Mathematical Biosciences 282 (2016) 1–15

Contents lists available at ScienceDirect

Mathematical Biosciences

journal homepage: www.elsevier.com/locate/mbs

Modeling antimicrobial tolerance and treatment of heterogeneous

biofilms

Jia Zhao

a , b , ∗, Paisa Seeluangsawat a , Qi Wang

a , c , d

a Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA b Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599, USA c Beijing Computational Science Research Center, Beijing 10 0 083, China d School of Mathematics, Nankai University, Tianjin 30 0 071, China

a r t i c l e i n f o

Article history:

Received 24 February 2016

Revised 23 July 2016

Accepted 6 September 2016

Available online 22 September 2016

Keywords:

Phase field

Biofilms

Persister

Hydrodynamics

a b s t r a c t

A multiphasic, hydrodynamic model for spatially heterogeneous biofilms based on the phase field for-

mulation is developed and applied to analyze antimicrobial tolerance of biofilms by acknowledging the

existence of persistent and susceptible cells in the total population of bacteria. The model implements a

new conversion rate between persistent and susceptible cells and its homogeneous dynamics is bench-

marked against a known experiment quantitatively. It is then discretized and solved on graphic processing

units (GPUs) in 3-D space and time. With the model, biofilm development and antimicrobial treatment

of biofilms in a flow cell are investigated numerically. Model predictions agree qualitatively well with

available experimental observations. Specifically, numerical results demonstrate that: (i) in a flow cell,

nutrient, diffused in solvent and transported by hydrodynamics, has an apparent impact on persister for-

mation, thereby antimicrobial persistence of biofilms; (ii) dosing antimicrobial agents inside biofilms is

more effective than dosing through diffusion in solvent; (iii) periodic dosing is less effective in antimicro-

bial treatment of biofilms in a nutrient deficient environment than in a nutrient sufficient environment.

This model provides us with a simulation tool to analyze mechanisms of biofilm tolerance to antimicro-

bial agents and to derive potentially optimal dosing strategies for biofilm control and treatment.

© 2016 Elsevier Inc. All rights reserved.

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. Introduction

In nature, as soon as bacteria colonize on moisture surfaces,

biofilm is likely to form thereafter, consisting of the micro-

rganisms aggregated by bacteria along with their self-produced,

lue-like exopolysaccharide matrix, also known as the extracellular

olymeric substance (EPS). It’s commonly perceived by the medi-

al community that biofilms are responsible for many diseases or

ilments associated with chronic infections, evidenced for exam-

le by the survey that biofilms are present in the removed tissue

f 80% of patients undergoing surgery for chronic sinusitis [37] .

nlike a planktonic bacterium, biofilms are hard to be eradicated

y the standard antimicrobial treatment [30] , which perhaps ex-

lains the frequent relapse of chronic diseases or ailments associ-

ted with bacterial infections.

Thus, an understanding of the mechanism that underlies biofilm

olerance/persistence to antimicrobial agents can greatly enhance

∗ Corresponding author at: Department of Mathematics, University of North Car-

lina at Chapel Hill, Chapel Hill, NC 27599, USA.

E-mail address: zhaojia@live.unc.edu (J. Zhao).

A

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ttp://dx.doi.org/10.1016/j.mbs.2016.09.005

025-5564/© 2016 Elsevier Inc. All rights reserved.

herapeutic treatment of diseases related to biofilms. Intensive re-

earch has been conducted, primarily in experiment, to try to un-

erstand biofilm structures and dynamics, but the detailed mech-

nism is still not fully known. Readers may refer to the review

apers [15,30] for overviews of current advances in treatment

f biofilms. One essential factor for antimicrobial persistence of

iofilms is perhaps the existence of persistent cells (persisters)

ithin the biofilm colony, which are consisted of a small portion

f dormant bacterial variants that are highly tolerant to antimicro-

ial agents [4] . Contrasting to persistent cells, the other bacteria

re collectively called susceptible bacteria since they respond to

ntimicrobial treatment sensitively.

From the clinical point of view, understanding the mechanism

f persister formation would be useful for biofilm control and

radication, which will impact on treatment of diseases related to

iofilms. For reviews on mechanisms underlying the persister for-

ation, readers are referred to the two papers by Lewis [29,30] .

s dormant variants of regular bacterial cells, which don’t undergo

enetic changes, it is perceived that persisters are converted from

egular cells due to stresses [4] , such as nutrient depletion [1–3,7] ,

xistence of antimicrobial agents [33] and so on. Later, when the

nvironment is tolerable, namely nutrient becomes sufficient or

2 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

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the concentration of antimicrobial agents drops under a threshold

value, biofilms can relapse [8] , which implies that persisters have

converted back into susceptible bacteria for regrowth. It is a com-

mon belief that persister cells are much slow growers compared to

susceptible cells.

Taking into account persister formation, researchers have con-

ducted research on therapeutic treatment of diseases induced by

biofilms. The review paper [41] provides some control strategies

for biofilms. The dosing strategy when administering antibiotics

to treatment of biofilms is also an important issue. There exists

an evidence that a concentrated dose of biocide is more effective

than using a prolonged dose of a lower concentration [22] . In addi-

tion, dosing by shocks is more effective than dosing in a persistent

manner [21] . To the best of our knowledge, there have not been

any optimal strategies derived for biofilm control or disease treat-

ment related to biofilms so far. Currently, the environmental im-

pact of biocide or side-effect of antibiotics have become common

concerns, which makes the development of an optimal antimicro-

bial strategy even harder.

From the modeling perspective, simple mathematical models

have been developed to test certain hypotheses of persister forma-

tion based on the experimental evidence that supports the concept

of persisters [12,23,35,40] . For instance in [35] , the author used a

simple mathematical model to show that persister formation can

lead to higher bacterial persistence with respect to antimicrobial

agents than those grown in planktonic culture. In [23] , a 3D agent-

based model for biofilm dynamics under antimicrobial treatment

was developed, in which it showed that the substrate limitation

can contribute to persistence of biofilms with respect to antimicro-

bial treatment. Notably, Cogan et al. have worked on some possible

mechanisms of persister formation using time-dependent, spatially

homogeneous models recently [11,12,28] .

Some mathematical models on dosing strategies for treating

diseases related to biofilms have also been derived. For instance,

Cogan et al. discussed effective dosing strategies using a simple

mathematical model in [12,28] . In [14] , he discussed the effect

of periodic disinfection using a one-dimensional model. In [44] ,

the adaptive response to dosing protocols for biofilm controls was

analyzed, which provided some sufficient conditions for eradicat-

ing biofilms using a constant dosing approach. In addition, models

analyzing other impact factors, which may contribute to biofilm’s

persistence to antimicrobial treatment were also proposed. For in-

stance, the author in [17] analyzed and simulated diffusive resis-

tance of bacterial biofilms to penetration of antibiotics.

Most of the modeling effort s in the past focused on reactive

kinetics of biofilm persistence and treatment. Very few consid-

ered the hydrodynamic effect and the spatio-temporal heteroge-

neous structures of biofilms in 3D space and time. It is well-

known that biofilms are of highly heterogeneous spatial structures

and rich temporal dynamics. The spatial heterogeneity can signif-

icantly impact on biofilm formation and its function, especially,

concerning biofilm’s persistence to antimicrobial agents. In this pa-

per, we develop a multiphasic hydrodynamic model for biofilms

of multiple bacterial phenotypes; in particular, we limit the phe-

notypes to the persister and susceptible type. This model extends

our previous model of biofilms based on biomass-solvent mixtures

[45] by distinguishing between the persister cell and the suscepti-

ble cell when biofilms are treated by antimicrobial agents. In this

model, the interplay among the various biomass components such

as various bacterial types, EPS and solvent is carefully taken into

account both hydrodynamically and chemically [46] . The model

shows that the dynamical interaction between persistent and sus-

ceptible phenotypes can impact dramatically on overall dynam-

ics of the biofilm. It provides the spatio-temporal resolution that

is needed for more details about antimicrobial treatment against

biofilm colonies in space and time than the previous models can,

roviding more insight into hydrodynamics of biofilms under an-

imicrobial treatment.

The paper is organized as follows. In Section 2 , we formu-

ate the hydrodynamic theory for the biofilm system based on

he phase field formulation. Then, an efficient numerical solver

or the governing partial differential equation system is developed

sing the semi-implicit finite difference strategy in Section 3 . In

ection 4 , numerical results are presented and discussed. Finally,

e summarize the result and draw a conclusion.

