Home -- Symptoms -- Cycles -- Compartment Models Displaying Lyme Disease Symptom Cycles

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.

• Their amplitude (the symptom severity) decreases when the niches inject less toxin, e.g. when they run out of toxin after their Bb populations have been killed.
• Their frequency (or the time between flares) depends on many parameters of the feedback control, not just on the generation time of the spirochete. In fact, even a population of dead Bb, of TI-1 antigens like LPS or perhaps some Osp, in the niches produces proinflammatory cytokines (Straubinger et al. 1998) and is therfore able to drive such oscillations of the immune system activity.

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

• the immune system switching function, f(C(t), t).
  It has properties 1 and 2 of the immune system .
• the compartments.
  They represent those parts of the host in which toxin levels develop. I will visualize the system of interconnected compartments as a non-linear system of first order differential equations.

Thus, the compartment model

• will not give an interpretation of the microbiological and medical mechanisms behind the immune switching function,
• but will illustrate how some properties of flare cycles depend on model parameters, such as (links point to Notation section for explanation of symbols used)
  • in vivo and in vitro Bb generation times (T_Bb and T_Bb^{in vitro}, respectively),
  • elimination kinetics and thresholds characterizing the immune system (T^I_Bb, T^I_F, f(C, t), C₁, lag time),
  • threshold concentrations of Bb or fragments of (killed) Bb above which symptoms are experienced (C₂),
  • rate at which a Bb niche population enters the subsystem of the host responsible for the considered symptom (r(t))

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.

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.

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 f_Bb(C_Bb(t), t), while keeping the Bb generation time T_Bb and the elimination half life T^I_Bb 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:

• When the Bb population grows (the zig-zag curve tending generally upward), flare cycles melt together with increasing time (days 60 - 100 in bottom diagram), and therefore conversely, when Bb population decreases, flare cycles "melt away" with time.
• Shifting the threshold C₂ changes the fraction of the cycle during which the symptom is felt, but does not change the length of the cycle.
• As is typical for the pre-treatment time of an infection, the exponential growth of the bacteria population is so fast, leading quickly to high concentrations before it is stopped by the immune system, that there is not much doubt in the patient's subjective impression as to whether a symptom occurs or not.

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 C_Bb(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 C_F(t).

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

• Left compartment: Bb population outside niche, concentration C_Bb(t),
• right compartment: Bb fragment inventory, concentration C_F(t),
• differential equations equivalent with compartment system, and
• the corresponding Mathematica program.
• The equation below the compartment model (top part of figure) gives the fragment concentration when the Bb compartment has reached equilibrium for slowly varying r(t), i.e. at times when the dashed curve approaches a horizontal line.

In the case depicted in Fig. 12,

• the Bb compartment receives continuous influx of spirochetes from niche population.
  • Bb source r is constant in upper diagram and
  • steadily decreasing in lower.
• Antibiotic kills fraction ln2/T_Bb of C_Bb(t) per unit time, creating Bb fragments (see arrow connecting compartments). For demonstration purposes it suffices here to oversimplify the relationship between number of killed Bb and clinical symptoms:
  1. each killed Bb releases one antigen molecule, e.g. LPS or Osp, (Tables 2 and 3 in Hurley 1992),
  2. the highly non-linear relationship between LPS or Osp concentration and clinical symptoms is replaced by a linear one.
• Both pools are subject to immune system clean-up (arrows pointing downward from compartments), and -if this is oscillatory- will go through cycles, much as the Bb pool in the system in Fig. 11 did.
• For illustration purposes, towards end of time coverd in upper diagram it is assumed that immune system loses track of Bb population (dashed curve approaches an equilibrium level). Thresholds C₁ and C₂ are assumed to coincide to simplify diagrams, as in Fig. 11.

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, C_F(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.

