Nanorobot Drug Delivery System for Curcumin for Enhanced Bioavailability During Treatment of Alzheimer’S Disease

Journal of Encapsulation and Adsorption Sciences, 2013, 3, 24-34 http://dx.doi.org/10.4236/jeas.2013.31004 Published Online March 2013 (http://www.scirp.org/journal/jeas)

Nanorobot Drug Delivery System for Curcumin for Enhanced Bioavailability during Treatment of Alzheimer’s Disease

Kal Renganathan Sharma Advanced Technology, Lone Star College, Houston, USA Email: [email protected]

Received December 29, 2012; revised January 29, 2013; accepted February 7, 2013

ABSTRACT Robotics has emerged as a collegiate course about 20 years ago at Stanford University, Stanford, CA. From the first IRB6, the electrically powered robot in 1974, the industry has grown over a 30-year period. A leading supplier of robots has put out over 100,000 robots by year 2001. Robot capable of handling 500 kg load was introduced in 2001, IRB 7000. A number of advances have been made in nanostructuring. About 40 different nanostructuring methods were re- viewed recently [1]. Nanobots can be developed that effect cures of disorders that are difficult to treat. Principles from photodynamic therapy, fullerene chemistry, nanostructuring, X-rays, computers, pharmacokinetics and robotics are applied in a design of nanorobot for treatment of Alzheimer’s disease. The curcuma longa that has shown curative effects in rats’ brain with Alzheimer’s is complexed with fullerenes. The drug is inactive when caged. It is infused intrathecally into the cerebrospinal system. Irradiation of the hypothalamous and other areas of the brain where Alzheimer’s disease is prevalent lead to breakage of fullerenes and availability of the drug with the diseased cells. Due to better mass transfer, better cure is affected. The other plausible reactions such as addition polymerization of fullerene, polycurcumin formation and other hydrolysis reactions are modeled along with the drug action under the Denbigh scheme of reactions. The fractional yield of drug-curcumin interaction is a function of intensity of radiation, frequency of radiation, patient demographics, age, gender, and other disorders etc. Chromophore in curcumin is used as a sensor and computer imaging and feedback control design can result in more bioavailability for curcumin therapeutic action to cure Alzheimer’s disease. This study examines the principles used in the design, the strategy of the design of the nanorobot drug delivery system with a specific target and pharamacokinetic formulation of the associated competing parallel reactions. The burrow and link capabilities at a nanoscopic level are also available if needed.

Keywords: Nanorobot; Alzheimer’s Disease; Curcuma Longa; Photodynamic Therapy; Intrathecal Infusion; Denbigh Reaction Scheme

1. Introduction nant cells treated; 3) fewer mistakes on account of computer control and automation; 4) reach remote areas in About 4000 mistakes are made by surgeons every year human anatomy not operatable at the surgeon’s operating [2]. There are regions such as the hypothalamus in the table; 5) as drug molecules are carried by nanorobots and cranium that the surgeon cannot reach in the operating released where needed the advantages of large interfacial table on account of high morbidity rates for patients who area during mass transfer can be realized; 6) non-inva- underwent open brain surgery. A number of patients have sive technique; 7) computer controlled operation with cancer that is terminal. Knowledge of the presence of nobs to fine tune the amount, frequency, time of release; malign cells wherever it is in the human anatomy is not 8) better accuracy; 9) drug inactive in areas where ther- sufficient to affect cures. Nanorobot drug delivery sys- apy not needed minimizing undesired side effects. tems can be a solution to these problems. Advances in Fullerenes were mass produced by FCC, Frontier Carbon the areas of robotics, nanostructuring, medicine, bioin- Corporation at 40 tons/year in 2003. formatics, computers can lead to the development of nanorobot drug delivery system. Nanorobots can offer a 1.1. Development of Robotics as Colelgiate number of advantages over current methods such as; 1) Course use of nanorobot drug delivery systems with increased bioavailability; 2) targeted therapy such as only malig- Industrial robots came about in the 1960s. The adoption

