on photodynamic therapy of alzheimer’s disease using … · * tel.: +1 281. 290. 2834. e-mail...

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SDI Paper Template Version 1.6 Date 11.10.2012 1 On Photodynamic Therapy of Alzheimer’s 2 Disease using Intrathecal Nanorobot Drug 3 Delivery of Curcuma Longa for Enhanced 4

Bioavailability the Paper 5 6

Kal Renganathan Sharma* 7 8

1Lone Star College University Park 9

20515 State Hwy 249 10 Houston,TX 77070 11

12 13 14 15 .16 ABSTRACT 17 18 Robotics was first instructed as a collegiate course about 20 years ago at Stanford University, Stanford, CA. From the first IRB6 the electrically powered robot in 1974 over a 30 year period the industry has grown. 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 reviewed recently [2]. 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 developing a strategy for nanorobot, treatment of Alzheimer’s disease. The curcuma longa that has shown curative effects in rats’ brain with Alzheimers 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 leads to breakage of fullerenes and availability of the drug with the diseased cells. Due to better mass transfer better cure is effected. 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, 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 is also available if needed.

19


Keywords: [nanorobots; drug delivery; intrathecal injection; pharmacokinetics; 20 photodynamics; Alzheimer’s disease; curcuma Longa} 21 22 23

1. INTRODUCTION 24 25 About 4000 mistakes are made by surgeons every year [1]. There are regions such as the 26 hypothalamous in the cranium that the surgeon cannot reach in the operating table on 27 account of high morbidity rates for patients who underwent brain surgery. A number of 28 patients have cancer that is terminal. Knowledge of the presence of malign cells wherever it 29 is in the human anatomy is not sufficient to effect cures. Nanorobot drug delivery systems 30 can be a solution to these problems. Advances in the areas of robotics, nanostructuring, 31 medicine, bioinformatics, computers can lead to the development of nanorobot drug delivery 32 system. Nanorobots can offer a number of advantages over current methods such as; (i) 33 use of nanorobot drug delivery systems with increased bioavailability; (ii) targeted therapy 34 such as only malignant cells treated; (iii) fewer mistakes on account of computer control and 35 automation; (iv) reach remote areas in human anatomy not operatable at the surgeon’s 36 operating table; (v) as drug molecules are carried by nanorobots and released where 37 needed the advantages of large interfacial area during mass transfer can be realized; (vi) 38 non-invasive technique; (vii) computer controlled operation with nobs to fine tune the 39 amount, frequency, time of release; (viii) better accuracy; (ix) drug inactive in areas where 40 therapy not needed minimizing undesired side effects. Fullerenes was mass produced by 41 FCC, Frontier Carbon Corporation at 40 Tons/year in 2003. 42

43

1.1 DEVELOPMENT OF ROBOTICS AS COLELGIATE COURSE 44

Industrial robots came about in the 1960s. The adoption of robotic equipment came 45 about in the 1980s. In the late 1980s there was a pullback in the use of robots. Use of 46 industrial robots have been found to be cheaper than manual labor. More sophisticated 47 robots are emerging. Nanorobot drug delivery systems can be one such robot. 78% of the 48 robots installed in the year 2000 were welding and material-handling robots. The Adept 6 49 mamipular had 6 rotational joints. The most important form of the industrial robot is the 50 mechanical manipulator. The mechanics and control of the mechanical manipulator is 51 described in detail in introductory robotics courses [2]. The programmability of the device is 52 a salient consideration. The math needed to describe the spatial motions and other 53 attributes of the manipulators is provided in the course. The tools needed for design and 54 evaluation of algorithms to realize desired motions or force applications are provided by 55 control theory. Design of sensors and interfaces for industrial robots is also an important 56 task. Robotics is also concerned with the location of objects in 3 dimensional space such as 57 its: (i) position and orientation; (ii) coordinate system and frames of reference such as tool 58 frame and base frames; (iii) transformation from one coordinate system to another by 59 rotations and translations. 60

Kinematics is the science of motion that treats motion without regard to the forces 61 that cause it. In particular attention is paid to velocity, acceleration and acceleration of the 62 end effector. The geometrical and time based properties of the motion is studied. 63 Manipulators consist of rigid links that are connected by joints that allow relative motion of 64 neighbouring links. Joints are instrumented with position sensors which allow the relative 65 motion of neighboring links to be measured. In case of rotory or revolute joints these 66 developments are called joint offset. The number of independent position variables that 67 would have to be specified in order to locate all parts of the mechanism are the number of 68 degrees of freedom. 69


End effector is at the end of the chain of links that make up of the manipulator. This 70 can be a gripper, a welding torch, an electromagnet, etc. Inverse kinematics is the 71 calculation of all possible sets of joint angles that could be used to attain the given position 72 and orientation of the end-effector of the manipulator. For industrial robots the inverse 73 kinematic algorithm equations are non-linear. Solution to these equations are not possible in 74 closed form. The analysis of manipulators in motion in the workspace of a given manipulator 75 includes the development of the Jacobian matrix of the manipulator. Mapping from velocities 76 in joint space to velocities in cartesian space is specified by the Jacobian. The nature of 77 mapping changes with configuration. The mapping is not invertibe at points called 78 singularities. 79

