# Automating Predictive Phage Therapy Pharmacology

## Abstract

**:**

## 1. Introduction

## 2. Predictive Phage Therapy Pharmacology

#### 2.1. Multiplicity of Infection

_{input}vs. MOI

_{actual}(Figure 1)—and a way of predicting the latter. An appreciation of these concepts can be useful toward the development of subsequent calculations of phage titer impacts on bacteria.

#### 2.1.1. MOI_{input} vs. MOI_{actual}

_{input}vs. MOI

_{actual}[14,18]. The simplest as well as easiest to use—but the one that is also often misleading [12]—is MOI

_{input}:

_{actual}(below). Phage therapy dosing based on MOI

_{input}, in other words, at best should be viewed as “hopeful” since in many cases MOI

_{input}does not guarantee nor necessarily even approximate MOI

_{actual}.

_{actual}instead is the more traditionally used meaning of MOI [19]. It is relevant to phage therapy first because it serves as the basis of Poisson distributions of adsorbed phages over susceptible bacteria and second because the extent of the impact of phages on bacteria also is Poissonal (Section 2.5).

_{actual}is similar to that of MOI

_{input}, though with a clear difference:

_{adsorbed}not the initial phage titer but instead the concentration, such as per mL, of phages that have adsorbed bacteria, especially as seen after some interval of incubation of free phages with those phage-susceptible bacteria. That is, whereas MOI

_{input}is defined in terms of the total number of phages added to bacteria (again, such as per mL), MOI

_{actual}is based only on those virions that succeed in adsorbing and, importantly regarding phage therapy, generally only adsorbed phages have an impact on targeted bacteria.

#### 2.1.2. Predicting MOI_{actual}

_{input}, nevertheless MOI

_{actual}still can be fairly easy to determine in vitro as

_{0}is the starting concentration (titer) of free phages and P

_{F}is the number free phages remaining unadsorbed following some interval of time (F standing for “Final”), assuming that all free phage losses are due to virion adsorption of targeted bacteria. Unfortunately, determining P

_{F}can be impractical in vivo. Consequently, it can be helpful instead to be able to predict MOI

_{actual}. In particular, it can be useful to possess some appreciation of the extent to which targeted bacteria may be impacted by treatment phages, with that impact, for a given phage type, generally being a function of MOI

_{actual}(e.g., Section 2.5).

^{7}/mL, and/or over shorter adsorption intervals, such as over a few minutes rather than over many tens of minutes. In contrast, at all bacterial concentrations or adsorption intervals, one can instead employ

^{−Nkt}goes to zero as Nkt becomes larger, i.e., given higher concentrations of targeted bacteria, higher rates of phage adsorption to individual targeted bacteria, and/or longer incubation and thereby longer adsorption times. In that case, to the extent that e

^{−Nkt}trends toward zero, then MOI

_{actual}will in fact come to approximate MOI

_{input}.

#### 2.1.3. Running the Calculator

^{7}phages/mL, 5 × 10

^{6}bacteria/mL, a 10 min adsorption period, and an adsorption rate constant [12,15] of 2.5 × 10

^{−9}mL

^{−1}min

^{−1}[20] yields an MOI

_{input}(as equivalent to MOI

_{addition}) of 2 but an MOI

_{actual}based on Equation (5) instead of 0.25, or 8-fold lower. Additionally, a total of only 1.1 × 10

^{6}phages of that original 1 × 10

^{7}will be expected to have adsorbed over that interval, while roughly 4 × 10

^{6}bacteria/mL will be expected to have remained unadsorbed out of that original 5 × 10

^{6}, i.e., about 80% of bacteria targeted will not have been phage adsorbed in this example.

#### 2.2. Bacterial Likelihood of Being Phage Adsorbed

_{actual}, and also solved using moi.phage.org, is simply the likelihood that a targeted bacterium will become phage adsorbed per unit of time, such as per min [15]. An appreciation of this likelihood can be helpful in gaining a better understanding of what may be accomplished upon achieving a given in situ phage titer during treatments. Here, I start with a model of phage adsorption over time and use this to derive the probability of adsorption to a single bacterium over a single unit of time.