Copyright © 2013 SciRes. JEAS K. R. SHARMA 25 of robotic equipment came about in the 1980s. In the late motion of manipulator. State space form of the Newton- 1980s, there was a pullback in the use of robots. Use of Euler equations can be used. Simulation is used to re- industrial robots has been found to be cheaper than man- formulate the dynamic equations such that the accelera- ual labor. More sophisticated robots are emerging. Na- tion is computed as a function of actuator torque. One norobot drug delivery systems can be one such robot. way to effect manipulator motion from here to there in a 78% of the robots installed in the year 2000 were weld- specified smooth fashion is to cause each joint to move ing and material-handling robots. The Adept 6 manipu- as specified by a smooth function of time. To ensure lator had 6 rotational joints. The most important form of proper coordination, each and every joint starts and stops the industrial robot is the mechanical manipulator. The motion at the same time. The computation of these func- mechanics and control of the mechanical manipulator are tions is the problem of trajectory generation. A spline is described in detail in introductory robotics courses [1]. a smooth function that passes through a set or via points. The programmability of the device is a salient considera- End effector can be made to travel in a rectilinear manner. tion. The math needed to describe the spatial motions and This is called Cartesian trajectory generation. other attributes of the manipulators is provided in the The issues that ought to be considered during the me- course. The tools needed for design and evaluation of chanical design of manipulator are cost, size, speed, load algorithms to realize desired motions or force applica- capability, number of joints, geometric arrangement, tions are provided by control theory. Design of sensors transmission systems, choice and location of actuators, and interfaces for industrial robots is also an important internal position and sensors. The more joints a robot task. Robotics is also concerned with the location of ob- arm contains, the more dexterous and capable it will be. jects in 3 dimensional space such as its: 1) position and But it will also be harder to build and be more expensive. orientation; 2) coordinate system and frames of reference Specialized robots are developed to perform specified such as tool frame and base frames; 3) transformation tasks and universal robots are capable of performing from one coordinate system to another by rotations and wide variety of tasks. 3 joints allow for the hand to reach translations. any position in 3 dimensional spaces. Stepper motors or Kinematics is the science of motion that treats motion other actuators can be used to execute a desired trajectory without regard to the forces that cause it. In particular, directly. This is called the linear positional control. Ki- attention is paid to velocity, acceleration and acceleration netic energy of the manipulator links can be calculated of the end effector. The geometrical and time based using the Langrangian formulation. Both the linear ki- properties of the motion are studied. Manipulators con- netic energy and rotational kinetic energies can be sist of rigid links that are connected by joints that allow tracked. relative motion of neighboring links. Joints are instru- The evolution of robots can be studied from the market mented with position sensors which allow the relative experience of one leading manufacturer of robots for motion of neighboring links to be measured. In case of example, ABB robotics, Zurich, Switzerland. This shall rotary or revolute joints, these developments are called be used later to place in perspective the increased interest joint offset. The number of independent position vari- in use of nanorobots in the hospital for drug delivery. ables that would have to be specified in order to locate all The different products from ABB Robotics during a 30 parts of the mechanism is the number of degrees of free- year period are given below in Table 1 [3]. dom. End effector is at the end of the chain of links that 1.2. Development of Nanotechnology make up of the manipulator. This can be a gripper, a welding torch, an electromagnet, etc. Inverse kinematics Submarine nanorobots are being developed for use in is the calculation of all possible sets of joint angles that branchy therapy, spinal surgery, cancer therapy, etc. could be used to attain the given position and orientation Nanoparticles have been developed for use in drug de- of the end effector of the manipulator. For industrial ro- livery systems and for cure in eye disorders and for use in bots, the inverse kinematic algorithm equations are non- early diagnosis. Research in nanomedicine is under way linear. Solution to these equations is not possible in in development of diagnostics for rapid monitoring, tar- closed form. The analysis of manipulators in motion in geted cancer therapies, localized drug delivery, and im- the workspace of a given manipulator includes the de- proved cell material interactions, scaffolds for tissue en- velopment of the Jacobean matrix of the manipulator. gineering and gene delivery systems. Novel therapeutic Mapping from velocities in joint space to velocities in formulations have been developed using PLGA based Cartesian space is specified by the Jacobean. The nature nanoparticles. Nanorobots can be used in targeted ther- of mapping changes with configuration. The mapping is apy and in repair work of DNA. Drexler and Smalley not invertible at points called singularities. debated whether “molecular assemblers” that are devices Dynamics is the study of actuator torque functions of capable of positioning atoms and molecules for precisely

Table 1. State of art in robots over a 40 year period.

Year Product

World’s First Microcomputer controlled Electric Industrial Robot, IRB6 from ASEA was delivered to a small mechanical 1974 engineering company in Southern Sweden.