Dynamics is the study of actuator torque funtions of motion of manipulator. State 80 space form of the Newton-Euler equations can be used. Simulation is used to reformulate 81 the dynamic equations such that the acceleration is computed as a function of actuator 82 torque. One way to effect manipulator motion from here to there in a specified smooth 83 fashion is to cause each joint to move as specified by a smooth function of time. To ensure 84 proper coordination, each and every joint starts and stops motion at the same time. The 85 computation of these functions is the problem of trajectory generation. A spline is a smooth 86 function that passes through a set or via points. End effector can be made to travel in a 87 rectilinear manner. This is called Cartesian trajectory generation. 88

The issues that ought to be considered during the mechanical design of manipulator 89 are cost, size, speed, load capability, number of joints, geometric arrangement, transmission 90 systems, choice and location of actuators, internal position and sensors, . The more joints a 91 robot arm contains the more dexterous and capable it will be. But it will also be harder to 92 build and be more expensive. Specialized robots are developed to perform specified tasks 93 and universal robots are capable of performing wide variety of tasks. 3 joints allow for the 94 hand to reach any position in 3 dimensional space. Stepper motors or other actuators can 95 be used to execute a desired trajectory directly. This is called the linear positional control. 96 Kinetic energy of the manipulator links can be calculated using the Langrangian formulation. 97 Both the linear kinetic energy and rotational kinetic energies can be tracked. 98

The evolution of robots can be studied from the market experience of one leading 99 manufacturer of robots for example. ABB robotics, Zurich, Switzerland. This shall be used 100 later to place in perspective the increased interest in use of nanorobots in the hospital for 101 drug delivery. The different products from ABB Robotics during a 30 year period [3] is given 102 below in Table 1.0. 103

104

105

Year Product

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

1975 IRB6 – First Robot for Arc Welding

1977 First Robots installed in France and Italy

1979 IRB 60 – First Electrical Robot for Spot Welding; First Robot installed in Spain

1982 Robots introduced in Japan

1983 S2, New Control System. Outstanding, HMI, Menu Programming, TCP (Tool Center Point) and the Joy Stick introduced. Allows control of several axes.


1986 ASEA bought Trallfa Robot operations, Bryne, Norway. Trallfa launched the world’s First Painting Robot in 1969. Sales boomed 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.

1990/1991 ABB acquired Cincinatii Milacron, USA, Graco, USA (robotic painting), Rarsburg Automotive (electrostatic painting atomizers). IRB 6000 – 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.

1998 Launch of Flex Picker Robot, the World’s fastest Pick and Place Robot. Robot Studio – First simulation tool based on virtual 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.

2002 ABB – First company in World to sell 100,000 Robots Virtual Arc – True arc welding simulation tool that gives robot welding 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.

Table 1.0 State of Art in Robots over a 40 Year Period 106

107

108

109

1.2 DEVELOPMENT OF NANOTECHNOLOGY 110

Submarine nanorobots are being developed for use in branchy therapy, spinal 111 surgery, cancer therapy, etc. Nanoparticles have been developed for use in drug delivery 112 systems and for cure in eye disorders and for use in early diagnosis. Research in 113 nanomedicine is under way in development of diagnostics for rapid monitoring, targeted 114 cancer therapies, localized drug delivery, and improved cell material interactions, scaffolds 115 for tissue engineering and gene delivery systems. Novel therapeutic formulations have been 116 developed using PLGA based nanoparticles. Nanorobots can be used in targeted therapy 117 and in repair work of DNA. Drexler and Smalley debated whether ‘molecular assemblers’ 118 that are devices capable of positioning atoms and molecules for precisely defined reactions 119 in any environment is possible or not. Feynman’s vision of miniaturization is being realized. 120 Smalley sought agreement that precision picking and placing of individual atoms through the 121 use of ‘Smalley-fingers’ is an impossibility. Fullerenes, C60, are the third allotropic form of 122 carbon. Soccer ball structured, C60, with a surface filled with hexagons and pentagons 123 satisfy the Euler’s law. Fullerenes can be prepared by different methods such as : (i) first 124 and second generation combustion synthesis; (ii) chemical route by synthesis of 125 corannulene from naphthalene. Rings are fused and the sheet that is formed is rolled into 126 hemisphere and stitched together; (iii) electric arc method. Different nanostructuring 127


