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A MATHEMATICAL “QUASI-STEADY” MODEL FOR
THE GROWTH OF MENINGITIS BACTERIUM IN THE
HUMAN BRAIN
Arnab Mukherjee,
Undergraduate student (7th semester)
Department of Biotechnology,
IIT Madras.
Reviewed by,
Prof G.K. Suraishkumar,
Head of Department,
Department of Biotechnology,
IIT Madras.
ABSTRACT:
Meningitis is a potentially fatal infection of the human brain that can be mediated by
bacterial or viral attacks, though the former tends to predominate. Initiated by pathogenic
colonization of the brain, meningitis spurs a cascade of physiological reactions that
ultimately lead to a massive damage of neuronal cells of the cerebral cortex. My attempt
here has been to mathematically model the pattern of bacterial growth in the brain in
meningitis infected patients that leads to this unique, yet deadly cascade of cell-damaging
events.
The pattern of growth has been modeled using principles of diffusive and convective
mass transfer, mass balances and microbial growth equations. The pattern was seen to
predict correctly the brain-cell damaging events that follow within days of meiningeal
infection. In other words, beginning with a set of mathematical equations, the model
attempted to explain why meningitis bacteria gave rise to the characteristic vicious
cascade that ultimately results in massive neuronal damage.
INTRODUCTION:
The brain stands out from all other organs in the human body in its oxygen and nutrition
requirements relative to its size. Although comprising no more than 2 % of the total body
weight, yet the brain tissue utilizes almost 20 % of our oxygen supply [1]
Bacterial meningitis can be mediated by N. meningitidis (Meningococcus),
Pneumococcus, as well as some strains of E. coli and M. tuberculosis. The mathematical
model that will subsequently be derived aims to provide approximate formulae to predict
the rate of multiplication (growth) of bacteria once it has colonized its target region. (the
meninges). Based on its growth rate pattern, the model should be able to explain the
cascade of events that has become idiosyncratic to meningitis bacteria-mediated cell
damage. The generalized model can be extended for different bacterial strains as well as
varying classes of patients (e.g. – immunocompromised patients) and might as well offer
an insight into the extreme significance of meningitis as a potent medical emergency.
THE BRAIN (an anatomical perspective):
To develop a model for meningitis-induced neuronal death, it would be necessary to
comprehend the normal anatomy of the human brain. For convenience I have accentuated
only those points about the cerebral anatomy that are necessary for an understanding of
the model.
The meninges:
Enclosed in a protective cranial box, the brain (cerebrum) is further protected by 3 layers
– an outer dura matter, inner pia matter and medial arachnoida matter; collectively
referred to as the meninges. The blood supply to the brain necessarily traverses these
layers before reaching the fissures and grooves in the cerebral cortex. Located between
the arachnoida and inner pia is a region relatively rich in vasculature, described as the
Sub Arachnoid Space (SAS). The SAS remains filled with a protective, cushioning fluid
labeled Cerebrospinal Fluid (CSF). The CSF exists to cushion the brain against physical
shocks as well as to act as a ‘sink’ for waste products released into the brain.
The brain tissue:
Since meningitis bacteria target primordially the cerebral cortex, in the interest of the
model, it would be sufficient to consider only the cerebral portion of the brain, which
comprises the major fraction of whole brain tissue. Circumscribed by the meninges, the
cerebral tissue lying within is imbued by the Interstitial Fluid (ISF) which in effect
makes up the volume inside the brain – and is the pathway for substances from the
bloodstream to the cerebral cells. Apart from this the cerebrum also has its own regulated
microvasculature – the major source of cellular nutrients and oxygen.
The pressure that exists within the brain is refered to as the Intracranial Pressure (ICP)
and any perturbation from its normal range of values can spell major injury for the brain.
The internal CSF source – choroid plexi:
The 3rd region of interest is a source of CSF located within the brain itself – known as the
Choroid Plexus. The CSF produced in this region is partly secreted from the endothelial
cells and partly from ultrafiltration of the blood that nourishes it. The CSF so produced is
circulated in the brain’s internal CSF circulatory system (Ventricular System), before
being drained away to the SAS to be exuded in the veinous flowstream.
The regulatory mechanism:
There are 2 main areas of contact between the cerebral vasculature (the network of
capillaries within the brain) and the brain – the first one is where the vasculature
permeates the cerebral cortex through a volume of interstitial fluid; and the second one is
where the vasculature interacts with the cerebrospinal fluid. The second interface exists
outside the cerebral cortex, at the level of the meninges (precisely, within the Sub
Arachnoid Space) as well as within the cortex, in the brain’s choroids plexus and
ventricular circulation network.
These 2 broad interfaces comprise respectively the BLOOD BRAIN BARRIER (BBB)
and the BLOOD CSF BARRIER (BCSFB). The word barrier in the nomenclature
essentially reflects that the endothelial cell lining of the capillary network in these regions
(the vasculature) provides high resistance to transport of solutes/solvents from the
bloodstream to the CSF or ISF. It is this enhanced resistance that keeps out unwanted
molecules from seeping into the brain. Since transport is primarily passive–diffusion
mediated this essentially implies low values of Ficksian diffusivity or mass transfer
coefficient as well as low values of membrane permeability.
THE INFIRMITY – MENINGITIS –
Innocuous levels of meningococcal bacteria are usually present in the upper respiratory
tract. The problem arises when these non-mobile gram-negative aerobic bacteria are
carried in the blood stream to the brain. The exact mode by which these microbes gain
entry into the brain has not yet been recognized. For my model, it has been assumed
that once inside the blood delivery system (vasculature) of the brain, it can gain
entry into the brain itself via mechanisms like endocytosis that do not require it to
overcome the aforestated restrictive blood brain and blood csf barriers. [2]
Once inside the brain they colonize and multiply particularly in the SAS, mediating the
release of pro-inflammatory and neutophil attractant cytokines, invoking a massive
neutrophil infiltration at the affected zones.[4] This is followed by an increase in BBB
and BCSFB permeability permitting potentially deadly chemicals released by the
accumulated neutrophils to find their way to the neuronal cells.
The reduced permeability also permits an osmotic influx of water into the brain, resulting
in increased Intracranial Pressure (vasogenic edema), which in its turn reduces cerebral
blood flow (cerebral ischemia) by vasoconstriction and microthrombosis. Thus apart
from necrotic and apoptotic chemicals, the cells are further stressed by increasing
physical pressure on them and Oxygen-Glucose Deprivation (OGD) [5] due to reduced
blood flow.(anoxia). [1,3]
Enhanced barrier permeability, vasogenic edema, cerebral ischemia and OGD are
the primary means by which this bacterium mediates massive neuronal necrosis and
apoptosis.
Neutrophil accumulation
Cytokine/ROS release
CELL
DEATH
Bacteria
Enhanced
permeability
of barriers
Intracerebral
accumulations
Ischemia
OGD
Increased
ICP
Figure 1 – The cell damaging mechanism of meningococcus bacteria
THE MODEL – (an overview)
As stated earlier, I have identified 3 primary regions of interest to my model, depending
on the regions of brain that the bacteria colonize and damage. These 3 regions are –



