Citation: Mônica H. M. Nascimento, Milena T. Pelegrino, Joana C. Pieretti, Amedea B. Seabra. How can nitric oxide help osteogenesis?[J]. AIMS Molecular Science, 2020, 7(1): 29-48. doi: 10.3934/molsci.2020003
[1] | Zeliha Korpinar, Mustafa Inc, Dumitru Baleanu . On the fractional model of Fokker-Planck equations with two different operator. AIMS Mathematics, 2020, 5(1): 236-248. doi: 10.3934/math.2020015 |
[2] | Zihang Cai, Chao Jiang, Yuzhu Lei, Zuhan Liu . Global classical solution of the fractional Nernst-Planck-Poisson-Navier- Stokes system in R3. AIMS Mathematics, 2024, 9(7): 17359-17385. doi: 10.3934/math.2024844 |
[3] | Shu-Nan Li, Bing-Yang Cao . Anomalies of Lévy-based thermal transport from the Lévy-Fokker-Planck equation. AIMS Mathematics, 2021, 6(7): 6868-6881. doi: 10.3934/math.2021402 |
[4] | Hassan Khan, Umar Farooq, Fairouz Tchier, Qasim Khan, Gurpreet Singh, Poom Kumam, Kanokwan Sitthithakerngkiet . The analytical analysis of fractional order Fokker-Planck equations. AIMS Mathematics, 2022, 7(7): 11919-11941. doi: 10.3934/math.2022665 |
[5] | Yan Ren, Yan Cheng, Yuzhen Chai, Ping Guo . Dynamics and density function of a HTLV-1 model with latent infection and Ornstein-Uhlenbeck process. AIMS Mathematics, 2024, 9(12): 36444-36469. doi: 10.3934/math.20241728 |
[6] | Shuqi Tang, Chunhua Li . Decay estimates for Schrödinger systems with time-dependent potentials in 2D. AIMS Mathematics, 2023, 8(8): 19656-19676. doi: 10.3934/math.20231002 |
[7] | Ying He, Bo Bi . Threshold dynamics and density function of a stochastic cholera transmission model. AIMS Mathematics, 2024, 9(8): 21918-21939. doi: 10.3934/math.20241065 |
[8] | Jingen Yang, Zhong Zhao, Xinyu Song . Statistical property analysis for a stochastic chemostat model with degenerate diffusion. AIMS Mathematics, 2023, 8(1): 1757-1769. doi: 10.3934/math.2023090 |
[9] | Wen Wang, Yang Zhang . Global regularity to the 3D Cauchy problem of inhomogeneous magnetic Bénard equations with vacuum. AIMS Mathematics, 2023, 8(8): 18528-18545. doi: 10.3934/math.2023942 |
[10] | Sen Ming, Jiayi Du, Yaxian Ma . The Cauchy problem for coupled system of the generalized Camassa-Holm equations. AIMS Mathematics, 2022, 7(8): 14738-14755. doi: 10.3934/math.2022810 |
In this paper, we are concerned on the Cauchy problem to a nonlinear Fokker-Planck equation as follows
{∂tF+v⋅∇xF=ρ∇v⋅(∇vF+vF),F(0,x,v)=F0(x,v), | (1.1) |
where the nonnegative unknown functions F(t,x,v) is the distribution function of particles with position x=(x1,x2,x3)∈R3 and velocity v=(v1,v2,v3)∈R3 at time t≥0, and the density ρ(t,x) is defined as ρ=∫R3Fdv.
In statistical mechanics, nonlinear Fokker-Planck equation is a partial differential equation which describes the Brownian motion of particles. This equation illustrates the evolution of particle probability density function with velocity, time and space position under the influence of resistance or random force. This equation is also widely used in various fields such as plasma physics, astrophysics, nonlinear hydrodynamics, theory of electronic circuitry and laser arrays, population dynamics, human movement sciences and marketing.
The global equilibrium for the nonlinear Fokker-Planck Eq (1.1) is the normalized global Maxwellian
μ=μ(v)=(2π)−32e−|v|22. |
Therefore, we can define the perturbation f=f(t,x,v) by
F(t,x,v)=μ+μ12f(t,x,v), |
then the Cauchy problem (1.1) of the nonlinear Fokker-Planck equation is reformulated as
{∂tf+v⋅∇xf=ρLf,ρ=1+∫R3μ12fdv,f(0,x,v)=f0(x,v)=μ−12(F0(x,v)−μ), | (1.2) |
where the linear Fokker-Planck operator L is given by
Lf=μ−12∇v⋅(μ∇v(μ−12f))=Δvf+14(6−|v|2)f. | (1.3) |
Let us define the velocity orthogonal projection
P:L2(R3v)→Span{μ12, viμ12(1≤i≤3)}. |
For any given function f(t,x,v)∈L2(R3v), one has
Pf=a(t,x)μ12+b(t,x)⋅vμ12, | (1.4) |
with
a=∫R3μ12fdv,b=∫R3v⋅μ12fdv. | (1.5) |
Then by the macro-micro decomposition introduced in [9], we get the decomposition of solutions f(t,x,v) of the nonlinear Fokker-Planck Eq (1.1) as follows
f(t,x,v)=Pf(t,x,v)+{I−P}f(t,x,v), | (1.6) |
where I denotes the identity operator, Pf and {I−P}f are called the macroscopic and the microscopic component of f(t,x,v), respectively.
Before the statement of main result, we need list some notations used in this paper.
● A≲B means that there is a constant C>0 such that A≤CB. A∼B means A≲B and B≲A.
● For a multi-index α=(α1,α2,α3), the length of α is |α|=α1+α2+α3. We denote ∂α=∂αx=∂α1x1∂α2x2∂α3x3 and use ∂i to denote ∂xi for each i=1,2,3.