methods are discussed in [4,5]. These include: (1) sputtering of molecular ions;(2) gas 128 evaporation; (3) process to make ultrafine magnetic magnetic powder; (4) triangulation and 129 formation of nanoprisms by light irradiation ; (5) nanorod production using condensed phase 130 synthesis method; subtractive methods such as; (6) lithography; (7) etching; (8) galvanic 131 fabrication; (9) lift-off process for IC circuit fabrication; (10) nanotips and nanorods formation 132 by CMOS process; (11) patterning Iridium Oxide nanostructures; (12) dip pen lithography; 133 (13) SAM, self assembled monolayers; (14) hot embossing; (15) nanoimprint lithography; 134 (16) electron beam lithography; (17) dry etching; (18) reactive ion etching; (19) quantum 135 dots and thin films generation by; (20) sol gel; (21) solid state precipitation; (22) molecular 136 beam epitaxy; (23) chemical vapor deposition; (24) CVD; (25) lithography; (26) nucleation 137 and growth; (27) thin film formation from surface instabilities; (28) thin film formation by spin 138 coating;(29) cryogenic milling for preparation of 100-300 nm sized titanium; (30) atomic 139 lithography methods to generated structures less than 50 nm; (31) electrode position method 140 to prepare nanocomposite; (32) plasma compaction methods; (33) direct write lithography; 141 (34) nanofluids by dispersion. Thermodynamic miscibility of nanocomposites can be 142 calculated from the free energy of mixing. The four thermodynamically stable forms of 143 Carbon are diamond, graphite, C60, Buckminster Fullerene and Carbon Nanotube. 5 different 144 methods of preparation of CNTs, carbon nanotubes were discussed. Thermodynamically 145 stable dispersion of nanoparticles into a polymeric liquid is enhanced for systems where the 146 radius of gyration of the linear polymer is greater than the radius of the nanoparticle. Tiny 147 magnetically-driven spinning screws were developed. Molecular machines are molecules 148 that can with an appropriate stimulus be temporarily lifted out of equilibrium and can return to 149 equilibrium in the observable macroscopic properties of the system. Molecular shuttle, 150 molecular switches, molecular muscle, molecular rotors, molecular nanovalves are 151 discussed. Supramolecular materials offers alternative to top-down miniaturization and 152 bottom-up fabrication. Self-organization principles holds the key. Gene expression studies 153 can be carried out in biochips. CNRs are a new generation of self-organizing collectives of 154 intelligent nanorobots. This new technology includes procedures for interactions between 155 objects with their environment resulting in solutions of critical problems at the nanoscopic 156 level. Biomimetic materials are designed to mimic a natural biological material. 157 Characterization methods of nanostructures include SAXS, small angle X-ray scattering, 158 TEM, transmission electron microscopy, SEM, scanning electron microscopy, SPM, 159 scanning probe microscope, Raman microscope, AFM atomic force miscroscopy, HeIM 160 helium ion microscopy. 161

162

1.3 ALZHEIMER’S DISEASE AND DRUGS USED TODAY 163

By the year 2050 the number of patients with Alzheimer’s disease will be 16 million. 164 The cost of memory loss is about $1.1 trillion. Currently 5 million patients suffer from 165 Alzheimer’s in this country, United States. The root curcuma longa, turmeric root has found 166 to have a positive effect in experiments with rats to cure Alzheimer’s disease [6]. The drugs 167 used such as bapinezumab from Eli Lily and solanezumab from Johnson and Johnson used 168 today are used to target the amyloid-β protein in the brain. This protein has been found to 169 misfold and results in clump formation in patients’ brains. Furthermore these drugs are 170 expensive. Turmeric root is used in curries in India and is readily available from farms at a 171 lower cost. It is yellow colored. This is recommended by Ayurveda and Siddha Medicine. It is 172 known to posess anti-inflammatory and antioxidant properties. 173


174

Figure 1.0 Structure of Curcumin 175

176

Curcumin has been found to decrease inflammation in the brain. It causes reduction in 177 oxyradicals formed in patients’ brain, 178

179

180

181

1.4 NANOMEDICINE - ADVANTAGES OF NANOROBOT DRUG DELIVERY 182 SYSTEM 183

Nanorobots can offer a number of advantages over current methods such as; (i) use 184 of nanorobot drug delivery systems with increased bioavailability; (ii) targeted therapy such 185 as only malignant cells treated; (iii) fewer mistakes on account of computer control and 186 automation; (iv) reach remote areas in human anatomy not operatable at the surgeon’s 187 operating table; (v) as drug molecules are carried by nanorobots and released where 188 needed the advantages of large interfacial area during mass transfer can be realized; (vi) 189


non-invasive technique; (vii) computer controlled operation with nobs to fine tune the 190 amount, frequency, time of release; (viii) better accuracy; (ix) drug inactive in areas where 191 therapy not needed minimizing undesired side effects (x) greater speed of drug action. 192

Frietas [6] defined nanomedicine as follows: 193

“Nanomedicine is the preservation and improvement of human health using 194 molecular tools and molecular knowledge of the human body." 195

Tiny magnetically-driven spinning screws were developed by Ishiyama et. al. [7]. These 196 devices were intended to swim along veins and carry drugs to infected tissues or even to 197 burrow into tumors and kill them with supply of heat. Untethered microrobot containing 198 ferromagnetic particles under forces generated by MRI magnetic fields were tested for travel 199 through the human anatomy at the NanoRobotic laboratory at Montreal, Canada [8] in 2003. 200

