Citation: Vladimir Zaichick, Sofia Zaichick, Maxim Rossmann. Intracellular calcium excess as one of the main factors in the etiology of prostate cancer[J]. AIMS Molecular Science, 2016, 3(4): 635-647. doi: 10.3934/molsci.2016.4.635
[1] | Qiuxia Hu, Bilal Khan, Serkan Araci, Mehmet Acikgoz . New double-sum expansions for certain Mock theta functions. AIMS Mathematics, 2022, 7(9): 17225-17235. doi: 10.3934/math.2022948 |
[2] | Suxia Wang, Tiehong Zhao . New refinements of Becker-Stark inequality. AIMS Mathematics, 2024, 9(7): 19677-19691. doi: 10.3934/math.2024960 |
[3] | Zhenhua Su, Zikai Tang, Hanyuan Deng . Higher-order Randić index and isomorphism of double starlike trees. AIMS Mathematics, 2023, 8(12): 31186-31197. doi: 10.3934/math.20231596 |
[4] | Waleed Mohamed Abd-Elhameed, Youssri Hassan Youssri . Spectral tau solution of the linearized time-fractional KdV-Type equations. AIMS Mathematics, 2022, 7(8): 15138-15158. doi: 10.3934/math.2022830 |
[5] | Tie-Hong Zhao, Zai-Yin He, Yu-Ming Chu . On some refinements for inequalities involving zero-balanced hypergeometric function. AIMS Mathematics, 2020, 5(6): 6479-6495. doi: 10.3934/math.2020418 |
[6] | Mustafa Inc, Mamun Miah, Akher Chowdhury, Shahadat Ali, Hadi Rezazadeh, Mehmet Ali Akinlar, Yu-Ming Chu . New exact solutions for the Kaup-Kupershmidt equation. AIMS Mathematics, 2020, 5(6): 6726-6738. doi: 10.3934/math.2020432 |
[7] | Aslıhan ILIKKAN CEYLAN, Canan HAZAR GÜLEÇ . A new double series space derived by factorable matrix and four-dimensional matrix transformations. AIMS Mathematics, 2024, 9(11): 30922-30938. doi: 10.3934/math.20241492 |
[8] | Waleed Mohamed Abd-Elhameed, Abdullah F. Abu Sunayh, Mohammed H. Alharbi, Ahmed Gamal Atta . Spectral tau technique via Lucas polynomials for the time-fractional diffusion equation. AIMS Mathematics, 2024, 9(12): 34567-34587. doi: 10.3934/math.20241646 |
[9] | Ling Zhu . New inequalities of Wilker's type for hyperbolic functions. AIMS Mathematics, 2020, 5(1): 376-384. doi: 10.3934/math.2020025 |
[10] | Bai-Ni Guo, Dongkyu Lim, Feng Qi . Series expansions of powers of arcsine, closed forms for special values of Bell polynomials, and series representations of generalized logsine functions. AIMS Mathematics, 2021, 6(7): 7494-7517. doi: 10.3934/math.2021438 |
In [18] Ramanujan showed a total of 17 series for 1/π but he did not indicate how he arrived at these series. The Borwein brothers [5] gave rigorous proofs of Ramanujan's series for the first time and also obtained many new series for 1/π. Till now, many new Ramanujan's-type series for 1/π have been published, (see, for example, [4,6,8]). Chu [7], Liu [15,16] and Wei et al. [21,22] gave many π-formula with free parameters by means of gamma functions and hypergeometric series. Guillera [10] proved a kind of bilateral semi-terminating series related to Ramanujan-like series for negative powers of π. Moreover, Guillera and Zudilin [11] outlined an elementary method for proving numerical hypergeometric identities, in particular, Ramanujan-type identities for 1/π. Recently, q-analogues of Ramanujan-type series for 1/π have caught the interests of many authors (see, for example, [9,12,13,14,20,21]).
Although various definitions for gamma functions are used in the literature, we adopt the following definition [23, p.76]
1Γ(z)=zeγz∞∏n=1(1+zn)e−zn |
where γ is the Euler constant defined as
γ=limn→∞(1+12+⋯+1n−logn). |
It is easy to verify that Γ(1)=1,Γ(12)=√π and Γ(z+1)=zΓ(z). It follows that for every positive integer n, Γ(n)=(n−1)!.
