physical–mathematical modelling of fluid and particle transportation for dna vaccination

International Journal of Engineering Science 44 (2006) 1037–1049

www.elsevier.com/locate/ijengsci

Physical–mathematical modelling of fluid and particletransportation for DNA vaccination

Yi Liu

The PowderJect Centre for Gene and Drug Delivery Research, Department of Engineering Science,

University of Oxford, Oxford OX2 6PE, United Kingdom

Received 28 January 2006; received in revised form 3 June 2006; accepted 10 June 2006Available online 28 August 2006

Abstract

The powdered injection system (PowderJect) is a novel needle-free device for the delivery of micron DNA vaccines. Theunderlying principle is to harness energy from compressed Helium gas to accelerate a pre-measured dose of DNA vaccinecoated in micro-gold particles to an appropriate momentum in order to penetrate the outer layer of the skin or mucosaltissue to achieve a biological effect. One of the latest PowderJect developments is the Venturi system, using the venturieffect to entrain micron-sized vaccines into an established quasi-steady supersonic jet flow and accelerate them towardsthe target. In this paper, we developed a physical–mathematical model to simulate fluid and particle transportation ofa prototype Venturi device. The key features of the gas dynamics and gas–particle interaction are presented. The overallcapability of the Venturi system for particle delivery is discussed.� 2006 Elsevier Ltd. All rights reserved.

Keywords: DNA vaccine; Fluid; Modelling; Over-expanded; Particle; Supersonic

1. Introduction

The gene gun was first designed as a unique means of introducing new genetic material into plant cells [1]. Ithas a wide range of uses today. In vaccination application, gene gun immunization propels DNA-coatedmicro-particles at a high velocity through the stratum corneum into the epidermis/dermis of human skin, evendeeper into tissue and is proven to achieve efficient transfection of a number of cell types, with intracellulardelivery of the DNA-coated particle [2]. DNA vaccination offers a promising new therapeutic interventionfor a wide range of diseases.

DNA vaccine administered via gene gun represents the most potent regimen for DNA administration [3,4].There are many advantages over a conventional needle and syringe, or the liquid jet injection technology, interms of effectiveness, cost and health risk. In addition, vaccines in powder form are reasonably stable, makingtransportation and storage uncomplicated. However, only very few data are available concerning the charac-teristics of particle delivery and evaluation of achievable performance [5–7].

0020-7225/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijengsci.2006.06.007

E-mail address: [email protected]

mailto:[email protected]

Fig. 1. A hand-held shock-tube based PowderJect device.

1038 Y. Liu / International Journal of Engineering Science 44 (2006) 1037–1049

We have been developing a controllable hand-held powdered injection system (PowderJect) at Oxford Uni-versity in association with PowderJect plc., which harnesses energy of compressed Helium gas to accelerate themicro-sized vaccines to a sufficient momentum in order to breach the outer layer of skin or mucosal tissue andto target the cells of interest [5,8–11]. The powderJect system can deliver particles locally with a velocity rangeand spatial distribution preferable for DNA vaccine administration, for example, administration of vaccine toa mucosal site may be necessary for vaccination against some respiratory or gastro-intestinal pathogens. It isneedle-free and shown to be painless and can be applicable to a wide range of pharmaceuticals.

The devices based on shock-tube design, shown in Fig. 1, utilizing a quasi-steady quasi-one dimensionaltransonic flow behind a propagating shock wave to accelerate particles and to impact the skin target, havebeen reported and used in various studies both in vitro and in vivo with satisfactory results [9,11,12]. It providescontrollable impact parameters, eliminating the uncertainty associated with other particle entrainment tech-niques. However, a considerable amount of powder was observed trapped by ruptured membranes and resid-ing on the wall of the shock-tube nozzle with this diaphragmed delivery system. The sound level associatedwith the membrane rupture is also quite high. Therefore, a complicated silencer is required.

In the meantime, we proposed a diaphragm-free delivery system (Venturi), an alternative mode of Powder-Ject, which bypasses the requirement of a membrane diaphragm. The key feature is the use of venturi effects toentrain powdered vaccines into an established quasi-steady high speed gas jet flow, so that no diaphragm isrequired to retain the particles, and potentially avoids the fragment carryovers. All vaccine particles are accel-erated by a continuous high speed jet flow in the Venturi device over a period of �1–3 ms, targeting specificskin or mucosal cells.