● For any function f and g, denote the inner product and norm as follows
⟨f,g⟩:=∫R3fgdv,|f|2L2v=∫R3f2dv,|f|2ν:=|f|2L2ν=∫R3(|∇vf|2+ν(v)|f|2)dv where ν(v):=1+|v|2,‖f‖2ν:=∫R3|f|2νdx=∫R3∫R3(|∇vf|2+ν(v)|f|2)dvdx,‖f‖2:=‖f‖2L2(R3x×R3v) or ‖a‖2:=‖a‖2L2(R3x). |
● Denoting the function spaces HNxL2v and HNxL2ν with the norm as
‖f‖2HNxL2v=∑|α|≤N‖∂αxf‖2,‖f‖2HNxL2ν=∑|α|≤N‖∂αxf‖2ν. |
The basic properties of the linearized Fokker-Planck operator L in (1.3) can be referred in [3,6,7,10,15] as follows
⟨f,Lg⟩=⟨Lf,g⟩,KerL=Span{μ12},L(vμ12)=−vμ12, | (1.7) |
and the Fokker-Planck operator L is coercive in the sense that there is a positive constant λ0 such that
−⟨f,Lf⟩=∫R3|∇vf+v2f|2dv≥λ0|{I−P}f|2ν+|b|2. | (1.8) |
There are a lot of results about the global existence and large time behavior of solutions to the Fokker-Planck type equation. Such as for the Fokker-Planck-Boltzmann equation, DiPerna and Lions [4] first obtained the renormalized solution and established global existence for the Cauchy problem with large data. Li and Matsumura [12] proved that the strong solution for initial data near an absolute Maxwellian exist globally in time and tends asymptotically in the L∞y(L1ξ)-norm to another time dependent self-similar Maxwellian in large time. The global existence and temporal decay estimates of classical solutions are established based on the nonlinear energy method developed in [9] under Grad's angular cut-off in [17] and without cut-off in [16], respectively.
As for the Vlasov-Poisson-Fokker-Planck equation, Duan and liu [6] obtained the time-periodic small-amplitude solution in the three dimensional whole space by Serrin's method. Hwang and Jang [10], Wang [18] obtained the global existence and the time decay of the solution. For the problem (1.1), the global existence is proved by combining uniform-in-time energy estimates and the decay rates of the solution is obtained by using the precise spectral analysis of the linearized Fokker-Planck operator as well as the energy method in [13]. Interested readers can refer to the references [2,7,8,12,14,19] for more related details.
For the nonlinear Fokker-Planck equation, Imbert and Mouhot [11] obtained the Hölder continuity by De Giorgi and Moser argument together with the averaging lemma. Liao et al.[13] deduced the global existence of the Cauchy problem to the equation based on the energy estimates and the decay rates of the solutions by using the precise spectral analysis of the linearized Fokker-Planck operator in Sobolev space HNx,N≥4. Also the new difficulty caused by the nonlinear term was resolved by additional tailored weighted-in-v energy estimates suitable for Fokker-Planck operators. However, in this paper, we find that we can deal with the difficult by using the definition of the linearized Fokker-Planck operator L in (1.3) which is not necessary to estimate the dissipation ‖Lg‖2HNxL2v.
The rest of this paper is organized as follows. In Section 2, we give the main result of this paper. In Section 3, we deduce the microscopic and macroscopic dissipation by a refined energy method, respectively. Section 4 is devoted to close the a priori estimate, then the proof of main theorem is completed based on the continuation argument.
Now we define the energy norm and the corresponding dissipation rate norm, respectively, by
E(t)∼∑|α|≤2‖∂αxf‖2, | (2.1) |
D(t)∼∑|α|≤2(‖∂αx{I−P}f‖2ν+‖∂αxb‖2)+∑|α|≤1‖∂αx∇xa‖2. | (2.2) |
With the above preparation in hand, our main result can be stated as follows.
Theorem 2.1. Assume there exist a sufficiently small positive constant ϵ0 such that F0(x,v)=μ+μ12f0(x,v)≥0 satisfies E(0)≤ϵ0, then the Cauchy problem (1.2) admits a unique global solution f(t,x,v) satisfying F(t,x,v)=μ+μ12f(t,x,v)≥0, and it holds that
E(t)+∫t0D(s)ds≤E(0), | (2.3) |
for any t>0. In particular, we have the global energy estimate
supt≥0‖f(t)‖H2xL2v≤‖f0‖H2xL2v. |
Remark 2.1. ● Compared with the integer Sobolev space H4x used in [13], the regularity assumption on the initial data in H2x is weaker by virtue of the Sobolev embedding in Lemma 3.1, especially the estimate of L6(R3).
● In order to overcome the difficulty from the nonlinear term, the authors in [13] need to estimate the dissipation ‖Lg‖2HNxL2v. However, it seems to be not necessary for our estimates.
In this section, we will derive the energy estimates for the nonlinear Fokker-Planck equation. The first part is concerned on the estimates of the microscopic dissipation and the second part is about the estimates of macroscopic dissipation by the macroscopic equations similar as [13]. We need list the following lemma about Sobolev inequalities which are very important to obtain the corresponding energy estimates.
Lemma 3.1. (See [1,5].) Let u∈H2(R3), then there is a constant C>0 such that
● ‖u‖L∞≤C‖∇u‖12‖∇2u‖12≤C‖∇u‖H1,
● ‖u‖L6≤C‖∇u‖,
● ‖u‖Lq≤C‖u‖H1, 2≤q≤6.
Firstly, we need the estimates of the microscopic dissipation for the solution f in (1.2).
Lemma 3.2. It holds that
12ddt∑|α|≤2‖∂αxf‖2+λ0∑|α|≤2‖∂αx{I−P}f‖2ν+∑|α|≤2‖∂αxb‖2≲E12(t)D(t), | (3.1) |
for any t>0.
