Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407

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Commun Nonlinear Sci Numer Simulat

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Control of pore radius regulation for electroporation-based drug delivery

Xiaopeng Zhao *, Mingjun Zhang, Ruoting YangMechanical, Aerospace and Biomedical Engineering Department, University of Tennessee, Knoxville, TN 37996-2030, USA

a r t i c l e i n f o

Article history:Received 30 January 2009Received in revised form 17 April 2009Accepted 27 May 2009Available online 6 June 2009

PACS:87.50.cj02.30.Oz05.45.�a

Keywords:ElectroporationControlBifurcation

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.05.059

* Corresponding author. Tel.: +1 865 974 7682; faE-mail address: xzhao9@utk.edu (X. Zhao).

a b s t r a c t

Electroporation uses electric pulses to create transient, nonselective pores in a cell’s mem-brane, allowing drugs to be delivered into the targeted cells. To ensure proper uptake ofdrug molecules, it is essential to control the radii of the pores and the time duration, forwhich the pores remain open. Electroporation is intrinsically a nonlinear dynamic process.A careful analysis of the electroporation dynamics reveals that, under variation of the mag-nitude of the input voltage, the equilibrium pore radius undergoes a pair of saddle-nodebifurcations. As a result, there exists a range of pore radii that is physically unstable andthus cannot be maintained in conventional experiments. The bifurcations and the associ-ated unstable regime impose restrictions on the operation of electroporation, limitingthe sizes of deliverable drug particles. To overcome these problems, we design a novel con-trol strategy to stabilize the originally unstable solutions. In contrast to the conventionalcontrol algorithms based on local stability analysis, the present control is globally stable.Numerical examples show that the control eliminates the original bifurcations and allowsone to achieve a wide range of pore radii. The robustness and effectiveness of the controlstrategy would potentially enhance the application of electroporation.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

Electroporation is a technique to create transient pores in a cell’s membrane by applying electric pulses. The techniquehas become a useful tool to deliver biologically active molecules into the cell [1–3]. As shown in Fig. 1, the procedure of elec-troporation utilizes electric pulses to temporarily disrupt the phospholipids bilayer of a cell to form pores, which arenonselective and allow any chemical species to penetrate the cell. Electroporation has many potential medical applications.For example, the technique can deliver chemotherapeutic drugs into tumor cells to treat cancers [4]. Recently, electropora-tion has been used in gene therapy to deliver DNA into a variety of tissues [5–9], which can consistently result in a 100- to1000-fold enhancement of gene expression.

One key issue in electroporation is to create pores of an appropriate size to the intended application for a sufficientamount of time. For example, the pores should be at least 10 nm in diameter and last for more than 1 ms, in order to admitsupercoiled DNA molecules [10]. On the other hand, electroporation may cause substantial tissue damage when the poressize is too large or the pores remain open for too long. Thus, the success of electroporation requires an appropriatelydesigned protocol for pore creation. This is a difficult task because electroporation is a nonlinear dynamical process, whoseinput–output relation is not transparent. Thus, understanding nonlinear dynamics of electroporation becomes a criticalfactor to design and control electric pulses.

The purpose of this work is to investigate nonlinear phenomena during an electroporation process and to explore possiblecontrol mechanisms to improve the effectiveness of the process. Using bifurcation analysis, we investigate the relation

. All rights reserved.

x: +1 865 974 7663.

Fig. 1. Schematic illustration of an electroporation process: a strong electric pulse temporarily disrupts the membrane of a cell to form pores, which allowany chemical species to penetrate the cell.

X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407 1401

between input voltage and the size of the created pores. The result demonstrates that a range of pore sizes is not achievablein experiments. Moreover, the bifurcation leads to a hysteresis phenomenon. These nonlinear phenomena bring a substantialchallenge to design electric pulses, and thereby, limit the operation of electroporation.

Currently, the strength and duration of the electric field in electroporation are mainly determined by trial-and-error.Moreover, the common pulses used in experiments are open-loop controls, which suffer bifurcations and hysteresis.Recently, Ma and Zhang [11] proposed a feedback-linearization based nonlinear controller to regulate a target transmem-brane potential. In this paper, we propose a new control algorithm to improve the two-phase protocol in regulating the poresize. A detailed analysis demonstrates the robustness and the efficacy of the algorithm. In addition, the control strategy isglobally stable and allows one to achieve a wide range of pore radii.