. Mathematical model formulation

We model the biofilm together with its surrounding aqueous

nvironment as a multiphase complex fluid. The biofilm consists

f the mixture of biomass and solvent, in which biomass is made

p of bacteria of various phenotypes and their products like ex-

polysaccharide (EPS). Nutrient and antimicrobial agents are small

olecule substances dissolved in solvent. Their mass and volume

ractions are negligibly small, which will therefore be neglected in

his model. However, their chemical effects are important and will

herefore be retained. Let φbs be the volume fraction of the bac-

eria that are susceptible to antimicrobial agents, φbp the volume

raction of the bacteria that are persistent to the agent, φbd the

olume fraction of the dead bacteria, φb the volume fraction of

he bacteria, and φp the volume fraction of EPS. We in addition de-

ote the concentration of the nutrient and the antimicrobial agent

s c and d , respectively, and define φn the volume fraction of the

iomass, consisting of all the volume fractions for the bacteria as

ell as EPS,

n = φp + φbs + φbp + φbd . (1)

n addition to the volume fractions introduced above, the volume

raction of the solvent is denoted as φs . The incompressibility of

he complex fluid mixture then implies

s + φn = 1 . (2)

o help the reader to better understand the structure of our biofilm

odel, we show a schematic for the biofilm colony in Fig. 1 .

In this model, we make a simplifying assumption that all com-

onents in the biomass including the bacteria and EPS share the

ame mass density, which is roughly correct. We denote ρn and

s the density of the biomass and solvent, and ηn and ηs the vis-

osity of the biomass and solvent, respectively. Then, the volume

veraged viscosity and density are given, respectively, by

= φn ηn + φs ηs , ρ = φn ρn + φs ρs . (3)

We assume the bacteria, regardless whether they are alive or

ead, and the EPS mix with the solvent owing to the osmotic

ressure. Then, we adopt the modified free energy introduced in

45] and denote it by F:

=

∫ �

dx

[γ1

2

k B T |∇φn | 2 + γ2 k B T

(φn

N

ln φn

+(1 − φn ) ln (1 − φn ) + χφn (1 − φn )

)]. (4)

his is the modified Flory–Huggins free energy with a conforma-

ional entropy, in which γ 1 and γ 2 parametrize the strength of

he conformational entropy and the bulk mixing free energy, re-

pectively, χ is the mixing parameter, N is the extended polymer-

zation index for the biomass, k B is the Boltzmann constant and T

s the absolute temperature.

.1. Transport equations for biomass components

Given the free energy density functional f in Eq. (4) , the ‘ex-

ended’ chemical potentials with respect to biomass components

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 3

Fig. 1. A schematic for a biofilm colony, where multiple key components are depicted.

a

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re summarized as follows

bs =

δF

δφbs

, μbp =

δF

δφbp

, μbd =

δF

δφbd

, μp =

δF

δφp . (5)

he transport equation for the volume fraction of each biomass

omponent is governed by a reactive Cahn–Hilliard equation,

t φi + ∇ · (v φi ) = ∇ ·(λi φi ∇μi

)+ g i , i = bs, bp, bd, p. (6)

here g bs , g bp , g bd and g p are the reactive terms (rates) for the sus-

eptible bacteria, persisters, dead bacteria, and the EPS component,

espectively, which will be given in details in the next subsection.

bs , λbp , λbd and λp are the motility parameters for the transport

f the four components, respectively, which can be functions of the

iomass volume fractions.

.2. Transport equations for nutrient and antimicrobial agents

As mentioned above, the nutrient and antimicrobial agents are

reated as phantom materials, in which their volume are com-

letely neglected, but their chemical effects are retained. This

s because nutrient and antimicrobial agents all consist of small

olecule materials compared to the biomass, which are made up

f the bacterial cells and macromolecular EPS.

In this paper, we denote the concentration of nutrient and an-

imicrobial agents by c and d , respectively. They are assumed to

onvect with the solvent velocity, to diffuse in the biofilm as well

s to react with biomass. The governing equations for the two

hemical components are given by the following two reaction–

onvection–diffusion equations, respectively,

∂(φs c)

∂t + ∇ · (cv φs ) = ∇ · (D c φs ∇c) + g c , (7)

∂(φs d)

∂t + ∇ · (dv φs ) = ∇ · (D d φs ∇d) + g d , (8)

here D c , D d are diffusion rates, and g c , g d are the reactive terms

or the nutrient and antimicrobial agents, respectively.

.3. Hydrodynamic equations for the complex fluid mixture

For the multiphase complex fluid mixture, the average velocity

is assumed to be solenoidal. Then, the continuity and the mo-

entum balance equations are given as follows,

(∂ t v + v · ∇v ) = −∇p + ∇ · τ − γ1 k B T ∇ · (∇φn � ∇φn ) ,

∇ · v = 0 , (9)

here ρ is the volume-averaged density, p is the hydrostatic pres-

ure, τ is the extra stress tensor. The last term in the momentum

quation is known as the Ericksen stress [45,47] , which is due to

he spatial inhomogeneity of the biomass distribution. The Ericksen

tress arises wherever there exists a density/concentration/volume

raction gradients in the mixture, which also contributes to the os-

otic pressure of the system. In this paper, we are interested in

rowth dynamics of the biofilm, whose time scale is significantly

arger than the relaxation time scale in the EPS. We therefore

odel all effective biofilm components as viscous fluids, namely,

= 2 ηD , where D =

1 2 (∇v + ∇v T ) is the rate of deformation ten-

or. We note that the viscoelastic effect is important only when we

nvestigate biofilm dynamics in shorter time scales.

.4. Reactive kinetics

In order to complete the model formulation, we need to derive

eactive kinetics for every effective biomass and chemical compo-

ent. We develop reactive kinetics as general as possible by taking

nto account all experimental evidences and hypotheses in the lit-

rature. Later in the numerical study section, we focus on a specific

odel with parameter values collected from the literature and our

est guesses.

Bacterial growth is fueled by the nutrient and inhibited by an-

imicrobial agents present in biofilms. The two different bacte-

ial phenotypes (the susceptible and persister cells) have different

rowth mechanisms; thus we assume that both the susceptible

nd the persister cell grow on their own at their own rates. The

rowth of persisters are perceived to be much slower than that

f susceptible cells. For both bacterial phenotypes, death rates due

o natural causes are taken into account as well. In addition, both

usceptible bacteria and persisters can be killed by antimicrobial

4 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

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agents, but persisters are killed in a much slower rate due to its

antimicrobial persistence. We allow all these features in the model

so that they can be turned on and off depending on the time scale

that we are interested in. It is perceived that susceptible bacteria

and persisters can be converted mutually, based on the stage of

their growth [30] and the environment [12] , such as accessibility

to nutrient and exposure to antimicrobial agents.

Based on the mechanisms mentioned above, reactive kinetics

for the two types of live bacteria are proposed as follows:

g bs =

C 2 c

K 1 + c

(1 − φbs

φbs 0

)φbs − b sp φbs + b ps φbp

−(

r bs K

2 sd

K

2 sd

+ c 2 +

C 3 d

K 3 + d

)φbs ,

g bp =

C 4 c

K 2 + c

(1 − φbp

φbp0

)φbp + b sp φbs − b ps φbp

−(

r bp K

2 pd

K

2 pd

+ c 2 +

C 12 d

K 3 + d

)φbp , (10)

where C 2 and C 4 are growth rates for the susceptible and the per-

sister, respectively; φbs 0 and φbp 0 are carrying capacities for sus-

ceptible bacteria and persisters due to growth, respectively; b sp and

b ps are conversion rates between the two types of bacteria; r bs and

r bp are natural death rates due to nutrient depletion for susceptible

bacteria and persisters, respectively, and C 3 and C 12 are death rates

for susceptible bacteria and persisters due to antimicrobial agents,

respectively. K 1 , K 2 , K 3 , K sd , and K pd are half saturation constants

in the Monod models adopted. In this model, the decay rates of

the live bacteria increase as the concentration of the antimicrobial

agents increase while decrease with respect to the concentration

of nutrient. Based on the experimental evidence, we assume r bs >

> r bp , C 2 > > C 4 and C 3 > > C 12 .

In some studies [3,20] , b sp and b ps are assumed constants. In

other studies [12] , nonlinear conversion rates were used

b sp = b sp1 c

c + k sp , b ps = b ps 1

(1 − 1

1 + e −d−d 0

ε

), (11)

where b sp 1 and b ps 1 are the maximum conversion rates and d 0 is

a threshold value for antimicrobial agents, ε is a small parameter.

These nonlinear functions improved the model prediction to some

extent. But, the improvement is still not satisfactory.

Here, we propose a new formulation for the switch function b sp

and b ps between the two types of bacteria, based on the experi-

mental evidence as well as latest interpretation about the conver-

sion in the literature. We assume conversion between the suscep-

tible and the persister cell is primarily by the “stress” [33] , which

includes two parts: the stress from exogenous sources, such as an-

timicrobial agents, and the self-imposed stress, such as starvation.