1975 IRB6—First Robot for Arc Welding

1977 First Robots installed in France and Italy

1979 IRB60—First Electrical Robot for Spot Welding; First Robot installed in Spain

1982 Robots introduced in Japan

S2, New Control System. Outstanding, HMI, Menu Programming, TCP (Tool Center Point) and the Joy Stick introduced. Allows 1983 control of several axes. ASEA bought Trallfa Robot operations, Bryne, Norway. Trallfa launched the world’s First Painting Robot in 1969. Sales boomed 1986 in the mid 1980s when the automotive industry started to paint bumpers and other plastic parts. IRB2000—10 Kg Robot. First to be driven by AC motors. Large working range. Great accuracy. ABB acquired Cincinatii Milacron, USA, Graco, USA (robotic painting), Rarsburg Automotive (electrostatic painting atomizers). 1990/1991 IRB6000—200 kg Robot introduced. First Modular Robot that is the fastest and most accurate spot welding robot on the market. Unique hollow wrist introduced on Painting Robots. Allows faster and more agile motion.

1994 S4—Breakthrough in user fiiendliness, dynamic models, gives outstanding performance. Flexible rapid language.

1996 Integrated Arc Welding Power Source in Robot Cabinet.

Launch of Flex Picker Robot, the World’s fastest Pick and Place Robot. Robot Studio-First simulation tool based on virtual 1998 controller identical to the real one revolutionize off-line programming.

2000 Pick and Place Software PickMaster introduced.

2001 IRB7000—First Industrial Robot to handle 500 kg.

ABB—First company in World to sell 100,000 Robots Virtual Arc-True arc welding simulation tool that gives robot welding 2002 engineers full “off-line” control of the MIG/MAG process. IRB 6000-Power Robot with bend over backwards flexibility.

2004 IRC5—Robot Controller. Windows interface unit.

2005 Launch of 55 new products and robot functions included with 4 new robots: IRB 660, IRB 4450, IRB1600, IRB260. defined reactions in any environment is possible or not. tructures; 12) dip pen lithography; 13) SAM, self-as- Feynman’s vision of miniaturization is being realized. sembled monolayers; 14) hot embossing; 15) nanoim- Smalley sought agreement that precision picking and print lithography; 16) electron beam lithography; 17) dry placing of individual atoms through the use of “Smalley- etching; 18) reactive ion etching; 19) quantum dots and fingers” is an impossibility. Fullerenes, C60, are the third thin films generation by; 20) sol gel; 21) solid state pre- allotropic form of carbon. Soccer ball structured, C60, cipitation; 22) molecular beam epitaxy; 23) chemical with a surface filled with hexagons and pentagons satisfy vapor deposition; 24) CVD; 25) lithography; 26) nuclea- the Euler’s law. Fullerenes can be prepared by different tion and growth; 27) thin film formation from surface methods such as: 1) first and second generation combus- instabilities; 28) thin film formation by spin coating; 29) tion synthesis; 2) chemical route by synthesis of coran- cryogenic milling for preparation of 100 - 300 nm sized nulene from naphthalene. Rings are fused and the sheet titanium; 30) atomic lithography methods to generated that is formed is rolled into hemisphere and stitched to- structures less than 50 nm; 31) electrode position method gether; 3) electric arc method. Different nanostructuring to prepare nanocomposite; 32) plasma compaction meth- methods are discussed in [4,5]. These include: 1) sput- ods; 33) direct write lithography; 34) nanofluids by dis- tering of molecular ions; 2) gas evaporation; 3) process persion. Thermodynamic miscibility of nanocomposites to make ultrafine magnetic magnetic powder; 4) triangu- can be calculated from the free energy of mixing. The lation and formation of nanoprisms by light irradiation; 5) four thermodynamically stable forms of Carbon are dia- nanorod production using condensed phase synthesis mond, graphite, C60, Buckminster Fullerene and Carbon method; subtractive methods such as; 6) lithography; 7) Nanotube. 5 different methods of preparation of CNTs, etching; 8) galvanic fabrication; 9) lift-off process for IC carbon nanotubes were discussed. Thermodynamically circuit fabrication; 10) nanotips and nanorods formation stable dispersion of nanoparticles into a polymeric liquid by CMOS process; 11) patterning Iridium Oxide nanos- is enhanced for systems where the radius of gyration of