This artcile is organized as follows: Section 2 introduces mathematical description of the physical processes of electro-poration. A simplified mathematical model is presented based on a commonly adopted protocol. Then, Section 3 exploresnonlinear dynamics of the model and demonstrates how bifurcations limite the operation of electroporation. A feedbackcontrol algorithm is introduced in Sectoin 4 to eliminate the bifurcations and to improve the efficiency of electroporation.Finally, Section 5 presents conclusions and discussions.

2. Problem description

Mathematical models have been developed to describe an electroporation process [12–14]. As discussed in detail in [14],the electroporation process can be represented as an electrical circuit model, see Fig. 2. Here, V0 is the external voltage, Rs

accounts for the resistance of the experimental setup, the cell’s membrane is represented as the capacitor C, the resistanceof the membrane is represented as the constant resistor R, Ip represents current through electropores. Then, the variation ofthe transmembrane potential Vm is governed by the following differential equation:

C _Vm þ1Rsþ 1

R

� �Vm þ Ip ¼

V0

Rs; ð1Þ

where the coefficients C = CmAs and R = RmAs stand for the total capacitance and resistance of the membrane, respectively.Cm and Rm are the surface capacitance and surface resistance of the membrane, and As represents the total area of pores.The current Ip can be formulated as follows:

Ip ¼ KVm

Rp þ Ri; ð2Þ

where K is the number of pores, Rp = h/(pgr2) represents the pore resistance, and Ri = 1/(2gr) represents the input resistance.Hydrophobic pores are initially created when the transmembrane potential grows sufficiently large. Most such pores are

quickly destroyed by lipid fluctuations; however, when a pore grows beyond a critical radius rp, it will spontaneously convertto a long-lived hydrophilic pore [14]. Hydrophilic pores will immediately expand to the minimum-energy radius rm [15]. Thepore density N is governed by a first-order differential equation [15]

_N ¼ aeðVm=VepÞ2 ½1� ðN=N0Þe�ðrmVm=rpVepÞ2 �; ð3Þ

where N0 is the pore density when Vm = 0mv. a represents the creation rate coefficient, and Vep stands for the characteristicvoltage of electroporation.

Fig. 2. Circuit representation of the electroporation process.

1402 X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407

The bilayer energy can be described in four components as follows:

Table 1Parame

Symbol

aVep

N0

r*

rm

DTa

bKB

cr0

r0

AFmax

Cm

Rm

Rs

hg

Fig. 3.pores t

X ¼ br�r

� �4þ 2pcr � preff r2 � FmaxV2

mr: ð4Þ

In the above equation, the bilayer energy X contains four contributions: steric repulsion energy b(r�/r)4, edge energy 2pcr,membrane tension preffr

2, and transmembrane potential FmaxV2mr. Here, b is the steric repulsion energy with minimum hydro-

philic pore radius r�, and c is the edge energy, and s stands for the combined area of the pores. The membrane tension is given by

reff ðAsÞ ¼ 2r0 � ð2r0 � r0Þð1� As=AÞ2

; ð5Þ

where As = Kpr2 represents the total area of pores, while A stands for the total area of membrane. r0 is the interfacial energyper unit area of the hydrocarbon–water interface, and r0 denotes the surface tension of the intact membrane wit As = 0. Thecoefficient Fmax is the maximum electric force.

Pores that initially have radii rm will change their radii to minimize the bilayer energy W. As a result, the average poreradius can be expressed as

_r ¼ � DkBTa

@X@r¼ � D

kBTa�4b

r4�

r5 þ 2pc� 2prreff � FmaxV2m

� �; ð6Þ

where D is the diffusion coefficient of the pore radius, kB is the Boltzmann constant, and Ta is the absolute temperature. Themodel parameters are summarized in the Table 1.

Sukharev et al. [16] proposed a two-pulse protocol, which consists of two electric pulses separated by a relaxation inter-val, see illustration in Fig. 3. The first pulse is a large voltage for a short duration, which quickly creates a number of pores.During the relaxation interval, all pores shrink to a minimum size. Then, a long yet weak pulse is applied to maintain the

ters of the electroporation model.