Based on this, we propose the conversion rates b sp and b ps as func-

tions of concentrations of both nutrient and antimicrobial agents,

b sp = b sp (c, d) , b ps = b ps (c, d) . (12)

For b sp , we note that there should be two separate leading

order terms, which represent the effect of nutrient and antimi-

crobial agents, respectively. For instance, nutrient depletion alone

can induce persister formation [7,8,31] ; without nutrient depletion,

persister formation can also be induced in response to the antimi-

crobial stress [19] . Thus, we propose the conversion rate b sp as fol-

lows:

b sp =

(b sp1

k 2 spc

k 2 spc + c 2 + b sp2

d 2

k 2 spd

+ d 2

)(1 − φbp

φbp0

), (13)

where the first term in the first multiplicative factor describes the

mechanism that bacteria turn into persisters due to nutrient de-

letion, the second term in the factor depicts the ‘stress’ induced

onversion by antimicrobial agents, and the second multiplicative

actor represents the carrying capacity for the persister to show

he conversion rate is limited with an upper bound φbp 0 . Here b sp 1

nd b sp 2 represent the maximum conversion rates from the sus-

eptible to the persister due to nutrient depletion and antimicro-

ial stress, respectively. In the formula, we use a Monod function

o model the transitional effect. Although it is plausible to propose

ne more term, which depends both on nutrient and antimicro-

ial agents, this correlation is perceived as a high-order effect and

s therefore omitted for simplicity. The second multiplicative fac-

or limits the conversion to the carrying capacity of persister cells,

eyond which there would be no more conversion going on.

For b ps , it is assumed nonzero only if both nutrient depletion

nd the antimicrobial stress are under certain threshold values.

amely, we expect it to possess the following properties: it is close

o zero when the level of antimicrobial agents is high enough,

ince a biofilm with persisters is tolerable to antimicrobial treat-

ent, and monotonically increases to a constant level as the con-

entration of antimicrobial agents drops below a threshold value,

ince biofilms are observed to recover after antimicrobial treat-

ent. In addition, the availability of nutrient should facilitate the

onversion process [8] . Based on these requirements, we propose

he following conversion rate:

ps = b ps 1 c 2

k 2 psc + c 2

k 2 psd

k 2 psd

+ d 2 , (14)

here b ps 1 is the maximum conversion rate, k psc and k psd are two

alf-saturation constants.

Besides the live bacteria, the volume fraction of the dead bacte-

ia is also tracked in this model. It is assumed that the dead bac-

eria stay within the biofilm in the time scale that this model is

onsidered. Meanwhile, we assume some dead bacterial cells are

ttached to the biofilm acting as EPS while others are converted

nto solvent due to cell lysis [5] at certain specified rates in the

ime scale of our interest,

bd =

(r bs K

2 sd

K

2 sd

+ c 2 +

C 3 d

K 3 + d

)φbs +

(r bp K

2 pd

K

2 pd

+ c 2 +

C 12 d

K 3 + d

)φbp

−r dp φbd − r bd φbd , (15)

here r bs , r bp are the natural death rates, C 3 , C 12 are the antimi-

robial killing rates, given before, r dp represents the conversion rate

rom dead bacteria to EPS, and r bd is the cell lysis rate of the de-

omposition of dead bacteria into solvent.

EPS is produced by susceptible bacteria and persisters at differ-

nt rates as well as converted from dead bacteria. Over times, EPS

an be dissolved in a slow rate into solvent. So, the reactive kinet-

cs for EPS is proposed as follows

p =

(C 5 c

K 5 + c φbs +

C 6 c

K 6 + c φbp

)(1 − φp

φp0

) + r dp φbd − r p φp , (16)

here C 5 , C 6 are the growth rate of EPS due to susceptible bacte-

ia and persisters, respectively, K 5 , K 6 are half saturation constants,

p 0 is the maximum volume fraction that EPS is produced by live

ells in the biofilm, and r p represents the dissolving rate of EPS

nto solvent.

We assume the nutrient is consumed by the live bacteria only

nd the nutrient decay rate is proportional to the bacterial volume

raction:

c = −C 7 (φbs + φbp ) c

K 7 + c , (17)

here K 7 is a saturation constant and C 7 parametrizes the con-

umption rate of nutrient.

The concentration of antimicrobial agents depends on concen-

rations of live bacteria and EPS. It is ‘absorbed/consumed’ by the

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 5

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ive bacteria and possibly diluted by the EPS and dead bacteria via

hemical reactions. Thus, the presence of all bacteria and EPS can

educe the concentration of antimicrobial agents. We thus propose

he decay rate of antimicrobial agents as follows:

d = −(C 8 φbs + C 9 φbp + C 10 φbd + C 11 φp ) d

K 8 + d , (18)

here K 8 is a half saturation constant, C 8 , C 9 , C 10 and C 11 are

he decay rates of antimicrobial agents due to the drug-susceptible

acteria interaction, drug-persister cell interaction, drug-dead bac-

eria interaction and drug-EPS interaction, respectively.

.5. Boundary conditions

In this study, we assume that the biofilm is confined in a cu-

ic domain in 3D space: [0, L x ] × [0, L y ] × [0, L z ] where L x ,

y , L z are the length in the x , y , and z direction, respectively. The

oundary conditions for the model are given based on the physical

ettings that we intend to simulate. In this paper, we consider two

hysical settings. One is for a quiescent aqueous long channel and

he other is for a short water tube or flow cell with a rectangular

ross-section, where flows can flow in and out of the domain.

To mimic the biofilm development in a long channel, both x and

directions are assumed periodic. In the y direction, zero wall ve-

ocity for v and no-flux boundary conditions are imposed for con-

entrations/volume fractions of biomass, nutrient and antimicrobial

gents, respectively,

v | y = 0 , L y = 0 ,

[ cv φs − D c φs ∇c] · n | y =0 ,L y = 0 ,

[ d v φs − D d φs ∇d ] · n | y =0 ,L y = 0 ,

∇φi · n | y =0 ,L y = 0 , i = bs, bp, bd, p,

v φi − λi φi ∇

δF

δφi

)· n | y =0 ,L y = 0 , i = bs, bp, bd, p. (19)

e can also impose a nutrient feeding condition c| y = L y = c ∗ in

lace of the zero-flux condition in certain parts of the boundary

f needed.

For the case of biofilms in a short flow cell, periodic boundary

onditions are imposed only in the z direction. The x -axis is as-

umed to align with the inlet–outlet direction. The inlet velocity at

= 0 is given by

0 = ( p 0 y (1 − y ) , 0 , 0 ) , (20)

here p 0 is a prescribed pressure gradient. By assuming that the

olvent has already reached a steady state while flowing out of the

ell at x = L x , we prescribe ∂v ∂x

= 0 at the outlet end. For the nu-

rient concentration c , we impose the feeding boundary condition

t x = 0 as c = c 0 (y ) and flux-free condition c x = 0 is assumed at

= L x . For the biomass components, we impose no-flux boundary

onditions in the x direction,

∇φi · n | x =0 ,L x = 0 , i = bs, bp, bd, p,

(v φi − λi φi ∇

δF

δφi

) · n | x =0 ,L x = 0 , i = bs, bp, p, bd. (21)

e remark that this boundary condition works for a limited time

rame before the biomass reaches the boundary. Beyond that mo-

ent, it must be modified. In the y direction, we impose no-

ux boundary conditions for nutrient, antimicrobial agents, and

iomass components, respectively, same as Eq. (19) .

.6. A reduced model and non-dimensionalization

In the development of reactive kinetics, we have proposed

general model by taking into account many experimental ev-

dences and theoretical hypotheses, which involves many time

cales. In the study to follow, we drop the terms which have weak

ffects on biofilm dynamics within the time scale that we’re inter-

sted in. Specifically, we set

4 = C 12 = r bp = r dp = r bd = r p = 0 . (22)

his means that we consider the persisters do not growth on their

wn fueled by nutrient nor they are killed by antimicrobial agents;

ead bacteria can’t be converted into EPS nor solvent in the time

cale that we consider; and EPS does not decay into solvent.

Since most of the model parameters are not directly measur-

ble by experiments and their specific functional form reflect the

rend of the reaction, we set some of the reactive rates and half-

aturation rate the same based on our best guesses,

8 = C 9 = C 10 = C 11 , K i = K spd = K spc = K psc = K psd , 1 ≤ i ≤ 8 . (23)

e believe that this choice of parameters may affect the model

rediction quantitatively, but not qualitatively.

Let t 0 and h represent the reference time and length scale in

he biofilm system. We use these two characteristic scales to non-

imensionalize the variables and equations. In this study, we as-

ume each biofilm component has the same constant mobility, de-

oted by λ. The dimensionless variables are then given by

˜ t =

t

t 0 , ̃ x =

x

h

, ̃ v =

v t 0 h

, ˜ τ =

τ t 2 0

ρ0 h

2 , ˜ p =

pt 2 0

ρ0 h

2 , ̃ c =

c

c 0 , ˜ d =

d

d 0 ,

=

λρ0

t 0 , Re i =

ρ0 h

2

ηi t 0 , i = s, b, p. �1 =

γ1 kT t 2 0

ρ0 h

4 ,

2 =

γ2 kT t 2 0

ρ0 h

2 , ˜ ρ = φs

ρs

ρ0

+ φn ρn

ρ0

,

˜ D c =

D c t 0 h

2 , ˜ D d =

D d t 0 h

2 , ˜ C i = C i t 0 , i = 2 , 3 , 5 , 6 , 7 , 8 ; ˜ b i = b i t 0 ,

i = sp1 , sp2 , ps 0 ;˜ K i =

K i

c 0 , i = 1 , spc, psc; ˜ K j =

K j

d 0 , j = 3 , 8 , spd, psd;

˜ r i = r i t 0 , i = bs, bd. (24)

here c 0 , and d 0 denote the characteristic concentration of nutrient

nd antimicrobial agents, respectively.