Copyright © 2013 SciRes. JEAS K. R. SHARMA 27 the linear polymer is greater than the radius of the heimer’s disease will be 16 million. The cost of memory nanoparticle. Tiny magnetically-driven spinning screws loss is about $1.1 trillion. Currently 5 million patients were developed. Molecular machines are molecules that suffer from Alzheimer’s in this country, United States. can with an appropriate stimulus be temporarily lifted out The root curcuma longa, turmeric root has found to have of equilibrium and can return to equilibrium in the ob- a positive effect in experiments with rats [9]. The drugs servable macroscopic properties of the system. Molecular used such as bapinezumab from Eli Lily and solanezu- shuttle, molecular switches, molecular muscle, molecular mab from Johnson and Johnson used today are used to rotors, molecular nanovalves are discussed. Supramo- target the amyloid-β protein in the brain. This protein has lecular materials offers alternative to top-down minia- been found to misfold and results in clump formation in turization and bottom-up fabrication. Self-organization patients’ brains. Furthermore these drugs are expensive. principles hold the key. Gene expression studies can be Turmeric root is used in curries in India and is readily carried out in biochips. CNRs are a new generation of available from farms at a lower cost. It is yellow colored. self-organizing collectives of intelligent nanorobots. This This is recommended by Ayurveda and Siddha Medicine. new technology includes procedures for interactions It is known to possess anti-inflammatory and antioxidant between objects with their environment resulting in properties. solutions of critical problems at the nanoscopic level. Curcumin (Figure 1) has been found to decrease Biomimetic materials are designed to mimic a natural inflammation in the brain. It causes reduction in biological material. Characterization methods of nanos- oxyradicals formed in patients’ brain, tructures include SAXS, small angle X-ray scattering, TEM, transmission electron microscopy, SEM, scanning 1.5. Nanomedicine-Advantages of Nanorobot electron microscopy, SPM, scanning probe microscope, Drug Delivery System Raman microscope, AFM atomic force miscroscopy, Nanorobots can offer a number of advantages over cur- HeIM helium ion microscopy. rent methods such as: 1) use of nanorobot drug delivery systems with increased bioavailability; 2) targeted ther- 1.3. Recent Developments in Nanorobots apy such as only malignant cells treated; 3) fewer mis- Some investigators refer to nanorobotics as molecular takes on account of computer control and automation; 4) robotics. A variety of definitions for nanorobots are reach remote areas in human anatomy not operatable at available in the literature. Per Ummat et al. [7] nanoro- the surgeon’s operating table; 5) as drug molecules are bots refer to any active structure with capability for ac- carried by nanorobots and released where needed the tuation, sensing, signaling, information processing, intel- advantages of large interfacial area during mass transfer ligence or swarm behavior at the nanoscale. Sandia Na- can be realized; 6) non-invasive technique; 7) computer tional Laboratories [8] provided a classification of nano- controlled operation with nobs to fine tune the amount, robotics into two main areas: 1) Design, simulation, con- frequency, time of release; 8) better accuracy; 9) drug trol and coordination of robots with nanoscale dimen- inactive in areas where therapy not needed minimizing sions. Naorobot drug delivery systems will fall into this undesired side effects; 10) greater speed of drug action. category/Much of the research conducted in this area Frietas [10] defined nanomedicine as follows: remains theoretical currently. This is primarily because “Nanomedicine is the preservation and improvement of of the difficulties in manufacturing these devices. Syn- human health using molecular tools and molecular thetic nanorobot development is driven by emulation of knowledge of the human body.” biological nanorobotic systems that exist in nature (Re- Tiny magnetically-driven spinning screws were de- quicha [9]). Lot of bionanorobots is protein based. 2) veloped by Ishiyama et al. [11]. These devices were in- Manipulation/assembly of nanoscale components with tended to swim along veins and carry drugs to infected microscale instruments of or robots/nanomanipulators. A tissues or even to burrow into tumors and kill them with number of research papers have appeared in the literature supply of heat. Untethered microrobot containing ferro- in this area. As can be seen from section 1.2 advances in magnetic particles under forces generated by MRI mag- nanostructuring methods have been made. Practical netic fields were tested for travel through the human technologies for the manipulation and assembly of nanoscale structures into functional nanodevices need be developed. Nanomanipulation and nanoassembly may be critical in the development of synthetic nanorobots.

1.4. Alzheimer’s Disease and Drugs Used Today By the year 2050 the number of patients with Alz- Figure 1. Structure of curcumin.