The SUB ARACHNOID SPACE and the BLOOD CSF BARRIER there
The BRAIN TISSUE + ISF COMPARTMENT and the BLOOD BRAIN
BARRIER there
The CHOROID PLEXUS+VENTICULAR SYSTEM and the BLOOD CSF
BARRIER there
I have modeled these 3 regions as three interconnected compartments each with a
microvasculature of its own. The microvasculature stems from the carotid arteries which
supply the brain and it proceeds from one compartment to the next and finally connects to
the veinous efflux line.
A schematic of the model is depicted below.
Figure 2 – Schematic Representation of the human brain
The labels are explained as follows –
MODEL COMPARTMENT
A
B
C
Arterial source
Veinous drainage
MODEL
COMPARTMENT
A
B
C
ANATOMICAL ANALOGUE in brain
Sub Arachnoid Space
Cerebral Cortex packed with brain neurons
Choroid plexus with ventricular system
Source of brain vasculature
Sink for brain vasculature
MODEL FLOWLINE
Arterial source  1
1X
12
2X
23
3X
3Y
4Y
ANATOMICAL
ANALOGUE
Vasculature in SAS
Veinous drainage from SAS
Vasculature in cortex+ISF
Veinous drainage from B
Vasculature in plexus
Veinous drainage from C
Filtrate CSF from plexus
vasculature
CSF produced in plexus
Label
Fcapillary
Fvein1
Fab
Fvein2
Fbc
Fvein3
Fcsf3
Fcsf4
YX
Veinous drainage of CSF
The labels in the last column indicate the variable (in volume units/time) that will be used
to represent the respective flows in subsequent equations - essentially the volumetric flow
rates.
Compartment A represents the CSF filled subarachnoid space. The flow-lines stand for
the vasculature or blood supply in the subarachnoid space. Thus we could interpret
compartment A along with the flow lines as “Fcapillary volumes of blood from the carotid
artery enter the subarachnoid space and at the point of exit, ramify into 2 branches – Fab
volumes go to the next compartment to constitute its vasculature while Fvein1 volumes is
drained out to the veinous drainage channel.
Similarly for Compartment B, the vasculature is constituted of Fab which at exit point
branches out into a venous stream Fvein2 and the vasculature for the next compartment.
Compartment B essentially stands for a packed mass of brain neurons embedded in a
volume of interstitial fluid or ISF.
Compartment C is essentially the CSF volume produced by choroid plexi and the
ventricular circulation system. The additional stream from 3  Y represents the volume
of CSF that is derived from the compartment’s microvasculature by cross-flow filtration,
while 4Y represents the CSF secreted by the endothelial lining of the cells here. The
final stream YX stands for the CSF that is absorbed into the veinous drainage network.
The residual blood in C after CSF filtration also passes into the veinous system as flow
line 3X.
THE MODEL EQUATIONS
In the subsequent sections I will attempt to develop the equations governing the model.
All the equations are developed in terms of algebraic symbols to prevent loss of
generality. For each set of equations the corresponding assumptions and their
justification have been stated.
1. MASS BALANCE FOR THE VARIOUS FLOW STREAMS
We begin our model development by considering a simple mass balance for the
individual brain compartments as well as an overall balance.
Assumptions:
1. The densities of blood and CSF are assumed nearly equal (sp.gr ~ 1.007) since both
are composed primarily of water
2. We in effect make a volume balance – this is not inaccurate however as the densities
of all inlet and outlet streams are considered same and unchanging.
3. The pressure inside the brain everywhere (i.e. in the CSF, ISF, vasculature) is
assumed to be at the Intracranial Pressure or ICP value. This would be necessary to
maintain all components in a state of homeostatic balance.
Fcsf
COMPARTMENT A:
Figure 3 – Vasculature in the subarachnoid space
Doing a mass balance around point 1,
Fartery  Fcap............(1)
Fcap  Fab  Fvein1....(2) (where Fcap = capillary  1; Fab = 12; Fvein1 = 1X)
The volumetric blood flow rate can be estimated using Bernoulli’s Equation.
1
1
Vartery 2  Pcap  Vcap 2
2
2

