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Compartment Models Displaying Lyme Disease Symptom Cycles*

by Joachim Gruber
An immune system ("oscillatory immune system") with the following two properties develops the above analyzed symptom flare cycles, i.e. self-organized oscillations between a symptom free and an unwell stage. These oscillations are a well known property of general feedback control systems without sufficient damping (Ball P, 1999). Oscillating immune responses have been observed in non-Lyme cases and have been modeled, the models being used to optimally direct the antibiotic intervention in a similar fashion as is done in this paper (in chronologcal order: Dibrov BF et al. 1976, 1978, Smirnova OA 1991, De Boer RJ et al. 1993, Muraille E et al. 1996, McKenzie FE, Bossert WH 1997, see also literature surveys 1 and 2).

  1. a delayed immune response.

    The immune system responds to what I will abbreviate as toxins such as

    • Borrelia burgdorferi (Bb),
    • or immune system stimulating membranous material from the outer surface of Bb, belonging to the class of thymus-independent antigens of type 1 (TI-1 antigens), like
      • perhaps some outer surface proteins (lipoproteins, Osp) and
      • lipopolysaccharides, LPS (Coyle 1997, LPS in B. burgdorferi - literature),
      • Bb nucleic acids,
      • other fragments from killed Bb or toxins released from these,
    e.g. by producing cytokines ( Beck et al. 1986 Ma et al. 1993, Tai et al. 1994, Sellati et al. 1996, Frieling et al. 1997, Burns et al. 1998, Giambartolomei et al. 1998, Straubinger et al. 1998, Zhang et al. 1998, see also the result of a Medline search). It is the cytokine levels that correlate with clinical responses ( Damas et al., 1992, Frieling et al., 1995, van Deuren et al., 1995).

    Via molecular mimicry, also autoimmune processes can be triggered by Bb proteins (Sigal 1997, Sigal and Williams 1997, Hemmer et al.1999, Klempner et al. 1999, an ongoing study headed by Adriana Marques, Laboratory of Clinical Investigation, National Institute of Allergy and Infectious Diseases, reported in NIAID's News). T-cell subpopulations (of short-lived T-cells) responsible for autoimmune processes might persist as long as sufficient levels of such proteins are present in the host (Kuby, chapter 12, S. 305). The existence of such autoimmune processes could bring about a decoupling of infection and inflammation, both in space and time. If such processes support e.g. symptom cycles, their period may differ from the periods of cycles triggered by infectious processes. The following description refers to an immune response directed against an infection.

    The immune response should be visualized as being twofold:

    1. clinical response, predominantly inflammatory. This first reaction happens within hours of the stimulation (Kuby, Kapitel 12, p 296, und Kapitel 13, 317 - 319).
    2. start-up of the toxin elimination process, apparently by the humoral response (Hu and Klempner, 1997). This follows the inflammatory response.
    Both reponses start some ("lag") times after the toxin levels exceeded their specific tolerance thresholds.

  2. too early an end of the immune response.

    The immune response ends when

    • the toxin concentration has fallen below a threshold and enough time has elapsed since then for the immune system to relax, or
    • sufficient time has elapsed for the toxin to disappear into a niche, e.g. by invading a site "invisible" to the immune system or -if the pathogen is an active Bb population- by changing its surface through antigenic variation.

These two steps are combined into a feedback control process aiming at the elimination of the toxin. Unlike with many other infections, the incubation time of the toxins is so short (i.e. some hours, like in viral influenza) that the immune system's memory is irrelevant (pp. 202, 447 in Kuby, 1997). Thus, steps 1 and 2 will be repeated in much the same form as long as the niches release new toxins into compartments under immune system surveillance. The feedback control system is locked into undamped oscillations.

As is illustrated in Fig. 10, the basic building blocks of the immune response model are

Simplest compartment system displaying oscillations seen in the symptom log

Fig. 10: A simple compartment system and an immune system control scheme that produces oscillations between an inflamed state and a symptom free state.

Summary of immune response model:

Thus, the compartment model

Sections V. 1 and V. 2 will give simple examples of possible feedback control cycles (cycles of type 1). The mechanism driving the cycles in the absence of antibiotics are different from the one responsible for cycles under the influence of antibiotics.

V. 1 No antibiotics present

Fig. 11 shows a compartment model and the symptom cycles produced by an oscillatory immune system. The concentration C(t) of the substance invoking immune response is assumed to be proportional to the Bb concentration.

