Abstract: Phage therapy, which can be described as a phage-mediated biocontrol of bacteria (or, simply, biocontrol), is the application of bacterial viruses—also bacteriophages or phages—to reduce densities of nuisance or pathogenic bacteria. Predictive calculations for phage therapy dosing should be useful toward rational development of therapeutic as well as biocontrol products. Here, I consider the theoretical basis of a number of concepts relevant to phage dosing for phage therapy including minimum inhibitory concentration (but also "inundation threshold"), minimum bactericidal concentration (but also "clearance threshold"), decimal reduction time (D value), time until bacterial eradication, threshold bacterial density necessary to support phage population growth ("proliferation threshold"), and bacterial density supporting half-maximal phage population growth rates (KB). I also address the concepts of phage killing titers, multiplicity of infection, and phage peak densities. Though many of the presented ideas are not unique to this chapter, I nonetheless provide variations on derivations and resulting formulae, plus as appropriate discuss relative importance. The overriding goal is to present a variety of calculations that are useful toward phage therapy dosing so that they may be found in one location and presented in a manner that allows facile appreciation, comparison, and implementation. The importance of phage density as a key determinant of the phage potential to eradicate bacterial targets is stressed throughout the chapter.
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(1996), tG ¼ 1=kN0 ð ÞþL; (1 (2) The larger the ratio of bacteria to phages, then the longer it will take to reduce bacterial densities a given amount (i (2000) along with Payne and Jansen (2001, 2003) use passive treatment and passive therapy interchangeably (and also use the terms active treatment and active therapy interchangeably) (2000) and Payne and Jansen (2001) (2000) and Payne and Jansen (2001), but my preference is to describe them simply as new or subsequent (2000) call it the minimum inundatory [or inundative] threshold (2000) suggest that passively effective therapy should be dependent on the concentration of phage (p (2000)—that is, a function of the actual phage burst size (BA) (2001), Pt ¼ P0Bt=tG ; (1 (A and B) Numerical solutions for NF ¼ 1 bacterium/ml and NF ¼ 10 _6 bacteria/ml, respectively (A and B) Numerical solutions for NF ¼ 1 bacterium/ml and NF ¼ 10 _6 bacteria/ml, respectively (A and B) Numerical solutions for NF ¼ 1 bacterium/ml and NF ¼ 10 _6 bacteria/ml, respectively (A) Numerical solutions to Eq (C) The ratio of D as defined by Eq , multiplicity of 10, above), and P0 is a description of added phage densities rather than adsorbed phage densities, it would seem that Eq , 2000) , 2000) or proliferation density threshold (Payne and Jansen, 2003), which Payne and Jansen (2001) point out is analogous to the eradication threshold used by epidemiologists (p , 2001) , 2002) , 2002; see also Abedon and Thomas-Abedon, 2010) , 2010) , 2010) , 2010) , 2011) and as cited below , 2011; Loc-Carrillo and Abedon, 2011) , a seen in Fig , active treatment), and also that this is simply the minimum bacterial density that will keep the phage population from declining over time , as can give rise to active treatment , as found at t ¼ 0), t is the timing of phage addition to a growing bacterial culture, and IP is a bacteria-independent rate of phage decay , as seen in the expression, N0/P0) , auto dosing) , density of added phages divided by density of bacteria present), MOI can be readily calculated 10 Stephen Abedon Authors personal copy independent of bacterial density by assuming that total phage adsorption is insufficient to significantly change phage density (Abedon, 2008a) , infected bacteria instead remain as individual, undivided Phage Therapy Pharmacology: Calculating Phage Dosing 29 Authors personal copy cells) , loss of reversibly adsorbed state) k2 Turnover rate Function equivalent to the length of time that an enzyme requires to convert a substrate to product, here conversion of a bacterium into new phages, approximated by the reciprocal of phage latent period (1/L) Authors personal copy KB Phage growth, half-maximal constant Bacterial density that supports phage population growth at a rate of one-half that rate defined by mmax L Latent period Time required by a phage to progress from adsorption to lysis (from within) of a bacterium m Phage population growth rate Slope of log-transformed phage density as a function of time for exponentially growing phage population mmax Maximum phage population growth rate That growth rate available were phages able to obtain new bacteria immediately following release from infected bacteria M Multiplicity of infection The ratio of adsorbed phages to phage-susceptible bacteria, also known as MOI or MOIactual MBC Minimum bactericidal density of phages Phage density necessary to kill some fraction of bacterial population, for example, such as down to NF < 1 MBCt MBC of phages necessary to remove bacteria over an interval, t Incorporates issues of rates of phage acquisition of bacteria, over some time, t, into MBC determination MIC Minimum inhibitory concentration of phages Phage density that balances bacterial death due to phage adsorption and bacterial growth m Bacterial growth rate Malthusian parameter, that is, bacterial exponential growth may be expressed as Nt ¼ N0em N Bacterial density Refers to phage susceptible bacteria in per milliliter units N0 Bacterial density, initial N, typically at the point of phage addition NF Bacterial density, final The bacterial density achieved as a consequence of therapeutic treatment with phages NP Proliferation threshold Bacterial density required to offset phage decay such that phage densities do not decline over time Nt Bacterial density at time, t Indicates either delays between initiation of bacterial population growth and phage addition or delays between phage addition and determinations of bacterial viability (continued) Authors personal copy TABLE 1 , m ¼ kP0 when dN/dt ¼ 0) , MBCt) , min); generally, k ¼ 2 , ml) over one unit time (e , seconds, minutes, hours, or days) is preferable to slow (e , see Abedon, 2010b; Abedon et al , see Figs , see the appendix of Goodridge, 2008; see also Abedon, 2011c; Hagens and Loessner, 2010) , should phage densities decline due to bacterial adsorption, even if they do not decline by much) , such as due to similarly rapid phage decay), then sufficient phage densities to effect treatment must be maintained by either high or repeated dosing , the phage proliferation level) , the prebactericidal end point) , they instead remain at the proliferation threshold) and, importantly, that phage-infected bacteria do not replicate as bacteria (i , weeks, months, or years) , when e _ kN0t 0, i , when kN0t 1) Dosing has both pharmacodynamic and pharmacokinetic aspects More recently, Cairns and Payne (2008) have described it instead as the analogue of the MIC, an analogue however where it is not so much individual bacterial growth that is prevented but instead bacterial population growth The calculation for a phage therapy clearance threshold is more complicated than that for a phage therapy MIC The second calculation is of the bacterial density that supports halfmaximal phage population growth rates The upshot is that practical MIC experimental determination for phages is best done kinetically, especially in terms of viable counts of bacteria along with low bacterial starting densities 0 0 0 1 0 1 0 1 0 101 102 103 104 106 107 108 109 1010 Bacterial density per ml (N0) te, Eq 0 13 0 2 0 2 0 2 0 9 0 9 0 A B 101 102 103 104 106 107 108 109 1010 Bacterial density per ml (N0) 100 105 0 0 for N0 ¼ 109 and 1010 bacteria/ml, respectively; see Table 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 (continued) Abbreviation Description Comment P0 Phage density, initial May or may not be held constant Pe Phage density, effective Minimum phage density required to achieve desired levels of bacterial eradiation; phage densities below this point will not achieve adequate bacterial killing efficacy over desired time frames Pt Phage density at time, t Indicates a delay of time, t, between phage addition and determination of subsequent phage density t Time Generally here considered to be in minute units tG Phage generation time Consists of the interval spanning from phage release from an infected bacterium through the phage extracellular search for new bacteria to infect, phage adsorption of a bacterium, and subsequent phage latent period V Volume Of environment containing a bacterial population, here considered to be in milliliter units Authors personal copy P0 ¼ m k : (1 1 10 1 104 9 1 107 16 1 109 20 1 11 1 32 1 6 1 6 1 for graphing of Eq 1 is unquestionably better predicted by P0kt than it is at that bacterial density at which kN0t ¼ 10, which is unquestionably better predicted by P0/N0 1 Minimum bactericidal concentrations (MBC) based on Eq 1 Terms used in this chapter Abbreviation Description Comment BA Burst size, actual Total number of phages produced per infected bacterium BE Burst size, effective Total number of phages produced per infected bacterium that survive to initiate new infections D Dvalue Time between phage addition, at a specific density, and 90% reduction in bacterial viability d Net phage decay Net phage losses taking into account not only all types of loss of free phages but also, as appropriate, gains in free phages due to auto dosing IP Phage decay Refers to free phage losses for reasons other than initiating productive phage infections and can include loss of adsorption ability or inactivation following adsorption k Phage adsorption rate constant Likelihood of one phage adsorbing one bacterium with both suspended in one unit volume (e 1) indicates is that phage density at which bacterial births are exactly balanced by bacterial deaths, where only phage adsorption leads to bacterial deaths and only bacterial replication leads to bacterial births 1) MIC ¼ P0 ¼ m/k NA Minimum bactericidal concentration (MBCt), a phage density, Eqs 1) Payne and Jansen (2001, 2003) describe this calculation as an inundation threshold; Payne et al 1), for phages it is not trivial to experimentally determine MIC 1), since at that point, Eq 1, 13 1, then D ¼ t ¼ 2:3=kP0: (1 10) 10) 105 106 107 108 109 1010 FIGURE 1 10) and (1 10) and (1 10) calculates a D value that is only 12% shorter than that calculated using Eq 10) likely