#### 2.2.1. Predicting Bacterial Adsorption Likelihood: p(A_{c})

_{t}), is as follows:

_{c}, with the “c” standing for “cell”) can be found simply by setting N, the bacterial concentration, to 1 (again keeping in mind that this is all considered as occurring within 1 mL; see Appendix A of [15] for additional detail):

_{actual}(Equation (4)). We can then approximate the probability of a single bacterium becoming adsorbed per mL and per min as

_{actual}as calculated over one min (see Equation (5) for the latter). Note in Equations (8) and (9) that the lowercase “p” stands for “probability” vs. the uppercase, italicized “P”, which stands for phage concentration, i.e., phage titer. In those equations, P is also implicitly equivalent to P

_{0}as to is N with N

_{0}.

#### 2.2.2. Running the Calculator

^{−9}mL

^{−1}min

^{−1}, for 10

^{6}phages/mL (=P) the probability that a given bacterium (N = 1) will become phage adsorbed over one min, p(A

_{c}), will be 0.0025. For P = 10

^{7}phages/mL, p(A

_{c}) is instead raised to 0.025. At P = 10

^{8}phages/mL, the probability is instead 0.25. This is all assuming that phages are adsorbing with replacement, i.e., as specified by Equation (8). If we assume that phages are not adsorbing with replacement, then bacterial concentration (N) will come to matter somewhat more. Thus, with P = 10

^{8}and N = 10

^{7}, the number of adsorptions per bacterium that are expected to occur over one min, which is the exponent in Equation (9), is 0.2469, while for N = 10

^{8}it is 0.2212, and for N = 10

^{9}, it is 0.0918. These correspond to p(A

_{c}) values of 0.2188, 0.1984, and 0.0877, respectively. The declines seen with greater bacterial numbers in turn are due to substantial losses of free phages to adsorption to the now substantial numbers of bacteria (the Nk term in Equation (9)) in combination with there simply being more bacteria for a given number of phages to adsorb (N as found in the exponent’s denominator).

^{8}, N = 10

^{8}, and t = 60 min, we have an expectation (Equation (5)) of a total (on average) of 1 phage adsorption per bacterium (i.e., in this case 1 = P/N vs. the 0.2212 indicated in the previous paragraph and p(A

_{c}) = 0.3679). With replacement of free phages following adsorption, however, the expectation (from Equations (4) or (7)) is instead an average of 15 phage adsorptions per bacterium over that same 60 min interval with p(A

_{c}) = 0.0000! Thus, unless phage concentrations can be sustained at high levels—e.g., by adding more phages, targeting smaller numbers of bacteria, or if phages are able sustain their numbers on their own such as due to in situ replication (Section 2.6)—then Equation (7)-type estimations can grossly overestimate expected per-bacterium levels of phage adsorption.

_{c}) too will be smaller. Alternatively, with phages that adsorb faster, the resulting p(A

_{c}) will be larger. These various ideas can be translated directly into what can be described as bacterial half-lives and related decimal reduction times (next section).

#### 2.3. Bacterial Reduction Times

_{0.1}). Nearly equivalent mathematically, we can speak of half-lives, which is the time it takes to reduce a target bacterial population by 50% (t

_{0.5}). Alternatively, we can consider reductions by 99% (t

_{0.01}), and so on. In addition, and also similar mathematically, is mean free time (t

_{MFT}), which for our purposes is the amount of time on average that it takes until a given bacterium becomes phage adsorbed. Overall, these constructs, as with likelihoods of bacteria being adsorbed by phages (above), can provide insight into the antibacterial utility of a given in situ phage titer.

_{0.5}< t

_{MFT}< t

_{0.1}< t

_{0.01}. This means that half of a bacterial population will become phage adsorbed faster than the average for single bacterium in a population to become phage adsorbed, and in turn it will take even longer for 90% of bacteria to become adsorbed, or indeed for 99% of bacteria to succumb to phage adsorption. In any case, for all of the presented equations in Section 2.3.1 and Section 2.3.2, it is assumed that phages adsorb with replacement, with the without-replacement case addressed instead in Section 2.3.3.