The objective of this study is to implement a fidelity physical–mathematical approach to characterize theperformance of prototype Venturi devices, with the primary emphasis on the over-expanded nozzle flowand the interaction between the gas and the particles.

2. The venturi device and configuration

One form of PowderJect system, called the venturi powdered vaccine delivery system (Venturi), has beendeveloped to utilize the venturi effect created in the venturi gallery to introduce powdered vaccines into thedelivery system [10]. A schematic diagram of the Venturi device, which is configured for pre-clinical use, isshown in Fig. 2. The key internal components of the device are: a BOC-cylinder, in which compressed Heliumis stored typically at a pressure of �1–6 MPa; an externally arranged cassette (unlike the arrangement in theshock-tube based PowderJect devices), where the powdered vaccines are retained; the combination of a sonicthroat, a venturi gallery and following parallel section, which is configured to form a supersonic jet and createa lower pressure cavity; and finally a diverging conical nozzle, used to further accelerate the gas and particlesto a large area, with a diameter of 10 mm at the nozzle exit. The large exit area is configured to deliver a largemount of powdered dose, and at the same time avoid local over-loading effects on the target, which inevitablycreates complicated gasdynamic phenomena as illustrated in this study.

When compressed Helium gas is released from the BOC-cylinder, a large pressure difference builds upacross of the sonic throat, which leads to the formation of a supersonic jet. The supersonic jet propagatesdown to the venturi gallery and the parallel section. Subsequently, a low pressure cavity is created. The areachange of the diverging nozzle initiates an unsteady gas flow, as in the classical starting process theory of thesupersonic nozzle. Later, a sustained bulk flow of Helium from the cylinder is established. In the course ofthese processes, when a sufficient pressure difference is reached in the venturi gallery, particles are entrainedinto the mainstream flow and accelerated towards the nozzle exit. As the gas–particle flow impinges on theskin target, gas is deflected away and vented to atmosphere through a silencer (which is not shown here).

Fig. 2. A schematic diagram of the Venturi system.

-.07 -.06 -.05 -.04 -.03 -.01 .00 .01 .02 .03 .04 .05-.010

-.005

.000

.005

.010

.015

Gas cylinder

Powder cassette

Throat

P*

Pitot

P1

P8P4

Φ10 mm at the nozzle exitΦ1.2 mm, the throatΦ 3.0 mm, parallel section P7P6P5P3P2

P0

m

m

02

Fig. 3. Configuration of the prototype Venturi device and experimental setup.

Y. Liu / International Journal of Engineering Science 44 (2006) 1037–1049 1039

The particles, with their relatively large inertia, maintain a high velocity and penetrate the stratum corneumand target the cells of interest.

For the vaccine delivery, the system needs to be optimized to ensure the particles are delivered within a quasi-steady supersonic jet flow (QSSJF), but not in the starting process. This can be achieved by choosing gasspecies, operating conditions and dimensions of the system geometry. The gas resource and practicalconstraints, designed for a hand-held device, limit the device length and the duration of the QSSJF for theparticles to be entrained and accelerated. Dedicated design and detailed characterization are therefore required.

The principle of the Venturi system has been investigated analytically and experimentally at the Universityof Oxford over the last few years. The over-expanded supersonic nozzle flow and resulting non-uniform par-ticle delivery were observed. These investigations provide a useful, but incomplete indication of important flowaspects. In this paper, we will implement a physical–mathematical model to simulate a complete gas andparticle dynamics of the prototype device. New insights into the Venturi system are explored numerically.

The prototype Venturi device under concern is illustrated schematically in Fig. 3. The BOC-cylinder with4 MPa Helium is used. The cassette is loaded with a powdered payload of �1.0 mg. The model particles fornumerical simulation and experimental studies were gold spheres with a density of 19,920 kg/m3 and diameterof 1.8 lm, which are representative of intracellular DNA vaccine delivery applications.

3. Physical–mathematical modelling

3.1. Governing equations

The two-dimensional axisymmetric Reynolds-averaged Navier–Stokes (RANS) equations for an unsteadycompressible flow in a conservative form are written as


o bWotþ obF i

oxi¼ 1

ReocFvi

oxiþ S

" #ð1Þ

where t, xi are the independent variables for time and the co-ordinates in the computational domain; bW is theconservative variable vector [q,quj,qe]T; bF i; cFvi and S are the inviscid, viscous flux vector and the source term,respectively. Further details can be found in the literature [12].