Copyright © 2013 SciRes. JEAS 28 K. R. SHARMA anatomy at the Nano Robotic laboratory at Montreal, may be planted above the stack. SMT stack consists of a Canada in 2003. top electrode, a free layer, a non-magnetic layer a pinned A nanorobot to measure surface properties was pat- magnetic structure and a bottom electrode. ented [12]. This technology is 10 - 20 years in the future. Nanowhiskers can be grown by control of nucleation The nanorobot unit has a manipulation unit and an end conditions [16]. Nanowhisker formation on substrates effector. The end effector can be a senor or made to can be made using the VLS, vapor-liquid-solid mecha- move as close as possible to the surface of interest. The nism. A particle of catalyst material is placed on a sub- drive device has piezoelectric drives. The resolution of strate is heated in the presence of gases until it melts. A the measurement can be in the nanometer range and the pillar is allowed to form under the melt. As the melt rises actual measurement in the centimeter range. The nano- up on top of the pillar and a nanowhisker is formed. robot is made sensitive in all directions, in multiple di- Miller direction (111) may be a preferred growth direc- mensions. It can operate under vacuum. Surface rough- tion of the whiskers. The catalytic property is present at ness can be measured using nanorobots. the interface of whisker and air. InP nanowhiskers, for A patent was developed [13] for minimally invasive example, were grown using metal-organic vapor phase procedure by means of DPR, Dynamic Physical Render- epitaxy. Characterization of nanowhiskers is by electron ing. Use of “intelligent”, “autonomous” particles were microscopy. MOVPF is low pressure metal-orgaic vapor made. An interventional aid is formed with the aid of phase epitaxy process. 50 nm aerosol gold particles were self-organizing nanorobots. These nanorobots were made deposited on InP substrate. This was placed on a hori- of catoms. C-arm angiography are used to monitor DPR zontal reactor cell heated by radio frequency heated procedures. A 3 dimensional image data record on target graphite suceptor. Hydrogen was used as carrier gas. region is obtained. The determined form was converted Temperatuer was ramped tp 420˚C for 5 min. The molar to a readable and executable program code for the catoms fraction of flow rate in the cell was 0.015. Nanowhisker of the nanorobots. The determined form was transferred growth was found to commence upon addition of TMI, to a storage unit. The program code was executed in or- trimethylindium. The molar fraction of TMI was 3 mil- der to achieve self-organization in the unstructured ca- lionth. Typical growth time for production of nanowhisk- toms that were introduced in the target region. The exe- ers was found to be 8 minutes. cution of the program was triggered by a timer or position sensor. The intervention aid is used as an endovascu- 2. Design and Control Strategy lar target region. Nanocrystal with motor properties was patented [14]. The strategy for design of nanorobot drug delivery sys- Reciprocating motor is formed by a substrate, atom res- tem among other things comprise of the following salient ervoir, nanoparticle ram and nanolever. The nanoparticle features: ram is contacted by the nanolever resulting in movement 1) Use fullerenes as carriers of curcumin. of atoms between the reservoir and the ram. Substrate 2) Complex the fullerene with curcumin and deactivate and nanolever are made of MWNTs, multi-walled nano- the drug action as in other cases of photodynamic ther- tubes made of Iridium. The substrate used was a silicon apy. chip. 3) Computer controlled irradiation can activate the Nanoscale oscillator [15] was patented by Sea Gate fullerene. Breakage of fullerene is expected and release technology, Scots Valley, CA. A microwave output is of curcumin at constant rate [17,18]. The intensity, generated by application of a DC current that is allowed wavelength and areas of brain where is irradiated is con- to pass through layers of magnetic structure separated by trolled using computer software. nanometer dimensions. Spin Momentum Transfer, SMT 4) Curcumin acts on the diseased cells and cure is af- is a phenomena realized to exist in 1989. It can be used fected. in MRAM devices. Phase-locked microwave spin trans- 5) Pharmacokinetic model with 4th order process for fer is the next advance of the technology. The electric irradiation and competing parallel reactions for fullerene current produced is in the GHz spectrum. The local breakage, polymerization of fullerene and drug action is magnetic field source is used in the application of a developed. magnetic field to a free layer of spin momentum transfer 6) Time for drug release can be computed from the stack. The magnetic source can be from a horseshoe drug delivery model. Drug delivery model for fullerene magnet with poles stationed above and below the SMT nanorobot curcumin delivery system can be developed stack. The magnetic source can take on other forms such including all the main features of the procedure. as helical coil that surrounds the SMT or pancake type 7) Chromophores in curcumin is used as a dye and coils above and below the SMT stack or an annular pole picked up from imaging. Information from the sensor on and a coil that surround the stack. A permanent magnet drug action is compared against model and feedback

