Journal of ISSN: 2377-4282JNMR

Nanomedicine Research
Short Communication
Volume 5 Issue 1 - 2017
A Comprehensive Theoretical Study of Drug Delivery at Nanoscale
RQ Sofi1, S Majeed2 and AH Sofi3*
1SSM College of Engineering and Technology, India
2PG Department of Electronics and Instrumentational Technology, University of Kashmir, India
3National Institute of technology, India
Received: December 27, 2016 | Published: February 15, 2017
*Corresponding author: Ashaq Hussain Sofi, National Institute of technology- Srinagar, J&K, India, Email:
Citation: Sofi RQ, Majeed S, Sofi AH (2017) A Comprehensive Theoretical Study of Drug Delivery at Nanoscale. J Nanomed Res 5(1): 00106. DOI: 10.15406/jnmr.2017.05.00106

Abstract

Nanotechnology, a network of technologies, has achieved momentous precedence worldwide over last few decades. Progression in nanoparticle synthesis and nanoparticle based drug delivery systems is extensively anticipated to change the scenario of pharmaceutical industries because of the remarkable change in the properties of materials at nanoscale typically by virtue of their increased specific surface area to volume ratio and reactivity, which in principal may increase their biomedical applications. In the present study, a theoretical study has been carried out about nanostructures acting as nano capsules for drug delivery. A mathematical model has been formulated for the delivery and removal rate of nano capsules using a simple differential approach.

Introduction

Nanotechnology, an interdisciplinary science that incorporates physics, chemistry, biology, etc. involves creation and utilization of materials, devices or systems on the nanometer scale and is currently undergoing unexpected development on many fronts. It finds numerous applications in material strengthening and fabrication, healthcare, agriculture, processing and storage of foods, robotics for human welfare, energy conservation and utilization, transport, manufacturing of safe and quality products and security at global level [1-11]. Besides, these unparalleled applications it has incredible potential for revolutionizing all fields of technology and is expected to create innovations and play a vital role in various biomedical applications like, drug delivery systems, molecular imaging, biomarkers, therapeutics, diagnostics, cancer therapy, biosensors, etc. Synthesis via benign route and building of orderly drug delivery systems are very important for medical and health-care. In this field, nano-based drug delivery system specifically nanoparticles have developed a great modification that provides new opportunities to move over hindrances and flaws of conventional drug delivery methods like enhance solubility of drugs, diminish drug toxicity, beshields drugs from deterioration, ineffective and benign delivery of drugs to specific target sites in the body, etc. [12-15].

In the present communication, a theoretical study has been carried out about nanostructures acting as nano capsules for targeting drugs to specific sites in the body in drug delivery systems without incorporating the effect of viscosity and friction of blood, gravity, and external guiding field. A mathematical model has been formulated for the delivery and removal rate of nano capsules using a simple differential approach.

Mathematical Model

In this section, we will try to find and solve the model for nano-particle injection (acting as nano capsules) into the blood stream of a patient. Let us assume that in the beginning i.e., at t = 0 a constant amount of A grams/min is injected into the blood stream of a patient. Upon reaching the target the removal of nano capsules is equally important in order to avoid the complications caused by their presence in the target body. Let us further assume that the nano-particles are simultaneously removed at a rate proportional to the amount of nano-particles present at time t [16-18]. Let y (t) represents the amount of nano-particles present at time t, then
Remaval rate y( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOuaiaadwgacaWGTbGaamyyaiaadAhacaWGHbGaamiBaiaa cckacaWGYbGaamyyaiaadshacaWGLbGaaiiOaiabg2Hi1kaadMhada qadaWdaeaapeGaamiDaaGaayjkaiaawMcaaaaa@482C@
Remaval rate=ky( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOuaiaadwgacaWGTbGaamyyaiaadAhacaWGHbGaamiBaiaa cckacaWGYbGaamyyaiaadshacaWGLbGaeyypa0Jaam4AaiaadMhada qadaWdaeaapeGaamiDaaGaayjkaiaawMcaaaaa@477E@
Where k is the constant of proportionality. Hence the model is, y ( t )=Aky( t ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGabmyEa8aagaqba8qadaqadaWdaeaapeGaamiDaaGaayjkaiaa wMcaaiabg2da9iaadgeacqGHsislcaWGRbGaamyEamaabmaapaqaa8 qacaWG0baacaGLOaGaayzkaaaaaa@41AC@
Where y'( t )= dy dt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEaiaacEcadaqadaWdaeaapeGaamiDaaGaayjkaiaawMca aiabg2da9maalaaapaqaa8qacaWGKbGaamyEaaWdaeaapeGaamizai aadshaaaaaaa@4001@