Since gas flow involves Helium and air, the mass fraction (Yk) is predicted by a transport equation

oqY k

otþ oqY kui

oxi¼ oJ k;i

oxiþ Rk þ Qk

� �ð2Þ

where Rk is the rate of creation or depletion by chemical reaction, and Qk is the rate of creation by additionfrom the dispersed phase plus any other source. Jk,i is the diffusion flux of the kth species, which arises due toconcentration gradients. Subscript k is the index of species.

Helium gas due to its high speed of sound and low molecular weight is generally employed as the driver gasin order to achieve a sufficient particle velocity in the PowderJect system.

3.2. Turbulence model

Considered the nature of the Venturi system and required accuracy, the standard k–� turbulence model,expression of Eq. (3), is used. It has been proven robust, economical and accurate for a wide range of turbulentflows in many industrial applications, and the shock-tube based PowderJect devices [8,12]

o bW k�

otþ oðbF iÞk�

oxi¼ 1

ReocF v

k�i

oxiþ Sk�

" #ð3Þ

A non-equilibrium wall function is employed to consider effects of the pressure gradient and strong non-equi-librium in the near-wall region of the Venturi devices, which may involve separation, possibly reattachment,and impingement. In this area, the mean flow and turbulence are subjected to severe pressure gradients andchange rapidly.

3.3. Particle dynamics

Noted that the particle cloud is sufficiently dilute in the PowderJect system, i.e., the particle–particle inter-action and the effects of the particle volume fraction on the gas phase are negligible, the particle motion ismodelled in a Lagrangian frame of reference. The force balance equates the particle inertia with the forcesacting on the particle, and reads

dup

dt¼ F Dðu� upÞ þ

gðqp � qÞqp

ð4Þ

in which u and up are the fluid velocity and the particle velocity, respectively. qp and Dp are the particle densityand the particle diameter. Rep is the particle Reynolds number, based on particle diameter and slip velocity,ju� upj. F Dðu� upÞ is the drag force per unit particle mass and

F D ¼3l

qpD2p

CDRep

4ð5Þ

3.4. Drag correlations

One of important parameters in the simulation of gas and particle interaction is the drag coefficient, CD. It isclearly shown that there is great discrepancy between drag correlations proposed for evaluating the particledrag coefficient. These disparities stem from the different ranges of Reynolds and Mach number covered,gas properties, particle size and density ranges and the particle concentration. Regarding our particular


application, which consists of both subsonic and supersonic flows, the correlation of Henderson [13], valid forall the flow regimes of interest, is firstly selected.

For Map 6 1, the drag coefficient is

CD ¼ 24 Rep þ ðc=2Þ1=2Map 4:33þ 3:65� 1:53T p=T1þ 0:353T p=T

� �exp �0:247ð2=cÞ1=2 Rep

Map

� �� 1

þ exp � 0:5Map

Re1=2p

!4:5þ 0:38ð0:03Rep þ 0:48Re1=2

p Þ1þ 0:03Rep þ 0:48Re1=2

p

þ 0:1Ma2p þ 0:2Ma8

p

" #

þ 0:6ðc=2Þ1=2Map 1� exp �Map

Rep

� ��
For Map P 1.75, it is given by
CD ¼0:9þ 0:34

Ma2p

þ 1:86Map

Rep

� �1=2

2þ 4

cMa2p

þ 1:0581

Map

2T p

cT

� �1=2

� 4

c2Ma4p

" #1þ 1:86 Map=Rep

� 1=2

In the Map between 1 and 1.75, the drag coefficient is linearly interpolated

CD ¼ CDð1:0;RepÞ þ ð4=3ÞðMap � 1Þ½CDð1:75;RepÞ � CDð1:0;RepÞ�
Given that the particle motion in the Venturi system is highly non-stationary, i.e., it experiences significantacceleration and deceleration. One simple expression considering the unsteady effects proposed by Kurianand Das [14], among others, applicable for a cloud of particles in a range of Reynolds numbers from 70 to4470, is explored to determine the drag coefficient
CD ¼ 11:77Re�0:278p