Copyright © 2013 SciRes. JEAS K. R. SHARMA 29 control affected by changing the intensity and wave- tors. PFR analysis will be a more reasonable approxima- length of irradiation. This is used to trigger feedback tion of these processes. Denbigh scheme is the general control or model based control of irradiation using com- scheme under which the parallel reactions of fullerene puters. breakage, polymerization of fullerene and polymerization 8) Itrathecal route of entry for nanorobot fullerene cur- of cucrumin may fall under. cumin complex can be preferred to sublingual entry. The choice of cerebrospinal fluid system may obviate any 3.1. Transient Dynamics of Plug Flow Reactor, digestive disorders from fullerene carbon. PFR Conventional treatments lack in their selectivity. Drugs The composition of the fluid in a PFR, plug flow reactor, administered generally become active in areas where varies from point to point along a flow path. The mass treatment is not needed. Sometimes negative side effects balance on the reacting species need be made over a of the drugs can be prevalent in those areas. Suppression of side effects is emerging as a salient consideration in differential volume Az as shown in Figure 2. A is the the development of new drugs and drug strategies. cross-sectional area of the PFR and z is the incremental Nanorobot drug delivery systems that are act only in tar- length of the slice considered in Figure 2. This is different geted areas can be more beneficial. from a CSTR where the mass balance on the reacting Complexing curcumin with fullerene may be more species is performed over the volume of the CSTR. For a advantageous compared with binding with a polymer reacting species A with a inlet composition of CAi −3 molecule as carrier. The dimensions of the macromole- (mol·m ) and v is the superficial velocity of the fluid −1 cule carrier are large. On account of its size it cannot (m·s ) the mass balance over the slice AzVd can be migrate to any part of the human anatomy. In such cases written as follows: the drug stays near the region it is administered during rate of reactant influx rate of reactant efflux the duration of treatment. This may call for separate ad- ministration of complex to each area requiring treatment.  disappearence of rectant reaction (1) This can consume lot more time and the procedure be-  rate accumulation reactant comes impractical given a great number of treatment AvC AvC AkC z A z C t (2) sites. The fullerene molecules as carriers can migrate to  AAzzz   A any region of the human anatomy such as the hypo- Dividing Equation (2) throughout by Az and obtaining thalamus in the brain and be a better choice for carrier. Attempts to design drug delivery system with lipo- the limit as z  0 Equation (2) becomes: some encapsulated drug involve the rupture of the mem- CC vkCAA (3) brane and drug release. This is triggered by irradiation zA t and pH changes. The response was found to be inconsis- tent. The pathway to drug release is indirect. i.e., the ir- Equation (3) can be made dimensionless as follows: radiation and pH changes affect a photosensitizer. A CCAi A kL z vt XDaZA  ;;; (4) compound in turn is generated that is capable of cleaving CvLL a specific entity in the liposome membrane. At tempera- Ai tures below the body temperature of 37˚C the drug re- Upon substitution of Equation (5) in Equation (6): lease drops. Thus at 15˚C the release rate was less than XX 15%. This limits the application of this design. AADaX  Da (5) Z  A 3. Pharmacokinetics of Curcumin Therapy Differentiating Equation (5) with respect to dimen- The irradiation is supplied to activate the fullerene and sionless time: effect detachment of the cur cumin from the fullerene 22 X AAX XA setting the onset of therapeutic action. The irradiation can 2 Da (6) Z   also trigger polymerization of fullerene. Formation of polyfullerene can soak up the energy from irradiation Differentiating Equation (3) with respect to dimen- that is needed for activation of the drug. Curcumin may polymerize to form polycurcumin. The selectivity of the breakage of fullerene cages and release of drug over formation of polyfullerene and formation of polycurcumin are studied using transient dynamic analysis of PFR, plug flow reactor and pharmacokinetics of Denbigh reaction. The therapeutic action of curcumin during Alz- herimer’s disease can be viewed as a set of micro reac- Figure 2. Transient dynamics in a plug flow reactor, PFR.