For any complex α, we define the general rising shifted factorial by
(z)α=Γ(z+α)/Γ(z). | (1.1) |
Obviously, (z)0=1. For every positive integer n, we have
(z)n=Γ(z+n)/Γ(z)=z(z+1)⋯(z+n−1) |
and
(z)−n=Γ(z−n)/Γ(z)=1(z−1)(z−2)…(z−n). |
For convenience, we use the following compact notations
(a1,a2,…,am)n=(a1)n(a2)n…(am)n |
and
(a)(n1,n2,…,nm)=(a)n1(a)n2…(a)nm. |
Following [1,3], the hypergeometric series is defined by
r+1Fs[a0,a1,…,arb1,…,bs;z]=∞∑k=0(a0,a1,…,ar)k(b1,…,bs)kzkk!, |
where ai,bj(i=0,1,…,r,j=1,2,…,s) are complex numbers such that no zero factors appear in the denominators of the summand on the right hand side.
We let Fp:r;uq:s;v (p,q,r,s,u,v∈N0={0,1,2,…}) denote a general (Kampé de Fériet's) double hypergeometric function defined by (see [2,19])
Fp:r;uq:s;v[α1,…,αp:a1,…,ar;c1,…,cu;β1,…,βq:b1,…,bs;d1,…,dv;x,y]=∞∑m,n=0(α1,…,αp)m+n(a1,…,ar)m(c1,…,cu)n(β1,…,βq)m+n(b1,…,bs)m(d1,…,dv)nxmm!ynn!, |
where, for convergence of the double hypergeometric series,
p+r≤q+s+1andp+u≤q+v+1, |
with equality only when |x| and |y| are appropriately constrained (see, for details, [19,Eq 1.3(29),p.27]).
There exist numerous identities for such series. For example, we have
Theorem 1.1 (See [17,(30)] ) If Re(e−d)>0 and Re(d+e−a−b−c)>0, then
F0:3;31:1;1[−:a,b,c;d−a,d−b,d−c;d:e;d+e−a−b−c;1,1]=Γ(e)Γ(e+d−a−b−c)Γ(e−d)Γ(e−a)Γ(e−b)Γ(e−c). |
In [15], Liu used the general rising shifted factorial and the Gauss summation formula to prove the following four-parameter series expansions formula, which implies infinitely many Ramanujan type series for 1/π and π.
Theorem 1.2 For any complex α and Re(c−a−b)>0, we have
∞∑n=0(α)a+n(1−α)b+nn!Γ(c+n+1)=(α)a(1−α)bΓ(c−a−b)(α)c−b(1−α)c−a⋅sinπαπ. |
Motivated by Liu's work, in this paper we derive the following result from Theorem 1.1 which enables us to give many double series expansions for 1/π and π. To the best of our knowledge, most of the results in this paper have not previously appeared.
Theorem 1.3 If d∈N0,Re(e−d+σ−δ)>0 and Re(d+e−a−b−c+δ+σ−α−β−γ)>0, then
∞∑m,n=0(α)a+m(β)b+m(γ)c+m(δ−α)d−a+n(δ−β)d−b+n(δ−γ)d−c+nm!n!(δ+d)m+n(σ)e+m(δ+σ−α−β−γ)d+e−a−b−c+n=(α)a(β)b(γ)c(δ−α)d−a(δ−β)d−b(δ−γ)d−c(σ−δ)e−d(σ−α)e−a(σ−β)e−b(σ−γ)e−c⋅Γ(σ)Γ(σ−δ)Γ(δ+σ−α−β−γ)Γ(σ−α)Γ(σ−β)Γ(σ−γ). |
Several examples of such formulae are
∞∑m,n=0(12)3m(12)2nm!n!(m+n)!(m+1)!(2n+1)=4π, |
∞∑m,n=0(−12)3m(32)3nm!n!(m+n)!(n+3)!(12)m+1=π, |
and
∞∑m,n=0(−23)2m(13)3nm!n!(n+1)!(2−3m)(−13)m+n=√3π3. |
The remainder of the paper is organized as follows. In section 2 we give the proof of Theorem 1.3. Sections 3 and 4 are devoted to the double series expansions for 1/π and π, respectively.