 Partery  Pcap  (Vcap 2  Vartery 2 )
2
Pcap  ICP
(refer assumption #3)
Partery 
Partery  Pcap  Pcpp
Vcap 
2
Pcpp  Vartery 2

 Fcap  Vcap * Acap.............(3)
where
 Acap represents the total cross sectional area of the sub-arachnoidal
microvasculature.
 Partery represents the arterial blood flow pressure
 Pcap represents the capillary (in SAS) flow pressure which is approximated to
ICP

Pcpp represents the cerebral perfusion pressure or the driving force for blood flow
to brain
COMPARTMENT B
Figure 4 – The brain tissue in ISF (compartment B)
Doing mass balance around point 2,
Fab  Fvein2  Fbc ……….(4) (where Fab represents 12; Fvein2 2X ; Fbc 23)
COMPARTMENT C :
Figure 5: Choroid Plexus + Ventricular CSF (compartment C)
Balancing around 3 and Y
Fbc  Fcsf 3  Fvein3.......(5)
Fcsf  Fcsf 3  Fcsf 4.......(6)
(where Fbc :23; Fcsf3:3Y; Fcsf4:4Y Fvein3:3X; Fcsf:YX)
The venous pressure can be related to flow rate Fbc using Hagen-Poiseulli Equation:
Fbc 
R 4 P
8L
where
 R = capillary radius
  =blood flow viscosity
 L = effective length of vasculature in compartment C
P  ICP  Pvein
 Pvein  ICP  P......(7)
where
 Pvein = pressure of blood in venous drainage
 ICP = intracranial pressure
2. TRANSPORT MECHANISM IN THE BRAIN
The second part of the model deals with the transport of solutes in the bloodstream into
the cerebral cortex across the BBB and BCSFB in a brain that’s already been infected by
meningitis bacterium. For evaluating transport phenomena in the brain, the following
assumptions were made:
1. The volume of each compartment is assumed to remain constant. This is relatively
valid as the brain enclosed inside the cranial box never has much scope for
volume expansion.
2. The volumetric flow-rates of the vasculature in each compartment are considered
constant.
3. For any compartment mass balance yields –
Accumulation rate of solute = rate of diffusion mediated entry – rate of bacterial
usage of solute – rate of host cell usage of solute.
It is evident that both accumulation rates and bacterial consumption rates are
functions of time, since bacterial growth rate is a function of time as well. In
addition bacterial growth rate is a function of accumulated substrate as well. To
simplify this unsteady state model we make a quasi-steady state assumption for
bacterial utilization of substrate:
“Bacteria do not use up substrate initially as it accumulates. Substrate utilization
commences only after substrate concentration has reached maximal levels of
accumulation.” This assumption is reasonable enough, because the bacterial
uptake of substrate usually is a lot faster than the restrictive entry of
solute/substrate across the diffusion barriers, necessitating a “waiting period” that
the bacteria spend for the substrate influx after having consumed the substrate
accumulated earlier. Also the maximal substrate concentration would give
bacteria the maximum growth rate (as predicted by Monod equation). It would
thus be optimal for bacterial multiplication to allow for substrate levels to reach
concentrations that permit it to multiply fastest than to utilize substrate as it
accumulates and grow at specific growth rates below the maximum. The “waiting
time” the bacteria spend for the substrate to accumulate can be regarded as
analogous to the “lag phase” in batch culture when bacteria adapt to their new
environment. In a way, the unsteadiness of the system is resolved by assuming the
existence of a batch-growth like lag phase each time substrate consumption leads
to substrate depletion. In this model, this assumption will be referred to as the
batch-growth like quasi-steady state assumption.
COMPARTMENT A:
The relevant assumptions are –
 Essentially we assume that the transfer of solute from the blood stream to the CSF
in compartment A is a diffusion phenomenon.
 In effect, both Ccap and Ca are variable concentrations,(refer equations) However
Ccap may be assumed constant. This is because owing to the barrier resistance
diffusion is restrictive. Hence at any instant concentration of solute in the CSF is
significantly lower than the concentration in the bloodstream. Also, not very
much diffusion can be expected to take place in the SAS as the primary objective
of blood transport is to nourish the brain cells in B and not the CSF in A. Hence
the concentration of solute in blood in A is assumed constant.
The ideal equation without quasi-steady state assumption would be –
dCa
Va *
 Ka.Sa.(Ccap  Ca)  f ( Xa)
dt
where
Va
dXa
f ( Xa)  (
)(
)
YX A / C A dt
where