Value Description

1 � 109 m�2 s�1 Creation rate coefficient [21]0.258 V Characteristic voltage [21]1.5 � 109 m�2 Equilibrium pore density [21]0.51 � 10�9 m Min. radius of hydrophilic pores [22]0.8 � 10�9 m Min. energy radius [22]5 � 1014 m2 s�1 Diffusion coefficient for pore radius[23]310 K Absolute temperature1.4 � 10�19 J Steric repulsion energy [15]1.38 � 10�23 m2 kg s�2 K�1 Boltzmann constant1.8 � 10�11 J m�1 Edge energy [23]1 � 10�2 J m�2 Bilayer tension without pores [23]2 � 10�2 J m�2 Hydrocarbon water interface tension[24]1.26 � 10�9 m2 Total area of membrane [14]7 � 10�10 N V�2 Max. electric force for Vm = 1 V [15]9.5 � 10�3 F m�2 Membrane surface capacitance [21]0.523 X Membrane surface resistance [21]100000 X Series resistance [18]5 � 10�9 m Membrane thickness [15]2 S m�1 Conductivity of the solution [26]

Illustration of the two-phase protocol. The first phase uses a strong and short pulse to create pores. A relaxation time follows the first phase to allowo recover. The second phase uses a low-magnitude pulse to expand the pores.

X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407 1403

growth of the pores to facilitate electrophoretic movement of drug molecules. Experiments proved that the two-pulse pro-tocol is more efficient in uptake of drug particles than a single pulse [17].

Besides its efficacy in drug delivery, the two-pulse protocol also allows significant simplifications to the model of electro-poration [18]. Specifically, since all pores are created in the first phase and they all shrink to the minimum size during therelaxation interval, the following conditions hold for pores in the second phase: (i) the number of pores is constant and (ii) allpores have the same size. As a result, the model (1, 5, 6) is reduced to the following form:

Fig. 4.(dashed

dVm

dt¼ 1

CV0

Rs� 1

Rsþ 1

R

� �Vm � Ip

� �; ð7Þ

drdt¼ � D

kBTa�4b

r4�

r5 þ 2pc� 2prreff � FmaxV2m

� �: ð8Þ

In the following, we will analyze the nonlinear dynamics of electroporation using this model.

3. Bifurcation and hysteresis

Dynamics of the model (7) and (8) can be understood from a phase-plane analysis based on nullclines. On a V-nullcline (orr-nullcline), the rate of change of Vm (or r) is zero, that is, dVm/dt = 0 (or dr/dt = 0). Thus, the intersections of a V-nullcline andan r-nullcline correspond to equilibria of the system. Fig. 4 shows the V-nullclines for V0 = 0.2, 1.0, 2.2, and 4.0 V and ther-nullcline. For purpose of illustration, we choose the number of pores K = 2500 in this paper although the results for othervalues of K will be similar.

It can be seen from (7) and (8), a V-nullcline is a function of V0 whereas an r-nullcline is independent of V0. Moreover, onecan show that when V0 increases (decreases), the V-nullcline shifts up (down). One can show that dr/dt < 0 in the region be-low the N-shaped r-nullcline whereas dr/dt > 0 when it is above the r-nullcline. Correspondingly, dVm/dt > 0 in the regionbelow the V-nullcline and dVm/dt < 0 in the region above the V-nullcline. As seen in Fig. 4, for a very small V0 (e.g.V0 = 0.2 V), the V- and r-nullclines intersect at the leftmost branch of the r-nullcline, corresponding to a small pore radiusin equilibrium. When V0 increases, the equilibrium value of r increases. For example, when V0 = 1.0 V, the V- and r-nullclinesintersect at a point in the middle branch of the r-nullcline. When V0 is further increased, the nullclines may have multipleintersections. For example, when. V0 = 2.2 V, there are two intersections at the middle branch and one at the right branch.Finally, when V0 is sufficiently large (e.g. V0 = 4.0 V), there is one intersection at the right branch of the r-nullcline. Fig. 4 indi-cates that the number of equilibriums varies as V0 changes, a phenomenon known as bifurcation [19].

When a constant V0 is applied, Vm and r reach their equilibriums values after a few microseconds. The equilibriums of Vm

and r depend on V0 in a nonlinear fashion. In particular, there exists a range of V0 characterized by the coexistence of multiplesolutions. Diagrams in Fig. 5 are known as bifurcation diagrams, which show that there coexist three solution branches for1.82V 6 V0 6 2.83V (the dashed curves). Here, the top and bottom branches are stable whereas the middle branch is unstable.The critical point where a stable and an unstable solution branches meet corresponds to a saddle-node bifurcation [19].