First of all, by making use of (1.1), Theorem 1.3 can be restated as follows:
∞∑m,n=0Γ(a+m)Γ(b+m)Γ(c+m)Γ(d−a+n)Γ(d−b+n)Γ(d−c+n)m!n!Γ(d+m+n)Γ(e+m)Γ(d+e−a−b−c+n)=Γ(a)Γ(b)Γ(c)Γ(d−a)Γ(d−b)Γ(d−c)Γ(e−d)Γ(d)Γ(e−a)Γ(e−b)Γ(e−c). | (2.1) |
From (1.1) it is easy to see that
Γ(a+α+m)=(α)a+mΓ(α), Γ(b+β+m)=(β)b+mΓ(β), Γ(c+γ+m)=(γ)c+mΓ(γ),Γ(d−a+δ−α+n)=(δ−α)d−a+nΓ(δ−α), Γ(d−b+δ−β+n)=(δ−β)d−b+nΓ(δ−β),Γ(d−c+δ−γ+n)=(δ−γ)d−c+nΓ(δ−γ), Γ(d+δ+m+n)=(δ)d+m+nΓ(δ)Γ(e+m+σ)=(σ)e+mΓ(σ), Γ(a+α)=(α)aΓ(α), Γ(b+β)=(β)bΓ(β), Γ(c+γ)=(γ)cΓ(γ),Γ(d−a+δ−α)=(δ−α)d−aΓ(δ−α), Γ(d−b+δ−β)=(δ−β)d−bΓ(δ−β),Γ(d−c+δ−γ)=(δ−γ)d−cΓ(δ−γ), Γ(e−d+σ−δ)=(σ−δ)e−dΓ(σ−δ),Γ(d+δ)=(δ)dΓ(δ),Γ(e−a+σ−α)=(σ−α)e−aΓ(σ−α),Γ(e−b+σ−β)=(σ−β)e−bΓ(σ−β), Γ(e−c+σ−γ)=(σ−γ)e−cΓ(σ−γ),Γ(d+e−a−b−c+δ+σ−α−β−γ)=(δ+σ−α−β−γ)d+e−a−b−cΓ(δ+σ−α−β−γ). |
and we realize that (δ)d+m+n=(δ)d(δ+d)m+n when d∈N0. Replacing(a,b,c,d,e) by (a+α,b+β,c+γ,d+δ,e+σ) in (2.1) and substituting above identities into the resulting equation, we get the desired result.
In this section we will use Theorem 1.3 to prove the following double series expansion formula for 1/π.
Theorem 3.1 If d∈N0,Re(e−d+1)>0 and Re(d+e−a−b−c+32)>0, then
∞∑m,n=0(12)(a+m,b+m,c+m,d−a+n,d−b+n,d−c+n)m!n!(d+1)m+n(2)e+m(32)d+e−a−b−c+n=(12)(a,b,c,d−a,d−b,d−c)(1)e−d(32)(e−a,e−b,e−c)⋅4π. |
Proof. Let (α,β,γ,δ,σ)=(12,12,12,1,2) in Theorem 1.3. We find that
∞∑m,n=0(12)(a+m,b+m,c+m,d−a+n,d−b+n,d−c+n)m!n!(d+1)m+n(2)e+m(32)d+e−a−b−c+n=(12)(a,b,c,d−a,d−b,d−c)(1)e−d(32)(e−a,e−b,e−c)⋅Γ(2)Γ(1)Γ(32)Γ3(32). | (3.1) |
Substituting Γ(32)=√π2 into (3.1) we obtain the result immediately. Putting (a,b,c)=(0,0,0) in Theorem 3.1 we get the following general double summation formula for 1/π with two free parameters.
Corollary 3.2 If d∈N0,Re(e−d+1)>0 and Re(d+e+32)>0, then
∞∑m,n=0(12)3(m,d+n)m!n!(d+1)m+n(2)e+m(32)d+e+n=4(12)3d(1)e−dπ(32)3e. |
Setting d=0 and e=k∈N0 in Corallary 3.2 we have the following result.