Va = volume of compartment A
Ca = concentration of transported molecule in A
Ccap = concentration of transported molecule in ‘A’ blood supply
Ka = mass transfer coefficient for transport across the BCSFB in A
Sa = total surface area of sub-arachnoidal microvasculature
f(Xa) = function describing bacterial uptake of substrate.
Xa = bacterial cell concentration in A
YXa / Ca = amount of bacterial cell yield per unit concentration of substrate utilized
With the quasi-steady state assumption for bacterial utilization of substrate the equation
simplifies todCa
 Ka.Sa.(Ccap  Ca) …..(8)
dt
Boundary Condition:
 Ca=0 at t=0
Va *
COMPARTMENT B:
Since significant diffusion does occur here (across the BBB), the variation in
concentration along the horizontal direction cannot be neglected.
We use a shell balance approach to derive the concentration profile.



The total cerebral vasculature in this region is approximated by a single capillary
in the shape of a cylinder as done earlier.
Convective transport occurs along the z-direction
Diffusive transport occurs in the radial direction but only across the thickness ‘p’
of the blood brain barrier
The convective transport balance yields
Vab.(2Rdr ).dCab  Input  Output
where
 Vab = blood flow velocity in compartment B = Fab/(total cross sectional area of
B vasculature)
 dCab = concentration gradient across the elemental shell in the z-direction
 R = capillary radius
 dr = differential thickness of capillary BBB varying from 0 to the capillary
thickness p.
(Note: the variable ‘dr’ varies over the BBB thickness from 0 to p, where p is the
capillary BBB thickness and not the capillary radius.)
The diffusive transport equation yields
(2Rdz ) Nab |0 (2Rdz ) Nab |0 r  Input  Output
Nab   Db




Cab
r
Nab = flux of solute across the BBB
Db = Diffusivity of solute across the BBB
Cab = concentration of solute in the B vasculature = f(r,z)
∆r is the elemental thickness of BBB
Equating the input-output values,
Vab.(2Rdr ).dCab  (2Rdz ) Nab | R (2Rdz ) Nab | R  r
Vab
Cab  ( Nab)

z
r
Vab
Cab
 2 Cab
 Db
...............(9)
z
r 2
Boundary Conditions –
 At r =0, Cab=Csat where Csat is the maximal concentration of solute in the blood
stream side of the BBB. The boundary condition arises because the rate of
diffusion across the BBB is significantly slow. Hence a saturation concentration
of molecules can always accumulate at the blood stream side of the BBB in the
time it takes to diffuse into the compartment B.
 At z=0, Cab = Ccap (where z=0 is the plane where A and B compartments meet)
 For the 3rd boundary condition, we can either assume a known value of Cab at
z=L where L is the total length of compartment. The value can be estimated from
the aforestated mass balance equations.
Cab
 Or we can assume
=0 at r=0. The latter recognizes the assumption that
r
within the capillary for a particular z, the concentration shows no variation in the
radial direction except within ‘p’ units thick blood brain barrier. Thus within the
capillary (0<r<R, for a constant z) no diffusion occurs.
From blood the solute enters the ISF in B only at the r=p surface, since diffusion occurs
only across the surface area of the capillary-ISF barrier.
zL
Rate 

z 0
where





Db (.
Cab
| r  p )dz (2 [ R  p]) …………….(10)
r
Rate= implies quantity of solute transferred to B per unit time from its vasculature
2π(R+p)dz = elemental surface area of vasculature
L = length of compartment B
p = capillary BBB thickness
A part of the solute that enters is used up for host cell maintenance while the rest
accumulates in ‘B’. The accumulated solute becomes substrate for the bacteria
colonized here. As earlier even here we consider the quasi-steady state
assumption for substrate utilization by bacteria.
In the absence of quasi-steady state assumption,
Doing a mass balance for ‘B’, the accumulation rate can be predicted by the equation –
Vb
Cb
 rate  f (cell )  f ( Xb) …………..(11)
t
where




the LHS represents the rate of accumulation of solute in ‘B’
f(cell) is a function defining the rate of consumption of resources by host cell
Xb=bacterial concentration in B
f(Xb) = rate of substrate utilization by bacteria.
Vb
dXb
)(
)
= (
YX B / CB dt
With pseudo steady assumption in place the last term of (11) RHS is 0 till Cb reaches
maximum.
A significant cellular uptake occurs by receptor mediated mechanisms which follows
saturation kinetics very similar to M/M kinetics for enzymes. Especially for Glucose (an
essential nutrient for the brain and a preferred substrate for most bacteria) a significant
portion of cellular intake occurs via the GLUT1 transporter following kinetics very
similar to M/M kinetics. [6]
For such systems f(cell) would assume the form
f (cell ) 
V max .Cb
.Vb
Km  Cb
Where
 Vmax is maximum rate of cellular uptake
 Cb is concentration of solute in B
 Km is the concentration of solute at half the maximal uptake rate.
The actual equation would be
zL
Vb
Cb
V max .Cb
Vb
dXb
Cab
.Vb - (
=  Db (.
)(
)
|r  p )dz (2 [ R  p]) t z 0
Km  Cb
YX B / CB dt
r
while with quasi-steady state assumption we could neglect the last term.
COMPARTMENT C:
In compartment the CSF as depicted in the model schematic (refer figure 2) comes from
2 sources – namely a secretory CSF from the ependymal and ventricular cells and CSF
from cross-filtration of blood in the plexus (compartment C) vasculature [7].
Keeping this in mind the following assumptions were made:
 The diffusion of solute molecules across the blood CSF barrier here is neglected –
as it is assumed to offer a much higher resistance than the alternate transport path
available to the solute molecules – the cross filtration route. Thus the major


transport in compartment C is believed to stem from the cross flow of CSF from
the blood stream.
The venous pressure is assumed constant
The blood inside compartment C vasculature is assumed to be at a constant
pressure very close to the Intracranial Pressure (ICP)
Thus the rate of solute accumulation in C would be given by :
Vc
dCc Pvein  ICP
Vc dXc