Proof. Step 1. α=0. Multiply (1.2)1 by f and integrate over R3v×R3x to obtain
12ddt∫R3∫R3f2dvdx−∫R3∫R3fLfdvdx=∫R3∫R3afLfdvdx. | (3.2) |
By using (1.5) and (1.7), we have
⟨LPf,f⟩=⟨L(aμ12),f⟩+⟨L(b⋅vμ12),f⟩=a⟨L(μ12),f⟩+b⟨L(vμ12),f⟩=−b⟨vμ12,f⟩=−|b|2. | (3.3) |
Similarly, we can get
⟨L{I−P}f,f⟩=⟨L{I−P}f,{I−P}f⟩+⟨L{I−P}f,Pf⟩=⟨L{I−P}f,{I−P}f⟩+⟨LPf,{I−P}f⟩=⟨L{I−P}f,{I−P}f⟩. | (3.4) |
Therefore, by (3.3) and (3.4) and the definition of L (1.3), we can obtain
⟨Lf,f⟩=⟨LPf,f⟩+⟨L{I−P}f,f⟩=⟨L{I−P}f,{I−P}f⟩−|b|2=⟨Δv{I−P}f,{I−P}f⟩+32⟨{I−P}f,{I−P}f⟩−⟨|v|2{I−P}f,{I−P}f⟩−|b|2=−|∇v{I−P}f|2L2v+32|{I−P}f|2L2v−|v{I−P}f|2L2v−|b|2, |
where we have used the integration by parts of v, i.e.,
⟨Δv{I−P}f,{I−P}f⟩=−⟨∇v{I−P}f,∇v{I−P}f⟩=−|∇v{I−P}f|2L2v. |
Consequently,
|⟨Lf,f⟩|=|∇v{I−P}f|2L2v+32|{I−P}f|2L2v+|v{I−P}f|2L2v+|b|2≤C(|{I−P}f|2ν+|b|2). |
Furthermore, Sobolev embedding in Lemma 3.1 yields
|∫R3∫R3afLfdvdx|≤∫R3|a||⟨f,Lf⟩|dx≲∫R3|a|(|{I−P}f|2ν+|b|2)dx≤‖a‖2L∞x(‖{I−P}f‖2ν+‖b‖2)≲‖∇xa‖H1xD(t)≲E12(t)D(t). |
Therefore, from (3.2) we have
12ddt‖f‖2+λ0‖{I−P}f‖2ν+‖b‖2≲E12(t)D(t). |
Step 2. 1≤|α|≤2. Taking ∂αx of (1.2)1 yields
∂t∂αxf+v⋅∇x∂αxf=L∂αxf+∂αx(aLf), | (3.5) |
Multiply above equation by ∂αxf and integrate over R3v×R3x to get
12ddt∫R3∫R3|∂αxf|2dvdx−∫R3∫R3∂αxfL(∂αxf)dvdx=∫R3∫R3∑|β|≤|α|Cβα∂α−βxaL(∂βxf)(∂αxf)dvdx. | (3.6) |
Case 1. β=0. The estimates of the last term in above equation is as follows by the definition of L (1.3):
∫R3∫R3∂αxaLf∂αxfdvdx=∫R3∫R3∂αxa(Δvf+14(6−|v|2)f)∂αxfdvdx=∫R3∫R3∂αxaΔvf∂αxfdvdx⏟J1+32∫R3∫R3∂αxaf∂αxfdvdx⏟J2−∫R3∫R3∂αxa|v|2f∂αxfdvdx⏟J3. | (3.7) |
Using the integration by parts of v, Hölder inequality and Sobolev embedding in Lemma 3.1 to get
|J1|=|∫R3∫R3∂αxa∇vf∂αx∇vfdvdx|≤∫R3|∂αxa||∇vf|L2v|∂αx∇vf|L2vdx≤‖∂αxa‖L2x‖∇vf‖L∞xL2v‖∂αx∇vf‖L2xL2v≲‖∂αxa‖L2x‖∇x∇vf‖H1xL2v‖∂αx∇vf‖L2xL2v≲E12(t)D(t), | (3.8) |
where we have used
‖∇x∇vf‖H1xL2v≲‖∇x∇vPf‖H1xL2v+‖∇x∇v{I−P}f‖H1xL2v≲‖∇xa‖H1x+‖∇xb‖H1x+‖∇x{I−P}f‖H1xL2ν≲D12(t), |
‖∂αx∇vf‖L2xL2v≲‖∂αx∇vPf‖L2xL2v+‖∂αx∇v{I−P}f‖L2xL2v≲‖∂αxa‖L2x+‖∂αxb‖L2x+‖∂αx{I−P}f‖L2xL2ν≲D12(t). |
Similarly, we can easily get
|J2|≲E12(t)D(t),|J3|≲E12(t)D(t). |
Case 2. β=1. The estimates of the last term in (3.6):
∫R3∫R3∂α−βxaL(∂βxf)(∂αxf)dvdx=∫R3∫R3∂α−βxaΔv∂βxf∂αxfdvdx⏟J4+32∫R3∫R3∂α−βxa∂βxf∂αxfdvdx⏟J5−∫R3∫R3∂α−βxa|v|2∂βxf∂αxfdvdx⏟J6. | (3.9) |
Using the similar techniques to estimate J1, we have
|J4|=|∫R3∫R3∂α−βxa∂βx∇vf∂αx∇vfdvdx|≤∫R3|∂α−βxa||∂βx∇vf|L2v|∂αx∇vf|L2vdx≤‖∂α−βxa‖L3x‖∂βx∇vf‖L6xL2v‖∂αx∇vf‖L2xL2v≲‖∂α−βxa‖H1x‖∇x∂βx∇vf‖L2xL2v‖∂αx∇vf‖L2xL2v≲E12(t)D(t). | (3.10) |
Similarly, we can easily get
|J5|≲E12(t)D(t),|J6|≲E12(t)D(t). |
Case 3. β=α. It holds that
∫R3∫R3aL(∂αxf)(∂αxf)dvdx=∫R3∫R3aΔv∂αxf∂αxfdvdx⏟J7+32∫R3∫R3a∂αxf∂αxfdvdx⏟J8−∫R3∫R3a|v|2∂αxf∂αxfdvdx⏟J9 | (3.11) |
Using the similar techniques to estimate J1, we have
|J7|=|∫R3∫R3a∂αx∇vf∂αx∇vfdvdx|≤∫R3|a||∂αx∇vf|L2v|∂αx∇vf|L2vdx≲‖a‖L∞x‖∂αx∇vf‖L2xL2v‖∂αx∇vf‖L2xL2v≲‖∇xa‖H1x‖∂αx∇vf‖2L2xL2v≲E12(t)D(t). | (3.12) |
Similarly, we can easily get
|J8|≲E12(t)D(t),|J9|≲E12(t)D(t). |
Take the summation over 1≤|α|≤2 to get
12ddt∑|α|≤2‖∂αxf‖2+λ0∑|α|≤2‖∂αx{I−P}f‖2ν+∑|α|≤2‖∂αxb‖2≲E12(t)D(t). |
Thus we complete the proof of Lemma 3.2.
Now we give the estimate of the macroscopic component a by the macroscopic equations.