Results and Discussion

In this section, we will try to analyze the results of mathematical model. Comparing it with dy dt + Py=Q MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSaaa8aabaWdbiaadsgacaWG5baapaqaa8qacaWGKbGaamiD aaaacqGHRaWkcaGGGcGaamiuaiaadMhacqGH9aqpcaWGrbaaaa@4066@ , whose solution is y  e Pdt =  Q e Pdt + C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEaiaacckacaWGLbWdamaaCaaabeqaamaavacabeqabeaa caaMb8oabaWdbiabgUIiYdaacaWGqbqcfaIaamizaiaadshaaaqcfa Oaeyypa0JaaiiOa8aadaqfGaqabeqabaGaaGzaVdqaa8qacqGHRiI8 aaGaamyuaiaadwgapaWaaWbaaeqabaWaaubiaeqabeqaaiaaygW7ae aapeGaey4kIipaaiaadcfajuaicaWGKbGaamiDaaaajuaGcqGHRaWk caGGGcGaam4qaaaa@529E@ , where P and Q are constants or functions of t. Here, P = k, Q = A and e Pdt = e kdt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyza8aadaahaaqabeaadaqfGaqabeqabaGaaGzaVdqaa8qa cqGHRiI8aaGaamiuaKqbGiaadsgacaWG0baaaKqbakabg2da9OGaam yza8aadaahaaWcbeqaamaavacabeadbeqaaiaaygW7a4qaa8qacqGH RiI8aaWccaWGRbqcbaIaamizaiaadshaaaaaaa@47C5@ we have
y e kdt =  Q e kdt + C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEaiaadwgapaWaaWbaaeqabaWaaubiaeqabeqaaiaaygW7 aeaapeGaey4kIipaaiaadUgajuaicaWGKbGaamiDaaaajuaGcqGH9a qpcaGGGcWdamaavacabeqabeaacaaMb8oabaWdbiabgUIiYdaacaWG rbGaamyza8aadaahaaqabeaadaqfGaqabeqabaGaaGzaVdqaa8qacq GHRiI8aaGaam4AaKqbGiaadsgacaWG0baaaKqbakabgUcaRiaaccka caWGdbaaaa@51B0@
y e kt =  A e kt k + C MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEaiaadwgapaWaaWbaaeqajuaibaWdbiaadUgacaWG0baa aKqbakabg2da9iaacckadaWccaWdaeaapeGaamyqaiaadwgapaWaaW baaeqajuaibaWdbiaadUgacaWG0baaaaqcfa4daeaapeGaam4Aaaaa cqGHRaWkcaGGGcGaam4qaaaa@4636@
y=  A k + C e kt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEaiabg2da9iaacckadaWccaWdaeaapeGaamyqaaWdaeaa peGaam4AaaaacqGHRaWkcaGGGcGaam4qaiaadwgapaWaaWbaaeqaju aibaWdbiabgkHiTiaadUgacaWG0baaaaaa@42C5@
Imposing the boundary conditions, y(0) = 0, at t = 0 we have, C=  A k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaam4qaiabg2da9iaacckadaWccaWdaeaapeGaamyqaaWdaeaa peGaam4Aaaaaaaa@3B91@ . Therefore, the level of the nanoparticles present in the blood stream of the patient at time t is given as
y( t )=  A( 1  e kt ) k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamyEamaabmaapaqaa8qacaWG0baacaGLOaGaayzkaaGaeyyp a0JaaiiOamaaliaapaqaa8qacaWGbbWaaeWaa8aabaWdbiaaigdacq GHsislcaGGGcGaamyza8aadaahaaqcfasabeaapeGaeyOeI0Iaam4A aiaadshaaaaajuaGcaGLOaGaayzkaaaapaqaa8qacaWGRbaaaaaa@479A@
Where A represents the amount of nanocapsules injected into the blood stream per minute. The removal rate is given as