will overestimate rates of reduction in bacterial densities 10) Note that this value is independent of bacterial density, which occurs because P0 is held constant in these calculations and which, in turn, implies that especially given higher bacterial densities then Eq 10) should provide a 20 Stephen Abedon Authors personal copy reasonable approximation of phage D values under many circumstances; for illustration, see the horizontal-line portions of curves found in Fig 10) to (1 10), is rearranged to solve for t, te ¼ t ¼ _ ln NF=N0 ð Þ=kP0: (1 10), while (B) is based on Eq 10-6 Phage proliferation threshold (Np) in bacteria/ml Phage decay rate(Ip) in phages per ml per min 10-5 10-4 10-3 10-2 10-1 100 1_100 1_101 1_102 1_103 1_104 1_105 1_106 BA=10 BA= 1000 1_107 FIGURE 1 11) 11) 11) are similar when the ratio of P0 to N0 is in the range of 100 11) can no longer calculate D since ln(0) ¼ ln(1_ (2 11) D ¼ 2 11) for various phage densities 11) Solving for various combinations of P0 and N0, it appears that Eqs 11) to that as defined by Eq 11), note that they are conveniently independent of k, that is, of the phage adsorption rate constant 11)/DEq 11); see Fig 12) which, as in the derivation of Eq 13) 13) 13) 13) and (1 13) and (1 13) FIGURE 1 13) In Fig 13) te, Eq 14 Stephen Abedon Authors personal copy (see Section V 14) Solving for t, te ¼ t ¼ _ ln ð1 þ ð ln ðNF=N0ÞN0=P0ÞÞ=kN0; (1 14)—intuitively, this occurs because, for example, if it takes 10 min to reduce a bacterial population by six orders of magnitude, then doubling of total incubation time or rate atwhich phages find bacteria should reduce bacterial populations by on the order of another six orders of magnitude rather than simply by twice as much on a linear scale (compare Fig 15) 15) 15) 15) for NF ¼ 1 and NF ¼ 10 _6 bacteria/ml 15) te ¼_ln (NF/N0)/kP0 te ¼_ln (1 þ (ln (NF/N0)N0/P0))/kN0 Maximum interval between doses (ti), Eq 15) to be valid 15) to that as defined by Eq 15) which of course is simply a generalization of Eq 15) would occur after 100 min (vs 15), that is, as seen in ln (NF/N0)N0/P0, also has the effect of making this expression more negative such that as N0 becomes larger then so too does te 15), that is, in the denominator of the larger expression 15), though certainly having some impact on te, nonetheless is not sufficiently robust to result in deviations from the intuitive result of greater bacterial densities requiring longer intervals to eradicate than lower bacterial densities 15), would have the paradoxical effect of reducing te 15)/te, Eq 15)/te, Eq 16 Stephen Abedon Authors personal copy course of phage adsorption to bacteria, through ongoing phage dosing or if phage densities are replenished through active phage replication (auto dosing) 16) NA ti ¼ ln (P0/Pe)/d Phage proliferation threshold (NP), a bacterial density, Eq 16) The larger ti, then the less frequently doses need to be administered 17) as based on more a general model provided by Payne and Jansen (2001) 17) for N0 when BE is set equal to 1 (meaning that bacterial density which sustains phage densities with no phage increase or decrease over time, i 17) where kN0 is rate of loss of free phages to the initiation of new productive, that is, phage-producing bacterial infections, whereas IP is the rate of free phage loss for reasons other than initiation of productive infections 18) 18) also is relevant only to circumstances in which k is a reasonable description of phage adsorption rates, that is, under those conditions where phage adsorption can be reasonably accurately described in terms of mass action 18) does not address is either the rapidity of phage population growth or the potential of phage population growth to increase phages to sufficient densities, that is, to minimum effective phage densities that are sufficient to appreciably reduce bacterial numbers 18) NA NP ¼ IP/kBA Half-maximal bacterial density (KB), Eq 18) that the proliferation threshold is highly dependent on the phage decay rate as that occurs independent of bacterial adsorption, a quantity that can be difficult to ascertain in vivo as so too can be determination of the phage actual burst size or adsorption rate constant in situ 18) which is derived by solving Eq 19) consequently overestimates the bacterial density defining KB, and that overestimation becomes greater the smaller that bacterial density 19) is only an approximation, versus an exact description, comes from at least two sources 19) NA KB _ 1/Lk Authors personal copy Assuring that the expression is equal to less than zero is the requirement that NF < N0, that is, that ending bacterial densities are lower than starting bacterial densities 19) which is derived from the Michaelis–Menten constant, KM ¼ (k_ 1 þ k2)/ k1 19), is N0 ¼ 1/Lk, where N0 is introduced into the expression via Eq 2 _ 108 phages/ml 2 _ 108, 8 2 ¼ 9 2 103 6 2 105 11 2 12 2 3 2 34 2 9 2 and 1 2 Comparing different approaches to calculating MOI 2 for comparisons between Eqs 2 for illustration 2 for summary) 2 for the row corresponding to N0 ¼ 106) 2 Minimum bactericidal concentrations (t ¼ 10 min and V ¼ 1 ml) NF ¼ 1 NF ¼ 10 _6 Constant P Declining P Constant P Declining P N0 a Mb MBC10 c MBC10 d Me MBC10 f MBC10 g 1010 23 2 Stephen Abedon Authors personal copy That is, phage therapy protocols minimally should be designed such that bacterial killing and/or eradication is at least theoretically likely 2 that I generalize Eqs 2 where I extrapolate the straight lines associated with the P ¼ 1010 phages/ml curve 2) 2) 2) and particularly so because assumptions of complete phage adsorption—if over relatively short periods to relatively low bacterial densities—in many instances can be outrageously unrealistic (i 2) can be simplified to MBC > Pt _ MIC þ Nt þ ln Nt ð ÞIP=k; (1 2) where Pt is the phage density at time, t, N0 is the initial bacterial density (i 2)–(1 2, 6 20) is equal to m ¼ (ln(P0Bt/tG)_ln(P0))/t, where tG is defined by Eq 20) where B is burst size and tG is the phage generation time 21) 21) can be set equal to 2L 21) such that 1/kN0 is the average time between phage release from an infected bacterium and phage adsorption 21), whereas the slope of maximal phage population growth, that is, assuming no delay between phage release and phage adsorption, is equal to mmax ¼ (ln(P0Bt/L)_ln(P0))/t 22 Stephen Abedon Authors personal copy type from an environment is sought 22) could hold in situations where reversible adsorption may be assumed 22) thus defines the Michaelis–Menton constant using phage-specific rather than enzyme-specific variables 227) 24 Stephen Abedon Authors personal copy approximately a halving of te, the time until bacterial eradication 2h 2302 3 (Continued) Phage Therapy Pharmacology: Calculating Phage Dosing 21 Authors personal copy B 3 _ 10 _7 s 3 _ 108, 7 3 0 3 10 3 102 4 3 207 3 29 3 8 3 Decimal reduction times 3 for summary) 3 Summary of formulae relevant to phage therapy dosing Constant phage density Nonconstant phage density Minimum inhibitory concentration (MIC), a phage density, Eq 3) 3) which is how the equation is presented by Payne and Jansen (2003) 3)) is not solvable 3, respectively (or 20 3, that is,_ ln(0 3/2 3/kP0 D ¼_ln (1_(2 3–1 33 _ 107 bacteria/ml 33 _ 107 bacteria/ml, rates of phage amplification at best can only be doubled by increasing bacteria availability to phages 3A) 3B for solutions to Eq 3C 3N0/P0))/kN0 Time until bacterial eradication (te), Eqs 4 (¼ ln(108) 4 _ 108, 5 4 _ 108, 6 4 101 2 4 108 18 4 20 4 6 4 7 4 7 4 7 4 7 4 a Bacterial density is in units of bacteria/ml 4 and 1 4 and 1 4 and 1 4 Bacterial eradication times based on Eq 4) so long as an excess of phages is required to assure substantial phage adsorption to bacteria 4) which again should be viewed as an approximation 40) 46 min for reduction to only a single bacterium; compare Fig 47): Although passive biocontrol can be effective, it will usually require large and repeated dosages for success to be ensured 4A and 1 4A and B) 4A, I calculate time until eradication for the phage and bacterial densities indicated 4AandB or Fig 5 _ 10 _5, or _1/20,000 bacteria remaining uninfected given a multiplicity of 10 5 _ 10 _9 and L ¼ 30 min, then 32 Stephen Abedon Authors personal copy KB ¼ 1 5 _ 10 _9 min/ml along with NF ¼ 1, then te ¼ 5 5 _ 10 _9 ml/min 5 _ 10 _9 ml/min 5 _ 10 _9 ml/min 5 _ 10 _9 ml/min 5 _ 10 _9 ml/min 5 _ 10 _9 ml/min and P0 ¼ 1010, 109, 108, 107, 106, 105, 104, 103, 102, or 101 phages/ml, then D _ 10 _1, 100, 101, 102, 103, 104 , 105, 106, 107 , and 108 min, respectively 5 _ 10 _9 ml/min and t ¼ 10 min, then only for N0 in the range of 107 bacteria/ml and greater are the results substantially different from those seen assuming instead phage replacement, that is, given no change in phage density over time (Table 1 5 _ 10 _9 ml/min and VN0 ¼ 1010, 109, 108 , 107 , 106 , 105, 104 , 103 , 102, and 101 bacteria/ml, then MBC is greater than 9 5 _ 10 _9 ml/min for examples k1 Phage adsorption constant Description of rate of formation of enzyme–substrate complexes, here as equivalent to initial, reversible phage-bacterium complex k_1 Phage desorption constant Constant describing rates of conversion of reversibly adsorbed phages back into free phages (i 5 _ 10 _9 ml/min, and t ¼ 10 min 5 _ 10 _9 ml/min, and t ¼ 10 min); a generalization of Eq 5 _ 10 _9 ml/min, t ¼ 10 min, and P0 ranges from 1010 phages/ml down to 101 phages/ml (as indicated) 5 _ 10 _9 ml/min, then 109 phages/ml should achieve bacterial extinction for cultures of 100% phage-sensitive bacteria even given nearly 1010 bacteria/ml, all in only 10 min 5 _ 10 _9 ml/min; also, implicitly, NF ¼ 1 bacterium/ml (i 5 _ 108, 4 5 1 5 1 5 1 5 100 101 102 103 104 Bacterial density (N0) in bacteria per ml DEq 5 101 102 103 104 105 106 107 108 109 1010 102 103 104 105 106 107 108 109 1010 2 5 101 102 103 104 105 106 107 108 109 1010 2 5 106 13 5 13 5 2 5 3 5 3 5 4 5 5 5 Bacterial eradication times based on Eq 5 except that what is shown is the ratio of te as defined by Eq 5 for graphical representation 5) 5) 5) as otherwise seen in full with a doubling of k, that is, as one can derive directly from Eq 5) where 1 refers to a single remaining bacterium 5, 9 5A and B and see, too, Appendix) 5A and B, and also see especially Table 1 5A) 5A), too results in much less than a doubling of te 5B, the 101 phages/ml curve is not present in (B) 6 _ 108, 3 6 1 6 11 6 25 6 27 6 36 6 4 6 Comparison of approaches to calculating bacterial eradication times 6 for comparison of the outputs of Eqs 6) 6) and solving for P0, then MBC10 ¼ P0 > lnðVN0Þ=k10; (1 6) and solving for P0, then MBCt ¼ P0 > N0 ln VN0 ð Þ= 1 _ e _kN0t _ _ : (1 6) Phage Therapy Pharmacology: Calculating Phage Dosing 9 Authors personal copy for P0 > MBC, that is, as holds for phage densities that exceed their minimum bactericidal density, keeping in mind that an important component of M is phage density (below) 6) with NF as an endpoint replacing 1 (see Tables 1 6) with V ¼ 1 ml 6) works out to M > 13 6)–(1 6)–(1 6), MBC is a function of bacterial density or, indeed, of total bacterial numbers 6), what MOI, M, is necessary to reduce 108 bacteria to 1 bacterium That answer is 18 6, and 2 63P0/N0, indicating that M as employed in Eq 7 _ 10 _7 s, which is an expectedly longer interval given that starting bacterial densities are higher 7 _ 108, 2 7 23 7 3 7 3684 7 8 7 8 7 and 23 7 for illustration) 7 Phage proliferation threshold as a function of hypothetical phage decay rates 7) 7)– (1 7) and (1 7) and (1 7) and (1 7) if bacterial densities are small, or 10-fold higher phage densities than those indicated by Eq 7) is predictive at N0 > 1/kt only if phage densities are not allowed to change over the 1_103 1_102 1_101 1_100 1_10-1 1_10-2 1_10-3 1_10-4 1_10-5 1_10-6 1_10-7 1_10-8 1_10-9 1_10-10 105 106 107 Bacterial density per ml (N0) N0=1/kt P0=101ml-1 P0=1010ml-1 MOlactual 108 109 1010 FIGURE 1 7) or (1 7) where phage densities are assumed to hold constant, that is, adsorption with replacement 7) which is meaningful for VN0 > 1 7) will become less predictive at increasing bacterial densities such as above N0 ¼ 1/kt 7) will provide greater predictive power under these conditions regardless of the bacterial density 7) with V ¼ 1 ml and k ¼ 2 7), (1 7), which approximates very low levels of phage adsorption, rather than Eq 7), which is based on M ¼ P0kt 8 8 _ 108, 1 8 _ 108, and 0 8 1 8 14 8 18 8 2 8 20 8 345 8 5 8) 8) 8) 8) 8) for bacterial reductions to densities of NF bacteria/ml 8) For k ¼ 2 8) for various volumes and for t ¼ 10 as well as for t ¼ 100 min 8) if bacterial densities instead are high 8) is slightly lower (by 37%) than both MOIinput (¼P0/N0) and MOIactual (¼P0kt) at that same bacterial density 8) nonetheless is more complicated than either Eq 8) when all phages adsorb (i 8) will remain fully predictive across all bacterial densities and degrees of phage adsorption so long as densities of free phages are allowed to decline and do so solely as a consequence of phage adsorption 8) with V ¼ 1 ml, k ¼ 2 8), and even (1 8), and then simply use more phages than indicated, such as 10-fold more (or greater) if bacterial numbers are uncertain, 10- fold higher than that indicated by Eq 8), but see also Eq 8), upon which these graphs are based, phages are assumed to decline in density as a function of bacterial adsorption, contrasting Eq 8), which is based on M ¼ P0(1_e _ kN0t)/N0, will provide a more accurate prediction of MBC, including better than Eq 8): MBC ¼ P0 > N0 ln ðVN0Þ: (1 8, 11 9 _ 108 phages/ml, respectively (see also Table 1 9 0 9 12 9 16 9 2 9 35 9) Another way of viewing this latter point is that if sufficient numbers of bacteria are present, such that the vast majority of added phages succeed in adsorbing, then what is known as MOIinput (¼P0/N0) can be a preferable calculation for M than MOIactual as determined via the formula P0kt and also, of course, a simpler calculation for MOIactual than using P0(1_e _ kN0t)/N0 9) less predictive and indeed increasingly poorly predictive the more below that point that bacterial densities are found 9) underestimate bacterial killing 9) would assume and thereby more bacterial killing than would otherwise be anticipated 9), so we might reasonably ask under what conditions we can use Eq 9), which approximates very high levels of phage adsorption, or vice versa 9), which is based on M ¼ P0/N0, meanwhile is equivalent to Eq 9); generalizations to reductions to NF are also shown (NF ¼ 1, top) MBCt > ln (VN0)/kt MBCt > N0 ln (VN0)/(1_e _ kN0t) MBCt >_ln (NF/VN0)/kt MBCt >_N0 ln (NF/VN0)/(1_e _ kN0t) Decimal reduction time (D), Eqs 9, 4 A A A A A 1-million-fold reduction in bacterial densities, taking into account these and other factors, consequently is not so unreasonable, particularly if complete elimination of a bacterial C 3 A B 1_109 1_108 1_107 1_106 1_105 1_104 1_103 1_102 1_101 1_100 1_10-1 1_10-2 1_109 1_108 1_107 1_106 1_105 1_104 1_103 1_102 1_101 1_100 1_10-1 1_10-2 100 101 102 103 104 Bacterial density (N0) in bacteria per ml Decimal reduction time (D) in min Decimal reduction time (D) in min 105 106 107 108 109 101ml-1 102ml-1 103ml-1 104ml-1 105ml-1 106ml-1 107ml-1 108ml-1 109ml-1 1010ml-1 101ml-1 102ml-1 103ml-1 104ml-1 105ml-1 106ml-1 107ml-1 108ml-1 109ml-1 1010ml-1 1010 100 101 102 103 104 Bacterial density (N0) in bacteria per ml 105 106 107 108 109 1010 FIGURE 1 A conservative alternative choice for NF is 10 _6, that is, 1-million-fold 100 1_10-3 1_10-2 1_10-1 1_100 1_101 1_102 1_103 1_104 1_105 1_106 1_107 1_108 1_109 1_1010 1_10-3 1_10-2 1_10-1 1_100 1_101 1_102 1_103 1_104 1_105 1_106 1_107 1_108 1_109 1_1010 101 102 103 104 105 106 107 108 109 1010 Bacterial density per ml (N0) 100 101 102 103 104 105 106 107 108 109 1010 Bacterial density per ml (N0) Time until bacterial eradication (te) in min Time until bacterial eradication (te) in min P0= 1010ml-1 P0= 1010ml-1 P0= 101ml-1 P0= 102ml-1 B A FIGURE 1 A doubling in bacterial densities, holding all other parameters constant, does not result in nor even come close to doubling the time it takes to reduce bacterial densities to the same endpoint, NF A further simplification that can be made is to assume that phages decline in density solely as a consequence of bacterial adsorption such that IP ¼ 0 A key advantage of phages as antibacterials, however, is their low toxicity (Abedon, 2012a; Abedon and Thomas-Abedon, 2010) which, to perhaps a large degree, obviates the need to precisely calculate minimal effective dosages because the risk to patients associated with employing greater doses is low A multiplicity of 10 therefore might be attained only locally but nevertheless result in successful local bacterial eradication, with achievement of locally higher phage densities due to phage population growth as it occurs adjacent to those bacteria Above, I considered the phage multiplicity of adsorbed phages that is required to reduce bacterial numbers from 1 million to 1 According to Wang et al Accurate MIC calculation also depends on knowledge of rates of bacterial replication in situ as well as in situ phage adsorption constants Achievement of a multiplicity of 10, however, requires sufficient phage numbers and/or sufficient time of bacterial exposure to phages actively, is not what is under discussion in this section) Again one finds that N0 ¼ 1/Lk Again using Eq Again, the more the phages available to adsorb per bacterium, then the faster each bacterium would be adsorbed Almost certainly in the real world, the presented calculation for MBCt will be an underestimation, but with even greater certainty, one can be assured that a failure to achieve MBCt will coincide with a failure to eradicate bacteria (at least so long as phages represent the sole mechanism giving rise to bacterial death) Alternatively, an interval of ti ¼ 1 can be tolerated if d ¼ 0 Alternatively, Eq Alternatively, if complete bacterial eradication is desired, then reducing bacterial viability only to a calculated just below one bacterium actually may be inadequate, or at least not conservative Alternatively, if either longer time spans or less bacterial killing is indicated, then lower phage densities may be adequate, though too much lower than 108 phages/ml will likely result in insubstantial bacterial killing (keeping in mind that how these phage densities are attained, passively vs Alternatively, in the course of solving for N0, Eq Alternatively, it is possible to calculate a minimum phage density necessary to achieve bacterial eradication over a given period of time (i Alternatively, phage host-range mutants can potentially arise, resulting in culture lysis even of resistant bacteria given growth of the phage-resistant bacterial population past its threshold density Alternatively, see Fig An additional and important consideration is that MIC is only that phage density that keeps bacterial populations from growing in number An additional and somewhat comprehensive as well as complementary review of phage therapy pharmacology also was recently accepted for publication (Abedon, 2012a) An experimentally determined ED50, however, might be more fully understood by considering such things as bacterial reduction times and phage proliferation thresholds (above) An important characteristic of these equations is that they are dependent on a half-maximal velocity constant, which is in units of substrate density Another way of stating this is that Phage Therapy Pharmacology: Calculating Phage Dosing 7 Authors personal copy while one of the important utilities of MIC measurement is its relative ease of in vitro, experimental determination, that ease would appear to be utterly lacking when dealing with antibacterial agents that, as is true for most phages used in phage therapy, are capable of reproducing when supplied to their bacterial targets APPENDIX As a result, it can be difficult to distinguish between phage densities that are initially below MIC from phage densities that are initially above MIC since both can give rise, following the typical overnight incubation of MIC assays, to an absence of culture turbidity As above, NF ¼ N0e _kP0t; (1 As an aside, in assessing the above ratios of Eqs As considered at the beginning of this section, margins of safety may be built into protocols by choosing values for NF that are much less than 1 As indicated in Eqs As MOI is equal to the number of phages adsorbed divided by total bacteria of the population being adsorbed, then M ¼ P0(1_e _ kN0t)/N0 As nearly 108 phages/ml are required to eradicate even as few as 10 bacteria/ml in 10 min, then perhaps 109 phages/ml may be viewed, given reasonably high phage adsorption constants, as a good default minimum for biocontrol dosing where both rapid phage action and complete bacterial eradication are desired or required As noted, VN0 is equal to the total bacterial number within an environment of volume, V As one increases bacterial densities further above the proliferation threshold, then rates of phage population growth too continue to increase Assuming a phage burst size (BA) that is sufficiently greater than one, such that