These correlations were evaluated with studies of a number of shock-tube based PowderJect systems. Partic-ularly, the drag correlation proposed by Igra and Takayama [15], which covers a wider range of Reynoldsnumbers (200 to 101,000) and also takes unsteady effects into account, provided the most satisfactory compar-ison [8].

log CD ¼ 7:8231� 5:8137 log Rep þ 1:4129ðlog RepÞ2 � 0:1146ðlog RepÞ3

3.5. Computational grid

Fig. 4 shows the computational domain (marked within the box in Fig. 3) and boundary conditions spec-ified for the prototype Venturi device. It is extended to 80D · 40D (where D is diameter of the sonic throat) toaccommodate a non-reflective boundary condition imposed. The grid has 22,444 cells with 80 points along the

axiswall

Non-reflect boundary

Out

let

Inle

t

Fig. 4. Computational domain and boundary conditions.


axis direction and 40 points in the radial direction within the diverging nozzle. The mesh is concentrated nearthe wall and well placed in the region of interest (e.g., more than 3000 cells are used in the venturi gallery toobserve flow details), with the first adjacent cells being y+ of �10–30 away from the wall. The magnified mesharound the corner of the venturi gallery is also shown in Fig. 4.

3.6. The initial and boundary conditions

The computational domain is initialized with the air at an atmospheric pressure and room temperature. Theparticle trajectory is initialized by defining the initial position, velocity, size and temperature of each modelparticle.

The mixture gas of high-pressure Helium initially stored in the BOC-cylinder with residual air at an atmo-spheric pressure is imposed as the inlet flow, as labelled in Fig. 4.

Atmospheric pressure is specified at the outlet boundary for the subsonic outflow. The device wall temper-ature is assumed to remain constant during the particle delivery (usually <5 ms of interest), with a non-slipvelocity condition imposed. The non-reflective boundary condition is set for the far-field boundary.

3.7. Numerical procedure

The transient gas flow and its interaction with particles are modelled simultaneously and interactively. Thesolution of multi-species gas phase is obtained by numerically solving the RANS equation (Eq. (1)), togetherwith turbulence model (Eq. (3)), and species transport equation (Eq. (2)).

The particle motion equations, in conjunction with the drag correlation, are advanced with gas flow sim-ulation. The inter-phase exchange of momentum and heat is also considered in each time step. Integration intime of Eq. (4) yields the velocities of the particle at each point along the trajectory, with the trajectory itselfpredicted by

dxdt¼ up ð6Þ

Equations similar to Eqs. (4) and (6) are solved in each co-ordinate direction to predict the trajectory of indi-vidual particle.

Fluent [16], a commercial-available application software, is chosen to model the gas and particle transpor-tation. A coupled explicit solver is selected in order to capture the main features of the unsteady motion of theshock wave process. An overall second-order accuracy is satisfied both spatially and temporally to accuratelypredict the interaction between the oblique shock and the turbulence boundary layer.

The numerical approaches described here have been validated and shown good agreement with experimen-tal measurements in a range of similar applications [8,12].

4. Numerical results and discussion

Simulations with different combinations of the computational grid, physical–mathematical model and par-ticle drag correlation are carried out to ensure grid-independent solutions, and to evaluate the drag correlationfor the micro-particle in the transient supersonic jet flow.

4.1. The transient gas flow

Fig. 5 shows calculated and experimental time histories of the static pressure, with the locations marked inFig. 3. The pressure measurements here are averaged value over an area of �1.8 mm in diameter (the size ofthe pressure transducer).

The calculated static pressures in positions P0 and P4 (Fig. 5b and c) before the time of �200 ls are higherthan those measured and are also quite oscillatory. This is probably due to the mis-match of the inlet bound-ary conditions, since no measurement of the inlet total pressure or temperature is available. Rather, the pres-sure trace from the pressure transducer in position P* (shown in Fig. 5a) was taken as the total pressure. This

Time (ms)

Pre

ssur

e (P

a)P

ress

ure

(Pa)

Pre

ssur

e (P

a)P

ress

ure

(Pa)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.00.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