Structure of flare cycles in absence of antibiotic

Fig. 11: Schematic of flare cycles driven by oscillatory immune response fBb(CBb(t), t), where

The immune response is calculated with the compartment model shown in top part of figure (with a program written in Mathematica): The mathematical model is displayed below the box model representation.

Note the logarithmic concentration scale in diagrams for C(t): A straight line up (down) represents exponential growth (decay).
Immune system always starts up (f = 1) when Bb concentration has reached a concentration C1. Thus, the immune system being triggered by Bb concentration, always lags behind Bb growth.
Immune system always shuts down (f = 0) at Bb concentration C2, i.e. before all Bb have been eliminated. To simplify the figure, thresholds C1 and C2 have been assumed to coincide.
f has been chosen symptom specific, assuming that immune system has localized properties. f's are chosen such that logs of symptoms 7 and 12 are reproduced (see symptom logs placed at level C2).
Data used in calculations for illustration purposes

The phases of a flare cycle are:

  1. Bb population grows unnoticed by immune system.
  2. Immune system selects suitable antibodies (lag phase).
  3. Immune system eliminates Bb until Bb population disappears from its sight.
  4. The new Bb population grows unnoticed by immune system (phase 1, again). Each zig-zag cycle represents a Bb population that "catches the immune system by surprise".

In Fig. 11 we have fitted the compartment model to the symptom cycles by allowing a variation of the location of the peaks of the immune system switching function fBb(CBb(t), t), while keeping the Bb generation time TBb and the elimination half life TIBb fixed (thus the widths of the peaks are constant). This results in shifting fixed zig-zag segments (one branch going up the other going down) around. We did not succeed fitting the data of a symptom log by doing the reverse, i.e. keeping the f-curve fixed while adjusting the slopes of the individual zig-zag branches. Thus, it seems that the times when the immune system loses track of a Bb population and starts seeing the next one are variable.

The geometry of the curves in Fig. 11 lets us see the following properties of symptom cycles before antibiotic treatment:

During antibiotic treatment the spirochete population growth is at most proportional to time. One can see this, when looking at the symptom log, as will be demonstrated with the following model.

V. 2 Presence of antibiotics

Fig. 12 shows the properties of a compartment system applicable when a cell wall antibiotic is present. The system consists of three coupled parts:
  1. a source r(t) of Bb. It resides in a niche that shields it from the antibiotic. This source feeds
  2. the pool of spirochetes, concentration of Bb is CBb(t). The pool is exposed to the cell wall antibiotic, which creates Bb fragments whenever a spirochete enters its cell division phase. This feeds
  3. the pool of Bb fragments, the concentration of which is CF(t).
If the source r(t) contains only Bb fragments, it feeds the fragment compartment directly, i.e the Bb pool in Fig. 12 is missing.

Structure of flare cycles in presence of antibiotic

Fig. 12: Concentration CBb(t) of Bb population outside niche (dashed line) and CF(t) of Bb fragments (heavy line) resulting from a Bb source r(t). Concentrations are calculated with compartment model shown in top of figure.

Like in Fig. 11, f is the immune switching function, while C1 and C2 are the immune response start up and shut down thresholds, respectively.

Superimposed on the concentration curves in the upper diagram at level C2 is a section of the symptom log of symptom 7, i.e. the vertical series of dots for symptom 7 (Light Hypersentisitivity) between day 272 and 292 in Fig. 1.

In the case depicted in Fig. 12,

Data used in computations
TBb = 5 days.
TIBb = 0.5 days.
TIF = 1 day.
r(t) as stated in upper right corner of diagrams.

Specific properties of this system are:

Thus, a Bb populations entering the system from niches drive flare cycles, much like dust entering into a room from an outside source makes periodic room cleaning necessary. As long as there exists the dust source outside, we need to periodically clean the room. Similarly, the Bb niche population is called "active" by J.J. Burrascano as long as the Bb fragment concentration oscillates across the threshold for a Herxheimer reaction, CF(t) > C2.