Proposition 3.3 Let k be a nonnegative integer. Then
∞∑m,n=0(12)3(m,n)m!n!(m+n)!(m+k+1)!(32+k)n=4k!π(32)2k. |
Example 3.1 (k=0 in Proposition 3.3).
∞∑m,n=0(12)3m(12)2nm!n!(m+n)!(m+1)!(2n+1)=4π. |
If d=e=k∈N0 in Corollary 3.2 we achieve
Proposition 3.4 Let k be a nonnegative integer. Then
∞∑m,n=0(12)3(m,n+k)m!n!(k+1)m+n(m+k+1)!(32)n+2k=4π(2k+1)3. |
If we put k=0 into Proposition 3.4, then we can also get Example 3.1.
In this section we will prove the following theorem, which allows us to derive infinitely double series expansions for π.
Theorem 4.1 If d∈N0,Re(e−d−σ+1)>0 and Re(d+e−a−b−c+2)>0, then
∞∑m,n=0(σ−1)(a+m,b+m,c+m)(σ)(d−a+n,d−b+n,d−c+n)m!n!(2σ+d−1)m+n(σ)e+m(2)d+e−a−b−c+n=(σ−1)(a,b,c)(σ)(d−a,d−b,d−c)(1−σ)e−d(1)(e−a,e−b,e−c)⋅πsinσπ. |
Proof. Let (α,β,γ,δ)=(σ−1,σ−1,σ−1,2σ−1) in Theorem 1.3. We obtain that
∞∑m,n=0(σ−1)(a+m,b+m,c+m)(σ)(d−a+n,d−b+n,d−c+n)m!n!(2σ+d−1)m+n(σ)e+m(2)d+e−a−b−c+n=(σ−1)(a,b,c)(σ)(d−a,d−b,d−c)(1−σ)e−d(1)(e−a,e−b,e−c)⋅Γ(σ)Γ(1−σ)Γ(2)Γ3(1). | (4.1) |
Combining Γ(σ)Γ(1−σ)=πsinσπ with (4.1) we get the desired result immediately. Putting a=b=c=0 in Theorem 4.1 we obtain the following equation.
Corollary 4.2 If d∈N0,Re(e−d−σ+1)>0 and Re(d+e+2)>0, then
∞∑m,n=0(σ−1)3m(σ)3d+nm!n!(2σ+d−1)m+n(σ)e+m(2)d+e+n=(σ)3d(1−σ)e−d(1)3e⋅πsinσπ. |
Letting σ=12 in Corollary 4.2, we get the following proposition.
Proposition 4.3 If d∈N0,Re(e−d+12)>0 and Re(d+e+2)>0, then
∞∑m,n=0(−12)3m(12)3d+nm!n!(d)m+n(12)e+m(2)d+e+n=(12)3d(12)e−d(1)3eπ. |
When we set d=1 and e=k∈N={1,2,3…} in Proposition 4.3 we obtain
Proposition 4.4 If k is a positive integer, then
∞∑m,n=0(−12)3m(32)3nm!n!(m+n)!(n+k+2)!(12)m+k=π(12)k−1(k!)3. |
Example 4.1 (k=1 in Proposition 4.4).
∞∑m,n=0(−12)3m(32)3nm!n!(m+n)!(n+3)!(12)m+1=π. |
Putting σ=13 in Corollary 4.2, we get the following proposition.
Proposition 4.5 If d∈N0,Re(e−d+23)>0 and Re(d+e+2)>0, then
∞∑m,n=0(−23)3m(13)3d+nm!n!(d−13)m+n(13)e+m(2)d+e+n=2√3π(13)3d(23)e−d3(1)3e. |
When we set d=0 and e=k∈N0 in Proposition 4.5 we obtain
Proposition 4.6 If k is a nonnegative integer, then
∞∑m,n=0(−23)3m(13)3nm!n!(−13)m+n(13)m+k(n+k+1)!=2√3π(23)k3k!3. |
Example 4.2 (k=0 in Proposition 4.6).