Lemma 3.3. It holds that
ddt∑|α|≤1∫R3∂αxb∇x∂αxadx+∑|α|≤1‖∇x∂αxa‖2≲∑|α|≤1‖∇x∂αxb‖2+∑|α|≤1‖∇x∂αx{I−P}f‖2+∑|α|≤1‖∂αxb‖2+E12(t)D(t). | (3.13) |
Proof. Firstly, multiplying (1.2)1 by μ12 and vμ12 respectively, then integrating with respect to v over R3 to obtain
∂ta+∇x⋅b=0, | (3.14) |
and
∂tb+∇xa+∇x⋅⟨v⊗vμ12,{I−P}f⟩+(a+1)b=0. | (3.15) |
Secondly, taking ∂αx of (3.15) for |α|≤1 to get
∂αx∂tb+∇x∂αxa+∇x⋅⟨v⊗vμ12,∂αx{I−P}f⟩+∂αx(ab)+∂αxb=0. |
Multiply the above equation by ∇x∂αxa and integrate with respect to x to obtain
‖∇x∂αxa‖2=−∫R3∂αx∂tb∇x∂αxadx−∫R3∇x⋅⟨v⊗vμ12,∂αx{I−P}f⟩∇x∂αxadx−∫R3∂αx(ab)∇x∂αxadx−∫R3∂αxb∇x∂αxadx. | (3.16) |
Using (3.14) to get
∫R3∂αx∂tb∇x∂αxadx=ddt∫R3∂αxb∇x∂αxadx−∫R3∂αxb∇x∂αx∂tadx=ddt∫R3∂αxb∇x∂αxadx−‖∇x∂αxb‖2. | (3.17) |
By Young's inequality, we have
|∫R3∇x⋅⟨v⊗vμ12,∂αx{I−P}f⟩∇x∂αxadx|≲∫R3|∇x∂αx{I−P}f|L2v|∇x∂αxa|dx≲η‖∇x∂αxa‖2+Cη‖∇x∂αx{I−P}f‖2, | (3.18) |
and
|∫R3∂αxb∇x∂αxadx|≲η‖∇x∂αxa‖2+Cη‖∂αxb‖2, | (3.19) |
where η>0 is a sufficiently small universal constant and Cη>0. Using Sobolev embedding in Lemma 3.1 to derive
|∫R3∂αxab∇x∂αxadx|≲‖∂αxa‖L2x‖b‖L∞x‖∇x∂αxa‖L2x≲‖∂αxa‖L2x‖∇xb‖H1x‖∇x∂αxa‖L2x≲E12(t)D(t), |
and
|∫R3a∂αxb∇x∂αxadx|≲‖a‖L∞x‖∂αxb‖L2x‖∇x∂αxa‖L2x≲‖∇xa‖H1x‖∂αxb‖L2x‖∇x∂αxa‖L2x≲E12(t)D(t). |
Thus we can obtain
|∫R3∂αx(ab)∇x∂αxadx|=|∫R3(∂αxab+a∂αxb)∇x∂αxadx|≤∫R3|∂αxa||b||∇x∂αxa|dx+∫R3|a||∂αxb||∇x∂αxa|dx≲E12(t)D(t). | (3.20) |
Combining (3.17)–(3.20) with (3.16) to derive, for |α|≤1
ddt∑|α|≤1∫R3∂αxb∇x∂αxadx+∑|α|≤1‖∇x∂αxa‖2≲∑|α|≤1‖∇x∂αxb‖2+∑|α|≤1‖∇x∂αx{I−P}f‖2+∑|α|≤1‖∂αxb‖2+E12(t)D(t), |
where we take η>0 sufficiently small enough. Thus the proof of Lemma 3.3 is completed.
This section is devoted to proving our main result based on the continuation argument. First, we need to close the a priori estimate.
Proposition 4.1. There is a small positive constant M>0 such that if
sup0≤t≤TE(f(t))≤M |
for any 0<T<∞, then it holds that
ddtE(t)+D(t)≤0. | (4.1) |
Proof. Taking the linear combination (3.1)+κ×(3.13) with κ>0 sufficiently small to get
ddt(∑|α|≤2‖∂αxf‖2+κ∑|α|≤1∫R3∂αxb∇x∂αxadx)+κ∑|α|≤1‖∇x∂αxa‖2+λ0∑|α|≤2‖∂αx{I−P}f‖2ν+∑|α|≤2‖∂αxb‖2≲E12(t)D(t). | (4.2) |
Noticing that
∑|α|≤1|∫R3∂αxb∇x∂αxadx|≤12∑|α|≤1[‖∂αxb‖2+‖∇x∂αxa‖2]≤∑|α|≤2‖∂αxf‖2, |
then we have
−κ∑|α|≤2‖∂αxf‖2≤κ∑|α|≤1∫R3∂αxb∇x∂αxadx≤κ∑|α|≤2‖∂αxf‖2, |
i.e.,
(1−κ)∑|α|≤2‖∂αxf‖2≤∑|α|≤2‖∂αxf‖2+κ∑|α|≤1∫R3∂αxb∇x∂αxadx≤(1+κ)∑|α|≤2‖∂αxf‖2. |
Consequently, let κ>0 be small enough, it holds
∑|α|≤2‖∂αxf‖2+κ∑|α|≤1∫R3∂αxb∇x∂αxadx∼∑|α|≤2‖∂αxf‖2∼E(t). |
By (4.2) and the definition of D(t) (2.2), it derives to
ddtE(t)+D(t)≲E12(t)D(t). |
Finally, choosing M>0 to be small enough, then the desired estimate (4.1) is obtained.
Proof of Theorem 2.1. Firstly, the local-in-time existence and uniqueness of the solutions to the Cauchy problem (1.2) can be established by performing the standard arguments as in [13]. To extend the local solution into the global one, we can deduce that
E(t)+∫t0D(s)ds≤E(0), |
from (4.1) in Proposition 4.1 by virtue of the smallness assumption on E(0). Combining this with the local existence, the global existence of solution and uniqueness follows immediately from the standard continuity argument. This completes the proof of the global existence and the uniform estimate of Theorem 2.1.
This paper proves the global existence to the Cauchy problem on a nonlinear Fokker- Planck equation near Maxwellian with small-amplitude initial data by a refined nonlinear energy method. And the regularity assumption on the initial data is much weaker by virtue of the Sobolev embedding inequalities.
The authors would like to thank the anonymous reviewers for providing useful comments and suggestions which help to strengthen the manuscript.
The research of Xingang Zhang is supported by the Key Scientific Research Projects of Colleges and Universities in Henan Province of China under contracts 23A520027, 23A520038 and Key Scientific and Technological Research Projects in Henan Province under contracts 222102320369. The corresponding author is supported by the National Natural Science Foundation of China under contracts 12026263, Research ability cultivation fund of Hubei University of Arts and Science (2020kypytd006), the Project of Hubei University of Arts and Science (XK2021022), the Humanities and Social Science Youth Foundation of Ministry of Education of China (17YJC630084), the Hubei Provincial Department of Education (B2021211).
The authors declare that they have no competing interests.