The model explains an interesting feature consistent with that analogy:

  1. Near the end of a successful antibiosis the patient automatically lowers threshold C2 on a logarithmic scale, soon arriving at a point where this threshold dives below the minima of the oscillating fragment concentration (lower part of Fig. 12), and the symptom free days disappear.
  2. If the illness is improving, the fragment concentration oscillations have the general downward tendency shown in the lower diagram of Fig. 12. Then, some time after the patient has inadvertantly lowered threshold C2, the oscillations will cross the lower level C2 again, and cycles reappear. If their severity is smaller than at earlier times, this confirms the assumption of illness improvement.
  3. When the patient lowers C2 again, the sequence (1) - (2) is repeated. It is important that the severity of symptoms decreases during this repetitive process, otherwise the reappearance of cycles would signal a failure to eradicate the Bb source (and with it the Bb niche population), as is exemplified in the upper diagram of Fig. 12.

VI. Conclusions

The presented method of statistical evaluation enhances the visibility of flare cycles if the data basis is sufficiently large. The structure of flare cycles is analyzed on the basis of simple compartment models. The parameters of the models are related to mechanisms in the medical microbiology of an active inflammationwith Bb, based on some rough understanding of the system. Once this link to modern medical research has been strengthened, possibly improving the compartment models in unison with the medical research, this analysis will probably be a further help in making decisions concerning the necessary medication during the course of the inflammation.

The method is exemplified with the data of a female patient's symptom log. Specifics of this log are:

Some medico-microbiological interpretation of the evolution in time of the patient's flare cycle structure is given with the help of two compartment models, and needs critique and improvement from the medical research community.

The models use as input

  1. functions f describing the activity of the immune system (f = 1: immune response is "on", f = 0: no immune response). The immune response f switches from 0 to 1 and reverse when compartment concentrations or times cross adjustable thresholds. These would also include the typical time span Bb needs for its antigenic variation.
  2. guesses of the time constants describing Bb in vivo growth and decay (TBb, TIBb) and the rate of leakage of Bb or Bb fragments from the niche into compartments under immune surveillance.
As long as the niches leak, the Bb (fragments) keep contaminating the compartments under immune surveillance and trigger a clean-up by the immune system.

The results of the model seem to be stable against reasonable variations of input parameters (2), but their relative contributions, i.e. the effect of the immune system relative to the antibiotic (expressed as arrows in Fig. 12), needs to be discussed further. Once the immune system switching function f thresholds can be deduced from medico-microbiological principles, therfore not needing to be adapted to get the modeled symptom logs fit the data (as done in this analysis), the presented -rather empirical- interpretation of the infection's flare cycles would be markedly improved.

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VIII. Appendix

VIII. 1. Notation

b = bioavailability of cefuroxime relative to its biolavailability with prior food intake (b =: 1 with prior food intake). b = 1/2.7 without prior food intake, which value was the ratio of the measured peak plasma concentrations in patient's blood after cefuroxime intake without and with a prior meal (dimensionless).

Bb = Borrelia burgdorferi.

C(t) = concentration of substance provoking immune response, i.e. of Bb (CBb(t)) and of Bb fragments (CF(t)) (units: number of spirochetes per system volume).

cCSF(t) = concentration of antibiotic in CSF (units: mg/L).

cGI(t) = concentration of antibiotic in gastro-intestinal tract at time t after drug intake (units: mg/L).

CHx = threshold concentration of Bb fragments starting Herxheimer like reaction (units: number of fragments per system volume).

cP(t) = concentration of antibiotic in blood plasma at time t after drug intake/infusion (units: mg/L).

CooBb = stationary concentration of Bb population outside the niche = r(t)/(ln2 (1/TBb + 1/TIBb)) or r(t) TBb/ln2, depending on whether immune system is assumed to be eliminting Bb or not (units: number of fragments per system volume).

CooF(t) = concentration of Bb fragments after concentration CBb(t) of Bb population outside the niche has reached its stationary value CooBb (see Fig. 12) (units: number of fragments per system volume).

CSF = cerebrospinal fluid.

compartment model = a visualization of a linear system of first order differential equations describing the growth of the number of cells in a system. A system may consist of several subsystems, each of which will be represented by a compartment. Compartments have in- ond outfluxes having the dimension cells per time (when the entities within a compartment are cells). What comes out of one compartment may go into some other compatrment, the two compartments being "coupled". Each compartment is represented by a differential equation which states how much goes in and out per unit time. The Mathematica code representing the compartment models used here is given in http://www.lymenet.de/symptoms/cycles/mathcode.htm.

cytokines: Plasma LPS concentrations usually do not correlate with clinical symptoms (Roumen et al. 1993). It is the induction of cytokines through cell wall components like LPS which mediates the biological responses during bacterial infections. Cytokine levels and types of cytokines have repeatedly been shown to correlate with clinical outcome (Damas et al., 1992, Frieling et al., 1995, van Deuren et al., 1995).