∞∑m,n=0(−23)2m(13)3nm!n!(n+1)!(2−3m)(−13)m+n=√3π3. |
Setting d=e=k∈N0 in Proposition 4.5, we get
Proposition 4.7 If k is a nonnegative integer, then
∞∑m,n=0(−23)3m(13+k)3nm!n!(n+2k+1)!(k−13)m+n(13)m+k=2√3π3k!3. |
Therefore, Example 4.2 can also be deduced by fixing k=0 in the above equation.
Example 4.3 (k=1 in Proposition 4.7).
∞∑m,n=0(−23)3m(43)3nm!n!(n+3)!(23)m+n(43)m=2√3π9. |
Double series expansions for 1/π and π with several free parameters are established and many interesting formulas are obtained. A point that should be stressed is that there is an important connection between the summation formulas for double hypergeometric functions and double series expansions for the powers of π.
The author was partially supported by the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (grant 19KJB110006).
The author declares that there is no conflict of interest in this paper.
[1] | Forootan SS, Hussain S, Aachi V, et al. (2014) Molecular mechanisms involved in the transition of prostate cancer cells from androgen dependent to castration resistant state. J Androl Gynaecol 2: 9. |
[2] | Tao ZQ, Shi AM, Wang KX, et al. (2015) Epidemiology of prostate cancer: current status. Eur Rev Med Pharmacol Sci 19: 805-812. |
[3] |
Center AA, Jemal A, Lortet-Tieulent J, et al. (2012) International variation in prostate cancer incidence and mortality rates. Eur Urol 61: 1079-1092. doi: 10.1016/j.eururo.2012.02.054
![]() |
[4] | Cohen LA (2002) Nutrition and prostate cancer: a review. Ann NY Acad Sci 963: 148-155. |
[5] |
Jones BA, Liu W-L, Araujo AB, et al. (2008) Explaining the race difference in prostate cancer stage at diagnosis. Cancer Epidemiol Biomarkers Prev 17: 2825-2834. doi: 10.1158/1055-9965.EPI-08-0203
![]() |
[6] |
Jemal A, Murray T, Samuels A, et al. (2003) Cancer statistics, 2003. CA Cancer J Clin 53: 5-26. doi: 10.3322/canjclin.53.1.5
![]() |
[7] |
Rebbeck TR (2006) Conquering cancer disparities: new opportunities for cancer epidemiology, biomarker, and prevention research. Cancer Epidemiol Biomarkers Prev 15: 1569-1571. doi: 10.1158/1055-9965.EPI-06-0613
![]() |
[8] | Strahan RW (1963) Carcinoma of the prostate: Incidence, origin, pathology. J Urol (Baltimore) 89(6): 875-880. |
[9] |
McNeal JE, Redwine EA, Freiha FS, et al. (1988) Zonal distribution of prostatic adenocarcinoma. Am J Surg Pathol 12: 897-906. doi: 10.1097/00000478-198812000-00001
![]() |
[10] | Giovanucci E, Ascherio A, Rimm E, et al. (1995) Intake of carotenoids and retinol in relation to risk of prostate cancer. J Natl Cancer Inst 87: 1767-1776. |
[11] | Song Y, Chavarro JE, Cao Y, et al. (2013) Whole milk intake is associated with prostate cancer-specific mortality among U.S. male physicians. J Nutr 143: 189-196. |
[12] |
Aune D, Rosenblatt DAN, Chan DS, et al. (2015) Dairy products, calcium, and prostate cancer risk: a systematic review and meta-analysis of cohort studies. Am J Clin Nutr 101: 87-117. doi: 10.3945/ajcn.113.067157
![]() |
[13] | Lin PH, Aronson W, Freedland SJ (2015) Nutrition, dietary interventions and prostate cancer: the latest evidence. MC Med 13: 3. |
[14] |
Wilson KM,Shui IM, Mucci LA, et al. (2015) Calcium and phosphorus intake and prostate cancer risk: a 24-y follow-up study. Am J Clin Nutr 101: 173-183. doi: 10.3945/ajcn.114.088716
![]() |
[15] |
Zaichick V, Tsyb A, Matveenko E, et al. (1996) Instrumental neutron activation analysis of essential and toxic elements in the child and adolescent diets in the Chernobyl disaster territories of the Kaluga Region. Sci Total Environ 192: 269-274. doi: 10.1016/S0048-9697(96)05321-1