[1] |
Feng X, Mcdonald JM (2011) Disorders of Bone Remodeling. Annu Rev Pathol 6: 121-145. doi: 10.1146/annurev-pathol-011110-130203
![]() |
[2] |
Van'T Hof RJ, Ralston SH (2001) Nitric oxide and bone. Immunology 103: 255-261. doi: 10.1046/j.1365-2567.2001.01261.x
![]() |
[3] |
Parfitt AM (1987) Bone remodeling and bone loss: Understanding the pathophysiology of osteoporosis. Clin Obstet Gynecol 30: 789-811. doi: 10.1097/00003081-198712000-00004
![]() |
[4] |
Diwan AD, Wang MX, Jang D, et al. (2000) Nitric Oxide Modulates Fracture Healing. J Bone Miner Res 15: 342-351. doi: 10.1359/jbmr.2000.15.2.342
![]() |
[5] |
Pandya CD, Lee B, Toque HA, et al. (2019) Age-Dependent Oxidative Stress Elevates Arginase 1 and Uncoupled Nitric Oxide Synthesis in Skeletal Muscle of Aged Mice. Oxid Med Cell Longev 2019: 1-9. doi: 10.1155/2019/1704650
![]() |
[6] |
Aguirre J, Buttery LDK, O'Shaughnessy M, et al. (2001) Endothelial nitric oxide synthase gene-deficient mice demonstrate marked retardation in postnatal bone formation, reduced bone volume, and defects in osteoblast maturation and activity. Am J Pathol 158: 247-257. doi: 10.1016/S0002-9440(10)63963-6
![]() |
[7] |
Ignarro LJ (1999) Nitric Oxide: A Unique Endogenous Signaling Molecule in Vascular Biology (Nobel Lecture). Angew Chemie Int Ed 38: 1882-1892. doi: 10.1002/(SICI)1521-3773(19990712)38:13/14<1882::AID-ANIE1882>3.0.CO;2-V
![]() |
[8] |
Koshland D (1992) The molecule of the year. Science 258: 1861. doi: 10.1126/science.1470903
![]() |
[9] |
MacMicking J, Xie Q, Nathan C (1997) Nitric oxide and macrophage function. Annu Rev Immunol 15: 323-350. doi: 10.1146/annurev.immunol.15.1.323
![]() |
[10] |
Culotta E, Koshland DE (1992) NO news is good news. Science 258: 1862-1865. doi: 10.1126/science.1361684
![]() |
[11] |
Tousoulis D, Kampoli A, Tentolouris C, et al. (2012) The Role of Nitric Oxide on Endothelial Function. Curr Vasc Pharmacol 10: 4-18. doi: 10.2174/157016112798829760
![]() |
[12] |
Basudhar D, Cheng RC, Bharadwaj G, et al. (2015) Chemotherapeutic potential of diazeniumdiolate-based aspirin prodrugs in breast cancer. Free Radic Biol Med 83: 101-114. doi: 10.1016/j.freeradbiomed.2015.01.029
![]() |
[13] |
Seabra AB, Duran N (2017) Nanoparticulated Nitric Oxide Donors and their Biomedical Applications. Mini Rev Med Chem 17: 216-223. doi: 10.2174/1389557516666160808124624
![]() |
[14] |
Rolim WR, Pieretti JC, Renó DLS, et al. (2019) Antimicrobial Activity and Cytotoxicity to Tumor Cells of Nitric Oxide Donor and Silver Nanoparticles Containing PVA/PEG Films for Topical Applications. ACS Appl Mater Interfaces 11: 6589-6604. doi: 10.1021/acsami.8b19021
![]() |
[15] |
Kalyanaraman H, Schall N, Pilz RB (2018) Nitric oxide and cyclic GMP functions in bone. Nitric Oxide Biol Chem 76: 62-70. doi: 10.1016/j.niox.2018.03.007
![]() |
[16] |
Yang C, Jeong S, Ku S, et al. (2018) Use of gasotransmitters for the controlled release of polymer-based nitric oxide carriers in medical applications. J Control Release 279: 157-170. doi: 10.1016/j.jconrel.2018.04.025
![]() |
[17] |
Seabra AB, Pelegrino MT, Haddad PS (2016) Can Nitric Oxide Overcome Bacterial Resistance to Antibiotics? Antibiotic Resistance Elsevier, 187-204. doi: 10.1016/B978-0-12-803642-6.00009-5
![]() |
[18] |
Andrew PJ, Mayer B (1999) Enzymatic function of nitric oxide synthases. Cardiovasc Res 43: 521-531. doi: 10.1016/S0008-6363(99)00115-7
![]() |
[19] |
Forstermann U, Sessa WC (2012) Nitric oxide synthases: regulation and function. Eur Hear Journa 33: 829-837. doi: 10.1093/eurheartj/ehr304
![]() |
[20] |
Hutchinson PJA, Palmer RMJ, Moncada S (1987) Comparative pharmacology of EDRF and nitric oxide on vascular strips. Eur J Pharmacol 141: 445-451. doi: 10.1016/0014-2999(87)90563-2
![]() |
[21] |
Pelegrino M, de Araujo Lima B, do Nascimento M, et al. (2018) Biocompatible and Antibacterial Nitric Oxide-Releasing Pluronic F-127/Chitosan Hydrogel for Topical Applications. Polymers (Basel) 10: 452. doi: 10.3390/polym10040452
![]() |
[22] |
Pelegrino MT, Silva LC, Watashi CM, et al. (2017) Nitric oxide-releasing nanoparticles: synthesis, characterization, and cytotoxicity to tumorigenic cells. J Nanoparticle Res 19: 57. doi: 10.1007/s11051-017-3747-4
![]() |
[23] |
Ferraz LS, Watashi CM, Colturato-Kido C, et al. (2018) Antitumor Potential of S-Nitrosothiol-Containing Polymeric Nanoparticles against Melanoma. Mol Pharm 15: 1160-1168. doi: 10.1021/acs.molpharmaceut.7b01001
![]() |
[24] |
Aranda E, Lopez-Pedrera C, De La Haba-Rodriguez JR, et al. (2012) Nitric Oxide and Cancer: The Emerging Role of S-Nitrosylation. Curr Mol Med 12: 50-67. doi: 10.2174/156652412798376099
![]() |
[25] |
González R, Molina-Ruiz FJ, Bárcena JA, et al. (2018) Regulation of Cell Survival, Apoptosis, and Epithelial-to-Mesenchymal Transition by Nitric Oxide-Dependent Post-Translational Modifications. Antioxidants Redox Signal 29: 1312-1332. doi: 10.1089/ars.2017.7072
![]() |
[26] |