Conversion XA can be assumed to be analytic in the s ln 1 X A  DaZ c space and time Z, . The order of the differentialion (17) does not matter. i.e., sD aZ XcA 1e 22 X AX A  (8) At ZX 0,A  0. So, c  1. Equation (17) becomes:  ZZ   s DaZ X A 1e  (18) Adding Equations (6) and (7) and realization of Equa- tion (8) leads to: It can be seen that at steady state the conversion varies as a f ez . Equaiton (15) can be multiplied throughout 2X XX2X   AA AA by emZ. The term emt X can be seen to group and can 22Da Da (9)  A  Z Z   be called as : Equation (9) is a hyperbolic PDE, partial differential 22 mWWm   equation that is second order with respect to space and ee  (19) Z 22Z  4 second order with respect to time. The exact form ana- Now, lytical solution to Equation (9) can be obtained as fol- n W  lows. Equation (9) is multiplied by e . Equation (9) be- em m (20) comes: ZZ

2 nn 22 eeXX 2 m  W 2   AA nnX AAX e2mm   (21) Da Daee(10) 22 Z 22Z   Z Z Z n Substituting Equaiton (19)-(21) in Equaiton (15), The term e X A  can be seen to group and can be called as the wave concentration [19] such that: Equaiton (15) becomes: 2 WX en (11)  2 A 2 Da 2m mDa m Z Z Now, (22) 22Da  W X 2 nXeennA (12)  4  A  For mDa 2 , Equaiton (15) becomes: X 222222 n A W   Da Da   e nW (12a) or  (23)  ZZ2244 2 2 Differentiating Equation (12) again with respect to Equaiton (23) is the wave equation. In the dimensional time,  form of independent variables: 2 X WW2 W 22 n A 2 2     e 22nW  n   n (13) v  (24)   zt22 Plugging Equaiton (13), Equaiton (12) and Equaiton It can be seen that the solution to Equaiton (24) can be (11) in Equaiton (10) written as (Bird, Stewart and Lightfoot [20]): 2WW 2π  Da  2 001sin  zvt  (25) Z Z  2 (14) 2 WW where  is the wavelength of a harmonic wave with nDa  n W 2  Da 2 n     amplitude 00  traveling in the z direction at a speed v. For nDa 2 , Equaiton (14) becomes: The solution for th e transient conversion can be written as follows: 222 Da Da WWWDa W  2π Da (15) 22 22 X A ee00 1 sinzvt (26) Z Z  4  The steady state conversion can be obtained from The Dankwert’s boundary condition at z = L can be Equaiton (15) as follows: applied as:

X dC A  0 (27) T kC  kC (38) z dt 25R S 12π 2π The reactions are considered to perform in the CSTR 0100sinLvt  cosLvt(28) 2 similar to the one shown in Figure 5. Component mass

At the wave front, L = vt, Equaiton (28) becomes:

1 0 (29)

kt zk  2 2v 2π X A ee0 1sinzvt (30)  At the entrance of the reactor, the initial reaction rate is given by: dX A  k (31) dt Differentiating Equaiton (30) with respect to t, X vv2π A  1  k tL2 00 (32) 2kL   0 vL4π  

zk vkt  22Da 2v π X A e1sinzvt (33) vL4π   The transient conversion of drug-target interaction in a PFR is plotted in Figure 3 for a set of Damkhohler number and other parameters such as space time, length of Figure 3. Conversion XA in PFR for Damkohler Number, the reactor, fluid velocity, reaction rate constant, wave- Da = 6.75, v = 0.55 m·s−1, L = 1.5 m;  = 2.7 hr, k = 2.5 hr−1, length of propagating wave.  = 0.33 m.

3.2. Pharmacokinetics of Curcumin Drug and Polymerization Parallel Reactions Denbeigh Scheme of Reactions Consider the Denbigh scheme of reactions [21] performed in the CSTR shown in Figure 4. A scheme of reactions as shown in Figure 4 was discussed in [17] as a special case of Denbigh reactions. A state space model is Figure 4. Denbigh scheme of reactions. developed in order to describe the dynamics of the 5 species, CA, CR, CT, CB and CS. The assumptions are that the inlet stream contains species A and B at a concentration of CAi and CBi and the initial concentrations of the other species are zero. The kinetics of the simple irreversible reactions shown in Figure 4 can be written as follows: dC A kkC  (34) dt 13A dC B kC (35) dt 4 B dC S kC kC kC (36) dt 435B AS dC R kC  kC (37) dt 12AR Figure 5. CSTR, Continuous stirred tank reactor.