For simplicity, we drop the tilde ̃ on the dimensionless vari-

bles and parameters. The non-dimensionalized PDEs governing

he biofilm system are summarized as follows:

ρ( ∂v ∂t

+ v · ∇v ) = ∇ · (2 ηD ) − ∇p − �1 ∇ · (∇φn � ∇φn ) ,

∇ · v = 0 , ∂ ∂t

φi + ∇ · (φi v ) = ∇ ·( φi ∇

δF δφi

)+ g i , i = bs, bp, bd, p,

∂φs c ∂t

+ ∇ · (cv φs ) = ∇ · (D c φs ∇c) + g c , ∂φs d ∂t

+ ∇ · (dv φs ) = ∇ · (D d φs ∇d) + g d ,

(25)

here η =

φb

Re b +

φs

Re s +

φp

Re p , and the reactive terms for the compo-

ent are given by

g bs =

C 2 c

K 1 + c

(1 − φbs

φbs 0

)φbs − b sp φbs + b ps φbp −

(r bs K

2 sd

K 2 sd

+ c +

C 3 d

K 3 + d

)φbs ,

bp = b sp φbs − b ps φbp ,

bd =

(r bs

K 2 sd

K 2 sd

+ c 2 +

C 3 d

K 3 + d

)φbs − r bd φbd ,

g p =

[ C 5 c

K 1 + c φbs +

C 6 c

K 1 + c φbp

] (1 − φp

φp0

),

g c = −(φbs + φbp ) C 7 c

K 1 + c ,

g d = −C 8 φn d

K + d , (26)

3

6 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

1 v n +

n +1 +

1 F(φ

1

F(φ

1

F(φ

1 F(φ

R

b

e

w

R

N

G

w

b

a

f

3

S

t

v

t

c

s

c

a

t

with the conversion rates

b sp =

(b sp1

k 2 spc

k 2 spc + c 2 + b sp2

d 2

k 2 spd

+ d 2

)(1 − φbp

φbp0

),

b ps = b ps 1 c 2

k 2 psc + c 2

k 2 psd

k 2 psd

+ d 2 . (27)

Remark. By ignoring spatial effects, the model reduces to a system

of coupled ordinary differential equations for a spatially homoge-

neous (bulk) biofilm system. It is then solved using a Matlab ODE

solver. For the inhomogeneous biofilm equations, we will develop

a new numerical method to solve them next.

3. Numerical schemes and GPU implementation

We develop a second-order finite difference scheme to solve

the coupled partial differential equations in the biofilm model and

then implement it in a CPU–GPU hybrid environment. In the fol-

lowing, we denote the extrapolated data using over-lines; for in-

stance, v n +1 = 2 v n − v n −1 . We first present the 2nd order finite dif-

ference scheme.

3.1. A second-order semi-discrete numerical scheme

Given the initial condition ( v 0 = 0 , s 0 = p 0 = 0 , φ0 bs

, φ0 bp

, φ0 bd

,

φ0 p , c 0 , d 0 ), we first compute ( v 1 , s 1 , p 1 , φ1

bs , φ1

bp , φ1

bd , c 1 , d 1 ) by

a first-order scheme. Having computed ( v n −1 , s n −1 , p n −1 , φn −1 bs

,

φn −1 bp

, φn −1 bd

, φn −1 p , c n −1 , d n −1 ) and ( v n , s n , p n , φn

bs , φn

bp , φn

bd , φn

p ,

c n , d n ), where n ≥ 2, we calculate ( v n +1 , s n +1 , p n +1 , φn +1 bs

, φn +1 bp

,

φn +1 bd

, φn +1 p , c n +1 , d n +1 ) in the following steps,

1. Update ρn +1 and ηn +1 :

ρn +1 = ( φn +1

bs + φn +1

bp + φn +1

bd + φn +1

p ) ρn + φn +1

s ρs ,

ηn +1 = ( φn +1

bs + φn +1

bp + φn +1

bd + φn +1

p ) ηn + φn +1

s ηs . (28)

2. Solve u

n +1 as the solution of

⎧ ⎨

⎩

ρn +1 3 u n +1 −4 v n + v n −1

2 δt + ρn +1 v

n +1 · ∇ v n +1 +

1 2 (∇ · (ρn +

− 1 Re a

∇

2 u

n +1 = −�1 ∇

2 φn +1

n ∇ φn +1

n + ∇ · (ηn +1 (∇ v

u

n +1 | y =0 ,L y = 0 ,

where Re a is the averaged Reynolds number, computed by1

Re a =

φb, max

Re b +

φp, max

Re p +

1 −φb, max −φp, max

Re s , Re b , Re p and Re s are the

Reynolds numbers for bacteria, EPS and solvent, respectively,

and φi, max = max x ∈ � φi (x ) , i = b, p.

3. Solve intermediate variable ψ

n +1 as the solution of {−∇ · ( 1 ρn +1 ∇ψ

n +1 ) = ∇ · u

n +1 ,

∂ψ

n +1

∂n | y =0 ,L y = 0 .

(30)

4. Update (v n +1 , s n +1 , p n +1 ) : ⎧ ⎨

⎩

v n +1 = u

n +1 + ρn +1 ∇ψ

n +1 ,

s n +1 = s n − ∇ · u

n +1 ,

p n +1 = p n − 3 ψ

n +1

2 δt +

1 Re a

s n +1 .

(31)

5. Update volume fractions of biomass (φn +1 bs

, φn +1 bp

, φn +1 bd

, φn +1 p ) : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

3 φn +1 bs

−4 φn bs

+ φn −1 bs

2 δt + ∇ · ( φn +1

bs v n +1 ) = ∇ ·(φ

n +bs

3 φn +1 bp

−4 φn bp

+ φn −1 bp

2 δt + ∇ · ( φn +1

bp v n +1 ) = ∇ ·(φ

n +bp

3 φn +1 bd

−4 φn bd

+ φn −1 bd

2 δt + ∇ · ( φn +1

bd v n +1 ) = ∇ ·(φ

n +bd

3 φn +1 p −4 φn

p + φn −1 p

2 δt + ∇ · ( φn +1

p v n +1 ) = ∇ ·(φ

n +p

1 )) v

n +1 +

1 Re a

∇s n + ∇p n

(∇ v n +1

) T )) − 1 Re a

∇

2 v n +1

, (29)

n +1 bs

+ φbp + φbd + φp

n +1 ) )

+ g n +1 bs

,

n +1 bp

+ φbs + φbd + φp

n +1 ) )

+ g n +1 bp

,

n +1 bd

+ φbp + φp + φbs

n +1 ) )

+ g n +1 bd

,

n +1 p + φbp + φbd + φbs

n +1 ) )

+ g n +1 p ,

(32)

where F

n +1 (φ) = ( 1 N 1

φn +1 n + ε

+

1

1 −φn +1 n

− 2 χ)�2 ∇φ − �1 ∇ ∇

2 φ,

and the reactive terms are discretized by

g n +1 bs 1

=

C 2 c n +1

K 2 + c n +1

(

1 − φn +1

bs

φbs 0

)

φn +1 bs

− b n +1

sp φn +1 bs

+ b n +1

ps φn +1

bp

−(

r bs K

2 sd

K

2 sd

+ ( c n +1

) 2 +

C 3 d n +1

K 3 + d n +1

)

φn +1 bs

,

g n +1 bp1

= −b n +1

ps φn +1 bp

+ b n +1

sp φn +1

bs ,

g n +1 bd

= (r bs

K

2 sd

K

2 sd

+ ( c n +1 ) 2 +

C 3 d n +1

K 3 + d n +1

) φn +1

bs − r bd φn +1 bd

,

g n +1 p =

(C 5 c

n +1

K 5 + c n +1

φn +1

bs +

C 6 c n +1

K 6 + c n +1

φn +1

bp

)(1 − φn +1

p

φp0

). (33)

6. Update functional components (c n +1 , d n +1 ) : ⎧ ⎪ ⎪ ⎨

⎪ ⎪ ⎩

3 φn +1 s c n +1 −4 φn

s c n + φn −1

s c n −1

2 δt + v n +1 · ∇(c n +1 φn +1

s )

= ∇ · (D c φn +1 s ∇c n +1 ) − (φn +1

bs + φn +1

bp ) C 7 c

n +1

K 7 + c n , 3 φn +1

s d n +1 −4 φn s d

n + φn −1 s d n −1

2 δt + v n +1 · ∇(d n +1 φn +1

s )

= ∇ · (D d φn +1 s ∇d n +1 ) − C 8 φ

n +1 n

d n +1

K 8 + d n .