![]() |
[16] | Berridge MJ, Bootman MD, Roderick HL (2003) Calcium signalling: dynamics, homeostasis and remodelling. Nat Rev Mol Cell Biol 4: 517-529. |
[17] |
Zaichick S, Zaichick V (2011) INAA application in the age dynamics assessment of Br, Ca, Cl, K, Mg, Mn, and Na content in the normal human prostate. J Radioanal Nucl Chem 288: 197-202. doi: 10.1007/s10967-010-0927-4
![]() |
[18] |
Zaichick V, Nosenko S, Moskvina I (2012) The effect of age on 12 chemical element contents in intact prostate of adult men investigated by inductively coupled plasma atomic emission spectrometry. Biol Trace Elem Res 147: 49-58. doi: 10.1007/s12011-011-9294-4
![]() |
[19] |
Zaichick V, Zaichick S (2013) The effect of age on Br, Ca, Cl, K, Mg, Mn, and Na mass fraction in pediatric and young adult prostate glands investigated by neutron activation analysis. Appl Radiat Isot 82: 145-151. doi: 10.1016/j.apradiso.2013.07.035
![]() |
[20] |
Zaichick V, Zaichick S (2013) NAA-SLR and ICP-AES Application in the assessment of mass fraction of 19 chemical elements in pediatric and young adult prostate glands. Biol Trace Elem Res 156: 357-366. doi: 10.1007/s12011-013-9826-1
![]() |
[21] | Zaichick V, Zaichick S (2014) Androgen-dependent chemical elements of prostate gland. Androl Gynecol Curr Res 2: 2. |
[22] | Zaichick V, Zaichick S (2014) Relations of the neutron activation analysis data to morphometric parameters in pediatric and nonhyperplastic young adult prostate glands. Adv Biomed Sci Eng 1: 26-42. |
[23] |
Zaichick V, Zaichick S (2014) Relations of the Al, B, Ba, Br, Ca, Cl, Cu, Fe, K, Li, Mg, Mn, Na, P, S, Si, Sr, and Zn mass fractions to morphometric parameters in pediatric and nonhyperplastic young adult prostate glands. BioMetals 27: 333-348. doi: 10.1007/s10534-014-9716-9
![]() |
[24] |
Zaichick V, Zaichick S (2014) INAA application in the assessment of chemical element mass fractions in adult and geriatric prostate glands. Appl Radiat Isot 90: 62-73. doi: 10.1016/j.apradiso.2014.03.010
![]() |
[25] | Zaichick V, Zaichick S (2014) Determination of trace elements in adults and geriatric prostate combining neutron activation with inductively coupled plasma atomic emission spectrometry. Open J Biochem 1: 16-33. |
[26] |
Zaichick V (2015) The variation with age of 67 macro- and microelement contents in nonhyperplastic prostate glands of adult and elderly males investigated by nuclear analytical and related methods. Biol Trace Elem Res 168: 44-60. doi: 10.1007/s12011-015-0342-3
![]() |
[27] | Zaichick V, Zaichick S (2016) Age-related changes in concentration and histological distribution of Br, Ca, Cl, K, Mg, Mn, and Na in nonhyperplastic prostate of adults. Eur J Biol Med Sci Res 4: 31-48. |
[28] | Zaichick V, Zaichick S (2016) Age-related changes in concentration and histological distribution of 18 chemical elements in nonhyperplastic prostate of adults. World J Pharm Med Res 2: 5-18. |
[29] |
Deering RE, Choongkittaworn M, Bigler SA, et al. (1994) Morphometric quantitation of stroma in human benign prostatic hyperplasia. Urology 44: 64-67. doi: 10.1016/S0090-4295(94)80011-1
![]() |
[30] |
Zaichick S, Zaichick V (2013) Relations of morphometric parameters to zinc content in paediatric and nonhyperplastic young adult prostate glands. Andrology 1: 139-146. doi: 10.1111/j.2047-2927.2012.00005.x