Kollau A, Russwurm M, Neubauer A, et al. (2016) Scavenging of nitric oxide by hemoglobin in the tunica media of porcine coronary arteries. Nitric Oxide 54: 8-14. doi: 10.1016/j.niox.2016.01.005
![]() |
[27] |
Larry KK, Raymond WN, Keith MD, et al. (1996) “NONOates” (1-Substituted Diazen-1-ium-1,2-diolates) as Nitric Oxide Donors: Convenient Nitric Oxide Dosage Forms. Methods Enzymol 268: 281-293. doi: 10.1016/S0076-6879(96)68030-6
![]() |
[28] |
Wang PG, Xian M, Tang X, et al. (2002) Nitric oxide donors: Chemical activities and biological applications. Chem Rev 102: 1091-1134. doi: 10.1021/cr000040l
![]() |
[29] |
Liang H, Nacharaju P, Friedman A, et al. (2015) Nitric oxide generating/releasing materials. Futur Sci OA 1: FSO54. doi: 10.4155/fso.15.54
![]() |
[30] |
da Silva RS, de Lima RG, de Paula Machado S (2015) Design, reactivity, and biological activity of ruthenium nitrosyl complexes. Adv Inorg Chem 67: 265-294. doi: 10.1016/bs.adioch.2014.11.001
![]() |
[31] |
Seabra AB, De Souza GFP, Da Rocha LL, et al. (2004) S-Nitrosoglutathione incorporated in poly(ethylene glycol) matrix: Potential use for topical nitric oxide delivery. Nitric Oxide Biol Chem 11: 263-272. doi: 10.1016/j.niox.2004.09.005
![]() |
[32] |
Seabra AB, Fitzpatrick A, Paul J, et al. (2004) Topically applied S-nitrosothiol-containing hydrogels as experimental and pharmacological nitric oxide donors in human skin. Br J Dermatol 151: 977-983. doi: 10.1111/j.1365-2133.2004.06213.x
![]() |
[33] |
Oliveira HC, Gomes BCR, Pelegrino MT, et al. (2016) Nitric oxide-releasing chitosan nanoparticles alleviate the effects of salt stress in maize plants. Nitric Oxide Biol Chem 61: 10-19. doi: 10.1016/j.niox.2016.09.010
![]() |
[34] |
Heinonen I, Boushel R, Hellsten Y, et al. (2018) Regulation of bone blood flow in humans: The role of nitric oxide, prostaglandins, and adenosine. Scand J Med Sci Sports 28: 1552-1558. doi: 10.1111/sms.13064
![]() |
[35] |
Borys J, Maciejczyk M, Antonowicz B, et al. (2019) Glutathione Metabolism, Mitochondria Activity, and Nitrosative Stress in Patients Treated for Mandible Fractures. J Clin Med 8: 127. doi: 10.3390/jcm8010127
![]() |
[36] | Sela JJ, Bab IA (2012) Healing of Bone Fracture: General Concepts. Principles of Bone Regeneration Boston, MA: Springer US, 1-8. |
[37] |
Helfrich MH, Evans DE, Grabowski PS, et al. (1997) Expression of Nitric Oxide Synthase Isoforms in Bone and Bone Cell Cultures. J Bone Miner Res 12: 1108-1115. doi: 10.1359/jbmr.1997.12.7.1108
![]() |
[38] |
Basso N, Heersche JNM (2006) Effects of hind limb unloading and reloading on nitric oxide synthase expression and apoptosis of osteocytes and chondrocytes. Bone 39: 807-814. doi: 10.1016/j.bone.2006.04.014
![]() |
[39] |
Bakker AD, Huesa C, Hughes A, et al. (2013) Endothelial Nitric Oxide Synthase is Not Essential for Nitric Oxide Production by Osteoblasts Subjected to Fluid Shear Stress In Vitro. Calcif Tissue Int 92: 228-239. doi: 10.1007/s00223-012-9670-x
![]() |
[40] |
Grassi F, Fan X, Rahnert J, et al. (2006) Bone Re/Modeling Is More Dynamic in the Endothelial Nitric Oxide Synthase (−/−) Mouse. Endocrinology 147: 4392-4399. doi: 10.1210/en.2006-0334
![]() |
[41] |
MacPherson H, Noble BS, Ralston SH (1999) Expression and functional role of nitric oxide synthase isoforms in human osteoblast-like cells. Bone 24: 179-185. doi: 10.1016/S8756-3282(98)00173-2
![]() |
[42] |
Zheng H, Yu X, Collin-Osdoby P, et al. (2006) RANKL Stimulates Inducible Nitric-oxide Synthase Expression and Nitric Oxide Production in Developing Osteoclasts. J Biol Chem 281: 15809-15820. doi: 10.1074/jbc.M513225200
![]() |
[43] |
Samuels A, Perry MJ, Gibson RL, et al. (2001) Role of endothelial nitric oxide synthase in estrogen-induced osteogenesis. Bone 29: 24-29. doi: 10.1016/S8756-3282(01)00471-9
![]() |
[44] |
Van'T Hof RJ, Macphee J, Libouban H, et al. (2004) Regulation of bone mass and bone turnover by neuronal nitric oxide synthase. Endocrinology 145: 5068-5074. doi: 10.1210/en.2004-0205
![]() |
[45] |
Kasten TP, Collin-Osdoby P, Patel N, et al. (1994) Potentiation of osteoclast bone-resorption activity by inhibition of nitric oxide synthase. Proc Natl Acad Sci U S A 91: 3569-3573. doi: 10.1073/pnas.91.9.3569
![]() |
[46] |
Löwik CWGM, Nibbering PH, Van De Ruit M, et al. (1994) Inducible production of nitric oxide in osteoblast-like cells and in fetal mouse bone explants is associated with suppression of osteoclastic bone resorption. J Clin Invest 93: 1465-1472. doi: 10.1172/JCI117124
![]() |
[47] |
Wimalawansa SJ (2007) Rationale for using nitric oxide donor therapy for prevention of bone loss and treatment of osteoporosis in humans. Ann N Y Acad Sci 1117: 283-297. doi: 10.1196/annals.1402.066
![]() |
[48] |
Nichols SP, Storm WL, Koh A, et al. (2012) Local delivery of nitric oxide: Targeted delivery of therapeutics to bone and connective tissues. Adv Drug Deliv Rev 64: 1177-1188. doi: 10.1016/j.addr.2012.03.002
![]() |
[49] |
Rosselli M, Imthurn B, Keller PJ, et al. (1995) Circulating nitric oxide (nitrite/nitrate) levels in postmenopausal women substituted with 17β-estradiol and norethisterone acetate: A two-year follow-up study. Hypertension 25: 848-853. doi: 10.1161/01.HYP.25.4.848