A nanorobot to measure surface properties was patented [9]. This technology is 10-20 201 years in the future. The nanorobot unit has a manipulation unit and an end effector. The 202 end effector can be a senor or made to move as close as possible to the surface of interest. 203 The drive device has piezoelectric drives. The resolution of the measurement can be in the 204 nanometer range and the actual measurement in the centimeter range. The nanorobot is 205 made sensitive in all directions, in multiple dimensions. It can operate under vacuum. 206 Surface roughness can be measured using nanorobots. 207 A patent was developed [10] for minimally invasive procedure by means of 208 DPR, Dynamic Physical Rendering. Use of “intelligent”, “autonomous” particles were made. 209 An interventional aid is formed with the aid of self-organizing nanorobots. These nanorobots 210 were made of catoms. C-arm angiography are used to monitor DPR procedures. A 3 211 dimensional image data record on target region is obtained. The determined form was 212 converted to a readable and executable program code for the catoms of the nanorobots. 213 The determined form was transferred to a storage unit. The program code was executed in 214 order to achieve self-organization in the unstructured catoms that were introduced in the 215 target region. The execution of the program was triggered by a timer or position sensor. The 216 intervention aid is used as a endovascular target region. 217 Nanocrystal with motor properties was patented [11]. Reciprocating motor is formed 218 by a substrate, atom reservoir, nanoparticle ram and nanolever. The nanoparticle ram is 219 contacted by the nanolever resulting in movement of atoms between the reservoir and the 220 ram. Substrate and nanolever are made of MWNTs, multi-walled nanotubes made of Iridium. 221 The substrate used was a silicon chip. 222 Nanoscale oscillator [12] was patented by Sea Gate technology, Scots Valley, CA. 223 A microwave output is generated by application of a DC current that is allowed to pass 224 through layers of magnetic structure separated by nanometer dimensions. Spin Momentum 225 Transfer, SMT is a phenomena realized to exist in 1989. It can be used in MRAM devices. 226 Phase-locked microwave spin transfer is the next advance of the technology. The electric 227 current produced is in the GHz spectrum. The local magnetic field source is used in the 228 application of a magnetic field to a free layer of spin momentum transfer stack. The magnetic 229 source can be from a horseshoe magnet with poles stationed above and below the SMT 230 stack. The magnetic source can take on other forms such as helical coil that surrounds the 231 SMT or pancake type coils above and below the SMT stack or an annular pole and a coil 232 that surround the stack. A permanent magnet may be planted above the stack. SMT stack 233 consists of a top electrode, a free layer, a non-magnetic layer a pinned magnetic structure 234 and a bottom electrode. 235 Nanowhiskers can be grown by control of nucleation conditions [13]. Nanowhisker 236 formation on substrates can be made using the VLS, vapor-liquid-solid mechanism. A 237 particle of catalyst material is placed on a substrate is heated in the presence of gases until 238 it melts. A pillar is allowed to form under the melt. As the melt rises up on top of the pillar and 239


a nanowhisker is formed. Miller direction may be a preferred growth direction of the 240 whiskers. The catalytic property is present at the interface of whisker and air. InP 241 nanowhiskers, for example, were grown using metal-organic vapor phase epitaxy. 242 Characterization of nanowhiskers is by electron microscopy. MOVPF is low pressure metal-243 orgaic vapor phase epitaxy process. 50 nm aerosol gold particles were deposited on InP 244 substrate. This was placed on a horizontal reactor cell heated by radio frequency heated 245 graphite suceptor. Hydrogen was used as carrier gas. Temperatuer was ramped tp 420

0 C 246

for 5 min. The molar fraction of flow rate in the cell was 0.015. Nanowhisker growth was 247 found to commence upon addition of TMI, trimethylindium. The molar fraction of TMI was 3 248 millionth. Typical growth time for production of nanowhiskers was found to be 8 minutes. 249 250

251 252 253 254 2.0 DESIGN AND CONTROL STRATEGY 255 The strategy for design of nanorobot drug delivery system among other things 256 comprise of the following salient features; 257

(i) Use fullerenes as carriers of curcumin. 258 259

(ii) Complex the fullerene with curcumin and deactivate the drug action as in other 260 cases of photodynamic therapy. 261 262

(iii) Computer controlled irradiation can activate the fullerene. Breakage of fullerene 263 is expected and release of curcumin at constant rate [14]. The intensity, 264 wavelength and areas of brain where is irradiated is controlled using computer 265 software. 266 267

(iv) Curcumin acts on the diseased cells and cure is effected. 268

269

270 (v) Pharmacokinetic model with 4

th order process for irradiation and competing 271

parallel reactions for fullerene breakage, polymerization of fullerene and drug 272 action is developed. 273 274

(vi) Time for drug release computed for model. Drug delivery model for fullerene 275 nanorobot curcumin delivery system. 276

277

(vii) Chromophores in curcumin is used as a dye and picked up from imaging. 278 Information from the sensor on drug action is compared against model and 279 feedback control effected by changing the intensity and wavelength of 280 irradiation. This is used to trigger feedback control or model based control of 281 irradiation using computers. 282

283

284


(viii) Itrathecal route of entry for nanorobot fullerene curcumin complex can be 285 preferred to sublingual entry. The choice of cerebrospinal fluid system may 286 obviate any digestive disorders from fullerene carbon. 287