(34)

Here are several remarks regarding this numerical scheme.

emark 3.1. Here we introduced a stabilizer term − 1 Re a

∇

2 v on

oth side of (29) [39,48] , treating one term implicitly and the other

xplicitly via extrapolation. The stabilizer introduces a O ( δt 2 ) error,

hich is consistent with our second-order scheme.

emark 3.2. A Gauge–Uzawa method [34] is used to solve the

avier–Stokes equation with s and ψ as intermediate variables.

auge–Uzawa method lies in the category of projection method,

here an intermediate variable u is solved which satisfies the

oundary condition, but not the divergence free constraint. Then,

project step is carried out to adjust u to obey the divergence

ree constraint.

.2. GPU implementation

The scheme above is presented as semi-discrete in time.

econd-order central finite difference with staggered grids is used

o discrete the equation system in space, where the velocity field

is discretized on the surface of each grid, with the pressure p ,

he phase variables φbs , φbp , φbd , φp and variables c, d being dis-

retized on the center of the grid [49–51] . The resulting algebraic

ystem is implemented on a GPU using CUDA for high performance

omputing. We use a preconditioned BiCG solver (using the pack-

ge cusp [6] ) to solve the sparse linear system generated from

he full discretization, where the preconditioner is constructed as

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 7

Table 1

Dimensional Parameters.

Symbol Description Value Unit Source

T Absolute temperature 303 Kelvin [45]

k B Boltzmann constant 1 . 38065 × 10 −23 m

2 kg s −2

K −1 [45]

ρn Biomass density 1 × 10 3 kg m

−3 [45]

ρs Water density 1 × 10 3 kg m

−3 [45]

h Characteristic length scale 1 × 10 −3 m [32]

t 0 Characteristic time scale 10 or 10 0 0 s [45]

L x , L y , L z Size of computational domain 1 − 2 × 10 −3 m [45]

ηb , ηp Dynamic viscosity of bacteria and EPS 2.7 × 10 2 kg m

−1 s −1 [26]

ηs Dynamic viscosity of solvent 1 . 002 × 10 −3 kg m

−1 s −1 [26]

c 0 Characteristic oxygen concentration 8 . 24 × 10 −3 kg m

−3 [32]

d 0 Characteristic antimicrobial concentration 1 . 0 × 10 −2 kg m

−3 [43]

γ 1 Distortional energy coefficient 8 × 10 6 m

−1 [9]

γ 2 Mixing free energy coefficient 3 × 10 17 m

−3 [9]

λ Mobility parameter 1 × 10 −9 kg −1

m

3 s [9]

χ Flory–Huggins parameter 0.55 [45]

N Generalized polymerization parameter 1 × 10 3 [45]

D c Oxygen diffusion coefficient 2 . 3 × 10 −9 m

2 s −1 [42]

D d Antimicrobial diffusion coefficient 6 × 10 −10 m

2 s −1 [43]

C 2 Susceptible bacteria growth rate 4 × 10 −4 s −1 [11]

C 3 Susceptible bacteria decaying rate 4 × 10 −2 s −1 [11]

r bs Flush-out rate for susceptible 4 × 10 −7 s −1 Estimated

r bd Flush-out rate for dead bacteria 1 . 0 × 10 −7 s −1 Estimated

b sp 1 Transfer rate 2 × 10 −5 s −1 Estimated

b sp 2 Transfer rate 1 × 10 −3 s −1 Estimated

b ps 0 Transfer rate from φbs to φbp 4 × 10 −5 s −1 [11]

C 5 EPS growth rate 4 × 10 −4 s −1 [13]

K 1 , K 3 Monod constant 3 . 5 × 10 −4 kg m

−3 [32]

C 7 Nutrient consumption rate 4 × 10 −2 kg m

−3 s −1 [38]

C 8 Antimicrobial agents consumption rate 4 × 10 −2 kg m

−3 s −1 [38]

φbs 0 Carrying capacity for susceptible bacteria 0.2 [35]

φbp 0 Carrying capacity for persister 0.02 Estimated

φp 0 Carrying capacity for EPS 0.2 [18]

t

a

[

T

s

2

T

c

4

4

w

t

t

t

t

u

t

c

s

s

c

r

t

t

i

Fig. 2. Benchmark of the model using experimental data of the total live bacte-

ria and persisters. The curves are model predictions and circles are experimen-

tal data taken from [25] . The initial conditions are (φbs , φbp , φd , φp , c, d) = (2 ×10 −4 , 2 × 10 −8 , 0 , 0 , 1 . 0 , 0) . All parameters used are given in Table 1 , except that

c 2 = 8 × 10 −4 , c 7 = 8 × 10 −3 and φp0 = 0 . 002 . Here we assume the volume of each

bacteria is about 2 . 71 μm

3 , i.e., we use the initial bacterial volume fraction as

2 × 10 −4 , which is equivalent to 10 7 CF U/ ml in the experimental measurement. We

skip the initial transient lag phase in biofilm growth due to growth factors in the

biofilm to focus our modeling on long time growth mechanisms. We note that our

model is not developed to describe the initial lag phase.

m

i

N

t

o

s

g

he inverse of a good approximation of the implicit operator by

pproximating the variable coefficient with a constant coefficient

51] . The preconditioner can then be computed by the Fast Fourier

ransform. The data are saved as HDF5 format, which can be vi-

ualized using VisIt [10] . Currently, one solver can deal with up to

56 3 meshes, limited by the global memory size of a single GPU.

he scheme is second order in space and time in theory, which is

onfirmed by numerical convergence tests.

. Numerical results and discussion

.1. Model parameters and benchmarks

In this study, we focus on dynamics of biofilm growth with and

ithout antimicrobial treatment. Thus, we choose two characteris-

ic time scales corresponding to these two scenarios: the growth

ime scale at t 0 = 10 3 s without antimicrobial treatment and the

ime scale at t 0 = 10 s under antimicrobial treatment. We remark

hat the time scale for antimicrobial treatment can vary from min-

tes to hours, or even days depending on the choice of the an-

imicrobial agents and the outcome that one is interested in. We

hoose the smaller time scale comparable with the diffusion time

cale of the antimicrobial agent in this study.

All the (dimensional) model parameters used in the current

tudy are summarized in Table 1 with their respective references

ited. In the following discussion, unless noted otherwise, the pa-

ameter values are chosen directly from Table 1 . Corresponding to

he two characteristic time scales alluded to earlier, we will use

wo sets of dimensionless parameters in our simulations next.

To demonstrate the capability of the model, we first benchmark

t against a known experiment [25] , in which we use the experi-

ental data on persisters and the total live bacteria in a develop-

ng biofilm to calibrate the model. The result is shown in Fig. 2 .

otice that there exists a lag phase in the biofilm experiment ini-

ially, which is supposedly regulated by growth factors [27,36] . In

ur current model, this transient factor is not considered. So, we

tart the simulation at t = 2 , i.e., we bypass the delayed transient

rowth stage and apply our model to the later stage of biofilm

8 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

Fig. 3. Antimicrobial treatment of biofilms. (a)–(g) Morphology of live bacteria aggregates (green) in S. Epidermis biofilm at different time slots treated by 50 mg/L HOCl [16] .

(h–n) Numerical predictions of live bacteria aggregates at the corresponding time slots treated by antimicrobial agents. All parameters used in the simulation are chosen

from Table 1 , except C 3 = 4 × 10 −4 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article).

4

a

m

c

a

i

c

c

m

t

w

t

growth. As shown in Fig. 2 , the model prediction agrees very well

with the experiment in the time frame that it is supposedly valid

[25] .

In addition, we carry out another benchmark on spatially in-

homogeneous biofilms to study dynamics of the biocide action

against biofilms. In this experiment, some biofilm colonies are ini-

tially attached to the substrate; antimicrobial agents are then ap-

plied to treat the biofilm. The results are recorded in time series

depicted in Fig. 3 . By taking the initial biofilm morphological pro-

file from the experiment, our model with the parameters given

in Table 1 is employed to predict biofilm dynamics in the same

time series. From Fig. 3 , we observe the model predicted time se-

quences compare qualitatively well with the experimental time se-

quences reported in [16] . This lend us additional support for using

the model to make dynamical predictions about biofilm dynamics.

.2. Persister formation

It is perceived that persister bacteria reproduce themselves in

low rate. Hence, the gain of persister bacteria in the biofilm is

ainly due to the conversion from the susceptible cells under the

ondition of nutrient depletion or the existence of antimicrobial

gents. Here, we conduct several numerical investigations to exam-

ne persister formation during biofilm development and to make

omparisons with other existing conversion rate models, e.g., the

onstant conversion rate model and the nonlinear conversion rate

odel in Eq. (11) .

One results on dynamics of biofilm development without an-

imicrobial treatment are shown in Fig. 4 . For a well-grown biofilm

ith depleted nutrient in the end, both the susceptible and

he persister cells gradually diminish in the model of constant

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 9

Fig. 4. Comparison of biofilm growth dynamics in an environment with deficient nutrient supply using the homogeneous model. This figure shows that our new conversion

rate model gives a better prediction for persister cells in the nutrient deficient case compared with the other existing models. (A) The volume fraction of susceptible cells.