BA _ BA _1, then the proliferation threshold, NP, may be defined as, NP ¼ N0 ¼ IP=kBA; (1 Assumptions in this derivation are that bacteria do not replicate nor decline (i At infinite bacterial density note that the mean free time is equal to zero and thus tG ¼ L At this point, if P0 1, then larger initial bacterial densities, if this were seen solely with the third N0 term in Eq B B B b Based upon Eq BACTERIAL DENSITY AND PHAGE POPULATION GROWTH Phages are unique among antibacterials in that they can increase in density not just in the course of but also because of their bactericidal activity Because phages kill bacteria with single-hit kinetics, the idea of a transition between bacteriostatic and bactericidal actions against individual bacteria is not applicable Both of these results are expected, that is, more phages or fewer target bacteria should reduce times until bacterial eradication (see especially Figs By contrast, a doubling of phage density does indeed result in approximately a halving of te (Figs By contrast, the impact of doubling bacterial densities, unless phage densities are limiting, is approximately a doubling of the amount of bacterial survival, that is, as would be seen when holding P0, k, or t constant By contrast, the third caveat is substantially more relevant to phage therapy success and that is the question of whether dosing with higher phage densities necessarily will result in greater phage therapy efficacy or, instead, lower levels of bacterial clearance than the application of lower phage densities (Abedon, 2012a,b; Abedon and Thomas-Abedon, 2010) By contrast, when N0 ¼ 2 _ 101 then te ¼ 6 By effective I mean that number of phages from an average burst that survive to infect new bacteria (Abedon, 2009d; Chan and Abedon, 2011) By way of orientation, note that for 106 phages/ml, D ¼ 920 (_103) min, whereas even for 107 phages/ml, D ¼ 92 (_102) min (Fig C C C c Based upon Eq C) Complete phage adsorption As the number of phages within a culture that have adsorbed approaches 100%, then P0(1_e _ kN0t)/N0 becomes reasonably approximated by simply P0/N0 CONCLUSION Development of the use of phages as therapeutics could be aided by an accessible theoretical understanding of especially bactericidal dosing consequences Consequently, here, I define MBC as that phage density capable of reducing bacterial densities to zero, leaving undefined for the moment just what one means by zero Consistent with previous calculations, my assumption is that killing is sufficiently fast that both phage and bacterial replication can be ignored (where replication by the former presumably would increase rates of killing, by supplying more phages, while replication by the latter, by supplying more bacteria, would increase the time until total killing has been achieved) Continuous dosing is equivalent to ti ¼ 0 and is necessary if P0 ¼ Pe, that is, should dosing supply just enough phages to be effective Contrasting the simplicity of Eq Curves differ in terms of actual phage burst size, ranging from 10 phages/burst (top curve) to 1000 phages/burst (lower curve) Curves vary in terms of phage density (per ml; as labeled) and are as presented in Figs Curves vary in terms of phage density, as labeled for top and lower curve; k ¼ 2 Curves vary in terms of phage density, as labeled for top and lower curve; k ¼ 2 Curves vary in terms of phage density, as labeled; k ¼ 2 D d Based upon Eq Decimal reduction time Decimal reduction time (D-value, D) is the time it takes to reduce bacterial density by 90% Depending on the system, phage densities attained can be the consequence of either traditional dosing only (passive treatment) or, instead, a combination of conventional dosing such as by the clinician with phage in situ population growth (active treatment) Despite these complications, it is possible to calculate what interval between doses, ti, would allow a maintenance of phage densities at or above Pe Determinations of TD50 or LD50 are similarly not currently addressable using mathematical modeling e e e e e E e e e e e e e e e e e e e e e e Based upon M >_ln(NF/VN0), where NF ¼ 10 _ 6; a generalization of Eq Effective burst size therefore is equal to the product of the actual burst size and the likelihood that a free phage is going to survive to successfully infect a bacterium; see Abedon and Thomas-Abedon (2010) for the derivation of Eq EFFECTIVE DOSE REVISITED What ultimately matters given any drug application is the effective dose versus the toxic dose, which we can described in terms of an ED50 versus TD50 or LD50 (these are the dosages, respectively, which achieve efficacy, toxicity, or lethality in 50% of the individuals so administered) Empirical determinations of ED50, if possible and however defined, nonetheless should be helpful as a guide toward informed dosage calculation Equation (1 Equation (1 Equation (1 Equation (1 Essentially, what Eq Even if the goal is to reduce bacterial densities to simply below one, then to achieve that goal for larger than unit volumes one must increase the level of killing sought by the actual volume F f Based upon MBC10 ¼ P0 >_ln(NF/VN0)/k10, where NF ¼ 10 _ 6; a generalization of Eq Failure to take these simple calculations into account not only can be wasteful in terms of time and resources but also irresponsible with regard to the health and well being of others Finally, k1 describes the affinity of enzyme for substrate, here equivalent to the phage adsorption constant, or k Finally, the calculations presented assume that specific unit volumes are being treated, such as a single milliliter First is that 1/L is only an estimation of k2 First, phage densities need to be maintained at or above some minimal level, in situ, to achieve a desired level of bacterial killing (Abedon, 2012a) First, the capacity of a bacterium for phage adsorption (Adams, 1959) might limit M, by which I mean that the total number of phages a bacterium is capable of adsorbing may be saturable First, under any of the first three dosing regimens auto dosing can occur For (A), t ¼ 10 min, and for (B), t ¼ 100 min For a summary of mathematical abbreviations used in this chapter, see Table 1 For active treatment, where phage replication is required to achieve an effective dose, then an ED50 in all likelihood will not be easily estimated through modeling, given the physiological and spatial complexities seen in situ versus in vitro For additional discussion of MIC, see Abedon (2011a, 2012a) For additional discussion of phage killing titers within a phage therapy context, see Abedon and Thomas- Abedon (2010) and Abedon (2011a), see also Abedon (2012a) For all curves, k ¼ 2 For both panels k ¼ 2 For example, at P0 ¼ 1010 phages/ml, MOIactual ¼ P0kt ¼ MOIinput ¼ P0/N0 ¼ 250 phages/bacterium at N0 ¼ 1/kt ¼ 4 _ 107 For example, excess phage adsorption can result in a phenomenon known as lysis from without, which results in a sudden curtailment of a bacteriums potential to adsorb additional phages (Abedon, 2011b) For example, for k ¼ 2 For example, if one starts with 108 phages/ml and 105 bacteria/ ml, then 1 million-fold killing according to Eq For example, without question if MIC is not reached either in the course of dosing or following subsequent phage population growth, then there can be no expectation that elimination of ongoing bacterial infections might be achieved For passive treatment, that is, phage therapy whose efficacy is independent of the phage potential to replicate, then the ED50 presumably will be some dosage that is less than or equal to MBC, that is, which is sufficient to reduce bacterial densities either substantially or completely For phages there effectively are four different strategies of dosing, though with some overlap For phages, these values are less useful than for standard antibiotics owing to a combination of the bactericidal nature of phages, their single-hit bacterial killing kinetics, and the phage propensity to replicate upon infection of their bacterial targets, though the latter can be eliminated through genetic or physical phage modification (Goodridge, 2010; Hudson et al For substantial phage replication to occur, however, one typically must also have substantial bacterial densities Phage Therapy Pharmacology: Calculating Phage Dosing 13 Authors personal copy 103 1_105 1_106 1_107 1_108 1_109 1_1010 1_1011 1_105 1_106 1_107 1_108 1_109 1_1010 1_1011 A B 104 105 V = 10-3 ml V = 106 ml Bacterial density per ml (N0) MBC10 in phages per ml (P0) 106 107 108 109 103 104 105 V = 10-3 ml V = 106 ml Bacterial density per ml (N0) MBC100 in phages per ml (P0) 106 107 108 109 FIGURE 1 For the case where phage density is not held constant, calculations are more complicated Four variables are relevant: phage density held at this constant level (P0), density of phage-adsorbable bacteria also held at a constant level (N0), the phage adsorption constant (k), and final bacterial density desired (NF) Frequency of phage dosing Since phages can decline in number over time due to adsorption to bacteria as well as for other reasons, to maintain phage densities in situ it can be necessary to supply additional phages Further comparison of MOIactual with MOIinput Unless phages adsorb with replacement, then Eq Further, and as noted, phage densities in situ can be achieved by more than just that number of phages added, as phages typically are also capable of replicating over the course of effecting their bactericidal action Further, at N0 ¼ 1/kt, then M ¼ P0(1_e _ kN0t)/N0 ¼ P0(1_e _ 1)/N0 ¼ 0 Further, in supplying additional phages, we often can assume that P0 _ Pe such that phage densities postdosing are mostly a function of P0 rather than being dependent also on in situ phage densities at the time of dosing Further, it assumes neither phage nor bacterial replication along with 100% phage adsorption Further, MBC is effectively independent of Nt—except at bacterial densities where MIC _ Nt does not hold—as, in fact, Nt will tend to be small relative to MIC in Eq g g g G g g g g Based upon MBCt ¼ P0 >_N0 ln(NF/VN0)/(1_e _ kN0t), where NF ¼ 10 _ 6 (V ¼ 1 ml, k ¼ 2 Given that caveat, MOI is simply the ratio of phages to bacteria with the qualification that the only phages that are counted are those that have adsorbed (and, strictly, the only bacteria that should be counted are the ones that can be adsorbed) Given these assumptions, then the fraction of surviving bacteria is equal to e _ M, where M is the ratio of adsorbed phages to total bacteria, that is, the phage multiplicity of infection (MOI) Given these