288 3.0 PHARMACOKINETICS OF CURCUMIN THERAPY 289

The irradiation is supplied to activate the fullerene and effect detachment of the curcumin 290 from the fullerene setting the onset of therapeutic action. The irradiation can also trigger 291 polymerization of fullerene. Formation of polyfullerene can soak up the energy from 292 irradiation that is needed for activation of the drug. Curcumin may polymerize to form 293 polycurcumin. The selectivity of the breakage of fullerene cages and release of drug over 294 formation of polyfullerene and formation of polycurcumin are studied using transient dynamic 295 analysis of PFR, plug flow reactor and pharmacokinetics of Denbigh reaction. The 296 therapeutic action of curcumin during Alzherimer’s disease can be viewed as a set of 297 microreactors. PFR analysis will be a more reasonable approximation of these processes. 298 Denbigh scheme is the general scheme under which the parallel reactions of fullerene 299 breakage, polymerization of fullerene and polymerization of cicrumin may fall under. 300

301

3.1 TRANSIENT DYNAMICS OF PLUG FLOW REACTOR, PFR 302

303

Figure 2.0 Transient Dynamics in a Plug Flow Reactor, PFR 304

(The composition of the fluid in a PFR, plug flow reactor, varies from point to point along a 305 flow path. The mass balance on the reacting species need be made over a differential 306

volume A∆z as shown in Figure 2.0. A is the cross-sectional area of the PFR and ∆z is the 307 incremental length of the slice considered in Figure 2.0. This is different from a CSTR where 308 the mass balance on the reacting species is performed over the volume of the CSTR. For a 309 reacting species A with a inlet composition of CAi(mol.m

-3) and v is the superficial velocity of 310

the fluid (m.s-1

) the mass balance over the slice A∆z = dV can be written as follows; 311


=

−

−

treac

onaccumulati

rate

reaction

treacof

ncedisappeare

efflux

treac

of

rate

lux

treac

of

rate

tan

tan)(tan

inf

tan (1) 312

( ) ( )( )t

CzAzkCAAvCAvC AAzzAzA ∆

∆∆=∆−−

∆+ (2) 313

Dividing Eq. (2) throughout by A∆z and obtaining the limit as ∆z→0 Eq. (2) becomes; 314

t

CkC

z

Cv AA

A

∂

∂=−

∂

∂− (3) 315

Eq. (3) can be made dimensionless as follows; 316

L

vt

L

zZ

v

kLDa

C

CCX

Ai

AAi

A ===

−= τ;;; (4) 317

Upon substitution of Eq. (5) in Eq. (6); 318

DaDaXX

Z

XA

AA =+∂

∂+

∂

∂

τ (5) 319

Differentiating Eq. (5) with respect to dimensionless time; 320

τττ ∂

∂+

∂

∂=

∂∂

∂− AAA

XDa

X

Z

X2

22

(6) 321

Differentiating Eq. (3) with respect to dimensionless distance; 322

Z

X

Z

XDa

Z

X AAA

∂∂

∂−=

∂

∂+

∂

∂

τ

2

2

2

(7) 323


Conversion XA can be assumed to be analytic in the space and time (Z,τ). The order of the 324 differentialion does not matter. i.e., 325

Z

X

Z

X AA

∂∂

∂=

∂∂

∂

ττ

22

(8) 326

Adding Eq. (6) and Eq. (7) and realization of Eq. (8) leads to; 327

2

2

2

2

ττ ∂

∂+

∂

∂=

∂

∂+

∂

∂ AAAA XXDaZ

XDa

Z

X (9) 328

Eq. (9) is a hyperbolic PDE, partial differential equation that is second order with respect to 329 space and second order with respect to time. The exact form analytical solution to Eq (9) 330 can be obtained as follows. Eq. (9) is multiplied by e

nτ. Eq. (9) becomes; 331

( ) ( )2

2

2

2

ττττ

ττ

∂

∂+

∂

∂=

∂

∂+

∂

∂ AnAnAn

A

nX

eX

DaeZ

XeDa

Z

Xe (10) 332

The term (enτXA) can be seen to group and can be called as the wave concentration [14] 333

such that; 334

An

XeWτ= (11) 335

Now, 336

nWWX

e

or

XeXne

W

An

An

A

n

−∂

∂=

∂

∂

∂

∂+=

∂

∂

ττ

ττ

τ

ττ

(12) 337

Differentiating Eq. (12) again with respect to time, τ 338


τττττ

∂

∂−

∂

∂+

∂

∂−+=

∂

∂ Wn

WWnWn

Xe A

n

2

22

2

2 (13) 339

Plugging Eq. (13), Eq. (12) and Eq. (11) in Eq. (10) 340

( ) ( )