(B) The volume fraction of persistent cells. (C) The volume fraction of the dead bacteria. (D) The volume fraction of EPS. (E) The concentrate of nutrient. (F) The volume

fraction of total live bacteria. In this case study, we examine an extreme case where the nutrient concentration is set to zero. The legend X, Y, Z represent the conversion

rate model of (27) , constant rates and (11) , respectively.

c

i

s

s

a

r

i

o

d

a

s

v

n

i

h

e

a

g

g

E

p

a

s

c

t

i

n

n

s

s

c

t

t

n

c

t

m

d

a

c

t

l

i

t

b

[

e

l

t

w

s

I

c

w

l

t

(

m

t

p

s

e

b

4

t

n

onversion rates and in the model of a nonlinear conversion rates

n Eq. (11) . These are in direct conflict with the experimental ob-

ervations where persisters get into a dormant state [30] and can

urvive in a nutrient depleted environment [7,31] at least for quite

long period of time. In comparison, our model for conversion

ates agree qualitatively well with the survival persister population

n the experiments in which the persister cell population decays

nly slightly over a long period of time.

We also conduct several 3D numerical simulations on biofilm

evelopment including persister formation in an infinite long tube

nd in a finite-length flow cell, respectively, taking into account the

patial inhomogeneity of the biofilm structure. In Fig. 5 , biofilm de-

elopment in a quiescent aqueous environment is simulated, with

utrient supplied through the top surface y = L y . Growth dynam-

cs of several randomly distributed biofilm colonies developing into

ighly heterogeneous biofilm colonies is shown in Fig. 5 (a–e). To

xamine the distribution of each component internally, 2D slices

t x = 0 . 5 and t = 200 are depicted in Fig. 5 (f–i). Notice that, in

eneral, the susceptible cells are more concentrated near the re-

ion where nutrient is amply supplied. The volume fraction of the

PS pattern correlates with that of the susceptible cells as EPS is a

roduct of live bacteria at the presence of nutrient. The persisters

re primarily distributed inside the biofilm.

To further investigate the effect of nutrient distribution on per-

ister formation in heterogeneous environment, we carry out a

omparative study with respect to four different nutrient concen-

rations supplied through the top surface. The results are shown

n Fig. 6 . In any case, regardless if it is the nutrient sufficient or

utrient depleted case, persister cells would grow. However, in the

utrient depleted case, persister cells gain a higher ratio with re-

pect to the total live bacteria than the case with more nutrient

upplied. This is a direct consequence of low survival rate for sus-

eptible cells in the biofilm. This perhaps explains why, in the nu-

rient depleted case, the biofilm is more persistent to antimicrobial

reatment, as reported in some experiments [1–3] .

In Fig. 7 , we examine biofilm development in a flow cell with

utrient rich solvent flowing in through the inlet. The morphologi-

al change of biofilm colony is shown in Fig. 7 (a–d) in the simula-

ion. 2D slices at x = 0 . 5 and t = 100 are shown in Fig. 7 (e–i). The

igratory deformation in the colonies is clearly due to flow in-

uced shear. The scenario for volume fractions of susceptible cells

nd EPS is similar to that depicted in Fig. 5 , i.e., they are more

oncentrated near nutrient rich regions within the biofilm. Persis-

er cells are more concentrated inside the biofilm, where they have

ess access to nutrient.

In the above discussion, we focus primarily on growth dynam-

cs of biofilms without the presence of antimicrobial agents. When

he biofilm is treated by antimicrobial agents, some works have

een carried out studying dosing strategies, notably by Cogan et al.

11,12] . However, there is one essential factor that has evaded mod-

lers, that is the role of the nutrient concentration. As we have al-

uded to earlier, the conversion dynamics between persister and

he susceptible cells relies heavily on the nutrient supply. Here,

e carry out a comparative study using periodic dosing with re-

pect to different nutrient conditions in the homogeneous model.

n the following discussion, we set the nutrient concentration at a

onstant level, ignoring its dynamics in time. As shown in Fig. 8 ,

hen the nutrient supply is high ( c = 1 ), the live bacteria re-

apse, whereas if the nutrient supply is relatively low ( c = 0 . 1 ),

he biofilm is well-controlled. However, when the nutrient is low

i.e., c 0 = 0 . 01 , c 0 = 0 . 025 ), the volume fraction of persisters drops

uch slower. When nutrient is deficient at c 0 = 0 . 0 0 01 , the persis-

er can hardly be treated. This study shows that the dependence of

ersister conversion on nutrient can affect antimicrobial treatment

ignificantly. We hope this mechanism can be further confirmed by

xperimentalists so that we can come up with a better antimicro-

ial treatment strategy.

.3. Antimicrobial treatment of biofilms in a flow cell

To develop a proper strategy for biofilm control and eradica-

ion, a clear understanding of antimicrobial dynamics in heteroge-

eous biofilms is important. We now investigate the antimicrobial

10 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

Fig. 5. Persister formation in an infinitely long channel, where nutrient is supplied through the top boundary. The 3D view of biofilm colonies at different time slots are

shown in (a–e). 2D slice view (at x = 0 . 5 ) of the volume fraction of the susceptible cells, the persisters and EPS at t = 200 are shown in (f–h), respectively. (i) The 2D slice

view of nutrient concentration at t = 200 .

Fig. 6. Dynamics of persister growth. This demonstrates a comparative study on persister formation dynamics with respect to various nutrient supply conditions through the

top boundary, namely, c 0 = 1 . 0 , 0 . 5 , 0 . 1 , 0 . 01 at L y . The persister distribution at t = 200 for the nutrient supply conditions are shown in (a–d), respectively. (e) The volume

of total live cell, the susceptible cells and the persisters, i.e., ∫ � φbs + φbp dx ,

∫ � φbs dx and

∫ � φbp dx , and the ratio of persister cells with respect to the total live bacteria, i.e., ∫

� φbp dx / ∫ �(φbs + φbp ) dx , are shown, respectively. � is the domain of the biofilm. The legend 1,2,3,4 represent the simulations with c 0 = 1 . 0 , 0 . 5 , 0 . 1 , 0 . 001 , respectively.

t

s

c

r

p

F

process with respect to various antimicrobial dosing strategies in a

long fluid channel as well as in a short flow cell.

In the first case, we examine antimicrobial treatment in a

flow-cell, where the antimicrobial agents are carried into the cell

through the inlet boundary by solvent. This mimics the antimicro-

bial treatment of biofilms in a pipe, by flowing disinfectant solu-

tions. We depict the result in Fig. 9 . We observe that more suscep-

ible bacteria are killed facing the inlet boundary than on the other

ide. Meanwhile, more susceptible bacteria in regions facing the in-

oming flow are converted into persister bacteria than those in the

egion facing the other way. This study also demonstrates the role

layed by antimicrobial agents in facilitating the conversion.

To show more details, 2D slices at z = 0 . 5 are depicted in

ig. 9 (g–e). As one can observe in Fig. 9 (i), the distribution of the

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 11

Fig. 7. Biofilm dynamics in a flow cell. This figure depicts the dynamical process of biofilm growth in a flow cell with fresh nutrient flowing through the cell.(a–d) show

biofilm colonies at different time slots. (e–g) show 2D slices at x = 0 . 5 of φbs , φbp , φp at t = 100 , respectively. The concentration of nutrient c at t = 100 is shown in (h). (i)

The velocity field at x = 0 . 5 and t = 100 . Flow induced deformation are observed.

Fig. 8. Dynamics of spatially homogeneous biofilms under antimicrobial treatment. This figure demonstrates how nutrient deficiency can reduce the efficiency of periodic

dosing strategies. Here, we administer a periodic dosing with respect to various nutrient supply conditions: c = 1(X1) ; c = 0 . 5(X2) ; c = 0 . 1(X3) ; c = 0 . 025(X4) , c = 0 . 01(X5) ,

c = 0 . 001(X6) . (A) The volume fraction of the susceptible cells. (B) The volume fraction of persisters. (C) The volume fraction of live bacteria. (D) The volume fraction of EPS.

When the nutrient supply level is very low, the persister cells can hardly be treated.

12 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

Fig. 9. Antimicrobial treatment of biofilms in a flow cell using antimicrobial solutions flowing through the inlet boundary. (a–c) The volume fraction of susceptible cells at

time t = 2 , 14 , 24 , respectively. (d–e) The volume fraction of persisters at time t = 2 , 24 , respectively. (f) The volume fraction of dead bacteria at t = 24 . 2D slices at z = 0 . 5

and time t = 2 are shown in (g-l). (g) The consumption rate of antimicrobial agents. (h) The total biomass volume fraction. (i) The concentration of antimicrobial agents.