assumptions, then we can approximate KB as KB _ 1=Lk; (1 Given these two assumptions, then we can define what I will call an effective phage burst size (BE)—or basic reproductive number as described by Payne et al Given this consistency, and generalizing from the definition of the Michaelis–Menten equation, then it is conceivable that the approximation, KB _ ðk_1 þ 1=LÞ=k; (1 h All MBC10 presented values have been divided by 108; thus, 9 Half-maximal phage population growth rate Elsewhere, I show that the Michaelis–Menten or Monod equations, which describe enzyme activity or bacterial growth rates as functions of substrate densities, provide a means of modeling phage population growth (Abedon, 2009c) Here, I choose t ¼ 10 min (thus MBC10), as this number (1) is less than bacterial doubling times, (2) represents a reasonable interval for situations where phages are being employed as a disinfectant, for example, such as of Listeria contamination of food or food-handling equipment (Hagens and Loessner, 2010), and/or (3) may allow reasonable phage survival under conditions where phages are subject to relatively rapid decay, such as following environmental application (Balogh et al Here, I provide discussion of various means by which such estimations may be relatively easily calculated Here, I provide two calculations Here, if rates of phage adsorption to bacteria are slow relative to phage latent period lengths, then once again fewer phages than anticipated will be lost in the course bacterial killing (Abedon, 1990) How many phages then should one supply to effect substantial phage killing over desired time periods This question I address, in this section, in terms of both phage numbers per dose and frequency of phage application How often, then, should one repeat dosing to sustain minimally effective phage densities within the vicinity of target bacteria in situ Payne and Jansen (2003) provide a formula to calculate this rate, which I discuss further in Abedon (2012a) However, either this capacity will be sufficiently high as to not substantially impact bacterial killing, for example, >100 adsorbed phages/bacterium, or it will have the effect of limiting phage inactivation as a consequence of secondary adsorption to bacteria However, if phage densities are allowed to decline as a consequence of phage adsorption to bacteria, then Eq However, phage densities necessary to attain this multiplicity need only be achieved within the immediate vicinity of target bacteria rather than necessarily throughout the organism or environment that is being treated However, when host densities are small, or when phage replication cannot otherwise be counted upon to give rise to higher phage densities (e I I also return to the issue of peak phage density, which must meet or exceed an effective phage density for phage therapy to be successful I discuss this concept of peak phage densities, concentrations, or titers in greater detail elsewhere (Abedon, 2012a) I then solve these equations for NF ¼ 10 _6 bacteria/ml (columns 5–7) versus NF ¼ 1 bacterium/ml Identifying this point of intersection in fact is trivial as all one does is to determine Phage Therapy Pharmacology: Calculating Phage Dosing 15 Authors personal copy at what bacterial density MOIinput (¼P0/N0) and MOIactual (¼P0kt) are equal, which occurs when kN0t ¼ 1 If greater killing is desired, if phage penetration to bacteria is inefficient, if phage replication to sufficient densities cannot be counted upon, if phage loss other than due to bacterial adsorption is a concern, or if more rapid bacterial killing is required, then greater numbers of phages must be supplied If initial doses can be very high and/or auto dosing can supply necessary additional phages as needed, then repeated dosing may not be necessary If that number is 1 million, for example, then Eq II III IMPACT OF PHAGE AND BACTERIAL DENSITIES ON RATES OF BACTERIAL ERADICATION Here, I address the relationships between initial phage and bacterial densities as well as phage affinity for bacteria on the interval required to attain bacterial eradication, that is, the relationships between P0, N0, k, and te as seen in Eq In addition, an alternative and pharmacologically very common approach to sustaining drug densities at necessarily high levels can be accomplished through either multiple or continuous dosing (Abedon, 2012a,b; Abedon and Thomas-Abedon, 2010) In addition, how one calculates MOI can be crucial (above and Fig In addition, time until bacterial eradication only can be calculated given application of sufficient phage numbers to achieve that eradiation (Appendix) In addition, what Eq In addition, whether or not the calculation also requires assumptions of phage constancy, that is, phage adsorption to bacteria with replacement, depends on how one definesMOI,M, which I consider in the following as well as in the subsequent sections In any case, Eq In any case, phage dosing always should be indicated as titers rather than as MOIs regardless of whether or how MOI calculations are used to calculate dosing (Abedon, 2010a, 2011a, 2012a,b; Abedon and Thomas-Abedon, 2010) In contrast, any biocontrol agent with in vivo activity need only be given as one dose, and typically that dose need only be small In fact, each change by itself such as a doubling of just t can reduce bacterial survival by approximately an order ofmagnitudewhen phage densities are 108/ml and bacterial densities are 106, as indicated using Eq In fact, Eq In other words, phage densities that are less than MIC can result in the generation of phage densities that are greater than MIC, and potentially also equal to or greater than MBC In other words, there is substantial bacteria-killing utility in phage therapy to employing more phages, better phages, or longer incubations of phages with target bacteria In particular, there might exist a bacterial density at which MOIinput ¼ P0/N0 and MOIactual ¼ P0kt are equivalently predictive In short, it is possible to calculate what is at least minimally necessary in terms of phage numbers, time, or perhaps even dosing repetition to achieve phage therapy efficacy (see Table 1 In short, lack of sufficient appreciation of phage biology, and especially that biology as it affects bacterial killing or phage population growth, can result in protocols in which either phage choice or dosing errors are naively interpreted as indications of inadequate phage therapy performance In words, phage densities must be at least large enough so that the desired reductions in bacterial densities are possible Increases in bacterial density, however, can allow for net phage population growth to occur, that is, once bacterial growth has increased bacterial densities to their threshold density (below) Indeed, doubling any two of these parameters, such as P0 and t, has the effect of reducing bacterial killing substantially further Indeed, except at relatively high bacterial densities where P0/N0 becomes a reasonable approximation of M (i Indeed, in the case of lysis from within not only may the number of phages added possess greater overall killing power than anticipated, but more phages may be produced Indeed, reductions of a further million-fold can require relatively small additions of time (compare Fig Initial bacterial density thus has three simultaneous impacts on rates of phage-mediated eradication: (1) the larger N0 then the more bacteria that must be adsorbed to reduce that density to NF, that is, as seen in the expression, ln(NF/N0), where adsorbing more bacteria clearly must take longer than adsorbing fewer bacteria Instead, MOI, strictly speaking, is the ratio of infecting or adsorbing phages to bacteria (Abedon, 2008a, 2011a, 2012a; Abedon and Thomas-Abedon, 2010; Hyman and Abedon, 2009) INTRODUCTION One of the most important medical advances of the twentieth century was the development of antimicrobials, especially selectively toxic agents applicable to living tissues It also requires sufficient phage penetration to bacteria, which may or may not be facile to achieve It also solely reflects the replacement of those phages that are lost to decay such that were no phages lost then literally no bacteria would need to be present in the environment to retain whatever phage 30 Stephen Abedon Authors personal copy densities are present currently It is based on the idea that phages kill bacteria with single-hit kinetics, that multiple phages can adsorb to a single bacterium (thereby wasting all but the first phage in terms of effecting bacterial killing), and that phage adsorption to bacteria can be described using a Poisson distribution (Carlson, 2005) IV Keep in mind that because of the phage dilution that can follow dosing, the actual phage densities within formulations may be substantially greater than P0 as measured in situ, which may be defined as the peak phage density that is achieved within the vicinity of target bacteria following dosing Keep in mind, further, that the phage proliferation threshold is relevant only to the extent that phage replication, in situ, is necessary to achieve phage therapy success (i Keep in mind, though, that comprehensive bacterial clearance does not always represent the desired outcome of phage therapy, as, for example, immune systems also can exert a bactericidal action (Levin and Bull, 2004) Keeping bacterial densities low also reduces the absolute likelihood of bacterial mutation to phage resistance, which if that does occur could result, over night, in culture turbidity even given phage densities exceeding MIC; that is, phage presence selects for phage resistance in bacteria (Hyman and Abedon, 2010) KILLING TITER AND MBC CALCULATION A killing titer is a phage density calculated from the phage ability to kill bacteria Lastly, (3) the larger kN0, then the shorter the interval between phage addition and bacterial eradication Levels of phage adsorption under real-world conditions are uncertain, however Local phage densities given bacterial clumping such as into biofilms or confinement to small volumes may be larger MBC with constant phage density When phage densities can be assumed to hold constant, then M can be approximated as simply P0kt, or P0k10, in determining MBC10 (Abedon, 2008a) MBC without constant phage density What happens if the assumption of phage adsorption with replacement is relaxed That is, if bacterial densities are not low, if additional phages are not supplied via repeated or continuous dosing, and/or if in situ phage replication does not cover free phage losses In these cases, phages remaining