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 Bacteria DEATH

Enhanced Intracerebral permeabilit Ischemia OGD y of barriers accumulation s

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 ANATOMICAL ANALOGUE in brain A Sub Arachnoid Space B Cerebral Cortex packed with brain neurons C Choroid plexus with ventricular system Arterial source Source of brain vasculature Veinous drainage Sink for brain vasculature

MODEL MODEL FLOWLINE ANATOMICAL Label COMPARTMENT ANALOGUE

A Arterial source  1 Vasculature in SAS Fcapillary

1  X Veinous drainage from SAS Fvein1

B 1  2 Vasculature in cortex+ISF Fab

2  X Veinous drainage from B Fvein2

C 2  3 Vasculature in plexus Fbc

3  X Veinous drainage from C Fvein3

3  Y Filtrate CSF from plexus Fcsf3 vasculature

4  Y CSF produced in plexus Fcsf4 Y  X Veinous drainage of CSF Fcsf

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. 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 Partery  Vartery 2  Pcap  Vcap 2 2 2  Partery  Pcap  (Vcap 2 Vartery 2 ) 2 Pcap  ICP (refer assumption #3) Partery  Pcap  Pcpp 2 Vcap  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: R 4 P Fbc  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)  ( )( ) Y dt X A /C A 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 to-

dCa Va*  Ka.Sa.(Ccap  Ca) …..(8) dt Boundary Condition:  Ca=0 at t=0

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 Cab Nab  D b 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 Cab (Nab) Vab  z r

Cab  2 Cab Vab  D ...... (9) z b 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

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 Cab Rate  D (. | )dz(2[R  p])  b r p …………….(10) z0 r where  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 – Cb Vb  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 ( )( ) = Y dt X B /CB

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

V max.Cb f (cell)  .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 Cb zL Cab V max.Cb Vb dXb Vb = D (. | )dz(2[R  p]) - .Vb -( )( )  b r p Y dt t z0 r Km  Cb X B /CB 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 :

dCc Pvein  ICP Vc dXc Vc   ( ) dt Rc Y dt ………(12) X c /Cc 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 / c . dt A A 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 / c . …………….(14) dt B B 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  Yx / c . ……………………(15) dt C C 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  .(Ccap  Ca)  A dt Va Yx / s dt Cb 1 zL Cab 1 dX  ( )[ D (. | )dz(2[R  p])  f (cell)] B  b rR p 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,

d Yx / s KaSa 1 zL Cab Pvein  ICP  .{ (Ccap  Ca)  ( )[ D (. | )dz(2[R  p])  f (cell)] }...... (16)  b r p 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  2Cab Vab  D z b 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 non- homogeneous 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 –

(Csat  CB ) N B  Db 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 )  f (cell).Vb dt B V max.C f (cell)  b Km  Cb

dCb Cb Vb.  (Ac.N B ) V max.Vb dt Km  Cb dC (Csat  C ) C Vb. b  (Ac.D) b V max.Vb b dt l Km  Cb 

for cases where Km>>Cb , the solution is obtained as –

 A .D V max   {( c ) }.t  1 e Vb .l Km A .D Cb(t)   . c .Ccap  Ac .D V max  V .l    b  Vb .l Km  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 quasi- steady 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 quasi- steady 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. -4 2 Ka = 1.8*10 cm /min 3 Va = 0.87 * 500 cm 2 Sa = 4685 cm 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) 3 Vb = 1256.5 cm 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. % function to study accumulation rate & bacterial growth in B 2. % k is a variable parameter which incorporates the effect of increased 3. % permeabiiity that the bacteria induce 4. function c=bacterialGrowth(k,c,t,cmin); 5. c=cmin+(1-exp((-0.3-1.5*k)*t))*(8.25*k)/(.3+1.5*k); 6. % the above function is used in another program that simulates the complete cycle 7. % comprising accumulation,bacterial growth,rapid depletion of substrate and increased 8. % 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. init_value=0 8. % the outer loop keeps a count of the number of total cycles - each cycle 9. % comprising bacterial "batch like lag phase" - while it waits for 10.% substrate to build up 11.% 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.init_value=y(a-1); 24.end 25.% the init_value specifies the pre-existing concentrations of bacteria 26.% before each cycle 27.plot(0:a-2,y)

“CellDamage.m” 1. program to pattern substrate accumulation profiles 2. a=1; 3. % a is the index variable for the array of concentration values at 4. % different times 5. k=1; 6. % k is a mathematical representation of the increase in permeability 7. % subs_redxn is a mathematical representation of the lowering in substrate 8. % concentration each time due to bacterial growth. 9. subs_redxn=0; 10.% the outer loop keeps a count of the number of total cycles - each cycle 11.% comprising substrate accumulation with dormant bacteria 12.% rapid consumption of substrate by bacteria 13.% 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);