(j) The biomass flux. (k) The hydrostatic pressure. There exists a few low pressure spots. But, they are not significant. (l) The velocity field. The flow above the biofilm is

laminar. The treatment is effective on susceptible cells, but has little impact on persisters.

g

n

w

fl

n

t

fl

t

s

antimicrobial agents becomes heterogeneous within the biofilm,

in particular, higher concentrations are observed near the inlet

boundary. The consumption rate of antimicrobial agents is highly

heterogeneous in the biofilm as well, shown in Fig. 9 (g). This cor-

roborates with the biofilm morphology depicted in Fig. 9 (d), where

more persisters are converted from the susceptible cells due to the

antimicrobial stress. We note that in the laminar flow field, there is

a biomass flux moving towards the outlet boundary in the simula-

tion (see Fig. 9 (j)) in which the disinfection process dominates. But,

persisters can survive this treatment so that this dosing method

fails to eradicate the biofilm colony satisfactorily.

With the development of nano-technology, new dosing strate-

ies for biofilm have been investigated [24] , in particular, targeted

ano-scale encapsulated drug delivery for biofilms. In this context,

e next conduct a study simulating targeted drug delivery in a

ow cell, in which we investigate the combined effect of hydrody-

amic environment and the interaction between biofilms and an-

imicrobial agents.

The first case we study is one single dose released in the

ow cell adjacent to the biofilm subject to different inlet veloci-

ies (i.e., different hydrodynamic fluxes). The numerical results are

hown in Fig. 10 . We observe that a higher inlet flow facilitates the

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 13

Fig. 10. Comparison of antimicrobial efficacy at various inflow velocities in a flow cell. (a–d) show the concentration of antimicrobial agents at a few selected time slots at

inflow velocity v 0 = (1 . 0 , 0 , 0) . (e–h) shows the volume fraction of dead bacteria at t = 15 with respect to inlet horizontal velocities v x 0 = 1 , 2 , 4 , 8 , respectively. (i) shows

the time series of the total mass of antimicrobial agents ∫ � d d x and total volume of dead bacteria

∫ � φbd dx with respect to the four inlet velocities. 1,2,3,4 correspond to

v x 0 = 1 , 2 , 4 , 8 , respectively. The slower the inlet velocity is, the better antimicrobial treatment is.

a

i

a

o

F

b

t

t

f

e

r

T

f

t

t

fl

b

i

a

t

c

l

t

r

5

t

t

c

t

w

s

f

d

i

h

t

s

n

o

m

d

ntimicrobial effect on the biofilm so that more bacteria are killed

nitially. However, a faster flow also convects the antimicrobial

gents out of the flow cell quicker so as to weaken the efficacy

f antimicrobial agents in term of the overall antimicrobial effect.

ig. 10 (i) depicts the time-dependent concentration of antimicro-

ial agents in the flow cell on average. As the velocity increases,

he antimicrobial agents are washed out quickly in the flow cell,

hereby reduces the efficacy of the treatment. This can also be seen

rom the dead bacterial population in the cell shown in Fig. 10 (i).

Finally, we study the effect of dosing positions on antimicrobial

fficacy, where we inject a single dose at several different positions

elative to the biofilm. The numerical results are shown in Fig. 11 .

he numerical study shows the second dosing position is more ef-

ective, where antimicrobial agents are released inside the biofilm

o maximize contact with bacteria. In comparison, the first posi-

ion is the least effective since the antimicrobial agents can easily

ushed away by the flow before they even make contact with the

acteria. The dead bacteria for each dosing position are also shown

n Fig. 11 . Overall, the targeted injection seems to be more effective

s it kills more bacteria than the treatment by antimicrobial solu-

ions. In particular, observed from Fig. 11 (c) and (g), the antimi-

robial agents are injected and exposed directly inside biofilms. It

eads to an instantaneously concentrated disinfection. This is due

o the fact that the reaction rate is much faster than the diffusion

ate of the antimicrobial agents in this study.

. Conclusion

In this paper, we develop a multiphasic field theory for biofilms

aking into account interactions among multiple bacterial pheno-

ypes, EPS, solvent, nutrient, and possibly antimicrobial agents. We

lassify the bacteria in biofilms into persistent cells and suscep-

ible cells based on their response to antimicrobial treatment as

ell as dead cells. A new conversion rate model between the per-

istent and susceptible cells is implemented. A numerical scheme

or solving the governing system of partial differential equations is

eveloped and implemented on GPU in 3D space and time.

By focusing on a spatially homogeneous case, the model is cal-

brated against an experiment, which yields an excellent fit. The

omogeneous model is then used to show its superiority over the

wo existing rate models in susceptible and persistent cell conver-

ion when compared with experiments. In addition, the homoge-

eous model is also used to probe the effect of nutrient deficiency

n the efficacy of antimicrobial treatment. Then, a series of 3D nu-

erical simulations are carried out to investigate biofilm growth

ynamics and the dynamical process of biofilm treatment by

14 J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15

Fig. 11. Comparison of antimicrobial efficacy at various dosing positions. The antimicrobial agents are released (or injected) at four distinct locations in a flow cell in which

biofilms grow, x 0 = (0 . 2 , 0 . 75 , 0 . 5) , (0 . 45 , 0 . 2 ., 0 . 5) , (0 . 2 , 0 . 2 , 0 . 5) and (1.75, 0.2, 0.5), respectively. (a) The initial profile of the biofilm and velocity field; (b–e) The four dosing

positions. (f–i) The volume fraction of dead bacteria at t = 15 with respect to the four dosing positions. (j) The time series of the total mass of antimicrobial agents ∫ � d d x

and the total volume of dead bacteria ∫ � φbd dx for all four cases.

a

Z

t

R

antimicrobial agents with respect to various dosing strategies.

These studies reveal the crucial role played by the conversion

mechanism between susceptible and persistent bacterial cells in

biofilm dynamics with and without the antimicrobial treatment.

Experimentally, it is difficult to eradicate biofilms completely by

conventional means. Thus, effort s in designing proper strategies are

in need for biofilm control and eradication. The numerical simula-

tions confirm that dosing by local release is much more effective

as well as environment-friendly than using a nebulizer, which de-

livers the antimicrobial agent to the surface of the biomass-solvent

mixture or through an antimicrobial solution to rinse the biofilm.

In particular, if antibiotics carrying nano-capsule can be delivered

into the biofilm, the antimicrobial effect will be much more effi-

cient. Besides dosing positions, dosing time controls are also vital

for effective antimicrobial treatment and biofilm control, for which

the model can also provide valuable insight.

Acknowledgment

Qi Wang is partially supported by AFOSR , NIH and NSF

through awards FA9550-12-1-0178 , DMS-1200487 , DMS-1517347

nd R01GM078994-05A1 as well as a SC EPSCOR GEAR award. Jia

hao is partially supported by an ASPIRE grant from the Office of

he Vice President for Research at the University of South Carolina.

eferences

[1] S.M. Amato , M.P. Brynildsen , Nutrient transitions are a source of persiters inescherichia coli biofilms, PLOS One 9 (3) (2014) e93110 .

[2] S.M. Amato , C.H. Fazen , T.C. Henery , etc. , The role of metabolism in bacterialpersistence, Front. Microbiol. 5 (70) (2014) 1–9 .

[3] S.M. Amato , M.A. Orman , M.P. Brynildsen , Metabolic control of persister for-

mation in escherichia coli, Mol. Cell 50 (4) (2013) 475–487 . [4] N.Q. Balaban , J. Merrin , R. Chait , L. Kowalik , S. Leibler , Bacterial persistence as

a phenotypic switch, Science 305 (5690) (2004) 1622–1625 . [5] K.W. Bayles , The biological role of death and lysis in biofilm development, Nat.

Rev. Microbiol. 5 (9) (2007) 721–726 . [6] S. Dalton, N. Bell, L. Olson and M. Garland, Cusp: Generic Parallel Algorithms

for Sparse Matrix and Graph Computations, 2014, http://cusplibrary.github.io/ . [7] S.P. Bernier , D. Lebeaux , A.S. DeFrancesco , A. Valomon , G. Soubigou , J. Copppee ,

J. Ghigo , C. Beloin , Starvation together with the sos response mediates high

biofilm-specific tolerance to the fluoroquinolone ofloxacin, PLoS Genet. 9 (1)(2013) .

[8] G. Borriello , L. Richards , G.D. Enrlich , P.S. Stewart , Arginine or nitrate enhancesantibiotic susceptibility of pseudomonas aeruginosa in biofilms, Antimicrob.

Agents Chemother. 50 (1) (2006) 382–384 .

J. Zhao et al. / Mathematical Biosciences 282 (2016) 1–15 15

[

[

[

[

[

[

[

[

[

[

[

[

[

[

[[

[

[

[

[

[

[

[9] C. Chen , M. Ren , A. Srinivansan , Q. Wang , 3D numerical simulations of biofilmflows, East Asian J. Appl. Math. 1 (3) (2011) 197–214 .