balances on each of the species assuming incompressible where Da44  k   flow and constant volume reactor can be written as fol- Species S lows: dC Species A CDaDaCDaC1   S (41) SB54 3Ad dC C C1 Da Da A (39) Species R Ai A 13d dC tV CDaDaC1  R (42) where Da k;;; Da k ,  with units RA21d 11 33  v of (hr) is the residence time of the species in the reactor, Species T V is the volume of the reactor, (liter) and v is the volu- dC −1 CDaCDaC   T (43) metric flow rate (lit·hr ) in and out of the reactor. TRS25d Species B The model equations that can be used to describe the dCB dynamics of the 5 reactant/product species in a CSTR CCBi B 1 Da4  (40) d can be written in the state space form as follows:

CCA 1Da13  Da 0000A CAi  CCC B 01Da4 000B Bi d  CCS Da Da 100Da S 0 (44) dt 345  CCR Da120010Da R 0  CCT 00Da52 Da  1T 0

The stability of the dynamics of the 5 reactants/prod- The characteristic equation for the eigenvalues is ob- ucts in the Denbigh scheme performed a CSTR can be tained by evaluation of the following determinant: studied by ob taining the eigenvalues of the rate matrix.

 1000Da1  Da3 0  0100 Da4 0 det Da Da 10 Da 0 34 5 (45) Da1 001  Da2 0  00Da52 Da  1

111110Da13  Da  Da 4  Da 5  Da 2  

The 5 eigenvalues are negative when Damkohler C Cs Bi (47) numbers are greater than zero. When eigenvalues are all B s sDa1  negative the system is considered to be stable. 4 Da C The Laplace transform of the model equations devel- 3 Ai CsS  oped in order to describe the transient dynamics (Equa- s sDaDasDa1113   5 (48) tions (39)-(43)) for the 5 species in the Denbigh scheme Da4 CBi in a CSTR can be written as follows:  ss11 Das45   Da

CAi Da1 CAi CsA  (46) CsR  (49) s sDaDa1 13  s 1Da21 s s 1Da  Da3

1  Da Da C Da Da C Cs 53Ai 45Bi T   ss11  s  Da13  Da s  1 Da 5 s  1 Da4 s  1 Da 5 (50)  Da12 Da CAi   s11 Da2 s  Da13  Da 

The inverse Laplace transform of Equaiton (46) can be seen from the time shift property to be: obtained by invocation of the convolution theorem and Ct  written as: B   1 1 Da4  1e  (52) CDaBi 1 4 Ct 1 1 Da Da A 1e 13  (51)  The inverse Laplace transform of Equaiton (48) can be CDaDaAi 113 obtained by look-up of Laplace inversion Tables in The inverse Laplace transform of Equaiton (47) can be Mickley, Sherwood and Reed [22].

 1 Da2  1 Da13  Da Ct e e  1 R  (53) CAi 11 DaDaDaDaDaDaDaDaDa2213 13213 1 DaDa 21 13Da

3.3. State Space Representation dCA kC12AA  kC (54) State space models are those that describe more than one dt variable at a given instant in time. Vector form for sev- dCR eral variables is used. A set of n differential equations is kC134AR kC kCR (55) dt represented by one equation in matrices and vectors. The coefficient matrix and input and output vectors are used. dC S kC kC  kC (56) Output vector can be solved for by matrix manipulations. dt 356R SS This would form the output response of the system. The stability of the system can be studied by looking at the dCT  kC5 S (57) Eigen values of the coefficient of the matrix. The differ- dt ent instabilities that may arise and the conditions under dC which they would arise are given in Table 2 [23]. U  kC (58) dt 2 A 3.3.1. Kinetics of Simultaneous Reactions in State dC V  kC (59) Space Form dt 4 R As an example a state space model is developed to de- dCW scribe the kinetics of the following set of reactions in  kC6 S (60) series and in parallel as suggested in Levenspiel [18]. dt The kinetics of the 7 simultaneous reactions shown in The above equations can be represented in the state Figure 6 is given below as follows: space form in one line as follows:

CA kk12  0 0 0000CA  CR kkk1340 0000CR C0 kkk0000C S 356 S d   C 0 0 k 0000C (61) dt T 5 T CU k2 0 0 0000CU  CCV 0 k4 0 0000V  CW 0 0 k6 0000CW

Or, DetIK   0 (63) dC  KxC (62) The K matrix is sparse. The polynomial form of the dt characteristic equation can be written as follows: The stability of the system of reactions can be studied  kk kk kk  4 0 (64) by obtaining the Eigen values of the K matrix. The Ei- 12 34 56 genvalues of the K matrix are obtained from the roots of The 7 Eigenvalues are 0 repeated 4 time and the characteristic polynomial kk12 , kk34 and kk56. This system can

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