at time t, here with t the interval since phage addition, may 12 Stephen Abedon Authors personal copy be defined by Pt ¼ P0e _ kNt (Stent, 1963) MIC for phages thus represents a difficult to determine metric that at best indicates what phage densities presumably are far too low to achieve desired levels of phage therapy efficacy Minimum bactericidal concentration MBC is the minimal antibacterial density necessary to kill bacteria, that is, bactericidal as opposed to merely bacteriostatic densities Minimum inhibitory concentration As bacteria replicate there presumably must exist some phage density (P0) at which the rate of bacterial replication (m), also known as the Malthusian parameter, is exactly balanced by the rate of phage adsorption (k) to those bacteria: Phage Therapy Pharmacology: Calculating Phage Dosing 3 Authors personal copy TABLE 1 More concretely, M at the bacterial density at which kN0t ¼ 0 More generally, Payne et al Much of the difference between the two concepts is time Multiplicity of 10 A modest rule of thumb in phage therapy is that substantial bacterial killing may be achieved if one can attain a multiplicity of 10, that is, a 10- fold ratio of adsorbed phages to bacteria (Kasman et al Multiplicity of infection In considering MOI, the first thing that is essential to do is to debunk a myth Multiplying the fraction of bacteria surviving by the total number of bacteria present within an environment of volume, V, that is, by VN0, yields 1 > VN0e _M; (1 N0emt simply is a description of Nt, that is, bacterial density at time t Nevertheless, as phage-to-bacteria ratios in phage therapy often should be in the vicinity of 10 (i Nevertheless, it is useful to quantitatively appreciate standard dosing metrics as they provide an indication of how many phages are necessary to achieve two important phage therapy milestones: keeping infections from getting worse (as described by MIC) and eradicating bacteria (as described by MBC) Nevertheless, this equation is a good first approximation of how many bacteria are needed to support relatively fast phage population growth No matter what, however, so long as phage densities decline to below minimum effective densities prior to the achievement of desired 26 Stephen Abedon Authors personal copy 100 105 0 Note also that just what constitutes a minimum effective density may be difficult to define but nonetheless consideration of that issue has implicitly been the goal of previous sections in this chapter Note also that the density of bacteria required to support phage population growth decreases, given a larger phage adsorption constant, because then phages are finding bacteria to infect faster than they are inactivated prior to finding bacteria Note also that, in practice, MIC would have to be measured at low bacterial densities so that phage densities are minimally affected by bacterial adsorption as well as resulting phage replication (this idea of effectively unchanging phage densities particularly given low bacterial densities will be a recurring theme in this chapter) Note especially from Eq Note first that the relationship between P0 and N0 is constrained to circumstances where _ 1 < ln(NF/N0)N0/P0 < 0 Note for clarificaiton that my use of the term secondary adsorption is equivalent to that of Doermanns (1948) secondary infection rather than to the usage of Payne et al Note in deriving this expression that 1_e _ kN0t is equal to the fraction of phages that have adsorbed Note in either case that this value has been presented as an approximation rather than as a precise definition Note in the footnotes to Table 1 Note that an alternative means of deriving KB can be achieved based upon the following equation presented in Abedon et al Note that from this equation we can predict that the higher the affinity of a phage for a bacterium (k) or the longer the interval over which adsorption is allowed to occur (t), then the lower the bacterial density (N0) at which MOIactual better approximates M than MOIinput Note that no indication is made by this calculation of how fast this bacterial adsorption or killing will occur Note that te is greater the smaller the adsorption rate constant, k; that is, the longer it takes phages to adsorb bacteria then the longer it will take phages to eradicate bacteria from a given environment Note that these ideas represent very basic pharmacological concepts if not necessarily ones that are always rationally applied in the course of phage therapy research and development (Abedon, 2012b) Note that this is a relative value, as it is independent of the phage burst size, which otherwise is assumed to be held constant in these considerations Note that, as in Fig Note three things: (1) the placement of N0 ¼ 1/kt ¼ 4 _ 107 bacteria/ml (dashed, vertical line), (2) the near identity between the graphed curve and that of MOIactual ¼ P0kt as found to the left (solid horizontal lines as found at lower bacterial concentrations), and (3) the near identity between the graphed curve and that of MOIinput ¼ P0/N0 as found to the right (solid-angled lines found at higher bacterial concentrations) Note, however, that phage densities were calculated based on an assumption that bacteria would be reduced to a single individual and hence the number of phages necessary to actually eradicate bacteria would need to be greater, though not much greater Note, however, that these calculations assume an ideal, homogeneous killing environment across all bacteria, and any deviations from that ideal could lead to a requirement for greater durations of phage exposure than as calculated using Eq Notwithstanding its utility, Eq Notwithstanding the above (third) caveat, it is not an unreasonable proposition that if dosing with fewer phages, or drugs in general, does not result in sufficient efficacy, then the establishment of greater drug densities in situ may be warranted, at least to the extent that higher dosing does not compromise patient safety, comfort, or convenience, and so long as at least some reasonable level of efficacy is attained at lower doses Notwithstanding the above-noted considerations, for the sake of simplicity I also calculate times to bacterial reductions to just below 1, though note that, as I will discuss, differences in the amount of time it takes to reduce bacterial populations to 1 cell/ml versus 10 _6 cells/ml are not large, especially when starting at higher bacterial densities Notwithstanding this caveat, for k ¼ 2 NP also decreases the larger the actual burst size Of course, if free phages are lost for reasons other than due to bacterial adsorption, then either more phages will need to be supplied or more time will be required to achieve the same result, or both Of course, this calculation is absolutely dependent on phage adsorption with free phage replacement, as otherwise 109 phages will have little overall impact on 1010 bacteria Of course, this number will be smaller to the extent that (1) BA > BE, (2) BA is less than anticipated in situ versus in vitro, or (3) bacterial densities are less than 107 bacteria/ml Of particular relevance is where lines deviate from 1, that is, where the two equations no longer provide the same output One can then set m equal to mmax/2 and solve for N Particularly, if P0 were very large relative N0, then the value of ln(NF/N0) N0/P0 would approach 0 meaning that ln(1 þ (ln(NF/N0)N0/P0)), and therefore te too, also would approach 0 Payne and Jansen (2001) also describe passive treatment as inundation therapy, while Payne et al Payne and Jansen (2001) define this as MBC > Pt _ MIC þ N0emt þ ln N0emt _ _ IP=k; (1 Payne and Jansen (2001) make the same point this way (p Payne and Jansen (2001, 2003) describe it instead as a clearance threshold Peak phage density How many phages can a bacterial culture produce A maximal estimation is simply peak bacterial density multiplied by phage burst size Phage enhancement and limitations There are three caveats to the above deliberations Phage Therapy Pharmacology: Calculating Phage Dosing 11 Authors personal copy constant along with the situation where phage densities instead decline due to phage adsorption to bacteria Phage Therapy Pharmacology: Calculating Phage Dosing 25 Authors personal copy killing Phage Therapy Pharmacology: Calculating Phage Dosing 27 Authors personal copy therapeutic or biocontrol endpoints, then additional phage dosing will be necessary Phage Therapy Pharmacology: Calculating Phage Dosing 31 Authors personal copy Keep in mind that phage population growth rates are expected to increase as a function of bacterial densities—ignored in such considerations, though, are that bacteria at very high densities may enter into physiological states, such as stationary phase, that are unable to support phage replication Phage Therapy Pharmacology: Calculating Phage Dosing 35 Authors personal copy TABLE 1 Phages adsorbed over the interval, t, therefore would be equal to P0(1_e _ kN0t) Principally, the fraction of bacteria that remain uninfected following the adsorption of a certain quantity of phages is equal to e _M, where M is multiplicity and e is the base of the natural logarithm Proliferation threshold The proliferation threshold is that bacterial density necessary to support per capita phage replication at a rate that is equal to the per capita rate of phage decay Qualitatively, the more bacteria or the slower phage adsorption, then the more phages that are needed to completely eradicate bacteria over a given time interval or, alternatively, the more time that is required Rearranging, M > _ ln 1=VN0 ð Þ¼ln ðVN0Þ; (1 Recall that with Eq Recent reviews and additional discussion of phage therapy pharmacology and related issues are provided elsewhere (Abedon, 2010a; Abedon and Thomas-Abedon, 2010; Abedon et al REDUCTION TIMES AND DOSING An important consideration in phage therapy is not just whether a certain phage density can reduce bacterial populations to extinction but also how long these reductions can take (Abedon, 2008b) Regardless of the impact of whether phage numbers do or do not decline over the course of therapy or biocontrol, the achievement of a given MOI will give rise to a specific extent of bacterial killing Relaxing the assumption that phage densities remain constant I start with NF ¼ N0e _M where M ¼ P0 1 _ e _kN0t _ _ =N0: (1 Second, one may not actually know the bacterial density that is being treated, so by building in greater levels of killing, then one might be able to kill as many bacteria as are actually present Second, phage dosing cannot