(
)
………(12)
dt
Rc
YX c / Cc dt
Boundary condition,
 Cc = 0 at t=0
Where
 Cc = concentration of solute in compartment C
 Pvein = venous pressure as estimated from equation (7)
 Rc = resistance to mass transfer offered in C between blood and CSF
In quasi-steady state this reduces to
dCc Pvein  ICP
Vc

dt
Rc
3. BACTERIAL GROWTH RATE IN THE COLONIZED AREAS OF THE
BRAIN
As stated earlier the major bacterial colonization is initiated in the subarachnoid space
(or compartment A), from where they mediate neuronal necrosis and apoptosis.
To prevent loss of generality, we consider bacteria to have colonized the A and B
compartments and find out their growth (multiplication) rate in each compartment.
The assumptions made are:
 Bacteria rely on host nutrition for its own growth. [2] However unlike a virus, this
exploitation is not potent enough to cause significant cell death damage.
 Bacteria invade in concentrations not high enough to mediate immediate damage;
because the meningococcal bacteria are non-mobile and hence not carried in
significant amounts by the blood stream.
 Following colonization, bacteria begin growing and mediate cell damage at a rate
directly proportional to their growth rate.
 Bacteria, for multiplication need both nutrients and oxygen. However it would be
a reasonable assumption to consider the nutrient and not oxygen to be a limiting
reactant, because the brain usually has a sufficient oxygen supply; it is the
nutrients whose concentrations are regulated by its various barriers.
 Batch-growth like Quasi-steady state assumption is not considered here as we do
not intend to solve the resulting complex equation.
COMPARTMENT A:
The bacterial growth rate is given by,
dXa
dCa
 Yx A / c A .
dt
dt ……………….(13)
boundary condition,
 Xa = 0 at t=0
Where
 Xa = concentration of bacteria in A (in cfu per unit volume)
 dCa/dt = accumulation rate of substrate in A
COMPARTMENT B:
In compartment B the solute (nutrient) diffusing across the BBB is used up for host cell
maintenance as well as bacterial growth, since this is the compartment harboring brain
tissue.
The bacterial growth rate here is given by,
dXb
dCb
 Yx B / cB .
…………….(14)
dt
dt
Boundary condition
 Xb=0 at t=0


.
Xb= concentration of bacteria in B (in cfu per unit volume)
dCb/dt = substrate accumulation rate in B
COMPARTMENT C:
Using the same approach, we obtain the following equation for bacterial growth rate,
dXc
dCc
 YxC / cC .
……………………(15)
dt
dt
Boundary condition Xc=0 at t=0
where


Xc= concentration of bacteria in C (in cfu per unit volume)
dCc/dt = substrate accumulation rate in C
OUR OBJECTIVE EQUATION – THE OVERALL EXPRESSION FOR
BACTERIAL CELL GROWTH IN MENINGITIS INFECTED BRAIN
The overall growth rate of meningitis bacteria is expressed as –
d
where  is the overall bacterial concentration at an instant.
dt
Assumption:
 YXi / Ci s are equal for i= A,B,C and equal to Yx / s
 As in previous we do not consider quasi-steady state assumption here since we
do not intend to solve the resulting equation.
Thus,
d dXa dXb dXc



dt
dt
dt
dt
d
dCa dCb dCc

 Yx / s.(


)
dt
dt
dt
dt
Substituting the expressions obtained for substrate accumulation in the various
compartments,
dCa Ka.Sa
1 dX A

.(Ccap  Ca) 
dt
Va
Yx / s dt
zL
Cb
1
Cab
1 dX B
 ( )[  Db (.
|r  R  p )dz (2 [ R  p])  f (cell )] 
t
Vb z 0
r
Yx / s dt
dCc Pvein  ICP
1 dX c


dt
Vc.Rc
Yx / s dt
substituting the above expressions in the expression for overall bacterial growth,
zL
d Yx / s KaSa
1
Cab
Pvein  ICP

.{
(Ccap  Ca)  ( )[  Db (.
|r  p )dz (2 [ R  p])  f (cell )] 
}.........(16)
dt
2
Va
Vb z 0
r
Vc.Rc
The above equation bases itself on the following assumption