C1 = threshold concentration triggering the immune system to start toxin elimination (apparently by its humoral branch). The immune response starts with a lag phase. Immune response subsides when toxin concentrations "visible" to the immune system have fallen below another threshold concentration. (units: number of spirochetes or fragments per system volume).

C2 = concentration threshold for inflammation, i.e. above which illness symptoms are perceived (units: number of spirochetes or fragments per system volume).

D = dosage of cefuroxime (gram per intake).

deltai = length of the ith menstrual cycle.

Delta t = time during which cefuroxime concentration in CSF is larger or equal a given inhibitory concentration (units: hours/day).

Delta tsubinh = time during which cefuroxime concentration in CSF is smaller than a given inhibitory concentration (units: hours/day).

equilibrium of a compartment = state in which influx to the compartment equals outflux out of it. At equilibrium the compartment is full, meaning that its concentration will no longer rise.

In particular, here are some properties of the 2-compartment system in Fig. 12:

F = subscript meaning Bb fragments.

f = free (i.e. Bb affecting) fraction of cefuroxime concentration in considered subsystem (here CSF) relative to its plasma concentration (f = : 1 for plasma), f = cCSF/cplasma = 0.1 for CSF (dimensionless). Data from

f(C, t) = dimensionless function describing the activity of the immune response ("immune switching function"). f ia either 0 ("no immune response") or 1 ("immune response"). Here, the immune response is assumed to be directed

  1. either against Bb, then the immune switching function will be called fBb, or
  2. against Bb fragments, the immune switching function then being called fF.
Depending on the system, f may depend on both C and t or on only one of these arguments, the corresponding concentration and time thresholds being C1 and lag phase tau, respectively.

flare = cluster of days with symptom occurrence.

follicular phase (here used sensu lato) = the first phase of the menstrual cycle, starting with the menstruation (menstrual bleeding) and ending with the ovulation, i.e. days 1 through 12 ... 14.

I = superscript meaning immune system.

incubation time = time between infection (entrance of the pathogen into host) and development of clinical symptoms.

Immune Response Interval = time interval of approximately 6 days duration, centered around the day of menses (beginning of menstrual bleeding), during which Barkley, Harris and Szantyr observed systematically high antigen concentrations in the urine of a Lyme patient (Barkley et al., 1997). The authors suggest that the immune system has a higher level of activity during this phase (see also testimony of M.S. Barkley before the New York State Assembly Standing Committee on Health, Public Hearing "Chronic Lyme Disease and Long-Term Antibiotic Treatment", Albany, NY, USA, 27.11.2001, pp. 199 - 227).

invisible = located in a compartment into which the immune system or the antibiotic penetrates only poorly. The table gives examples of such locations in which Bb were found.

ki = ln 2/Ti (units: 1/hour).

i =

kPCSF = transition rate for drug from plasma to CSF compartment.

lag time = lag phase (symbol: tau)

ln 2 = (natural logarithm of 2 =) 0.69.

LPS = lipopolysaccharide.

luteal phase = phase of the menstrual cycle after the ovulation. memory = the attribute of the immune system mediated by memory cells whereby a second encounter with an antigen induces a faster start and a heightened state of immune reactivity (Kuby pp 397 - 399)

menses = day of onset of menstrual bleeding.

MIC = Minimum Inhibitory Concentration. Definition: MIC is the minimum level necessary to inhibit bacterial growth. It depends on the bacterial isolate. MIC50 and MIC90 are the levels at which 50 % and 90 % of the tested isolates are inhibited, respectively.
MIC for cefuroxime = 0.13 mg/L, as determined by Agger et al. 2 MIC = 0.26 mg/L is used in the computations, which corresponds to MIC90 = 0.25 mg/L as determined by Preac-Mursic (1987). A critical discussion of the concept behind the MIC can be found in Mattie H 2000) See also Preac-Mursic et al 1996.

ni = number of days of a constant cefuroxime regimen (units: day).

net growth of Bb population = growth of population remaining when decay of population has been subtracted.

niches protect Bb from the immune system or the antibiotic (Preac-Mursic et al., 1989) or render toxin released by Bb "invisible" to the immune system. The protection may wane with time and so will the size of the spirochete or toxin population in the niche.