be so high that toxicity or other side effects occur See Abedon (2011a, 2012a) for further discussion See also Table 1 See Fig See Fig See Fig See inset for enlargement Shown is MOIactual calculated as M ¼ P0(1_e _ kN0t)/N0, where k ¼ 2 Similarly, a halving of NF, except whenN0 andNF are fairly similar (i Similarly, and for the same reason, N0 cannot be too large relative to P0 Similarly, for 107, 106, 105, 104, 103, 102, and 101 bacteria, the answers are M ¼ 16 Smaller values of P0 clearly have this effect so result in a larger te, that is, longer times are required to eradicate bacteria if fewer phages are available (i So too the more negative simply ln(NF/N0),whichoccurs ifNF is smaller orN0 larger, then the larger te (Figs Solving for t, assuming that Nt/N0 ¼ 0 Solving this numerically, for k ¼ 2 Specifically, P0 must not be too small, that is, P0 >_ln(NF/N0)N0 must hold for Eq Specifically, when bacterial densities are lower than N0 ¼ 1/kt, where kN0t ¼ 1, then MOIinput overestimates M, rendering Eq STANDARD ANTIBACTERIAL IN VITRO METRICS Metrics that are commonly applied to antibacterials include minimum inhibitory concentrations (MICs) and minimum bactericidal concentrations (MBCs) Substituting for M in Eq Substituting into Eq Such estimations, though, are based on average bacterial densities That is, declines in phage densities can occur via two distinct routes, dilution as well as decay, and peak phage density at the site of phage activity will be the result of some balance between phage numbers supplied, dilution of phages between the point of application and reaching that target site, and decay in phage numbers also that can occur in the course reaching these sites That is, for the phage equivalent to these equations, we can describe a constant, here KB, which is the bacterial density that supports phage population growth that occurs at half of the maximum rate it possibly can, where the maximal rate assumes that phages find new bacteria as soon as these phages are released from infected bacteria That is, from N0 ¼ 1 That is, it does not take many phage infections to replace very few phages, given very low decay rates, nor many bacteria (see Fig That is, MOI is not necessarily synonymous with the ratio of phages added to bacteria That is, rather than offering a cure to bacterial infections, MIC at best prevents those infections from getting worse That is, the added ratio represents the limit, given infinite time for adsorption, of the infecting ratio That is, twice the minimum generation time defined as L That is, were phage decay to not occur, then the actual phage density in situ would remain either equal to or larger than P0 and treatment success should occur so long as these phage densities are greater than or equal to Pe The above two caveats are both conservative in the sense that they have the effect of making the calculations provided by Eqs The appearance of various limitations to antibiotic efficacy nonetheless is allowing a resurgence of less broadly active antibacterial agents such as phages as antibiotic substitutes (e The associated derivations are present in the following two sections where I consider the situation during which densities of free phages remain TABLE 1 The basic premise of Payne and Jansens derivation is that a clearance threshold, here MBC, is equal to the minimum phage density necessary to reduce bacterial concentrations to below one The constant k_ 1, as traditionally considered, describes the reversibility of interactions between enzyme and substrate The constant, k2, describes the enzyme turnover rate, here approximated by 1/L, and which can be thought of as the phage-infection turnover rate The converse is also true, however, such that if substantial numbers of bacteria are not present, then M is preferably calculated using MOIactual rather than MOIinput The derivation comes from dN/dt ¼ mN_kP0N ¼ 0, where N is the bacterial density, mN is the bacterial birth rate, and kPN is the rate at which phages adsorb bacteria (i The difference between where the two dotted lines intersect with the dashed line and the solid line below is 37% (see inset for enlargement) The first calculation, that of proliferation threshold, actually represents a lack of phage population growth and otherwise does not consider growth rates were population growth to occur, in each case as a function of bacterial densities The first is practical consideration over the duration of therapeutic protocols where, in most instances, fast (e The first is the minimal bacterial density that is necessary to sustain phage populations in light of phage losses The first simply is for the sake of being conservative in ones calculations of bacterial killing The great success of antibiotics, however, thrust these other categories of antibacterial agents out of the limelight The impact of modifying N0, by contrast, is much less The impact of modifying P0, phage density, on te approaches the relative impact of modifying k, the phage adsorption rate constant—with a doubling resulting in either case in actually or 100 101 102 103 1_105 1_104 1_103 1_102 1_101 1_100 1_10-1 1_10-2 1_10-3 1_106 1_107 1_108 1_109 1_1010 1_105 1_104 1_103 1_102 1_101 1_100 1_10-1 1_10-2 1_10-3 1_106 1_107 1_108 1_109 1_1010 A B P0=101ml-1 P0=1010ml-1 P0=101ml-1 P0=1010ml-1 104 105 Bacterial density per ml (N0) Time until bacterial eradication (te) in min Time until bacterial eradication (te) in min 106 107 108 109 1010 100 101 102 103 104 105 Bacterial density per ml (N0) 106 107 108 109 1010 FIGURE 1 The important point of the above paragraph, therefore, is that if substantial phage numbers are lost to adsorption, for example, _1/2, then that multiplicity, M, is better approximated by P0(1_e _ kN0t)/N0 than it is by P0kt The key take home messages from this equation thus are that MBC > MIC and, furthermore, that how much larger MBC is than MIC is dependent on bacterial density at the time of phage addition The larger point is that whenever single dosing alone is inadequate to sustain phage densities at sufficiently high levels to result in adequate bacterial eradication then multiple, continuous, or auto dosing will be required to replace those phages that are lost or diluted The latter difficulties are a consequence of the ability of phages to replicate, and the difficulty in determining an ED50, except empirically, may be seen as analogous to difficulties associated with determining MICs experimentally (above), which, in turn, are also a consequence of the potential of both phages and bacteria to replicate The latter I present in columns 2–4 and which is equivalent to NF as defined implicitly in Eqs The latter means increases in phage numbers that are a consequence of in situ phage replication, i The latter will result in effectively more phages being present than Eqs The latter, alternative approaches to calculating MOI, I have extrapolated (dotted lines) toward the N0 ¼ 1/kt point for the P0 ¼ 1010 phages/ml curve (top) The log-transformed slope of Eq The math underlying these claims as to the importance of achieving a multiplicity of 10 comes from (1) the propensity of phages to adsorb already infected bacteria, thereby wasting many adsorbed phages so far as bacterial killing is concerned and (2) assumptions of a Poisson distribution for those adsorptions The maximal phage population growth rate, again meaning that phages find new bacteria as soon as they are released from their parental infections, can be estimated as 1/L, where L is the phage latent period The overlaps are as follows The phage densities defining these metrics may be attained, in vivo, in situ, or in vitro, either directly as a consequence of conventional dosing (passive treatment) or instead as a function of phage replication while in association with target or related bacteria (active treatment) (Abedon and Thomas-Abedon, 2010) The pharmacodynamic component is twofold The practical consequence of this is that when phages are present in large excesses relative to bacteria (P0 _ N0), then doubling phage numbers, using phages that adsorb twice as fast, or doubling incubation times canmuchmore thandouble the degree of bacterial killing The premise of this study is that at least one form of misunderstanding or misinformation, that associated with the mathematics of phage therapy dosing, may be corrected by supplying phage workers with simple equations upon which phage dosing decisions or even the choice of phage products may be based The reason that Eq The relationship between te and both P0 and N0 is less straightforward in comparison to that for k, but still intuitively what one would Phage Therapy Pharmacology: Calculating Phage Dosing 23 Authors personal copy expect, that is, particularly te is longer the smaller P0 or the larger N0 (or, indeed, the smaller NF) The result is a straightforward formula, as provided by Payne and Jansen (2003): ti ¼ ln P0=Pe ð Þ=d: (1 The result is the following variation on Eq The result, as above, that is, Eq The resulting increase in phage numbers can give rise to subsequent phage-mediated bacterial eradication The same is true for lysis from within, which is the normal phage-induced Phage Therapy Pharmacology: Calculating Phage Dosing 17 Authors personal copy bacterial lysis, that is, that has the effect of releasing a phage burst The second and related concern is that remaining bacteria can replicate, slowing the decline of their populations The second calculation by contrast describes a relative growth rate that is a function of phage properties, that is, of phage latent period length, burst size, and adsorption constant as well as bacterial density (Abedon, 2009b, 2012b; Abedon and Thomas-Abedon, 2010) The second is that 1/k provides a flawed approximation of the contribution of phage adsorption to phage population growth rates, particularly at lower bacterial densities (Abedon et al The second issue is similar and has to do with phage-associated declines in bacterial adsorptive capacity The second overlap is between multiple and continuous dosing where continuous dosing is simply multiple dosing with the time interval between doses reduced to zero The second presence ofN0 inEq The two extremes are ti ¼ 0 and ti ¼ 1 There are three components to the formula: (1) What are the phage densities that either are or can be applied per dose (P0), (2) what phage density is minimally effective (Pe), and (3) how fast do phages decay in situ Note that this decay rate is by any means other than just due to phage adsorption to bacteria and also