∂

∂−+

∂

∂++−=

∂

∂+

∂

∂

ττ

WnDa

WWnnDa

Z

WDa

Z

W2

2

22

2

2

(14) 341

For n = Da/2, Eq. (14) becomes; 342

4

2

2

2

2

2WDaW

Z

WDa

Z

W−

∂

∂=

∂

∂+

∂

∂

τ (15) 343

The steady state conversion can be obtained from Eq. (15) as follows; 344

( )ZDa

X

Xs

A

s

A ∂=−

∂

1 (16) 345

Integration of both sides of Eq. (16) yields; 346

( )DaZs

A

s

A

ecX

cDaZX

−−=

+−=−

'"1

"1ln (17) 347

At Z = 0, XA = 0 348

So, c”’ = 1. Eq (17) becomes; 349

( )DaZsA eX −−= 1 (18) 350

It can be seen that at steady state the conversion varies as a f(e-z). Eq. (15) can be 351

multiplied throughout by emZ

. The term (emτXA) can be seen to group and can be called as ω; 352


42

2

2

2 ω

τ

ωττ −∂

∂=

∂

∂+

∂

∂

Z

We

Z

We

mm (19) 353

Now, ωωτ

mZZ

We

m −∂

∂=

∂

∂ (20) 354

2

22

2

2

2ZZ

mmZ

We

m

∂

∂+

∂

∂−+=

∂

∂ ωωωτ

(21) 355

Substituting Eq. (19-21) in Eq. (15), Eq. (15) becomes; 356

( ) ( )4

22

2

22

2

2 ω

τ

ωω

ωω DammDa

ZmDa

Z−

∂

∂=+−+

∂

∂−+

∂

∂ (22) 357

For m = Da/2, Eq. (15) becomes; 358

2

2

2

2

2

2

22

2

2

44

τ

ωω

ω

τ

ωωω

∂

∂=

∂

∂

−∂

∂=−

∂

∂

Z

or

DaDa

Z

(23) 359

Eq. (23) is the wave equation. In the dimensional form of independent variables; 360

2

2

2

22

tzv

∂

∂=

∂

∂ ωω (24) 361

It can be seen that the solution to Eq. (24) can be written as (Bird, Stewart and Lightfoot 362 [15]); 363

( )

−+= vtz

λ

πφωω

2sin1

00 (25) 364


where λ is the wavelength of a harmonic wave with amplitude (ω0φ0) traveling in the z 365 direction at a speed v. The solution for the transient conversion can be written as follows; 366

( )

−+=

−−

vtzeeX

ZDaDa

Aλ

πφω

τ2

sin1 0022 (26) 367

The Dankwert’s boundary condition at z = L can be applied as; 368

0=∂

∂

z

X A (27) 369

( ) ( )

−+

−+−= vtLvtL

λ

πφ

λ

πφ

2cos

2sin1

2

10

00 (28) 370

At the wave front, L = vt, Eq. (28) becomes; 371

01 φ=− (29) 372

( )

−−=

−−

vtzeeX vzkkt

Aλ

πω

2sin10

22 (30) 373

At the entrance of the reactor, the initial reaction rate is given by; 374

kdt

dX A = (31) 375

Differentiating Eq. (30) with respect to t, 376

( )

( )λπλ

ω

ωλ

πω

−=

=++−=∂

∂

Lv

Lk

kv

L

v

t

X A

4

2

21

2

0

00

(32) 377


( )( )

−−

−=

+−

vtzeL

DaX v

vktzk

Aλ

π

λπ

λ 2sin1

4

2 2 (33) 378

379

380

Figure 3.0 Conversion XA in PFR for Damkohler Number, Da =6.75, v = 0.55 m.s-1

, L = 381 1.5 m; θθθθ = 2.7 hr, k = 2.5 hr

-1, λλλλ = 0.33 m 382


3.2 PHARMACOKINETICS OF CURCUMIN DRUG AND POLYMERIZATION 383 PARALLEL REACTIONS 384

3.2.1 Denbeigh Scheme of Reactions 385

386

Figure 4.0 Denbigh Scheme of Reactions 387

Consider the Denbigh scheme of reactions performed in the CSTR shown in Figure 4.0. A 388 scheme of reactions as shown in Figure 4.0 was discussed in [13] as a special case of 389 Denbeigh reactions. A state space model that can be developed to describe the dynamics 390 of the 5 species, CA, CR, CT, CB and CS. The assumptions are that the inlet stream 391 contains species A and B at a concentration of CAi and CBi and the initial concentrations of 392 the other species are zero. The kinetics of the simple irreversible reactions shown in Figure 393 4.0 can be written as follows; 394

( ) AA Ckk

dt

dC31

+−= (33) 395

BB Ck

dt

dC4

−= (34) 396

SABS CkCkCk

dt

dC534

−+= (35) 397

RAR CkCk

dt

dC21

−= (36) 398


SRT CkCk

dt

dC52

+= (37) 399

400

Figure 5.0 CSTR, Continuous Stirred Tank Reactor 401

The reactions are considered to peformed in the CSTR similar to the one shown in 402 Figure 5.0. Component mass balances on each of the species assuming incompressible 403 flow and constant volume reactor can be written as follows; 404

405

406

Species A 407

( )τd

dCDaDaCC AAAi =++− 311 (38) 408

Where v

VtkDakDa ==== θ

θτθθ ;;;;