Niches are provided by the host in the form of physical compartments, but they can also be produced by Bb itself in the form of chemical or microbiological defense mechanisms (see also overview in Chapter Background Information of J.J. Burrascano's essay "Managing Lyme Disease".

In the model simulating the flare cycles in the presence of antibiotics, the term "niche" is used in this generalized sense.

The basic concept underlying the model is that the niche has the following properties:

The literature gives examples of such locations (i.e. intracellular locations) in which Bb were found. See also literature on mechanisms of cell invasion.

Osp = variable, plasmid encoded Outer sphere protein of Bb. The Osp's labeled OspA (30 ... 32 kD), OspB (34 ... 36 kD), OspC (21 ... 24 kD) are unique for Bb, as are the proteins p39 (39 kD) and p93 (93 kD).

ovulation = day on which the ripe ovum (egg cell) leaves the ovarian follicle.

prostaglandine E2 = a lipid inflammatory mediator with diverse biological activities, including increased vascular permeability and dilation, and induction of neutrophil chemotaxis (Kuby, p. 368-371).

r(t) = time variable Bb source term in compartment model (units: spirochetes per day entering unit system volume). It is assumed that r(t) varies much more slowly than concentrations C(t).

symptom = consequence of an inflammation of glial or neural tissue.

system = infected organ or tissue responsible for symptom. Basic systems are defined after Bleiweiss and have been further expanded here into subsystems characterized by the symptoms in Fig. 1.

ta = part of flare cycle during which immune system is active (units: day).

tb = part of flare cycle during which immune system is not yet active (flare cycle duration is tb + ta) (units: day).

TBb = in vivo Bb generation time (units: day).

TBbinvitro = in vitro Bb generation time. Values extracted tentatively from kill kinetics published by Agger et al. and Preac-Mursic et al. are 11 hours and 10 hours, respectively.

TBbI = Bb elimination half life characterizing immune system (units: day).

TCSF = elimination half life of antibiotic from CSF compartment (units: hour).

TFI = Bb fragment elimination half life characterizing immune system (units: day).

TGI = elimination half life for GI-tract resorption.

total concentration of drug = concentration of all chemical species of drug. Chemical species are the free drug and chemical complexes containing drug. Total concentrations are determined by breaking up all chemical complexes. (units: mg/l).

toxins are

These toxins produce cytokines (Ma et al. 1993, Tai et al. 1994, Sellati et al. 1996, Frieling et al. 1997, Burns et al. 1998, Giambartolomei et al. 1998, Straubinger et al. 1998, Zhang et al. 1998, see also the result of a Medline search). It is the cytokine levels that correlate with clinical responses (Damas et al., 1992, Frieling et al., 1995, van Deuren et al., 1995).

Via molecular mimicry (Kuby, Ch. 20, S. 497), also autoimmune processes can be triggered by Bb proteins (Sigal 1997, Sigal and Williams 1997). T-cell subpopulations (of short-lived T-cells) responsible for autoimmune processes might persist as long as a sufficient level of such proteins exists.

TP = renal elimination half life from plasma compartment.

t = time variable (units: hour in pharmacokinetic model, units: day in models for flare cycles, Figs. 11 and 12).

tinet = time for net growth of Bb population.

t0 = time of bolus infusion of cephalosporin (units: hour).

tau = lag time for resorption from GI tract. tau = 1.4 h, fitted from experimentally determined plasma concentrations (units: hour).
(see also lag phase in immune response.)

22.1 mg/l = peak plasma concentration measured in patient's plasma after intake of 2 gram of cefuroxime with prior meal.

xyz =: n this equation means xyz is by definition equal to n.

(*) This chapter is part of a draft: "Lyme Disease: Statistical Evaluation of a Symptom Log and an Empirical Theory of Flare Cycles
  • There is also an overview ("A Tentative Interpretation of Lyme Flare Cycles and a Corresponding Therapy")
  • and a summary ("Summary of the Empirical Theory of Flare Cycles").

  • version: January 8, 2004.
  • Location of this document is http://www.lymenet.de/symptoms/cycles/models.htm.
  • The home page of this website is http://www.lymenet.de.
  • Send comments to Dr. Joachim Gruber.