the existing results (Haroon et al. [12]) which validates the current research. The impact of reabsorption and viscosity parameter on the physical quantities of the flow field are observed through graphs. The numerical results of mean pressure and constant reabsorption against filtration rate show that constant viscosity of the plasma in slit requires less amount of average pressure but the biofluid having variable viscosity along the slit requires high average pressure for the fluid flow.

Conflict of interest

The authors declare no conflict of interest.

References

[1]	Carranza FA, Newman MG, Takei H, et al. (2006) Carranza's Clinical periodontology 10th edition. Philadelpia: Elsevier Inc, 2006.
[2]	Ross R, Glomset J, Kariya B, et al. (1974) A platelet-dependent serum factor that stimulates the proliferation of arterial smooth muscle cells in vitro. Proc Natl Acad Sci USA 71: 1207-1210. https://doi.org/10.1073/pnas.71.4.1207
[3]	Jurk K, Kehrel BE (2005) Platelets: physiology and biochemistry. Semin Thromb Hemost 31: 381-392. https://doi.org/10.1055/s-2005-916671
[4]	Schwertz H, Koster S, Kahr WH, et al. (2010) Anucleate platelets generate progeny. Blood 115: 3801-3809. https://doi.org/10.2174/157016112801784648
[5]	Amaral RJ, Balduino A (2015) Platelets in tissue regeneration. The Non-Thrombotic Role of Platelets in Health and Disease. London: IntechOpen 221-237.
[6]	Kiran NK, Mukunda KS, Tilak Raj TN (2011) Platelet concentratess: A promising innovation in dentistry. J Dent Sci Res 2: 50-61.
[7]	Stellos K, Kopf S, Paul A, et al. (2010) Platelets in regeneration. Semin Thromb Hemost 36: 175-184. https://doi.org/10.1055/s-0030-1251502
[8]	Ghanaati S, Booms P, Orlowska A, et al. (2014) Advanced platelet-rich fibrin: a new concept for cell-based tissue engineering by means of inflammatory cells. J Oral Implantol 40: 679-689. https://doi.org/10.1563/aaid-joi-D-14-00138
[9]	Whitman DH, Berry RL, Green DM (1997) Platelet gel: an autologous alternative to fibrin glue with applications in oral and maxillofacial surgery. J Oral Maxil Surg 55: 1294-1299. https://doi.org/10.1016/S0278-2391(97)90187-7
[10]	Roubelakis MG, Trohatou O, Roubelakis A, et al. (2014) Platelet-rich plasma (PRP) promotes feta mesenchymal stem/stromal cell migration and wound healing process. Stem Cell Rev Rep 10: 417-428. https://doi.org/10.1007/s12015-013-9494-8
[11]	Marx RE, Carlson ER, Eichstaedt RM, et al. (1998) Platelet-rich plasma: growth factor enhancement for bone grafts. Oral Surg Oral Med Oral Pathol Oral Radiol Endodontics 85: 638-646. https://doi.org/10.1016/S1079-2104(98)90029-4
[12]	Rozman P, Bolta Z (2007) Use of platelet growth factors in treating wounds and soft-tissue injuries. Acta Dermatoven APA 16: 156-165.
[13]	Kilian O, Flesch I, Wenisch S, et al. (2004) Effects of platelet growth factors on human mesenchymal stem cells and human endothelial cells in vitro. Eur J Med Res 9: 337-344.
[14]	Fréchette JP, Martineau I, Gagnon G (2005) Platelet-rich plasmas: growth factor content and roles in wound healing. J Dent Res 84: 434-439. https://doi.org/10.1177/154405910508400507
[15]	Kour P, Pudakalkatti PS, Vas AM, et al. (2018) Comparative evaluation of antimicrobial efficacy of platelet-rich plasma, platelet-rich fibrin, and injectable platelet-rich fibrin on the standard strains of Porphyromonas gingivalis and Aggregatibacter actinomycetemcomitans. Contemp Clin Dent 9: S325-S330. https://doi.org/10.4103/ccd.ccd_367_18
[16]	Rouwkema J, Khademhosseini A (2016) Vascularization and angiogenesis in tissue engineering: beyond creating static networks. Trends Biotechnol 34: 733-745. https://doi.org/10.1016/j.tibtech.2016.03.002
[17]	Dambhare A, Bhongade ML, Dhadse PV, et al. (2019) A randomized controlled clinical study of autologous platelet rich fibrin (PRF) in combination with HA and beta-TCP or HA and beta-TCP alone for treatment of furcation defects. J Hard Tissue Biol 28: 185-190. https://doi.org/10.2485/jhtb.28.185
[18]	Marx RE (2004) Platelet-rich plasma: evidence to support its use. J Oral Maxillofac Surg 62: 489-496. https://doi.org/10.1016/j.joms.2003.12.003
[19]	Meheux C, McCulloch P, Lintner D, et al. (2016) Efficacy of intra-articular platelet-rich plasma injections in knee osteoarthritis: a systematic review. Arthroscopy 32: 495-505. https://doi.org/10.1016/j.arthro.2015.08.005
[20]	Choukroun J, Adda F, Schoeffler C, et al. (2001) Uneopportunite enparo-implantologie: Le PRF. Implantodontie 42: e62.
[21]	Ghanaati S, Booms P, Orlowska A, et al. (2014) Advanced platelet-rich fibrin: a new concept for cell-based tissue engineering by means of inflammatory cells. J Oral Implantol 40: 679-689. https://doi.org/10.1563/aaid-joi-D-14-00138
[22]	Dohan DM, Choukroun J, Diss A, et al. (2006) Platelet-rich fibrin (PRF): a second-generation platelet concentration. Part I: technological concepts and evolution. Oral Surg Oral Med Oral Pathol Oral Radiol Endodontics 101: e37-e44. https://doi.org/10.1016/j.tripleo.2005.07.008
[23]	Dohan DM, Choukroun J, Diss A, et al. (2006) Platelet-rich fibrin (PRF): a second-generation platelet concentration. Part II: platelet-related biologic features. Oral Surg Oral Med Oral Pathol Oral Radiol Endodontics 101: e45-e50. https://doi.org/10.1016/j.tripleo.2005.07.009
[24]	Dohan DM, Choukroun J, Diss A, et al. (2006) Platelet-rich fibrin (PRF): a second-generation platelet concentration. Part III: leucocyte activation: a new feature for platelet concentrations?. Oral Surg Oral Med Oral Pathol Oral Radiol Endodontics 101: e51-e55. https://doi.org/10.1016/j.tripleo.2005.07.010
[25]	Farid Shehab M, Hamid NM, Askar NA, et al. (2018) Immediate mandibular reconstruction via patient-specific titanium mesh tray using electron beammelting/CAD/rapid prototyping techniques: One-year follow-up. Int J Med Robot Comp 14: e1895. https://doi.org/10.1002/rcs.1895
[26]	Pimentel T, Ritto F, Canellas JV, et al. (2020) Re: Randomized double-blind clinical trial evaluation of bone healing after third molar surgery with the use of leukocyte- and platelet-rich fibrin. Int J Oral Maxillofac Surg 49: 692. https://doi.org/10.1016/j.ijom.2019.10.018
[27]	Choukroun J, Ghanaati S (2018) Reduction of relative centrifugation force within injectable platelet-rich-fibrin (PRF) concentrates advances patients' own inflammatory cells, platelets and growth factors: the first introduction to the low speed centrifugation concept. Eur J Trauma Emerg Surg 44: 87-95. https://doi.org/10.1007/s00068-017-0767-9
[28]	Wang X, Zhang Y, Choukroun J, et al. (2017) Behavior of gingival fibroblasts on titanium implant surfaces in combination with either injectable-PRF or PRP. Int J Mol Sci 18: 331. https://doi.org/10.3390/ijms18020331
[29]	Lei L, Yu Y, Ke T, et al. (2019) The application of three-dimensional printing model and platelet-rich fibrin technology in guided tissue regeneration surgery for severe bone defects. J Oral Implantol 45: 35-43. https://doi.org/10.1563/aaid-joi-D-17-00231
[30]	Kyyak S, Blatt S, Pabst A, et al. (2020) Combination of an allogenic and a xenogenic bone substitute material with injectable platelet-rich fibrin–A comparative in vitro study. J Biomater Appl 35: 83-96. https://doi.org/10.1177/0885328220914407
[31]	Shah R, Thomas R, Mehta DS (2017) An update on the protocols and biologic actions of platelet rich fibrin in dentistry. Eur J Prosthodont Re 25: 64-72. https://doi.org/10.1922/ejprd_01690shah09