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

• Pre-treatment (98 days): Flare cycles between 24 and 28 days length are present prior to treatment.
• Cefuroxime (99 days):
  • The length of the cycles shifted from 24 days to about one half of that (15 + or - 5 days).
  • According to the symptom log, symptom severity and duration did not generally decrease.
  • Borrelia burgdorferi (Bb) in the CSF were exposed to twice the Minimum Inhibitory Concentration of cefuroxime (MIC = 0.13 mg/l) for less than 30 % of the day.
  • If the supposed treatment failure is traced back to a growth of the Bb population in the CNS due to subMIC cefuroxime exposure, our analysis allows us to extract an upper boundary for the Bb generation time (5 days).
• Ceftriaxone (150 days), Cefepime (126 days) : During these infusions
  • the lengths of the cycles were in the same range as during cefuroxime intake,
  • the duration and severity of the symptoms decreased as compared with the cefuroxime regimen.

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 (T_Bb, T^I_Bb) and the rate of leakage of Bb or Bb fragments from the niche into compartments under immune surveillance.

• The reason for the ongoing immune system's oscillation between "on" and "off" is the missing memory effect, a charcteristic of the LPS and perhaps some Osp's.
• After the long therapy with cephalosporins the flare cycles synchronize with the menstrual cycle of the patient, the symptoms occurring predominantly in the luteal phase, in which the level of the proinflammatory prostaglandin E2 is elevated (Leslie et al 1994). This is interpreted as an indication of the pathogen level being very low and possibly within the immune system's self-help range.

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.

Ball P, The self-made tapestry: pattern formation in nature, Oxford University Press, Oxford, New York, Tokyo, 1999.

Barkley M, Harris N, Szantyr B, 1997. The Lyme Urine Antigen Test (LUAT) during antibiotic therapy in women with recurrent menstrual cycles. 10th Annual Int. Conf. on Lyme Borreliosis, National Institute of Health, Bethesda, MD, April 28-30.

Beck G, Habicht GS, Benach JL, Coleman JL, Lysik RM, O'Brien RF, A role for interleukin-1 in the pathogenesis of Lyme disease. Zentralbl Bakteriol Mikrobiol Hyg [A] 263(1-2):133-6 , 1986.

Bingen E, Goury V, Bennani H, Lambert-Zechovsky N, Aujard Y, Darbord JC, Bactericidal activity of beta-lactams and amikacin against Haemophilus influenzae: effect on endotoxin release. J. Antimicrob. Chemother. 30:165-172, 1992.

Bleiweiss JD, When to Suspect Lyme Disease?

Bransfield RC, Sex & Lyme.

Bristol-Myers Squibb, Pharmaceutical Information, 777 Scudders Mill Road Plainsboro, NJ 08536, Tel.: +1 (609) 252-4000 or (800) 332 - 2056.

Brorson O, Brorson SH, In vitro conversion of Borrelia burgdorferi to cystic forms in spinal fluid and transformation to mobile spirochetes by incubation in BSK-H medium. Infection 26:144-150, 1998.

Bericht des Landeshygiene-Instituts Mecklenburg-Vorpommern, Schloßstraße 8, D - 17235 Neustrelitz, Tel. + 49 -3981- 272 - 100 (Dr. Kober), 1997.

Burns MJ, Furie MB, Borrelia burgdorferi and Interleukin-1 Promote the Transendothelial Migration of Monocytes In Vitro by Different Mechanisms, Infect Immun 66: 4875-4883, 1998.

Burrascano JJ, The new Lyme disease: diagnostic hints and treatment guidelines for tick-borne illnesses.

Buxton Hopkin DA., Too rapid destruction of gram-negative organisms. Lancet i:603-604, 1977.

Chambers HF, Hadley WK, Jawetz E, "Beta-lactam antibiotics & other inhibitors of cell wall synthesis", pages 725-726 in: Basic & Clinical Pharmacology (Katzung BG, ed.), Appleton & Lange, Stamford, CT, USA, 1998.

Cohen J, and McConnell JS, Antibiotic-induced endotoxin release. Lancet i:1089-1090, 1985. (Letter).

Cohen, J, and McConnell JS, Release of endotoxin from bacteria exposed to ciprofloxacin and its prevention with polymixin B. Eur. J. Clin. Microbiol. 5:13-17, 1986.

Crosby HA, Bion JF, Penn CW, and Elliott TSJ, Antibiotic-induced release of endotoxin from bacteria in vitro. J. Med. Microbiol. 40:23-30, 1994.