3311 , θ with units of (hr) is the residence time of 409

the species in the reactor, V is the volume of the reactor, (liter) and v is the volumetric flow 410 rate (lit.hr

-1) in and out of the reactor. 411

Species B 412

( )τd

dCDaCC BBBi =+− 41 (39) 413


Where Da4 = (k4θ) 414

Species S 415

( )τd

dCCDaCDaDaC SABS =+++− 3451 (40) 416

Species R 417

( )τd

dCCDaDaC RAR =++− 121 (41) 418

Species T 419

τd

dCCDaCDaC TSRT =++− 52 (42) 420

421

The model equations that can be used to describe the dynamics of the 5 reactant/product 422 species in a CSTR can be written in the state space form as follows; 423

424

( )( )

( )( )

+

−

+−

+−

+−

++−

=

0

0

0

100

0100

001

00010

00001

25

21

543

4

31

Bi

Ai

T

R

S

B

A

T

R

S

B

A

C

C

C

C

C

C

C

DaDa

DaDa

DaDaDa

Da

DaDa

C

C

C

C

C

dt

d425

426

427 (43) 428

The stability of the dynamics of the 5 reactants/products in the Denbigh scheme performed 429 a CSTR can be studied by obtaining the eigenvalues of the rate matrix. The characteristic 430 equation for the eigenvalues are obtained by evaluation of the following determinant; 431

432 433

( )( )

( )( )

( )

+

++

++

++

+++

10

0100

001

00010

00001

det

25

21

543

4

31

λ

λ

λ

λ

λ

DaDa

DaDa

DaDaDa

Da

DaDa

434


435

( )( )( )( )( ) 011111 25431 =++++++++++ λλλλλ DaDaDaDaDa (44) 436 437

The 5 eigenvalues are negativewhen Damkohler numbers are greater than zero. When 438 eigenvalues are all negative the system is considered to be stable. 439

The Laplace transform of the model equations developed in order to describe the 440 transient dynamics (Eqs. 38 – 42) for the 5 species in the Denbigh scheme in a CSTR can 441 be written as follows; 442

( )( )( )

311 DaDass

CsC AiA

+++= (45) 443

( )( )

41)( Dass

CsC BiB

++= (46) 444

( )( ) ( )

( )( )( )54

4

531

3

1111

DasDass

CDa

DasDaDass

CDasC BiAiS

+++++

++

+++

=

(47) 445

( )( )( )( )

312

1

11 DaDassDas

CDasC AiR

+++++= (48) 446

( )( ) ( )( ) ( )( ) ( )(

++++++

+++++

+++++

+=

12

21

54

54

531

35

1111111

1

DasDas

CDaDa

DasDas

CDaDa

DasDaDas

CDaDa

sssC AiBiAiT

447

448 (49) 449 450

The inverse Laplace transform of Eq. (45) can be obtained by invocation of the convolution 451 theorem and written as; 452

453

( ) ( )( )311

31

11

1 DaDa

Ai

A eDaDaC

tC ++−−

++= τ (50) 454

455

456

The inverse Laplace transform of Eq. (46) can be seen from the time shift property to be; 457


458

( ) ( )( )τ41

4

11

1 Da

Bi

B eDaC

tC +−−

+= (51) 459

The inverse Laplace transform of Eq. (47) can be obtained by look-up of Laplace inversion 460 Tables in Mickley, Sherwood and Reed [17]461 462

( ) ( )( )( )( )

( )( )( )( ) ( )( )31231231

1

3122

1

11

1

11

312

DaDaDaDaDaDaDaDa

e

DaDaDaDa

e

C

tC DaDaDa

Ai

R

++++

−−++

−−

+−+=

++−+− ττ

463

464 465 (52) 466

3.3 STATE SPACE REPRESENTATION 467

State space models are those that describe more than one variable at a given 468 instant in time. Vector form for several variables is used. A set of n differential equations is 469 represented by one equation in matrices and vectors. The coefficient matrix and input and 470 output vectors are used. Output vector can be solved for by matrix manipulations. This 471 would form the output response of the system. The stability of the system can be studied 472 by looking at the Eigen values of the coefficient of the matrix. The different instabilities that 473 may arise and the conditions under which they would arise are given in Table 2.0 [18]. 474

475 476 477

Eigen values Characterization of Stability

Stability Type

λλλλ2 < λλλλ1 < 0 Asymptotically Stable

Improper Node

λλλλ2 > λλλλ1 > 0 Unstable Improper Node

λλλλ2 =λλλλ1 = λλλλ Asymptotically

Stable if λ < 0, Unstable if λ > 0

Proper or Improper Node

λλλλ1


486 Figure 6.0 Scheme of 7 Simultaneous Reactions 487

The kinetics of the 7 simultaneous reactions shown in Figure 7.0 is given below as follows; 488