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AIMS Bioengineering

An overview on platelet concentrates in tissue regeneration in periodontology

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Abstract

1. Introduction

2. Materials and methods

2.1. Exponential viscosity

2.1.1. Solution of the problem

2.1.2. Velocity distribution

2.1.3. Pressure

2.1.4. Wall shear stress

2.1.5. Fractional reabsorption and leakage flux

2.2. Linear viscosity

2.2.1. Solution of the problem

2.2.2. Velocity field

2.2.3. Pressure distribution

2.2.4. Wall shear stress

3. Constant viscosity

4. Application to kidneys flow

5. Results

6. Discussion

7. Conclusions

References

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AIMS Bioengineering

An overview on platelet concentrates in tissue regeneration in periodontology

Related Papers:

Abstract

1. Introduction

2. Materials and methods

2.1. Exponential viscosity

2.1.1. Solution of the problem

2.1.2. Velocity distribution

2.1.3. Pressure

2.1.4. Wall shear stress

2.1.5. Fractional reabsorption and leakage flux

2.2. Linear viscosity

2.2.1. Solution of the problem

2.2.2. Velocity field

2.2.3. Pressure distribution

2.2.4. Wall shear stress

3. Constant viscosity

4. Application to kidneys flow

5. Results

6. Discussion

7. Conclusions

References

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