Coyle, PK, Borrelia burgdorferi infection: clinical diagnostic techniques. Immunol Invest 1997 Jan; 26(1&2):117-128

Damas P Ledoux D, Nys M , Vrindts Y, De Groote D, Franchimont P, and Lamy M., Cytokine serum level during severe sepsis in human IL-6 as a marker of severity. Ann. Surg. 215:356Ð362, 1992.

De Boer RJ, Perelson AS, Kevrekidis IG, Immune network behavior--I. From stationary states to limit cycle oscillations., Bull Math Biol 55(4):745-80, 1993.

Dever LL, Torigian CV, and Barbour AG, In Vitro activities of the everninomicin SCH 27899Êand other newer antimicrobial agents against Borrelia burgdorferi, Antimicrob Agents Chemother. 43(7): 1773-17, 1993.

Dibrov BF, Livshits MA, Vol'kenshtein MV, Mathematical model of the immune response, Biofizika 1976 Sep-Oct;21(5):905-9, 1976.

Dibrov BF, Livshits MA, Vol'kenshtein MV, Mathematical model of the immune response. IV. Threshold character of the infectious process, Biofizika 1978 Jan-Feb;23(1):143-7, 1978.

Dotevall L, Alestig K, Hanner P, Norkrans G, Hagberg L, The use of doxycycline in nervous system Borrelia burgdorferi infection. Scand J Infect Dis Suppl 1988;53:74-9.

Dotevall L, Hagberg L, Penetration of doxycycline into cerebrospinal fluid in patients treated for suspected Lyme neuroborreliosis. Antimicrob Agents Chemother 1989 Jul;33(7):1078-80.

Eng RHK, Smith SM, Fan-Havard P, and Ogbara T, 1993. Effect of antibiotics on endotoxin release from gram-negative bacteria. Diagn. Microbiol. Infect. Dis. 16:185-189.

Evans ME, Pollack M, Effect of antibiotic class and concentration on the release of lipopolysaccharide from Escherichia coli. J. Infect. Dis. 167:1336-1343, 1993.

Fallon BA, Nields JA, Microbiology of Borrelia burgdorferi, in: "Lyme Disease: A Neuropsychiatric Illness", an overview, The American Journal of Psychiatry 1994;151,11:1571-1583.

Friedland IR, Jafari H, Ehrett S, Rinderknecht S, Paris M, Coulthard M, Saxen H, Olsen K, and McCracken GH, Comparison of endotoxin release by different antimicrobial agents and the effect on inflammation in experimental Escherichia coli meningitis. J. Infect. Dis. 168:657-662, 1993.

Frieling JTM, Mulder JA, Hendrics T , Curfs JHAJ, van der Linden CJ, Sauerwein RW, Differential Induction of Pro- and Anti-Inflammatory Cytokines in Whole Blood by Bacteria: Effects of Antibiotic Treatment, Antimicrobial Agents and Chemotherapy 35:1439Ð1443, 1997.

Frieling JTM, Van Deuren M, Wijdenes J, Van der Meer JWM, Clement C, Van der Linden C, and Sauerwein RW, Circulating interleukin-6 receptor in patients with sepsis syndrome. J. Infect. Dis. 171:469Ð472, 1995. (see also Frieling et al 1997)

Giambartolomei GH, Dennis VA, Philipp MT, Borrelia burgdorferi stimulates the production of interleukin-10 in peripheral blood mononuclear cells from uninfected humans and rhesus monkeys, Infect Immun Jun;66(6):2691-7, 1998.

Goodman and Gilman's "The Pharmacological Basis of Therapeutics, 9th ed., McGraw Hill, 1995, abstracting this value from Schaad et al., 1990.

Groer M, Carr J, Younger MS, Relationships between self-reported symptoms of infection, menstrual-cycle-related distress, and cycle phase. Behav Med 1993 Spring;19(1):13-9, 1993.

Hemmer B, et al. Identification of candidate T-cell epitopes and molecular mimics in chronic Lyme disease. Nature Medicine 5(12):1375-82, 1999.

Hu LT, Klempner MS, Host-pathogen interactions in the immunopathogenesis of Lyme disease. J Clin Immunol 17(5):354-65, 1997.

Hurley JC, Antibiotic-induced release of endotoxin: a reappraisal. Clinical Infectious Diseases 1992;15:840-854.