![]() |
[50] |
Ralston SH, Ho L, Helfrich MH, et al. (1995) Nitric Oxide?: A Cytokine-Induced Regulator of. J Bone Miner Res 10: 1040-1049. doi: 10.1002/jbmr.5650100708
![]() |
[51] |
Ralston SH, Grabowski PS (1996) Mechanisms of cytokine induced bone resorption: Role of nitric oxide, cyclic guanosine monophosphate, and prostaglandins. Bone 19: 29-33. doi: 10.1016/8756-3282(96)00101-9
![]() |
[52] |
Wimalawansa SJ (2010) Nitric oxide and bone. Ann NY Acad Sci 1192: 394-406. doi: 10.1111/j.1749-6632.2009.05230.x
![]() |
[53] |
Turner CH, Owan I, Jacob DS, et al. (1997) Effects of nitric oxide synthase inhibitors on bone formation in rats. Bone 21: 487-490. doi: 10.1016/S8756-3282(97)00202-0
![]() |
[54] |
Mancini L, Moradi-Bidhendi N, Becherini L, et al. (2000) The biphasic effects of nitric oxide in primary rat osteoblasts are cGMP dependent. Biochem Biophys Res Commun 274: 477-481. doi: 10.1006/bbrc.2000.3164
![]() |
[55] |
Chambers TJ, Fox S, Jagger CJ, et al. (1999) The role of prostaglandins and nitric oxide in the response of bone to mechanical forces. Osteoarthritis Cartilage 7: 422-423. doi: 10.1053/joca.1998.0231
![]() |
[56] |
Chow JWM, Fox SW, Lean JM, et al. (1998) Role of nitric oxide and prostaglandins in mechanically induced bone formation. J Bone Miner Res 13: 1039-1044. doi: 10.1359/jbmr.1998.13.6.1039
![]() |
[57] |
Wimalawansa SJ, De Marco G, Gangula P, et al. (1996) Nitric oxide donor alleviates ovariectomy-induced bone loss. Bone 18: 301-304. doi: 10.1016/8756-3282(96)00005-1
![]() |
[58] |
Park YG, Kim KW, Song KH, et al. (2009) Combinatory responses of proinflamamtory cytokines on nitric oxide-mediated function in mouse calvarial osteoblasts. Cell Biol Int 33: 92-99. doi: 10.1016/j.cellbi.2008.09.012
![]() |
[59] |
Cuzzocrea S, Mazzon E, Dugo L, et al. (2003) Inducible Nitric Oxide Synthase Mediates Bone Loss in Ovariectomized Mice. Endocrinology 144: 1098-1107. doi: 10.1210/en.2002-220597
![]() |
[60] |
Robling AG, Castillo AB, Turner CH (2006) Biomechanical and Molecular Regulation of Bone Remodeling. Annu Rev Biomed Eng 8: 455-498. doi: 10.1146/annurev.bioeng.8.061505.095721
![]() |
[61] |
Mcallister TN, Du T, Frangos JA (2000) Fluid Shear Stress Stimulates Prostaglandin and Nitric Oxide Release in Bone Marrow-Derived. Biochem Biophys Res Commun 270: 643-648. doi: 10.1006/bbrc.2000.2467
![]() |
[62] |
Abnosi MH, Pari S (2019) Exogenous nitric oxide induced early mineralization in rat bone marrow mesenchymal stem cells via activation of alkaline phosphatase. Iran Biomed J 23: 142-152. doi: 10.29252/ibj.23.2.142
![]() |
[63] |
Ocarino NM, Boeloni JN, Goes AM, et al. (2008) Nitric Oxide Osteogenic differentiation of mesenchymal stem cells from osteopenic rats subjected to physical activity with and without nitric oxide synthase inhibition. Nitric Oxide 19: 320-325. doi: 10.1016/j.niox.2008.08.004
![]() |
[64] |
Lee JS, Lee HJ, Lee JW, et al. (2018) Osteogenic Effect of Inducible Nitric Oxide Synthase (iNOS)-Loaded Mineralized Nanoparticles on Embryonic Stem Cells. Cell Physiol Biochem 51: 746-762. doi: 10.1159/000495330
![]() |
[65] |
Fuseler JW, Valarmathi MT (2016) Nitric Oxide Modulates Postnatal Bone Marrow-Derived Mesenchymal. Front Cell Dev Biol 4: 1-20. doi: 10.3389/fcell.2016.00133
![]() |
[66] |
Takahashi TA, Johnson KM (2015) Menopause. Med Clin North Am 99: 521-534. doi: 10.1016/j.mcna.2015.01.006
![]() |
[67] |
Daripa M, Paula FJA, Rufino ACB, et al. (2004) Impact of congenital calcitonin deficiency due to dysgenetic hypothyroidism on bone mineral density. Brazilian J Med Biol Res 37: 61-68. doi: 10.1590/S0100-879X2004000100009
![]() |
[68] |
Wimalawansa SJ (2000) Restoration of ovariectomy-induced osteopenia by nitroglycerin. Calcif Tissue Int 66: 56-60. doi: 10.1007/s002230050011
![]() |
[69] |
Wimalawansa SJ (2000) Nitroglycerin therapy is as efficacious as standard estrogen replacement therapy (premarin) in prevention of oophorectomy-induced bone loss: A human pilot clinical study. J Bone Miner Res 15: 2240-2244. doi: 10.1359/jbmr.2000.15.11.2240
![]() |
[70] |
Visser JJ, Hoekman K (1994) Arginine supplementation in the prevention and treatment of osteoporosis. Med Hypotheses 43: 339-342. doi: 10.1016/0306-9877(94)90113-9
![]() |
[71] |
Fiore CE, Pennisi P, Cutuli VM, et al. (2000) L-arginine prevents bone loss and bone collagen breakdown in cyclosporin A-treated rats. Eur J Pharmacol 408: 323-326. doi: 10.1016/S0014-2999(00)00800-1
![]() |
[72] |
Veeriah V, Zanniti A, Paone R, et al. (2016) Interleukin-1β, lipocalin 2 and nitric oxide synthase 2 are mechano-responsive mediators of mouse and human endothelial cell-osteoblast crosstalk. Sci Rep 6: 1-14. doi: 10.1038/srep29880
![]() |
[73] |
Alcaide M, Serrano MC, Pagani R, et al. (2009) Biocompatibility markers for the study of interactions between osteoblasts and composite biomaterials. Biomaterials 30: 45-51. doi: 10.1016/j.biomaterials.2008.09.012