While deriving the above expression we have neglected any effect of factors that
mediate bacterial cell death – which would essentially involve host defense
mechanisms and nutrient limitation. The former is negligible because the stable
pneumococcal cell wall (PCW) and capsular layers of these bacteria make them
sufficiently resistant to immune attacks. The second consideration is omitted
because nutrient limitation is a highly unlikely mechanism that the body would
adopt to attenuate bacterial growth as that would jeopardize its self cells as well.
What actually causes bacterial population depletion at a later stage is toxins like
pneumolysin that lyse these cells. However the products released on lysis (e.g.
PCW) usually are capable of stimulating host cell damage in a manner similar to
the bacterial cells. Hence the derived equation essentially models the actively
growing phase of the bacteria.
AN ATTEMPT AT A GENERALIZED SOLUTION TO THE DERIVED MODEL
EQUATIONS –
We attempt solutions only for compartments A and B because that is where the maximum
growth and destructive effects are centred.
The equations derived so far call for intensive Mathematics if we are to head towards a
solution. What makes solving more difficult is the fact that exact values of many of the
constant terms (e.g., capillary cross section area, total surface area of vasculature,
permeability coefficients, diffusivities, mass transfer resistance, f(cell) parameters, etc)
are difficult to collect from standard literature and journals. However it may be pointed
out that all the constants can be evaluated with relatively high accuracies in standard in
vitro models. (Refer Appendix A for brief discussion on approximate values of
parameters that were used in solving the differential equations.)
Thus, solutions to the differential equations have been attempted in terms of the constants
themselves, to prevent loss of generality.
The solutions to the set of differentials governing the accumulation of solutes in the
various compartments were derived using Mathematica 5.1.
Note: While predicting the concentration profiles, we donot consider the decrease in
concentration due to bacterial consumption. All profiles are derived assuming the
batch-like pseudo or quasi-steady state criterion.
Concentration Profile in Compartment A (SAS)
dCa
Va *
 Ka.Sa.(Ccap  Ca)
dt
Ka.Sa.t
Ca(t )  Ccap(1  exp( 
))
Va
Assuming approximate values for the constant terms, it has been attempted to plot a
pattern of variation of substrate concentration with time in the first compartment. The
graphs were plotted using Matlab 7.0.
Figure 5 :Variation of Concentration in A vs time following bacterial
colonization
Concentration Profile in Compartment B (parenchyma+ISF)
Cab
 2 Cab
Vab
 Db
z
r 2
The above differential equation is similar in form to the one dimensional heat equation.
However a closer look reveals that the boundary conditions in this case are nonhomogeneous implying that any possible solution can be arrived at only by CFD tools
like finite difference analysis. However for the purpose of plotting we can approximate
diffusion in B by an equation similar in form to the one we wrote for A –
N B  Db
(Csat  C B )
l
where
 NB is the molar flux from bloodstream to the compartment
 Db is Ficksian diffusivity for the solute across BBB
 Csat is maximum concentration of solute at the capillary wall. Since diffusion
across capillary wall is a lot slower than dispersion of solute molecules within the
capillary, we can assume that the wall always remains saturated with maximal
concentration of the diffusing solute.


l is BBB thickness
CB is the (varying) concentration of solute in B
Keeping in mind cell consumption of solute resource we derive the equation for
accumulation using mass balance as earlier
dC
Vb. b  ( Ac.N B )  f (cell ).Vb
dt
V max .C b
f (cell ) 
Km  C b
Vb.
Vb.
dC b
Cb
 ( Ac.N B )  V max .Vb
dt
Km  C b
dC b
(Csat  C b )
Cb
 ( Ac.D)
 V max .Vb
dt
l
Km  Cb

for cases where Km>>Cb ,
the solution is obtained as –

A . D V max
{( c
)
}. t

Vb .l
Km
1

e
Cb(t )  
 Ac .D V max
 V .l  Km
b



. Ac .D .Ccap
 Vb .l


The plot reveals the following similar variation pattern –
Figure 6 :Variation of concentration in B with time following bacterial colonization
DISCUSSION (Interpreting the graphs)
Both the graphs predict concentration profiles in the human brain following
meningococcal bacterial colonization. The following key points are to be kept in mind
while studying the plots –
The graphs only depict that period of time when the bacteria are in the lag-phase (of this
quasi-steady state ‘batch growth’ like system) – and thus are in a period of “waiting” for
the accumulation to reach concentrations that are sufficient to support its multiplication.
As soon as the accumulated substrate reaches that level further accumulation stops. (as
shown in the flat portions of the plot). This ‘steadying’ is explained as follows –
 Compartment A: Further diffusion is restricted by the increasing solute
concentration in A which disrupts the concentration gradient that constituted the
driving force of diffusion. However this should not be taken to imply that the CSF
in SAS reaches as state of static equilibrium. What actually happens in the real
world model is that the CSF is in constant circulation, with the CSF in SAS being
replaced by fresh sources of CSF from certain ependymal cells in the brain. To
prevent complications in the model, this internal regeneration was neglected,
primarily because the CSF circulation occurs at an active but greatly slow rate.
However it should be kept in mind that this CSF in compartment B doesnot reach
static equilibrium and then stays there. In effect it is in a state of active circulation
and regeneration that ensures sustained transport of solutes from blood to CSF via
BCSFB
 Compartment B: The steadying of concentration here is more easily explained as
the state when diffusion mediated entry rate equals the host cell solute utilization
rate resulting in no net accumulation.
Now we consider bacterial growth (multiplication):
The pattern of bacterial utilization of substrate that the model is based on can be
summarized as –
 Bacteria utilize the diffused solute molecules in SAS and ISF as substrate for
growth
 The utilization for multiplication commences only after substrate levels reach a
maximum stable value as this would allow optimal growth.
 Bacterial utilization of substrate is very fast compared to the barrier restricted
diffusive influx – thus as soon as substrate reaches its maximal accumulation
value, it is almost instantaneously dragged down to a minimal value.
 To make up for this sudden drop in concentration in SAS/ISF the BBB and
BCSFB accommodate by permitting an increased influx of solute to make up for
the sudden establishment of the diffusion concentration gradient. This is
manifested as increase in BBB permeability – a crucial step in the cascade of
events leading to massive neuronal damage.
 The newly grown bacteria again wait in a “batch growth-like lag phase” for the
accumulating substrate to reach a maximal – this time at a faster rate owing to