AAA CkCk

dt

dC21

−−= (53) 489

RRAR CkCkCk

dt

dC431

−−= (54) 490

SSR

S CkCkCkdt

dC653

−−= (55) 491

S

T Ckdt

dC5

= (56) 492

A

U Ckdt

dC2

= (57) 493

R

V Ckdt

dC4

= (58) 494

S

W Ckdt

dC6

= (59) 495

The above equations can be represented in the state space form in one line as follows; 496

497

498

( )( )

( )

+−

+−

+−

=

W

V

U

T

S

R

A

W

V

U

T

S

R

A

C

C

C

C

C

C

C

k

k

k

k

kkk

kkk

kk

C

C

C

C

C

C

C

dt

d

000000

000000

000000

000000

00000

00000

000000

6

4

2

5

653

431

21

499

(60) 500

501

Or, 502


KxCdt

dC= (61) 503

The stability of the system of reactions can be studied by obtaining the Eigen values 504 of the K matrix. The Eigenvalues of the K matrix are obtained from the roots of the 505 characteristic polynomial 506

Det(λI-K) = 0 (62) 507

The K matrix is sparse. The polynomial form of the characteristic equation can be 508 written as follows; 509

( )( )( ) 04654321 =++++++ λλλλ kkkkkk (63) 510

The 7 Eigenvalues are 0 repeated 4 time and –(k1+k2), -(k3+k4) and –(k5+ k6). This system 511 can be viewed as an integrating system since all but Eigen values are negative with 4 Eigen 512 values 0. 513

514

515

516

517

518

519

520

521

522

523

524

525

526

527

REFERENCES 528

1. D. Sawyer, ABC World News, December 20th 2012. 529

2. J. J. Craig, Introduction to Robotics: Mechanics and Control, Third Edition, 530 Pearson Prentice Hall, 2005, Upper Saddle River, NJ. 531

3. http://labintsis.com/roboti/abb-roboti/?lang=en&output=json 532 4. K. R. Sharma, “Nanostructuing of Nanorobots for use in Nanomedicine”, 533

International Journal of Engineering & Technology, Vol. 2, 2, (2012), 116-134. 534 http://ietjournals.org/archive/2012/feb_vol_2_no_2/6686111325866989.pdf 535


5. K. R. Sharma, Nanostructuring Operations in Nanoscale Science and Engineering, 536 McGraw Hill Professional, 2010, New York. 537

6. S. Shishodia, G. Sethi, B. G. Aggarwal, “Getting Back to the Roots”, Annals. of the 538 New York Academy Sci., Vol. 1056, 3005, 206-217. 539

7. R. A. Frietas Jr., Nanomedicine, Vol I;Basic Capabilities, Landes Bioscience, 1999, 540 Georgetown, TX. 541

8. K. Ishiyama, M. Sendoh and K. I. Arai, “Magnetic Micromachines for Medical 542 Applications”. J Magnetism Magnetic Mater 242-245 (2002), 1163-1165. 543

9. J. B. Mathieu, S. Martel, L. Yahia, G. Soulez, G. Beaudoin G,”MRI Systems as a 544 mean of Propulsion for a Microdevice in Blood Vessels”, Proc. 25th Ann. Intl. Conf., 545 IEEE Engineering in Medicine and Biology; 2003 Sep 17-21; Cancun, Mexico; 2003. 546

10. V. Klocke, “Nanrobot Module, Automation and Exchange”, US Patent 547 2010/0140473 A1, 2010. 548

11. B. C. Regan, A. K. Zettl and S. Aloni, “Nanocrystal Powered Nanomotor”, US 549 Patent 7,863, 798 B2, 2011. 550

12. D. V. Dimitrov, X. Peng, S. S. Xue and D. Wang D, “Spin Oscillatory Device”, US 551 Patent 7,589, 600 B2, Seagate Technology, LLC, 2009. 552

13. W. Seifert, L. I. Samuelson, B. J. Ohlsson and L. M. Borgstrom, “Directionally 553 Controlled Growth of Nanowhiskers”, US Patent 7,911,035 B2, 2011. 554

14. K. R. Sharma, “Nanorobot Drug Delivery System for Cicumin for Treatment of 555 Alzheimer’s Disease with Increased Bioavailability during Treatment of Alzheimer’s 556 Disease”, 68

th Southwest Regional Meeting of the American Chemical Society, 557

SWRMACS, Baton, Rouge, LA, Oct/Nov, 2012. 558 15. K. R. Sharma, Damped Wave Transport and Relaxation, Elsevier, 2005, 559

Amsterdam, Netherlands. 560 16. R. B. Bird, W.E. Stewart and E. N. Lightfoot, Transport Phenomena, Revised 561

Second Edition, 2007, John Wiley, Hoboken, NJ. 562 17. O. Levenspiel. Chemical Reaction Engineering, 1999, John Wiley, Hoboken, NJ. 563 18. H. S. Mickley, T. S. Sherwood, C. E. Reed, Applied Mathematical Methods in 564

Chemical Engineering, McGraw Hill Professional, 1957, New York, NY. 565 19. A. Varma and M. Morbidelli., Mathematical Methods in Chemical Engineering, 566

1997, Oxford University Press, Oxford, UK. 567

568 569

570

571

572

573

574

575

576

577

578


579

580

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