![]() |
[31] |
Zaichick V, Zaichick S (2014) Age-related histological and zinc content changes in adult nonhyperplastic prostate glands. Age 36: 167-181. doi: 10.1007/s11357-013-9561-8
![]() |
[32] | Tvedt KE, Kopstad G, Haugen OA, et al. (1987) Subcellular concentrations of calcium, zinc, and magnesium in benign nodular hyperplasia of the human prostate: X-ray microanalysis of freeze-dried cryosections. Cancer Res 47: 323-328 |
[33] |
Banas A, Banas K, Kwiatek WM, et al. (2011) Neoplastic disorders of prostate glands in the light of synchrotron radiation and multivariate statistical analysis. J Biol Inorg Chem 16: 1187-1196. doi: 10.1007/s00775-011-0807-6
![]() |
[34] | Zaichick V, Zaichick S (1996) Instrumental effect on the contamination of biomedical samples in the course of sampling. J Anal Chem 51: 1200-1205. |
[35] | Zaichick V (1997) Sampling, sample storage and preparation of biomaterials for INAA in clinical medicine, occupational and environmental health. In: Harmonization of Health-Related Environmental Measurements Using Nuclear and Isotopic Techniques, Vienna: IAEA, 123-133. |
[36] | Avtandilov GG (1973) Morphometry in pathology, Moscow: Medicina. |
[37] |
Zaichick V (1995) Applications of synthetic reference materials in the medical Radiological Research Centre. Fresenius J Anal Chem 352: 219-223. doi: 10.1007/BF00322330
![]() |
[38] | Burgos MH (1974) Biochemical and functional properties related to sperm metabolism and fertility. In: Brandes D., Ed, Male accessory sex organs. New York: Academic press, 151-160. |
[39] |
Homonnai ZT, Matzkin H, Fainman N, et al. (1978) The cation composition of the seminal plasma and prostatic fluid and its correlation to semen quality. Fertil Steril 29: 539-542. doi: 10.1016/S0015-0282(16)43281-4
![]() |
[40] | Zaneveld LJD, Tauber PF (1981) Contribution of prostatic fluid components to the ejaculate. In: Murphy G.P., Sandberg A.A., Karr J.P., Eds. Prostatic Cell: Structure and Function. New York: Alan R. Liss, Part A, 265-277. |
[41] | Kavanagh JP, Darby C, Costello CB, et al. (1983) Zinc in post prostatic massage (VB3) urine samples: a marker of prostatic secretory function and indicator of bacterial infection. Urol Res 11: 167-170. |
[42] | Daniels GF, Grayhack JT (1990) Physiology of prostatic secretion. In: Chisholm G.D., Fair W.R., Eds. Scientific Foundation in Urology. Chicago: Heinemann Medical Books, 351-358. |
[43] |
Romics I, Bach D (1991) Zn, Ca and Na levels in the prostatic secretion of patients with prostatic adenoma. Int Urol Nephrol 23: 45-49. doi: 10.1007/BF02549727
![]() |
[44] |
Mackenzie AR, Hall T, Whitmore WFJr (1962) Zinc content of expressed human prostate fluid. Nature 193: 72-73. doi: 10.1038/193072a0
![]() |
[45] | Fair WR, Cordonnier JJ (1978) The pH of prostatic fluid: A reappraisal and therapeutic implications. J Urol 120: 695-698. |
[46] |
Kavanagh JP (1983) Zinc binding properties of human prostatic tissue, prostatic secretion and seminal fluid. J Reprod Fert 68: 359-363. doi: 10.1530/jrf.0.0680359
![]() |
[47] |
Zaichick V, Sviridova T, Zaichick S (1996) Zinc concentration in human prostatic fluid: normal, chronic prostatitis, adenoma and cancer. Int Urol Nephrol 28: 687-694. doi: 10.1007/BF02552165
![]() |
[48] | Iyengar GV, Kollmer WE, Bowen HGM (1978) The Elemental Composition of Human Tissues and Body Fluids. A Compilation of Values for Adults. Weinheim: Verlag Chemie. |
[49] |