The existence of saddle-node bifurcations leads to hysteresis, which is a path-dependent phenomenon and occurs in mag-netic and ferromagnetic materials [20]. The output of a hysteresis system depends not only on the input but also on the inputhistory. To demonstrate, let us assume that one starts from a small value of V0 that corresponds to the leftmost point onFig. 5. Then when the value of V0 is quasi-statically increased, the response of Vm (r) follows the upper (lower) solid curveshown in the top (bottom) panel of Fig. 5 until V0 reaches the intersection between the upper (lower) solid curve and themiddle dashed curve. When the value of V0 is further increased, the solution of Vm (r) jumps to and stays on lower (upper)solid curve shown in the top (bottom) panel. Correspondingly, if one starts from a large value of V0 and slowly decreases V0,the solution of Vm (r) will experience the opposite jump from the lower (upper) branch to the upper (lower) branch. Theabove process suggests that, during ‘‘loading” and ‘‘unloading” processes, the response of the system follows two different

Phase plane analysis showing V-nullclines (solid, from top to bottom) corresponding to V0 = 0.2, 1.0, 2.2, and 4.0 V, respectively, and r-nullcline).

Fig. 5. Bifurcation diagrams showing equilibrium Vm vs. V0 (top) and equilibrium r vs. V0 (bottom). Solid curves correspond to stable solutions whereasdashed curves correspond to unstable solutions.

1404 X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407

paths, i.e. hysteresis. To illustrate the hysteresis of electroporation, we use a stimulus of V0 = 2.5 + sin(t) to simulate a quasi-static loading and unloading process; see Fig. 6. Here, after a short transient, the system’s response falls on a hysteresis loop,which consists of a loading (increasing in V0) branch, an unloading branch, and quick transitions between the two branches.The hysteresis phenomenon results from the existence of saddle-node bifurcations. Indeed, the loading and unloadingbranches in the hysteresis loops correspond to the two equilibrium branches in the bifurcation diagrams (cf. Figs. 5 and6). And jumps between the loading and unloading branches are due to the bifurcation points. The corresponding time his-tories of V0, Vm, and r are shown in Fig. 7.

The existence of multiple stable solutions in the range 1.82V 6 V0 6 2.83V indicates that different initial conditions maylead to different equilibrium solutions. It is interesting to ask which solution can be achieved from a certain initial condition.To answer this question, we plot the stable manifold of the unstable saddle node corresponding to V0 = 2.5 V, see the reddotted curve in Fig. 8. This manifold divides the (r,Vm) state space into two regions. It can be concluded that any point on

Fig. 6. Hysteresis loops of Vm and r under a slowly varying pulse.

Fig. 7. Response of Vm (blue dotted) and r (black dashed) to a slowly varying sinusoidal pulse (red solid) showing hysteresis. (For interpretation of thereferences to color in this figure legend, the reader is referred to the web version of this paper.)

Fig. 8. When V0 = 2.5 V, the V-nullcline (solid, black) and the r-nullcline (dashed, blue) intersect to produce three equilibria. The left and right intersectionsare stable equilibria whereas the middle intersection is an unstable saddle node. The red dotted curve represents the stable manifold of the saddle node.(For interpretation of the references to color in this figure legend, the reader is referred to the web version of this paper.)

X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407 1405

the left of the manifold will lead to the stable equilibrium at r � 10 nm and any point on the right will lead to the stableequilibrium at r � 29 nm.

4. Controller design

Electroporation has been used to deliver drugs and other biological molecules into cells [4–9]. When electroporation isused for drug delivery, different drugs would need different pore sizes. While smaller pores are sufficient for delivering manytypes of anticancer drugs, large pores are needed to deliver DNA. Thus, functionality of electroporation greatly depends on itsability in creating pores of different sizes. However, as shown in Fig. 5, a pore size in the range between 15 and 25 lm isunstable. Thus, it is not possible to achieve pores in this range using the conventional open-loop input. To overcome thisproblem and to eliminate the hysteresis phenomenon described earlier, we design a control mechanism by choosing V0

as follows:

V0 ¼ Rs1Rsþ 1

R

� �Vm þ RsIp þ KpðVcontrol � VmÞ; ð9Þ

where Kp is a control gain. This controlled input renders Eq. (7) as

dVm

dt¼ 1

CRsKpðV control � VmÞ: ð10Þ

We now design two different control strategies: (1) voltage targeted control; and (2) radius targeted control. The first controlstrategy is easy to achieve. Because when a constant Vtarget is chosen such that Vcontrol = Vtarget, the dynamics of Vm becomesindependent of r. Then, it is trivial to show that this control mechanism is stable as long as the gain Kp is chosen to bepositive. Under this control, the equation of r becomes a ‘‘slave equation” in that the dynamics of r is determined by Vm

but r has no influence to Vm. Therefore, although this strategy can eliminate the bistability and hysteresis problems [11],it is not possible to know the resulted value of r in advance. Thus, the control has limited applications.