#### 2.3.1. Bacterial Half-Lives: t_{0.5}, and Also t_{MFT}

_{MFT}by −ln(0.5) (the 0.5 for half-life), which is equivalent to ln(2). Thus,

#### 2.3.2. Decimal Reduction Times: t_{0.1}, plus t_{0.01}

#### 2.3.3. Phage Adsorption without Replacement

_{0}> ln(10)N

_{0}to achieve decimal reduction or P

_{0}> ln(2)N

_{0}phages to reduce unadsorbed bacterial numbers by half.

#### 2.3.4. Running the Calculators

^{6}/mL, and an adsorption rate constant as above, then t

_{MFT}is calculated as 400 min vs. 277 min for t

_{0.5}. Raise the phage titer to 10

^{7}/mL and these numbers are reduced to 40 and 28 min, respectively, or 4 and 2.8 min given 10

^{8}phages/mL (all holding phage titers constant over time). An equivalent calculator, but instead determining phage half-lives as a function of bacterial concentrations, can be found at p-half-life.phage.org. The latter can be used to gain an appreciation of how rapidly a given titer of supplied phages will be expected, as a function of bacterial concentrations (N), to become explicitly antibacterial as they adsorb, e.g., such as 50% of those phages adsorbing per min vs. instead 50% per hour. See also Bull and Regoes [21] for an extension of phage half-life calculations to also include phage losses for reasons other than adsorption to phage-infected bacteria.

^{8}/mL and bacterial concentrations of 10

^{6}/mL. With no decline in phage numbers over time, output is t

_{0.1}= 9.2 min while t

_{0.01}= 18.4 min. At such a low bacterial concentration, the equivalent numbers, if assuming instead phage losses to adsorption, are only 9.3 min and 18.9 min, respectively, keeping in mind that total reductions in numbers of unadsorbed bacteria is ten times that for the latter (t

_{0.01}) vs. the former (t

_{0.1}). Raise bacterial concentrations to 10

^{7}/mL and the equivalent numbers again assuming phage adsorption without replacement instead are 10.5 min and 24.7 min. Then, raise phage titers to 10

^{9}(while keeping N at 10

^{7}/mL) and we find that t

_{0.1}= 0.9 min while t

_{0.01}= 1.8 or 1.9 min (these latter two values are without losses due to phage adsorption and with losses due to phage adsorption, respectively).

#### 2.4. Inundative Phage Quantities

^{3}bacteria in total is sought. In terms of required starting phage titers, I have dubbed this an “inundative phage density” (IPD

_{min}), with “density” and “titer” here being used synonymously. Alternatively, there is an “inundative phage number” (IPN

_{min}), which is the starting absolute number of phages required, that is, rather than starting phage concentrations (the latter again equivalent to “titer” and “density”). As with the other calculations already considered, an implicit assumption is that all targeted bacteria are equally available to phages for adsorption.

_{actual}> IPN

_{min}. Thus, a failure to successfully predict the extent of reductions in bacterial viability in the presence of predicted inundative quantities of phages can be used to indicate the presence of additional phenomena not considered by models. For example, less bacteria killing than expected can be due to not all targeted bacteria being equally available to phages, such as due to the presence of spatial or physiological refuges from phage attack [22]. Lower levels of killing than expected can also be a consequence of outright genetic bacterial resistance to phages and/or instead underestimations of phage adsorption rate constants. Alternatively, greater bacteria killing than expected can be due to the presence of additional antibacterial mechanisms and/or because new phages have been generated in situ (for the latter, see “Active treatment”, below; Section 2.6). In any case, calculations of inundative phage quantities can provide an appreciation of what phage titers should be required to reduce phage-susceptible bacteria to a given total number of remaining bacteria, over a desired length time, particularly as based on the antibacterial action of dosed phages alone.

#### 2.4.1. Inundative Phage Densities: IPD_{min}

_{min}, can be calculated either assuming or not assuming that these titers remain constant over time (Figure 2). As with the approaches considered above, assuming a constant phage titer simplifies calculations but becomes less valid the higher bacterial concentrations or the longer the time frame over which adsorption is allowed to occur. In any case, both phage and bacterial replication are ignored for these IPD

_{min}, or IPN

_{min}, determinations.