Removal rate( R )=ky=  KA( 1  e kt ) k = A( 1  e kt ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeGaamOuaiaadwgacaWGTbGaam4BaiaadAhacaWGHbGaamiBaiaa cckacaWGYbGaamyyaiaadshacaWGLbWaaeWaa8aabaWdbiaadkfaai aawIcacaGLPaaacqGH9aqpcaWGRbGaamyEaiabg2da9iaacckadaWc caWdaeaapeGaam4saiaadgeadaqadaWdaeaapeGaaGymaiabgkHiTi aacckacaWGLbWdamaaCaaajuaibeqaa8qacqGHsislcaWGRbGaamiD aaaaaKqbakaawIcacaGLPaaaa8aabaWdbiaadUgaaaGaeyypa0Jaai iOaiaadgeadaqadaWdaeaapeGaaGymaiabgkHiTiaacckacaWGLbWd amaaCaaajuaibeqaa8qacqGHsislcaWGRbGaamiDaaaaaKqbakaawI cacaGLPaaaaaa@61BC@
Now, the ratio of level present (L) and the removal rate (R) is L R =  1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaWdbiaadYeaa8aabaWdbiaadkfaaaGaeyypa0Ja aiiOamaaliaapaqaa8qacaaIXaaapaqaa8qacaWGRbaaaaaa@3CB7@ , or R L = k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaWdbiaadkfaa8aabaWdbiaadYeaaaGaeyypa0Ja aiiOaiaadUgaaaa@3BAC@ . The value of k will determine the behavior of level present and removal rate. Figure 1 shows the graphs between L R = 1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaWdbiaadYeaa8aabaWdbiaadkfaaaGaeyypa0Za aSGaa8aabaWdbiaaigdaa8aabaWdbiaadUgaaaaaaa@3B93@ and R L = k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaWdbiaadkfaa8aabaWdbiaadYeaaaGaeyypa0Ja aiiOaiaadUgaaaa@3BAC@ . It is clear that the value of k should be chosen in such a manner so that the removal rate of the nano capsules should be equivalent to the level present in the target body in order to avoid the complications created by its presence in the body.

Figure 1: The graphs between L R =  1 k MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaacbmWdbiaa=Xeaa8aabaWdbiaa=jfaaaGaeyyp a0Jaa8hOamaaliaapaqaa8qacaaIXaaapaqaa8qacaWFRbaaaaaa@3CB4@ , and R L = k MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbacbaaaaaaa aapeWaaSGaa8aabaacbmWdbiaa=jfaa8aabaWdbiaa=XeaaaGaeyyp a0Jaa8hOaiaa=Tgaaaa@3BA8@ .

Conclusion

In the present communication, a theoretical study has been carried out about nanostructures acting as nano capsules for targeting drugs to specific sites in the body in drug delivery systems without incorporating the effect of viscosity and friction of blood, gravity, and external guiding field. A mathematical model has been formulated for the delivery and removal rate of nanocapsules using a simple differential approach. It would be interesting to analyze the modeling results in terms of their connection with real parameters like viscosity and friction of blood, gravity and external guiding field.

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