increased permeability – and once more begin multiplying by rapid substrate
utilization.
With time bacterial growth can not cope up with the diffusive flux as permeability
keeps on increasing. Hence instead of instantaneous utilization of entire substrate,
bacteria are able only to partially utilize the accumulated solutes, and fail to drag
down the hiked concentrations to 0. This eventually results in steady
accumulation of increasing solute/water concentrations in the cerebral tissue.
Thus the model successfully explains another significant step in the meningitis
cascade – the increase in Intracranial Pressure and edema due to the steady
accumulation. The increased ICP reduces blood flow (ischemia) – as is evident
from the mass balance equations derived earlier and hence oxygen supply to brain
(anoxia).
The proposed pattern of bacterial growth and bacteria induced cell damage has been
modeled with Matlab 7.0. Refer Appendix B for the simulation code.
SIMULATION MODELLING OF BACTERIAL GROWTH AND SUBSTRATE
ACCUMULATION PROFILES WITH BATCH-LIKE QUASI-STEADY STATE
ASSUMPTION:
We examine a few plots depicting bacterial growth and substrate utilization in the quasisteady state situation. The plots are the result of the simulated model code.
Simulated Bacterial Growth Curve:
Figure 7:Bacterial growth profile
The pattern is in accordance with our assumption – that bacterial growth in brain is
composed of 2 essential parts –