_{0}). The final number of bacteria is independent of volume. That is, often when reducing bacterial presence, you want to reduce the number of bacteria to a given lower amount (N

_{F}) rather than to a given lower concentration. Thus, the fraction of bacteria that are expected to survive given the application of some inundative titer of phages will be N

_{F}/VN

_{0}(total ending bacterial numbers divided by total starting numbers of bacteria) and this fraction, or at least its inverse, is used in the same manner as for, e.g., decimal reduction time calculations (Section 2.3.2). If phage titers can be held more or less constant over time, then the minimum titer of phages required to achieve that fraction of surviving bacteria can be descried as

_{min}representing some phage concentration, i.e., P. For a 10-fold decline in bacterial numbers—a decimal reduction and thus x = 0.1—this would be P = ln(10)/kt. With rearranging and modifying the abbreviation for time, this is equivalent to the t

_{0.1}= ln(10)/Pk, as seen above in Equation (12). The quantity ln(10) in turn is equal to the MOI

_{actual}required to achieve this 10-fold reduction in concentrations of viable bacteria, i.e., 2.3.

_{actual}required to achieve the desired degree of reduction in bacterial numbers] divided by [fraction of added phages which succeed in adsorbing over time, t].

#### 2.4.2. Inundative Phage Number: IPN_{min}

_{min}, but without prior knowledge of bacterial concentrations. This approach can be relevant if numbers of bacteria are known or at least can be estimated, but where treatment volumes are less easily determined. There are two ways of going about this. One is to assume that phage titers are known and remain more or less constant or, alternatively, that 100% adsorption of added free phages can be assumed. Missing is the case where phage numbers are instead declining to some intermediate extent, due to phage adsorptions of bacteria, as that extent cannot be calculated without knowledge of bacterial concentrations.

_{T}is initial, unadsorbed bacterial numbers (“T” standing for “Total”), and this is rather than initial bacterial concentrations. As indicted though, this is again a calculation of IPD

_{min}, rather than of a minimum inundative phage number (IPN

_{min}), and this is because required phage titers rather than just phage numbers would be calculated; note also that the numerator again is equivalent to ln(1/x). If phage titers are not easily predicted, i.e., as due to phage application to volumes that are not well defined, we need to resort to assuming instead the noted 100% adsorption of added free phages:

_{min}is thus the total number of phages that need to be supplied, but again assuming 100% adsorption. Note that ln(N

_{T}/N

_{F}) is equal to that MOI

_{actual}(Section 2.1) required to reduce bacterial numbers from N

_{T}to N

_{F}, which, in turn, is an N

_{T}/N

_{F}-fold reduction, and equivalently this is (1/x)-fold. For example, with a 10-fold reduction, ln(N

_{T}/N

_{F}) = 2.3 = MOI

_{actual}. IPN

_{min}, as described by this equation, is therefore equal to that MOI

_{actual}multiplied by the total number of bacteria targeted, i.e., by N

_{T}.

_{actual}should fail to approximate MOI

_{input}, and the degree of discrepancy is known, then one can modify the previous equation as

_{input}/MOI

_{actual}≥ 1 to hold under all circumstances, since it is impossible to adsorb more phages than there are phages (as above, assuming no in situ phage replication). Therefore, the less extensively that phage adsorption occurs, e.g., MOI

_{input}≫ MOI

_{actual}, even assuming ideal adsorption conditions, then the more phages that will be required to reduce bacterial numbers to an equivalent extent.

#### 2.4.3. Running the Calculator

^{6}bacteria/mL (=N

_{0}), and consider only 1 mL of volume (V), then reductions to 10

^{3}bacteria in total (N

_{F}) over one hour (t) requires 4.5 × 10

^{7}phages/mL, assuming via Equation (15) that there are no phage losses (=IPD

_{min}). This changes to 5.0 × 10

^{7}phages/mL given phage losses to adsorption, as per Equation (16) (=IPD

_{min}). Alternatively, via Equation (18), a starting number of only 6.9 × 10

^{6}phages (=IPN

_{min}) is required if 100% phage adsorption is assumed. (Note in the example that 10

^{6}is both the starting bacterial concentration and starting bacterial number since only 1 mL is being considered.)