Karlsson M, Hammers S, Nilsson-Ehle I, Malmborg AS, Wretlind B, Concentrations of doxycycline and penicillin G in sera and cerebrospinal fluid of patients treated for neuroborreliosis. Antimicrob Agents Chemother 1996 May;40(5):1104-7.

Kessler RE, Fung-Tomc J, Susceptibility of bacterial isolates to beta-lactam antibiotics from U.S. clinical trials over a 5-year period, Am J Med 1996;100(suppl 6A):13S-19S.

Klempner MS and Huber BT, Is it thee or me?-autoimmunity in Lyme disease. Nature Medicine 5(12):1346-7, 1999.

Kossmann T, Hans V, Stocker R, Imhof HG, Joos B, Trentz O, Morganti-Kossmann MC, Penetration of cefuroxime into the cerebrospinal fluid of patients with traumatic brain injury. J Antimicrob Chemother 37:161-7, 1996.

Kuby J, Immunology, 3rd Ed., WH Freeman and Co, New York, 1997.

Lawrence C, Lipton RB, Lowy FD, Coyle PK, Seronegative chronic relapsing neuroborreliosis. Eur Neurol 1995;35(2):113-117.

Leslie CA, Dubey DP, Increased PGE2 from human monocytes isolated in the luteal phase of the menstrual cycle. Implications for immunity?, Prostaglandins 1994 Jan;47(1):41-54.

Ma Y, Weis JJ, Borrelia burgdorferi outer surface lipoproteins OspA and OspB possess B-cell mitogenic and cytokine-stimulatory properties. Infect Immun 1993 Sep;61(9):3843-53.

Mattie H, Antibiotic efficacy in vivo predicted by in vitro activity, Internat J Antimicrob Agents 14:91-98, 2000.

Mattman, LH, Cell Wall Deficient Forms - Stealth Pathogens, 2nd. ed., CRC Press, 1993.

McConnell, J. S., and J. Cohen, Release of endotoxin from Escherichia coli by quinolones. J. Antimicrob. Chemother 18:765-766, 1986. (Letter).

McKenzie FE, Bossert WH, The dynamics of Plasmodium falciparum blood-stage infection, J Theor Biol. 188(1):127-40, 1997.

Mertsola, J., O. Ramilo, M. M. Mustafa, X. Saez-LLorens, E. J. Hansen, and G. H. McCracken, Release of endotoxin after antibiotic treatment of gram-negative bacterial meningitis. Pediatr. Infect. Dis. J. 8:904-906, 1989.

Mohler, J. B. Fantin, J. L. Mainardi, and C. Carbon, Influence of antimicrobial therapy on kinetics of tumor necrosis factor levels in experimental endocarditis caused by Klebsiella pneumoniae. Antimicrob. Agents Chemother. 38:1017-1022, 1994.

Muraille E, Thieffry D, Leo O, Kaufman M, Toxicity and neuroendocrine regulation of the immune response: a model analysis, J Theor Biol. 183(3):285-305, 1996.

Nau R et al., Passage of cefotaxime and ceftriaxone into cerebrospinal fluid of patients with uninflamed meninges. Antimicrob Agents Chemother 37:1518-1524, 1993.

Northern AL, Rutter SM, Peterson CM, Cyclic changes in the concentrations of peripheral blood immune cells during the normal menstrual cycle. Proc Soc Exp Biol Med Oct;207(1):81-8, 1994.

Phillips SE, Mattman LH, Hulinska D, Moayad H, A Proposal for the reliable culture of Borrelia burgdorferi from patients with chronic Lyme disease, even from those previously aggressively treated, Infection 26 (1998) 364-367.

Preac-Mursic V, Wilske B, Schierz G, Holmburger M, Süß E, In vitro and in vivo susceptibility of Borrelia burgdorferi, Europ J Clin Microbiol 6:424-426, 1987.

Preac-Mursic V, Weber K, Pfister HW, Wilske B, Gross B, Baumann A, Prokop J, Survival of Borrelia burgdorferi in antibiotically treated patients with Lyme borreliosis. Infection Nov-Dec;17(6):355-9, 1989.

Preac-Mursic V, Wilske B, Schierz G, Süß E, Comparative antimicrobial activity of new macrolides against Borrelia burgdorferi. Europ J clin Microbiol Infect Dis 8:651-653, 1989.