represents a net decay, that is, phage overall losses taking into account also any gains that might occur due to phage in situ replication (i Therefore, if only passive treatment is anticipated—that is, either no requirement for phage population growth in situ to attain desired phage densities or no potential for phages to do so—then one might simply employ more phages than would otherwise be expected to achieve desired levels of phage adsorption rather than obsessing over calculation subtleties These are (1) single dosing, (2) multiple dosing, (3) continuous dosing, and what I have described as (4) auto dosing (Loc-Carrillo and Abedon, 2011) These criteria, however, are better captured, operationally, by the concept of the phage killing titer, which can then be employed to provide a more useful MBC calculation These increases, however, are expected to diminish with ever higher bacterial densities, resembling an enzyme saturation curve, and these declines are not necessarily a function of bacterial physiology but instead can be due solely to diminishing returns associated with increasing bacterial availability These increases, however, can be countered by phage decay/inactivation These latter relationships are discussed further in Appendix These tendencies are opposed to some extent by the third presence of N0 in Eq These two observations can be restated as follows: (i) more phages are required to drive bacteria to extinction than are needed solely to prevent bacterial populations from growing and (ii) if bacteria are displaying a net decline in density, such as may be induced by phage presence, then, by extrapolation, eventually phage-sensitive bacteria will go extinct They also can lead to misleading assumptions about the potential for bacterial populations to support the degree of amplification of phage populations that is necessary to achieve those effective phage densities This approach would be instead of adding fewer phages than are required to achieve complete bacterial eradication This better predictive power is true even if phage adsorption is insubstantial but still results in a depletion of free phages (i This concern comes about for at least two reasons This difference expands to a limit where the P0 toN0 ratio is 2 This difference is seen at the top of Fig This I consider in the following section This idea was also expressed by Murray and Jackson (1992): To maintain its population, a virus must, on average, have at least one of its progeny successfully infecting a member of its host species This increase in rates of phage population growth with higher bacterial densities also serves as the basis of proliferation thresholds, that is, below the proliferation threshold phage replication does not keep up with phage decay, at the proliferation threshold phage replication and decay balance, and above the proliferation threshold rates of phage replication exceed rates of phage decay, where proliferation threshold in all cases is measured in bacterial densities, that is, see Eq This inequality, however, is simply P0/N0 >_ln(NF/N0), which, in turn, is equivalent to Eq This is because the standard indication of not achieving MIC is an increase in culture turbidity, which is a sign that bacterial growth has not been inhibited This is because, as above, we employ M ¼ P0(1_e _ kN0t)/N0 rather than M ¼ kP0t This is equivalent to the reversible interaction of phage with bacteria, a phenomenon which, at least up to this point, I ignore This is spontaneous phage desorption, as indicated by k_ 1, that is, as is thought to occur prior to the achievement of irreversible adsorption (Ackermann, 2007) This is the so-called proliferation threshold (Payne et al This latter effect could dominate outcomes at best only under extreme conditions This latter issue without question is something that is in need of rigorous characterization both in the course of developing phage therapy protocols (are higher phages doses preferable to lower ones, or vice versa ) and, should lower doses prove more efficacious, then gaining a mechanistic understanding of why this might be so should be of obvious utility toward ongoing phage therapy development This latter propensity, however, is impacted by the value of P0 but not that of k This result, though strange, does possess a modicum of biological plausibility since the expression kN0 actually is a description of phage adsorption rates and the larger the value then the greater number of phage adsorptions that occur within a culture per unit time This retention of adequate phage titers in situ, for example, could be a consequence of phage auto dosing countering any phage decay that might otherwise occur This value thus is equal to what can be described as a phage mean free time Those further infections are called secondary infections in the parlance of Payne et al Though such phenomena are not mechanistically identical to the concept of adsorption capacity, nonetheless fewer phages end up being lost to bacterial adsorption, again resulting in a potential for more bacterial killing than one otherwise might have anticipated (since lysis from without in fact results in bacterial killing) Thus KB _ (0 þ (1/L))/k ¼ 1/Lk Thus the potential tendency introduced by the third N0 term in Eq Thus, 107 bacteria/ml combined with a burst size (BA) of 100 should yield, maximally, a peak phage density of 109 phages/ml Thus, 2L ¼ (1/kN0) þ L Thus, 8 Stephen Abedon Authors personal copy MBC > Pt _ MIC þ Nt; (1 Thus, assuming constant phage densities, and an adsorption constant of 2 Thus, bacterial densities of 105/ml will support the Phage Therapy Pharmacology: Calculating Phage Dosing 33 Authors personal copy production of only 107 phages/ml, maximum (assuming BA ¼ 100) Thus, BE ¼ BA kN0 kN0 þ IP _ _ ; (1 Thus, D ¼ t ¼ _ ln ð1 _ ð2:3N0=P0ÞÞ=kN0: (1 Thus, e _10 ¼ 4 Thus, Eq Thus, estimations based on one-million-fold killing should result in a greater likelihood of completely eradicating a given density of bacteria Thus, for example, at a biologically implausible phage density of 1017 ml _1 in combination with N0 ¼ 101 bacteria/ml, and an adsorption Phage Therapy Pharmacology: Calculating Phage Dosing 37 Authors personal copy constant of 2 Thus, however you define minimum effective phage densities in situ, if you supply just that amount per dose, 28 Stephen Abedon Authors personal copy then you have to make sure that amount is always present, which must be achieved via continuous dosing if net decay too is continuous Thus, if phage densities are too low, that is, below MIC, then culture densities of bacteria will increase Thus, in considering MOIs, one must always remember that time is a variable (Abedon, 1990; Kasman et al Thus, my advice would be to employ both formulae, Eqs Thus, there are limits on how many phages may be applied at one time along with what levels phages can decline to before additional dosing becomes necessary Thus, to eliminate 1 million bacteria from a culture, assuming that neither phages nor bacteria replicate, then approximately 14 phages must adsorb for every bacterium targeted (see next section for additional calculation) Thus, to reduce bacterial densities to below one in 1 l, one must seek to reduce bacterial per ml densities to 1000-fold lower than one Thus, while the calculation of MBC at higher bacterial densities such as greater than 107 bacteria/ml can employ an approximation of P0 ¼ M/N0, that is, M ¼ P0/N0, over even modest time frames (see, however, below), at lower bacterial densities it is important to choose the specific time interval over which phage adsorption is being considered, that is, t toward defining a value that I will call MBCt Time until eradication For a given phage and bacterial density, it is possible to calculate the time, te, necessary to reduce bacterial populations to any level, for example, such as to below one or, alternatively, to some excess beyond one such as to one-millionth of one To a degree, such theory is complicated, for phages, 34 Stephen Abedon Authors personal copy by a number of properties including their single-hit kinetics, the permanence of phage adsorption, the propensity of phages to multiply adsorb individual bacteria, and the phage potential to replicate in concert with bacterial killing To calculate this for phages, one can start with the equation Nt ¼ N0e _ kP0t, where e _ kP0t is the fraction of bacteria not adsorbed assuming a multiplicity defined by kP0t To perhaps a large degree, these concerns can be mitigated by phage population growth, which by increasing phage densities will give rise to faster declines in bacterial densities (Abedon, 2009a; Levin and Bull, 2004; Phage Therapy Pharmacology: Calculating Phage Dosing 19 Authors personal copy Payne and Jansen, 2001, 2003; Payne et al Together these properties can obscure the need to supply sufficient phage numbers to effect substantial bacterial killing V VI VII Volumes of treated environments range from 106 ml (top-most curves) to 10 _3 ml (lower-most curves), both as indicated We can therefore picture a system in which the initial, postdosing condition is P0 but which then declines down toward and then potentially below Pe at some rate, d What, then, are the phage densities required to attain these multiplicities As noted, MOI is a function of not just phage density but also of time Whether the expression is greater than _1, however, is dependent especially on the value of P0 While antibiotics are the most recognizable category of such agents, these microorganism-produced compounds were predated by synthetic antibacterial compounds as well as the use as antibacterial agents of bacterial viruses (bacteriophages or phages; Abedon, 2012a; Kutter et al While attainment of such an MOI may or may not be technically demanding, plus may or may not involve phage population growth in situ, the salient point is that failure to realize a multiplicity of approximately 10 could coincide with insufficient killing of 18 Stephen Abedon Authors personal copy bacteria to achieve reasonable phage therapy efficacy Why calculate for reductions to less than one bacterium There are three important reasons With a ratio of 10 (P0 to N0), then Eq With the resulting (or coming) resurgence of phage therapy—or related phage-mediated biocontrol of nonhuman organisms and environments (Abedon, 2009c)—there comes a need to more formally and consistently define relevant dosing metrics, especially as measured in vitro Within the above-noted constraints, the more negative the value of ln (NF/N0)N0/P0, then the larger te