_{min}determinations are found in Table 3, all assuming a value for k of 2.5 × 10

^{−9}mL

^{−1}min

^{−1}. Notice how nearly the same numbers of phages are required to reduce bacterial numbers to the same amount, e.g., 1 (=10

^{0}), regardless of starting bacterial numbers. Thus, starting with 10

^{6}bacteria/mL in 100 mL requires 1.2 × 10

^{8}phages per mL (assuming no phage losses over time) but still half as many phages starting with only 10

^{2}bacteria/mL despite the 10,000-fold difference in numbers of starting bacteria. Thus, reducing bacterial populations to a substantial extent requires relatively high phage titers and this is so even if starting bacterial concentrations are relatively low. The explanation for why this is the case has to do with the statistics of Poisson distributions (next section).

#### 2.5. Poisson Distributions

_{actual}(Section 2.1). Thus, the fraction of bacteria expected within each of the r categories is defined for a given MOI

_{actual}by a Poisson distribution [16].

#### 2.5.1. Predicting Bacterial Survival

^{0}. Rearranging, then n = MOI

_{actual}= −ln(p(0)) = −ln(N

_{F}/N

_{T}) = −ln(x), keeping in mind that ln(1/x) = −ln(x). N

_{T}is the starting number of unadsorbed bacteria, i.e., as found prior to phage addition, while N

_{F}is the ending or “Final” number of unadsorbed bacteria. MOI

_{actual}is thus equal to the negative natural log of the fraction of bacteria remaining unadsorbed following some extent of phage exposure, or the positive natural log of the fold-decrease in bacterial numbers.

#### 2.5.2. Killing Titers: P_{K}

_{K}) calculations [12] take the above prediction of bacterial survival and literally rearrange it. This, in contrast to much of the above, is therefore a phage titer determination that is based on bacterial survival rather than a prediction of bacterial survival that is determined, at least in part, by knowledge of initial phage titers. As equivalently seen with Equation (18), MOI

_{actual}is multiplied by the initial bacterial concentration, but here with MOI

_{actual}calculated based on the fraction of bacteria that have survived, assuming that all added phages have adsorbed:

_{actual}= −ln(p(0)). Thus, if 10

^{8}bacteria per mL are reduced to 10

^{7}, then the calculated killing titer is −ln(0.1) × 10

^{8}= 2.3 × 10

^{8}. This would be equal to P

_{0}, i.e., the starting phage concentration, assuming that all free phages initially present adsorbed (and that no phage replication has occurred).

_{actual}must equal MOI

_{input}, that is, in order for P

_{K}to be an actual phage titer determination. If insufficient time is allowed for adsorption, however, then MOI

_{actual}will be lower than MOI

_{input}, resulting in the calculated P

_{K}being less than P

_{0}. Consequently, killing titer determinations will always underestimate starting phage titers unless complete phage adsorption is allowed to occur, keeping in mind though that often a small fraction of phages will fail to adsorb seemingly no matter what [23,24,25,26]. Of course, for killing titer calculations to hold true, then bacterial replication also must be insubstantial during phage application. Nevertheless, killing titers can provide at least an approximation of what phage titers would have been necessary to achieve the amount of bacteria killing observed, which can in turn be compared with what phage titers actually had been present at the start of phage treatments of a bacterial population.

#### 2.5.3. Running the Calculators

_{actual}. Note that this need not be an integer. For example, for MOI

_{actual}= 1.5, the app indicates that the fraction of bacteria expected to not have been phage adsorbed is 0.22 (or 0.37 for MOI

_{actual}= 1). Additionally, relevant for certain phage biology experiments is the fraction of bacteria which are singly vs. multiply adsorbed [12]. For MOI

_{actual}= 1.5, these fractions are 0.33 and 0.44, respectively, such that, though with rounding error, 1 = 0.22 + 0.33 + 0.44. Also calculated are the fraction of bacteria, of those that have been adsorbed at all, which have been singly vs. multiply adsorbed. For this same example (MOI

_{actual}= 1.5), those fractions are 0.43 and 0.57, respectively, which also add up to 1. That is, 43% of bacteria that have been adsorbed in this example are predicted to have been singly adsorbed.

^{7}bacteria/mL and end up with 10

^{4}bacteria/mL, then your calculated killing titers would be 6.9 × 10

^{7}phages/mL. Additionally, MOI

_{actual}is calculated, which in this example, would be 6.9. The greatest utility of such killing titer determinations is for use toward establishing the titers of phages—or other agents such as phage tail-like bacteriocins—which, for whatever reason, are unable to form plaques on the bacterial strain being targeted, while still possessing single-hit kinetics of those bacteria [21]. Nevertheless, it is also useful to compare calculated killing titers (P

_{K}) with actual titers (P

_{0}) to assess treatments, with P

_{K}< P

_{0}implying a less-than-ideal phage impact while P

_{K}> P

_{0}would imply instead a greater-than-expected phage impact.