Preac-Mursic V, Marget W, Busch U, Pleterski D, Rigler S, Hagl S , Kill Kinetics of Borrelia burgdorferi, Infection, 24: 9 - 16, 1996.

Preac Mursic V, Wanner G, Reinhardt S, Busch U, Marget W, Formation and cultivation of Borrelia burgdorferi spheroplast-L-form variants. Infection 24 (1996): 218-226.

Roumen RMH., Frieling JT M , van Tits HWHJ , van der Vliet JA, and Goris RJA, Endotoxemia following major vascular operations. J. Vasc. Surg. 18:853-857, 1993.

Samples taken from 43 year old female whose data were analyzed in this report.

Schaad UB, Suter S, Gianella-Borradori A, Pfenninger J, Auckenthaler R, Bernath O, Chesaux JJ, and Wedgwood J, A comparison of ceftriaxone and cefuroxime for the treatment of bacterial meningitis in children. N. Engl. J. Med. 322:141-147, 1990.

Sellati TJ, Abrescia LD, Radolf JD, Furie MB. Outer surface lipoproteins of Borrelia burgdorferi activate vascular endothelium in vitro. Infect Immun Aug;64(8):3180-7, 1996.

Schutzer SE, Coyle PK, Belman AL, Golightly MG, Drulle J, Sequestration of antibody to Borrelia burgdorferi in immune complexes in seronegative Lyme disease. Lancet 1990 Feb 10;335(8685):312-315.

Schutzer SE, Coyle PK, Dunn JJ, Luft BJ, Brunner M, Early and specific antibody response to OspA in Lyme Disease. J Clin Invest Jul;1994(1):454-457, 1994.

Shenep, JL, Flynn PM , Barrett FF, Stidham GL, and Westenkirchner DF, Serial quantitation of endotoxemia and bacteremia during therapy for gram-negative bacterial sepsis. J. Infect. Dis. 157:565-568, 1988.

Shenep, JL, Barton RP, and Mogan KA, Role of antibiotic class in the rate of liberation of endotoxin during therapy for experimental gram-negative bacterial sepsis. J. Infect. Dis. 151:1012-1018, 1985.

Shmuklarsky MJ, Boudreau EF, Pang LW, Smith JI, Schneider I, Fleckenstein L, Abdelrahim MM, Canfield CJ, Schuster B, Failure of doxycycline as a causal prophylactic agent against Plasmodium falciparum malaria in healthy nonimmune volunteers. Ann Intern Med 1994 Feb 15;120(4):294-9.

Sigal LH, Immunologic mechanisms in Lyme neuroborreliosis: the potential role of autoimmunity and molecular mimicry, Semin Neurol 1997 Mar;17(1):63-8

Sigal LH, Williams S, A monoclonal antibody to Borrelia burgdorferi flagellin modifies neuroblastoma cell neuritogenesis in vitro: a possible role for autoimmunity in the neuropathy of Lyme disease, Infect Immun 1997 May;65(5):1722-8.

Smirnova OA, Study of cyclic kinetics of immunity by mathematical modeling methods, Kosm Biol Aviakosm Med. 25(5):53-6, 1991.

Spangler SK, Jacobs MR, Apfelbaum PC, MIC and Time-Kill Studies of Antipneumococcal Activity of 118819X (Sanfetrinem) Compared with Those of Other Agents, Antimicrob Agents Chemother 41:141-155, 1997.

Straubinger RK, Straubinger AF, Summers BA, Erb HN, Härter L, Appel MJG, Borrelia burgdorferi induces the production and release of proinflammatory cytokines in canine synovial explant cultures. Infect Immun 66(1):246-258, 1998.

Straubinger RK, PCR-Based Quantification of Borrelia burgdorferi Organisms in Canine Tissues over a 500-Day Postinfection Period. J Clin Microbiol 2000;38:2191.

Tai KF, Ma Y, Weis JJ, 1994. Normal human B lymphocytes and mononuclear cells respond to the mitogenic and cytokine-stimulatory activities of Borrelia burgdorferi and its lipoprotein OspA. Infect Immun Feb;62(2):520-8, 1994.