Costello LC, Franklin RB (2009) Prostatic fluid electrolyte composition for the screening of prostate cancer: a potential solution to a major problem. Prostate Cancer Prostate Dis 12: 17-24. doi: 10.1038/pcan.2008.19
![]() |
[50] | Costello LC, Lao L, Franklin R (1993) Citrate modulation of high-affinity aspartate transport in prostate epithelial cells. Cell Mol Biol 39: 515-524. |
[51] |
Roderick HL, Cook SJ (2008) Ca2+ signalling checkpoints in cancer: remodelling Ca2+ for cancer cell proliferation and survival. Nat Rev Cancer 8: 361-375. doi: 10.1038/nrc2374
![]() |
[52] |
Flourakis M, Prevarskaya N (2009) Insights into Ca2+ homeostasis of advanced prostate cancer cells. Biochim Biophys Acta 1793: 1105-1109. doi: 10.1016/j.bbamcr.2009.01.009
![]() |
[53] |
Feng M, Grice DM, Faddy HM, et al. (2010) Store-independent activation of orai1 by SPCA2 in mammary tumors. Cell 143: 84-98. doi: 10.1016/j.cell.2010.08.040
![]() |
[54] | Yang H, Zhang Q, He J, et al. (2010) Regulation of calcium signaling in lung cancer. J Thorac Dis 2: 52-56. |
[55] | Berridge MJ, Lipp P, Bootman MD (2000) The versatility and universality of calcium signalling. Nat Rev Mol Cell Biol 1: 11-21. |
[56] |
Monteith GR, Davis FM, Roberts-Thomson SJ (2012) Calcium channels and pumps in cancer: changes and consequences. J Biol Chem 287: 31666-31673. doi: 10.1074/jbc.R112.343061
![]() |
[57] |
Marcelo KL, Means AR, York B (2016) The Ca2+/Calmodulin/CaMKK2 Axis: Nature's Metabolic CaMshaft. Trends Endocrinol Metab 27: 706-718. doi: 10.1016/j.tem.2016.06.001
![]() |
[58] |
Dubois C, Vanden Abeele F, Lehen'kyi V, et al. (2014) Remodeling of channel-forming ORAI proteins determines an oncogenic switch in prostate cancer. Cancer Cell 26: 19-32. doi: 10.1016/j.ccr.2014.04.025
![]() |
[59] |
Yamaguchi M, Mori S (1989) Activation of hepatic microsomal Ca2+-adenosine triphosphatase by calcium-binding protein regucalcin. Chem Pharm Bull 37: 1031-1034. doi: 10.1248/cpb.37.1031
![]() |
[60] |
Takahashi H, Yamaguchi M (1993) Regulatory effect of regucalcin on (Ca(2+)-Mg2+)-ATPase in rat liver plasma membranes: comparison with the activation by Mn2+ and Co2+. Mol Cell Biochem 124: 169-174. doi: 10.1007/BF00929209
![]() |
[61] |
Laurentino SS, Correia S, Cavaco JE, Oliveira PF, Rato L, et al. (2011) Regucalcin is broadly expressed in male reproductive tissues and is a new androgen-target gene in mammalian testis. Reproduction 142: 447-456. doi: 10.1530/REP-11-0085
![]() |
[62] |
Maia C, Santos C, Schmitt F, Socorro S (2009) Regucalcin is under-expressed in human breast and prostate cancers: effect of sex steroid hormones. J Cell Biochem 107: 667-676. doi: 10.1002/jcb.22158
![]() |
[63] |
Yamaguchi M (2012) Role of regucalcin in brain calcium signaling: involvement in aging. Integr Biol (Camb) 4: 825-837. doi: 10.1039/c2ib20042b
![]() |
[64] |
Galheigo MR, Cruz AR, Cabral ÁS, et al. (2016) Role of the TNF-α receptor type 1 on prostate carcinogenesis in knockout mice. Prostate 76: 917-926. doi: 10.1002/pros.23181
![]() |
[65] |
Prevarskaya N, Ouadid-Ahidouch H, Skryma R, et al. (2014) Remodelling of Ca2+ transport in cancer: how it contributes to cancer hallmarks? Philos Trans R Soc Lond B Biol Sci 369: 20130097. doi: 10.1098/rstb.2013.0097
![]() |
[66] |
Panov A, Orynbayeva Z (2013) Bioenergetic and antiapoptotic properties of mitochondria from cultured human prostate cancer cell lines PC-3, DU145 and LNCaP. PLoS One 8: e72078. doi: 10.1371/journal.pone.0072078
![]() |