A dormancy period while substrate accumulation takes up substrate levels to
concentrations that will permit best growth
An exponential growth period rapidly using up the substrate.
With time the exponential phase is seen to become steeper indicating that bacteria
grow at a faster rate as increasing concentrations of substrate accumulate.
Simulated Substrate Accumulation Curves:
Figure 8:Concentration variation under the influence of bacterial growth in brain
For analysis, we consider one of the four cycles shown in the graph and divide it into the
following sections –
1. The positive slope part –This is the part depicting substrate accumulation in the
absence of bacterial utilization. In other words, in accordance with our quasisteady state bacterial growth, this is the rate at which bacteria allow the substrate
to accumulate each time before commencing utilization.
2. The flat part – Represents the maximal substrate concentration – time is now
favorable for bacteria to begin multiplying.
3. The steep negative slope part –Represents rapid exhaustion of substrate due to
bacterial consumption at rates many times faster than the barrier restricted
diffusion.Thus the accumulated substrate is used up and the cycle repeats itself
giving the next peak.
Now consider the effect of time – as we observe for a larger number of cycles –
Figure 9:Concentration profile – on a larger time range
As we increase the time range two things can be clearly observed –
 The flatness of the peaks reduces as the increasing permeability greatly
enhances diffusion rates, permitting a rapid influx of solute molecules.
 Following the first 3 peaks, with each subsequent peak, there is a little substrate
accumulation – a direct consequence of the permeability enhancement which
makes the bacteria unable to consume the excess substrate as fast as it is allowed
in by steadily increasing diffusion rates. Thus the newly multiplied bacteria enter
the “batch-like lag phase” even before the excess substrate in the brain is
exhausted. This accumulation leads to increased ICP and edema.
This proposed pattern of enhanced diffusion making complete substrate utilization by
bacteria stressful should not be confused with washout phenomenon of a chemostat.
What actually happens in washout is that substrate is supplied at too fast a rate to
allow cell growth. What happens here is that the solute accumulates in too large a
quantity for complete utilization. Hence there is only a partial exhaustion while the
rest accumulates.
Now with further increase in time,
Figure10 – accumulation with time
The concentration of accumulating substrate shows a steady increase.
Figure 11 – accumulation with time
The black arrow indicated the steady rise in solute concentration with time in the brain.
Since the volume of the compartments remains approximately constant this results in
increasing intracranial pressure and cerebral edema – ultimately mediating direct physical
cell damage as well as cell death due to vasoconstriction and OGD (oxygen glucose
deprivation)
CONCLUSION:
The mathematical model serves to predict the rate of growth of bacteria in the brain in
meningitis infected patients.(refer equation 16) The task was made difficult by the
unsteady nature of the substrate-limited bacterial growth in the affected areas of the brain,
the substrate for bacterial multiplication being in an unsteady flux increasing with time.
To complicate the mathematics, the rate of substrate permeation also increases with time
as the bacteria work to enhance capillary membrane permeability.
To simplify the modeling I had to consider either of two options – assume chemostat like
growth conditions neglecting substrate accumulation and assuming steady state at all
times. This was not a very viable option as uncontrolled substrate permeation is a key
event in the meningitis cascade. The alternate option was to consider batch-like growth
where the bacterial growth and substrate accumulation are essentially events disjoint in
time. Essentially the bacteria fluctuate between a dormant phase and a steady growth
phase. The dormant phase corresponding to the time lag between initiation of bacterial
growth and substrate concentration build-up to maximal levels. Though this “batch-like
quasi-steady state growth” was initially a simplifying assumption, subsequent modeling
using a MATLAB simulation code was seen to very well explain the cell-damaging
cascade of events that the meningitis bacteria spur.(refer DISCUSSION). This tempted
me to believe that this hypothesized pattern of growth might effectively model the real
world growth pattern of the meningitis bacteria that is still an intriguing phenomenon, as
well as explain the curious yet deadly cascade of events it unleashes.
APPENDIX A:
The constants used for plotting the substrate accumulation profiles are listed below.
Ka = 1.8*10-4 cm2/min
Va = 0.87 * 500 cm3
Sa = 4685 cm2
Ccap = 5.5 mM
The 0.87 factor in the 2nd parameter comes from the fact that the SAS produces 87 % of
the total CSF in brain, and at any instant the total CSF volume in brain is ~500 cm3.
Hence I have calculated the SAS volume as above.
The vasculature surface area is calculated by multiplying the brain density (~1400g/1300
cm3) with the calculated SAS volume to obtain a rough idea of the compartment A mass.
The result is then multiplied by 100 as approximately 100 cm2 of capillary tissue is
believed to exist per gram of brain mass. [9]
For the 2nd plot the parameters [6] are
D=0.673*10-5 cm2
Km = 17mM ,Vmax = .53 m/s (GLUT1 transport,Johnson et al,1990)
Vb = 1256.5 cm3
Capillary surface area = 135,315.38 cm2 (calculated as in previous)
BBB thickness = 2*10-5 cm [8]
APPENDIX B:
The following are the simulation codes.
“bacterialGrowth.m”
1.
2.
3.
% function to study accumulation rate & bacterial growth in B
% k is a variable parameter which incorporates the effect of increased
% permeabiiity that the bacteria induce
4.
5.
6.
7.
8.
function c=bacterialGrowth(k,c,t,cmin);
c=cmin+(1-exp((-0.3-1.5*k)*t))*(8.25*k)/(.3+1.5*k);
% the above function is used in another program that simulates the complete cycle
% comprising accumulation,bacterial growth,rapid depletion of substrate and increased
% accumulation with time due to loss in diffusion resistance
The above program is the same as the one used to profile the accumulation pattern in B.
However this function incorporates an extra ‘k’ factor – that increases with time and is a
mathematical representation of the enhanced permeability that the bacteria result in.
“Bac_growth.m”
1. % program to pattern bacterial growth
2. a=1;
3. % a is the index variable for the array of concentration values at
4. % different times
5. k=0;
6. % k is a mathematical representation of the increase in permeability
7.
8.
9.
10.
11.
init_value=0
% the outer loop keeps a count of the number of total cycles - each cycle
% comprising bacterial "batch like lag phase" - while it waits for
% substrate to build up
% exponential growth
12. for i=0:5
13. % The inner loop keeps a track of the time taken for substrate to reach
14. % critical concentrations when bacteria can commence growth
15. for t=0:8
16. y(a)=bacterialGrowth(k,0,t,init_value);
17. a=a+1;
18. end
19. k=k+.2;
20. % The above statement tries to incorporate the qualitative rise in
21. % growth rate due to enhanced permeability which leads to greater
22. % substrate availability.
23.
24.
25.
26.
init_value=y(a-1);
end
% the init_value specifies the pre-existing concentrations of bacteria
% before each cycle
27. plot(0:a-2,y)
“CellDamage.m”
1.
program to pattern substrate accumulation profiles
2.
3.
4.
a=1;
% a is the index variable for the array of concentration values at
% different times
5.
6.
7.
8.
9.
10.
11.
12.
13.
k=1;
% k is a mathematical representation of the increase in permeability
% subs_redxn is a mathematical representation of the lowering in substrate
% concentration each time due to bacterial growth.
subs_redxn=0;
% the outer loop keeps a count of the number of total cycles - each cycle
% comprising substrate accumulation with dormant bacteria
% rapid consumption of substrate by bacteria
% Enhanced diffusion to make up for sudden establishment of conc gradient
14. for i=0:30
15. % The inner loop keeps a track of the time taken for substrate to reach
16. % critical concentrations before being reduced instantly by rapid
17. % bacterial growth
18. for t=0:8
19. y(a)=bacterialGrowth(k,0,t,subs_redxn);
20. if (y(a)>=max(bacterialGrowth(k,0,[0:8],subs_redxn)))
21. y(a)=subs_redxn;
22. end
23. a=a+1;
24. end
25. k=k+4;
26. % The above statement tries to incorporate the qualitative rise in
27. % permeability by an increase in k by a hypothetical constant number
28. if (i>=2)
29. subs_redxn=subs_redxn+1;
30. end
31. % the above condition incorporates the prediction that with time the
32. % reduced permeability results in a diffusion rate too fast for bacteria
33. % to utilize at once. The above equation incorporates the gradual
34. % accumulation of excess solutes+water in brain with time.
35. end
36. pot(0:a-2,y);