1406 X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407

The second control strategy, although much more involved, is more desired in practical applications. To achieve this con-trol goal, we select the control input to be

Vcontrol ¼ Vnull þ Krðrtarget � rÞ; ð11Þ

where Vnull is implicitly defined as the nullcline function of r, Kr is a control gain, and rtarget represent the desired pore size.With this control, one can easily show that (rtarget, Vnull(rtarget)) is a fixed point of the controlled system (8, 10). To analyze thestability of the control, we compute the Jacobian matrix as follows:

J ¼@f@r

@f@Vm

� 1CRs

Kp1

CRsKp

@Vnull@r � Krr

� �264

375; ð12Þ

where f ¼ � DkBTa

�4b r4�

r5 þ 2pc� 2prreff � FmaxV2m

� �. To have stable solutions, both eigenvalues of the Jacobian matrix must

have negative real parts [19]. One shall then choose the proper values of gains Kp and Kr to satisfy this condition.However, this criterion of the gains based on eigenvalues can only guarantee local stability. As such, the system may still

have other stable solutions and an arbitrarily given initial condition may not converge to the desired target value. To resolvethis issue, we design a globally stable control algorithm based on the nullclines to achieve any desired pore size with arbi-trarily given initial conditions. From Eq. (8), one can show that dr/dt < 0 in the region below the N-shaped r-nullcline whereasdr/dt > 0 when it is above the r-nullcline. Thus, a simple yet effective control strategy is to place V0 above the r-nullcline(pores will grow) when r < rtarget and below the r-nullcline (pores will shrink) when r > rtarget. We can then choose the controlgains according to this criterion based on the nullcline.

One problem remains if the control gains are set to be constants. Because the control input Vcontrol is linearly proportionalto rtarget � r, it may become exceedingly large when r is too far away from rtarget and thus causes damage to the tissue. Toensure a realistic control effort and avoid potential harms, one can saturate the magnitude of Vcontrol when jrtarget � rj is over

Fig. 9. Relation between Vtarget and Vnull (i.e. the r-nullcline) when rtarget = 20 nm and Kr = 0.05 V/nm.

Fig. 10. Controlled input V0 (top) and the response in Vm (middle) and r (bottom) under control.

X. Zhao et al. / Commun Nonlinear Sci Numer Simulat 15 (2010) 1400–1407 1407

a certain limit. For example, we let Vcontrol ¼ Vnull þ K�r 1 nm wehn r < rtarget � 1 nm and V target ¼ Vnull � K�r 1 nm whenr > rtarget � 1 nm. The concept of the control algorithm is illustrated in Fig. 9 using Kr = 0.05 V/nm and rtarget = 20 nm. Here,the control algorithm leaves the r-nullcline unchanged but alters the V-nullcline to the shape of the black solid curve (cf.Figs. 4 and 9).

To demonstrate the effectiveness of the control algorithm, we plot the control result for rtarget = 20 nm, as shown inFig. 10. Recall from Fig. 5 that this specific pore radius is not achievable under a conventional constant-magnitude pulse.However, it is easily achieved with the proposed control. Fig. 10 shows the successful control response, where simulationsare based on Kp = 1 ls�1 and Kr = 0.05 V/nm. We test the robustness of the control algorithm by choosing various values ofrtarget and the control is always able to stably achieve the desired pore size.

5. Conclusions

We have investigated nonlinear dynamics of electroporation under a commonly adopted two-pulse protocol. Resultsbased on phase portrait and bifurcation analysis demonstrate limitations in the conventional constant-voltage input. Specif-ically, under variation of the magnitude of the applied voltage, the equilibrium pore radius undergoes a pair of saddle-nodebifurcations, which are connected through an unstable branch of solution corresponding to a range of pore radii that cannotbe maintained in conventional experiments. These undesirable nonlinear characters restrict the operation of electroporation,limiting the sizes of deliverable drug particles. Moreover, bifurcations may also induce hysteresis phenomena under quasi-static loading and unloading inputs. To overcome these shortcomings, we have explored both voltage- and radius-targetedcontrol mechanisms. Since the conventional control mechanisms based on local stability analysis are not able to guaranteeglobal stabilities, we have designed a novel control strategy based on phase space analysis. The designed algorithm elimi-nates the original bifurcations and allows one to achieve any desired pore radius starting from arbitrary initial conditions.Since the control algorithm is globally stable, it ensures robust operation of electroporation. This control strategy wouldpotentially enhance the effectiveness of electroporation, making it a more attractive technique for drug delivery and otherbio-molecular applications.

Acknowledgement

X. Z. would like to thank the financial support of the NSF under Grant No. CMMI-0845753.

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