![]() |
[74] |
Bielemann AM, Marcello-Machado RM, Del Bel Cury AA, et al. (2018) Systematic review of wound healing biomarkers in peri-implant crevicular fluid during osseointegration. Arch Oral Biol 89: 107-128. doi: 10.1016/j.archoralbio.2018.02.013
![]() |
[75] | Sugiatno E, Samsudin AR, Sosroseno W (2009) Effect of exogenous nitric oxide on the proliferation of a human osteoblast (HOS) cell line induced by hydroxyapatite. J Appl Biomater Biomech 7: 29-33. |
[76] |
Adhikari U, Rijal NP, Khanal S, et al. (2016) Magnesium incorporated chitosan based scaffolds for tissue engineering applications. Bioact Mater 1: 132-139. doi: 10.1016/j.bioactmat.2016.11.003
![]() |
[77] |
Herculano RD, Tzu LC, Silva CP, et al. (2011) Nitric oxide release using natural rubber latex as matrix. Mater Res 14: 355-359. doi: 10.1590/S1516-14392011005000055
![]() |
[78] |
Özmeriç N, Elgün S, Uraz A (2000) Salivary arginase in patients with adult periodontitise. Clin Oral Invest 4: 21-24. doi: 10.1007/s007840050108
![]() |
[79] |
Leitão RFC, Rocha FAC, Chaves HV, et al. (2004) Locally Applied Isosorbide Decreases Bone Resorption in Experimental Periodontitis in Rats. J Periodontol 75: 1227-1232. doi: 10.1902/jop.2004.75.9.1227
![]() |
[80] |
Lee SK, Choi HI, Yang YS, et al. (2009) Nitric oxide modulates osteoblastic differentiation with heme oxygenase-1 via the mitogen activated protein kinase and nuclear factor-kappaB pathways in human periodontal ligament cells. Biol Pharm Bull 32: 1328-1334. doi: 10.1248/bpb.32.1328
![]() |
[81] |
Jönsson D, Nebel D, Bratthall G, et al. (2011) The human periodontal ligament cell: a fibroblast-like cell acting as an immune cell. J Periodontal Res 46: 153-157. doi: 10.1111/j.1600-0765.2010.01331.x
![]() |
[82] |
de Menezes AMA, de Souza GFP, Gomes AS, et al. (2012) S-Nitrosoglutathione Decreases Inflammation and Bone Resorption in Experimental Periodontitis in Rats. J Periodontol 83: 514-521. doi: 10.1902/jop.2011.110332
![]() |
[83] | Martins CS, Leitão RFC, Costa DVS, et al. (2016) Topical HPMC/S-Nitrosoglutathione Solution Decreases Inflammation and Bone Resorption in Experimental Periodontal Disease in Rats. PLoS One 11: 1-19. |
[84] |
Chae HJ, Park RK, Chung HT, et al. (1997) Nitric oxide is a regulator of bone remodelling. J Pharm Pharmacol 49: 897-902. doi: 10.1111/j.2042-7158.1997.tb06132.x
![]() |
[85] | Lin YJ, Chen CC, Chi NW, et al. (2018) In Situ Self-Assembling Micellar Depots that Can Actively Trap and Passively Release NO with Long-Lasting Activity to Reverse Osteoporosis. Adv Mater 30: 1-6. |
[86] |
Tai YT, Cherng YG, Chang CC, et al. (2007) Pretreatment with low nitric oxide protects osteoblasts from high nitric oxide-induced apoptotic insults through regulation of c-Jun N-terminal kinase/c-Jun-mediatedBcl-2 gene expression and protein translocation. J Orthop Res 25: 625-635. doi: 10.1002/jor.20365
![]() |
[87] |
Differ C, Klatte-Schulz F, Bormann N, et al. (2019) Is NO the Answer? The Nitric Oxide Pathway Can Support Bone Morphogenetic Protein 2 Mediated Signaling. Cells 8: 1273. doi: 10.3390/cells8101273
![]() |
[88] |
Bandara N, Gurusinghe S, Lim SY, et al. (2016) Molecular control of nitric oxide synthesis through eNOS and caveolin-1 interaction regulates osteogenic differentiation of adipose-derived stem cells by modulation of Wnt/β-catenin signaling. Stem Cell Res Ther 7: 1-15. doi: 10.1186/s13287-015-0253-4
![]() |
[89] |
Chu L, Jiang Y, Hao H, et al. (2008) Nitric oxide enhances Oct-4 expression in bone marrow stem cells and promotes endothelial differentiation. Eur J Pharmacol 591: 59-65. doi: 10.1016/j.ejphar.2008.06.066
![]() |
[90] | Felka T, Ulrich C, Rolauffs B, et al. (2014) Bone marrow stromal cells Nitric oxide activates signaling by c-Raf, MEK, p-JNK, p38 MAPK and p53 in human mesenchymal stromal cells and inhibits their osteogenic differentiation by blocking expression of Runx2. J stem cell Res Ther 4: 195. |
[91] |
Jamal SA, Cummings SR, Hawker GA (2004) Isosorbide mononitrate increases bone formation and decreases bone resorption in postmenopausal women: A randomized trial. J Bone Miner Res 19: 1512-1517. doi: 10.1359/JBMR.040716
![]() |
[92] |
Jamal SA, Hamilton CJ (2012) Nitric oxide donors for the treatment of osteoporosis. Curr Osteoporos Rep 10: 86-92. doi: 10.1007/s11914-011-0087-7
![]() |
[93] |
Lazzarato L, Rolando B, Lolli ML, et al. (2005) Synthesis of NO-donor bisphosphonates and their in-vitro action on bone resorption. J Med Chem 48: 1322-1329. doi: 10.1021/jm040830d
![]() |
[94] |
Kalyanaraman H, Ramdani G, Joshua J, et al. (2017) A Novel, Direct NO Donor Regulates Osteoblast and Osteoclast Functions and Increases Bone Mass in Ovariectomized Mice. J Bone Miner Res 32: 46-59. doi: 10.1002/jbmr.2909
![]() |
[95] |
Wimalawansa SJ, Chapa MT, Yallampalli C, et al. (1997) Prevention of corticosteroid-induced bone loss with nitric oxide donor nitroglycerin in male rats. Bone 21: 275-280. doi: 10.1016/S8756-3282(97)00125-7
![]() |