[10] H. Childs , E. Brugger , B. Whitlock , J. Meredith , S. Ahern , D. Pugmire , K. Bia-gas , M. Miller , C. Harrison , G.H. Weber , H. Krishnan , T. Fogal , A. Sanderson ,

C. Garth , E.W. Bethel , D. Camp , O. Rübel , M. Durant , J.M. Favre , P. Navrátil ,Visit: An end-user tool for visualizing and analyzing very large data, High Per-

formance Visualization–Enabling Extreme-Scale Scientific Insight, Taylor andFrancis, 2012, pp. 357–372 .

[11] N.G. Cogan , Effects of persister formation on bacterial response to dosing, J.

Theor. Biol. 238 (2006) 694–703 . [12] N.G. Cogan , J. Brown , K. Darres , K. Petty , Optimal control strategies for disin-

fection of bacterial populations with persister and susceptible dynamics, An-timicrob. Agents Chemother. 56 (9) (2012) 4 816–4 826 .

[13] N.G. Cogan , J.P. Keener , The role of the biofilm matrix in structural develop-ment, Math. Med. Biol. 21 (2) (2004) 147–166 .

[14] N.G. Cogan , B. Szomolay , M. Dindos , Effects of periodic disinfection on persis-

ters in a one-dimensional biofilm model, Bull. Math. Biol. 75 (2013) 94–123 . [15] D. Davies , Understanding biofilm resistance to antibacterial agents, Nat. Rev.

Drug Discov. 2 (2003) 114–122 . [16] W.M. Davison , B. Pitts , P.S. Stewart , Spatial and temporal patterns of bio-

cide action against staphylococcus epidermidis biofilms, Antimicrob. AgentsChemother. 54 (7) (2010) 2920–2927 .

[17] L. Demaret , J.H. Eberl , M.A. Efendiev , R. Lasser , Analysis and simulation of a

meso-scale model of diffusive resistance of bacterial biofilms to penetration ofantibiotics, Adv. Math. Sci. Appl. 18 (2008) 269–304 .

[18] R.M. Donlan , Biofilms microbial life on surfaces, Emerg. Infect. Dis. 8 (9) (2002)881–890 .

[19] T. Dorr , M. Vulic , K. Lewis , Ciprofloxacin causes persister formation by inducingthe tisb toxin in escherichia coli, PLoS Biol. 8 (2) (2010) .

20] O. Gefen , N.Q. Balaban , The importance of being persistent: heterogeneity

of bacterial populations under antibiotic stress, FEMS Microbiol. Rev. 33 (4)(2009) 704–717 .

[21] D.M. Grant , T.R. Bott , Biocide dosing strategies for biofilm control, Heat Transf.Eng. 26 (1) (2005) 44–50 .

22] K. Grobe , J. Zahller , P. Stewart , Role of dose concentration in biocide effi-cacy against pseudomonas aeruginosa biofilms, J. Ind. Microbiol. Biotechnol.

29 (2002) 10–15 .

23] S.M. Hunt , M.A. Hamilton , P. Stewart , A 3d model of antimicrobial action onbiofilms, Water Sci. Technol. 52 (7) (2005) 143–148 .

[24] R.K. Kasimanickam , A. Ranjan , G.V. Asokan , V.R. Kasimanickam , J.P. Kastelic ,Prevention and treatment of biofilms by hybrid and nanotechnologies, Int. J.

Nanomed. 8 (2013) 2809–2819 . 25] I. Keren , N. Kaldalu , A. Spoering , Y. Wang , K. Lewis , Persister cells and tolerance

to antimicrobials, FEMS Microbiol. Lett. 230 (2004) 13–18 .

26] I. Klapper , C.J. Rupp , R. Cargo , B. Purvedorj , P. Stoodley , Viscoelastic fluid de-scription of bacterial biofilm material properties, Biotechnol. Bioeng. 80 (3)

(2002) 289–296 . [27] N. Larsen , M. Boye , H. Siegumfeldt , M. Jakobsen , Differential expression of pro-

teins and genes in the lag phase of lactococcus lactis subsp lactis grown insynthetic medium and reconstituted skim milk, Appl. Environ. Microbiol. 72

(2) (2006) 1173–1179 . 28] P.D. Leenheer , N.G. Cogan , Failure of antibiotic treatment in microbial popula-

tions, J. Math. Biol. 59 (4) (2009) 563–579 .

29] K. Lewis , Persister cells dormancy and infectious disease, Nat. Rev. Microbiol.5 (2007) 48–56 .

30] K. Lewis , Persister cells, Ann. Rev. Microbiol. 64 (1) (2010) 357–372 . [31] D. Nguyen , A. Joshi-Datar , F. Lepine , E. Bauerie , O. Olakanmi , K. Beer , G. McKay ,

R. Siehnel , J. Schafhauser , Y. Wang , B. Britigan , P. Singh , Active starvation re-sponses mediate antibiotic tolerance in biofilms and nutrient-limited bacteria,

Science 334 (2011) 982–986 . 32] C. Picioreanu , M.C.M. van Loosdrecht , J.J. Heijnen , Mathematical modeling of

biofilm structure with a hybrid differential discrete cellular automaton ap-proach, Biotechnol. Bioeng. 58 (1) (1998) 101–116 .

[33] K. Poole , Bacterial stress responses as determinants of antimicrobial resistance,

J. Antimicrob. Chemother. 20 (5) (2012) 227–234 . 34] J.H. Pyo , J. Shen , Gauge-Uzawa methods for incompressible flows with variable

density, J. Comput. Phys. 221 (2007) 181–197 . [35] M.E. Roberts , P.S. Stewart , Modeling protection from antimicrobial agents in

biofilms through the formation of persister cells, Microbiology 151 (2005)75–80 .

36] M. Rolfe , C. Rice , S. Lucchini , C. Pin , A. Thompson , A. Cameron , M. Alston ,

M. Stringer , R. Betrs , J. Baranyi , M. Peck , J. Hinton , Lag phase is a distinctgrowth phase that prepares bacteria for exponential growth and involves tran-

sient metal accumulation, J. Bacteriol. 194 (3) (2012) 686–701 . [37] J.A. Sanclement , P. Webster , J. Thomas , H.H. Ramadan , Bacterial biofilms in sur-

gical specimens of patients with crhonic rhinosinusitis, Laryngoscope 115 (4)(2005) 578–582 .

38] P. Seeluangsawat , 3d computational investigation of viscoelastic biofilms using

gpus, 2011 . Ph.D. Thesis 39] J. Shen , X. Yang , Numerical approximation of Allen-Cahn and Cahn-Hilliard

equations, Discret. Contin. Dyn. Syst. Ser. B 28 (4) (2010) 1669–1691 . 40] Y. Shen , J. Zhao , C. Fuentz-Nunez , Z. Wang , R.E.W. Hancock , C.R. Roberts , J. Ma ,

J. Li , M. Haapasalo , Q. Wang , Development and experimental validation of amodel for oral multispecies biofilm recovery after chlorhexidine treatment, Sci.

Rep. 6 (2016) e27537 .

[41] M. Simões , L.C. Simões , M.J. Vieira , A review of current and emergent biofilmcontrol strategies, LWT - Food Sci. Technol. 43 (4) (2010) 573–583 .

42] P.S. Stewart , Diffusion in biofilms, J. Bacteriol. 185 (5) (2003) 1485–1491 . 43] P.S. Stewart , Theoretical aspects of antibiotic diffusion into microbial biofilms,

Antimicrob. Agents Chemother. 40 (11) (1996) . 2517–22 44] B. Szomolay , I. Klapper , M. Dindos , Analysis of adaptive response to dosing

protocols for biofilm control, SIAM J. Appl. Math. 70 (8) (2010) 3175–3202 .

45] T. Zhang , N.G. Cogan , Q. Wang , Phase-field models for biofilms I. Theory andsimulations, SIAM J. Appl. Math. 69 (2008) 641–669 .

46] J. Zhao , Y. Shen , M. Happasalo , Z.J. Wang , Q. Wang , A 3d numerical study of an-timicrobial persistence in heterogeneous multi-species biofilms, J. Theor. Biol.

392 (2016) 83–98 . [47] J. Zhao , Q. Wang , A 3D hydrodynamic model for cytokinesis of eukaryotic cells,

Commun. Comput. Phys. 19 (3) (2016) 663–681 .

48] J. Zhao , Q. Wang , Modeling cytokinesis of eukaryotic cells driven by the acto-myosin contractile ring, Int. J. Numer. Methods Biomed. Eng. (2016) e02774 .

49] J. Zhao , Q. Wang , Semi-discrete energy-stable schemes for a tensor-based hy-drodynamic model of nematic liquid crystal flows, J. Sci. Comput. 68 (2016)

1241–1266 . 50] J. Zhao , Q. Wang , X. Yang , Numerical approximations to a new phase field

model for two phase flows of complex fluids, Comput. Methods Appl. Mech.Eng. 310 (2016b) 77–97 .

[51] J. Zhao , X. Yang , J. Shen , Q. Wang , A decoupled energy stable scheme for a

hydrodynamic phase field model of mixtures of nematic liquid crystals andviscous fluids, J. Comput. Phys. 305 (2016a) 539–556 .