Time (ms)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0E+00

5.0E+05

1.0E+06

1.5E+06

2.0E+06

2.5E+06

3.0E+06

3.5E+06

ModellingExperiment

Time (ms)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

0.0E+00

2.5E+04

5.0E+04

7.5E+04

1.0E+05

1.3E+05

1.5E+05

ModellingExperiment

Time (ms)0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

5.0E+04

7.5E+04

1.0E+05

1.3E+05

1.5E+05

ModellingExperiment

a

b

c

d

Fig. 5. Comparison of simulated and experimental pressure histories, with positions (a) P*, (b) P0, (c) P4 and (d) P8 marked in Fig. (3).



oscillation is also attributed to the assumption of a constant total temperature (taken as a room temperature).However, as indicated in Fig. 5b and c, this discrepancy, due to the boundary condition treatment, is unno-ticeable from �250 ls when the supersonic jet flow is established. Furthermore, from the light obscurationexperiment, it is observed for particle delivery commencing from �1 ms. Hence, the implemented inlet bound-ary conditions are considered acceptable as preliminary studies for the Venturi system.

In general, it shows good agreement between numerical simulations and experimental measurements.It is further investigated by monitoring pressure histories over various positions of interest, gathered in

Fig. 6. The pressure difference between atmosphere and P1 (shown in Fig. 6a) creates a low pressure cavity

P5

P4

P3

PPitotP8

P6

Time (ms)

Pres

sure

(Pa

)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time (ms)0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

Time (ms)

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00

0.0E+00

2.5E+04

5.0E+04

7.5E+04

1.0E+05

1.3E+05

1.5E+05

Pres

sure

(Pa

)

0.0E+00

2.5E+04

5.0E+04

7.5E+04

1.0E+05

1.3E+05

1.5E+05

Pres

sure

(Pa

)

5.0E+04

7.5E+04

1.0E+05

1.3E+05

1.5E+05

P1

Fig. 6. The computed static pressure histories, with various positions marked in Fig. 3. (a) position at P1, (b) position at P3, P4 and P5, and(c) position at P6, P8 and PPitot.


to allow particles to be drawn into the mainstream gas flow. The optimization of dimensions of the sonicthroat, the parallel section and the vaccine cassette (and also operation condition), ensures that particlesare entrained within an established quasi-steady supersonic jet flow (QSSJF), but not in the starting process.From a time of 300 ls, the pressure at P5 is constantly higher than the one at P4, which indicates a shock isgenerated in between. In fact, there is a complicated flow phenomena near this region (as shown in later 2Dfigures). The supersonic nozzle behaves partly like a diffuser, starting from a position between P3 and P4. Fromposition P4, the pressure gradually recovers to the atmospheric as it approaches the nozzle exit due to the over-expanded nozzle flow.

The calculated velocity profiles at 1 mm downstream of the nozzle exit plane are presented in Fig. 7 at arange of times. The measured mean velocity and standard deviation are plotted for a comparison. The exper-imental data were taken from the pressure measurements of Pitot probe and nearby pressure transducer P8 atthe time of 1 ms, which is considered in the QSSJF regime.

It can be observed that from a time of 1 ms, the coring feature becomes evident. The gas velocity above500 m/s, which is considered to have a sufficient momentum to accelerate the particles (as it will be demon-strated by the particle simulation results), is found within a center area of no more than 5 mm in diameter.This is largely due to the over-expanded nozzle operation condition.

Fig. 8 reveals time sequence of the two-dimensional flow field, in the form of velocity contours. We canclearly see the supersonic core flow region, with a high velocity in the central area and a quite low velocitytowards the nozzle wall. A shock-cell structure is formed as the supersonic flow develops. The oblique shockrepeatedly interacts with and is reflected from the wall in the parallel section and subsequent diverging nozzle.This interaction becomes progressively weaker as it propagates to the nozzle exit. The equivalent PIV imageand the derived velocity map, by comparison, are shown in Fig. 9. They are coincident in terms of the velocityrange and its contours.

4.2. The particle dynamics

The QSSJF characteristics, generated within the prototype Venturi device, have significant effects on theparticle delivery. We extract the calculated particle velocity data, and plot their variations against the radialposition at 1 mm downstream of the nozzle exit in Fig. 10, the same position as shown in Fig. 7. The modelparticles are gold spheres of 1.8 lm in diameter, with a density of 19,920 kg/m3. They are representative ofintracellular DNA vaccine delivery applications.