Thornsberry et al., In vitro activity of cefepime and other antimicrobials: survey of European isolates, J. Antimicrobial Chemother 32, Suppl. B, 55-62, 1993.

van den Berg C, De Neeling AJ, Schot CS, Hustinx WMN, Wemer J, and de Wildt DJ, Delayed antibiotic-induced lysis of Escherichia coli in vitro is correlated with enhancement of LPS release. Scand. J. Infect. Dis. 24:619-627, 1992.

van Deuren, M, van der Ven-Jongekrijg J, Bartelink AK, van Dalen R, Sauerwein RW, and van der Meer JW, Correlation between proin-flammatory cytokines and antiinflammatory mediators and the severity of disease in meningococcal infections. J. Infect. Dis. 172:433Ð439, 1995.

Wagner D private communication, Pharmaceutical Info Div., 777 Scudders Mill Road, Plainsboro, NJ 08536, Tel.: +1 (609) 897 - 2776, Fax: +1 (609) 897 - 6042.

Yim CW, Flynn NM, Fitzgerald FT, Penetration of oral doxycycline into the cerebrospinal fluid of patients with latent or neurosyphilis.Antimicrob Agents Chemother 1985 Aug;28(2):347-8.

Zhang JR, Hardham JM, Barbour AG, Norris SJ, Antigenic variation in Lyme disease borreliae by promiscuous recombination of VMP-like sequence cassettes, Cell Apr 18;89(2):275-285, 1997.

Zhang H, Niesel DW, Peterson JW, Gary R, Lipoprotein release by bacteria: potential factor in bacterial pathogenesis, Infect Immun 66(11):5196-5201, 1998.

Bb = Borrelia burgdorferi.

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

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

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

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

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

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

C^oo_F(t) = concentration of Bb fragments after concentration C_Bb(t) of Bb population outside the niche has reached its stationary value C^oo_Bb (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).

delta_i = 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 t^subinh = 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:

• The set of differential equations is

  (1)
  C_Bb(t)' = r(t) - ln2 C_Bb(t) (1/T_Bb + f_Bb/T^I_Bb)

  (2)
  C_F(t)' = ln2 (C_Bb(t)/T_Bb - f_F C_F(t)/T_F)
  

• The half life of a compartment can be read from its differential equation: It is related to the coefficient in front of the concentration in the compartment:
  1. (3)
     Bb compartment: (1/T_Bb + f_Bb/T^I_Bb)^-1.
  2. (4)
     Bb fragment compartment: T_F for f_F = 1, infinity else.

• Equilibrium levels of the Bb compartment: The equality of in- and outflux gives us its equilibrium concentrations with (f_Bb = 1) and without immune system (f_Bb = 0):

  (5)
  r = ln2 C_Bb^eq (1/T_Bb + f_Bb/T^I_Bb)

  These are the values between which the dashed curve in Fig. 12 oscillates.

• A compartment has arrived at 75 % of its equilibrium concentration after 2 half lives.

• At equilibrium of the Bb compartment, one can imagine the Bb compartment as being full and the source feeding the fragment compartment directly,. The 2-compartment system degenerates into the simpler 1-compartment system, the differential equation of which is given in the box at the top of Fig. 12:

  (6)
  C^eq_F(t)' = r(t) - f_F ln2 C^eq_F(t)/T_F.

• In the simplest case, a constant amount of Bb is emerging per unit time from the niche: r(t) = constant. Then also the fragment compartment has its equilibrium, which can be calculated equating in- and outflux as described above:

  (7)
  r = ln2 C^eq_F/T_F

  • During an off-time of the immune system (f_F = 0) the fragment concentration starts with this equilibrium value and increases linearly with time.
  • During the following on-time of the immune system (f_F = 0) the concentration relaxes roughly exponentially back towards this equilibrium value.

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 = c^CSF/c^plasma = 0.1 for CSF (dimensionless). Data from

• Goodman and Gilman's "The Pharmacological Basis of Therapeutics" and
• Kossmann et al., 19967.

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 f_Bb, or
2. against Bb fragments, the immune switching function then being called f_F.

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/T_i (units: 1/hour).

i =

• CSF,
• GI (gastro-intestinal tract),
• P (blood plasma).