The velocity distributions, calculated by three drag correlations described in this paper, display a quite sim-ilar velocity range (probably this good agreement is partly due to the range of velocity distribution itself). The

Radial position (mm)

Vel

ocit

y (m

/s)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 60

200

400

600

800

1000

1.33 ms

1.16 ms

375 μs

1000 μs

685 μs

2.00 ms

Fig. 7. The computed velocity profiles at 1 mm downstream of the nozzle exit plane, with measured mean velocity and standard deviationmarked.

Fig. 8. Sequence of computed velocity contours of the flow field.


particle velocity in the free jet exhibits a wide range, �400–1000 m/s, and with less particles distributed in theoutside area. This is consistent with the gas velocity distribution shown in Fig. 7.

Statistically, the mean velocity of 745.0 m/s and standard deviation of 172.50 m/s are obtained with thedrag correlation of Igra and Takayama.

In Fig. 11, we examine the drag correlation applicability and capability by plotting the computed particlevelocity against the cumulative particle percentage. Although the predicted particle velocity with Kurian andDas’s correlation [14] is higher than others, quite similar velocity range and distribution are found with all

Fig. 9. PIV image and derived velocity map within the QSSJF, time = 1.1 ms.

Radial position (mm)

Axi

al v

eloc

ity

(m/s

)

0 1 2 3 4 5 60

200

400

600

800

1000

1200Drag laws

Kurian & Das (1997)Igra & Takayama (1993)Henderson (1976)

Fig. 10. The computed particle velocity distribution at 1 mm downstream of the nozzle exit.


three drag laws. Approximately 30% particles (of total about 2000 sampled) are accelerated to velocity of<600 m/s, which is considered to be too low for vaccine delivery.

4.3. The characteristics of gas and particle interaction

Finally, we accumulate simulated key gas and particle flow features and plot them together in the typicalspace–time (x–t) diagram, Fig. 12. The representation of the gas flow for the Venturi system was achieved byvisualizing the variation of the calculated flow parameter (e.g., velocity magnitude) with time along thecentral-axis of the device. The shock-cell structure patterns are clearly displayed in the figure. The terminationof the nozzle starting process is less than 200 ls, during which the low pressure cavity is not yet created in theventuri gallery. As a result, the particles still reside in the cassette. The QSSJF is well established from the timeof �1.0 ms, which is desirable to the particle delivery.

Fig. 12 also plots the particle trajectories. It can be seen that all the particles are accelerated to the nozzleexit in <�100 ls.

Cumulative percentage (%)

Vel

ocity

(m

/s)

0 10 20 30 40 50 60 70 80 90 1000

200

400

600

800

1000

1200

Drag lawsHenderson (1976)

Igra & Takayama (1993)Kurian & Das (1997)

Fig. 11. The particle velocity distribution analysis.

Fig. 12. Space–time (x–t) diagram of the prototype Venturi device. A: the sonic throat, B: the start of parallel section, C: the nozzle exit.


The space–time (x–t) diagram provides us with the guideline to optimize the Venturi system, ensuring thatpowdered DNA vaccines are delivered within the QSSJF windows, not in the starting process, in order toachieve a sufficient momentum to breach the out layer of the skin.

5. Conclusions

Fluid and particle transportation within a prototype Venturi system has been investigated. A physical–mathematical methodology has been implemented to gain new insights into the behavior of the over-expandedsupersonic nozzle, which is configured to accelerate vaccines in micro-particle form to impact the human skin.

Simulated pressure histories and velocity distributions agree well with the corresponding pressure trans-ducer and PIV measurements. Comparisons demonstrate that the shock waves and important features ofthe over-expanded supersonic nozzle flow are accurately captured by the proposed physical–mathematicalapproach.

By incorporating the drag correlations with the modelled gas flow field, we further model the action of thegas flow accelerating the particles. The calculations show that the pressure losses and large area ratio haveresulted in non-uniform particle velocity distribution (400–1000 m/s). The statistical analyses show that a


mean velocity of 745 m/s has achieved for the modelled gold spheres (1.8 lm in diameter), representative ofintracellular DNA vaccine delivery.

Acknowledgements

This work is financially supported by PowderJect Plc and Chiron Vaccines. Professor B.J. Bellhouse isacknowledged for his technical insights of the PowderJect system and enormous encouragements to this study.Dr G. Costigan and Dr F. Carter are thanked for the permission of using their experimental data and PIVimages.

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