#### 2.6. Active Treatments

- Low bacterial concentrations without clumping and lower starting phage titers. In the case of low bacterial concentrations and no bacterial clumping, phage population growth likely is mostly irrelevant, since in situ phage replication will not be expected to have a substantial impact on more “global” phage titers. That is, bacteria are present in insufficient quantities to produce relatively large concentrations of new phages across environments. Still, these circumstances, given sufficient environmental mixing, are easily modelled mathematically.
- Low bacterial concentrations with clumping and lower starting phage titers. With spatial structure in combination with bacteria being found in clonal clusters—but bacteria nonetheless overall found at low concentrations—phage in situ replication could in fact be relevant, though not globally, and the mathematics portraying such situations is not straightforward. I describe this latter scenario as a locally active treatment [32].

- 3.
- Higher bacterial concentrations without clumping and higher starting phage titers. First is the noted passive treatment in which phage in situ replication is not required to achieve desired levels of bacterial eradication, e.g., as due to the employment of inundative phage concentrations (Section 2.4). This is because sufficient quantities of phages have been supplied via phage dosing alone.
- 4.
- Higher bacterial concentrations without clumping and lower starting phage titers. Second is what I have described as globally active treatment [32]. Here, the assumption is that phage virions are free to diffuse relatively rapidly about environments or otherwise be readily moved about, such as within blood. Therefore, phages produced in one location can give rise to sufficient increases in phage titers, i.e., to inundative densities (Section 2.4.1) throughout a phage-treated environment.

#### 2.6.1. Considering Phage Population Growth

_{t}

_{+1}) is equal to the just-previous phage concentration (P

_{t}) plus new phages generated upon phage-induced bacterial lysis (B meaning burst size) of those bacteria infected one latent period (L) earlier (BkP

_{t}

_{−L}N

_{t}

_{−L}). Subtracted from this are those phages lost to adsorption (kP

_{t}N

_{t}) along with any free phages lost for any additional reasons (I

_{P}P

_{t}). In addition is the construct, ${\mathrm{e}}^{-L{I}_{N}}$, which has the effect of removing phage-infected bacteria that have been lost to non-phage-related decay over the course of one latent period. I

_{N}is defined as the rate of loss of bacterial cells for non-phage-related reasons, as is also employed in the following section.

#### 2.6.2. Considering Bacterial Population Growth

_{t}

_{+1}) as equal to the just-previous unadsorbed bacterial concentration (N

_{t}) but also with new bacteria being added due to bacterial binary fission (μN

_{t}). Bacteria are lost to phage adsorption (kP

_{t}N

_{t}) as well as to phage-unrelated forms of inactivation (I

_{N}N

_{t}). As with modeling phage population dynamics, inactivation can be ignored (that is, by setting the parameter, I

_{N}, to zero). Alternatively, by setting I

_{N}and I

_{P}to the same value, then a chemostat-like system can be modeled, with both parameters thereby describing outflow. Inflow in any case can be ignored because, as noted, nutrient concentrations are not being considered.

#### 2.6.3. Running the Calculator

^{®}, to run these sorts of simulations [13], which involves stepping through the equations vertically in columns, with each row corresponding to one interval. Alternatively, an online calculator for modeling globally active treatment can be found at active.phage-therapy.org, though there this is described simply as “Active treatment”.

^{7}/mL), a phage inactivate rate (0.00001 as a per min fractional loss), an initial bacterial concentration (10

^{3}/mL), the Malthusian parameter (0.013), a phage-independent rate of bacterial loss (also set to 0.00001 and which, as also for phages, is set there deliberately small by default), and a simulation duration (60 min). Running the calculator using those inputs yields an only minor, log

_{10}0.003 increase in phage titers, owing to the very small starting bacterial concentration. This is roughly a 1% increase in phage numbers. In contrast, bacterial concentrations over this span are reduced by log

_{10}0.317, which corresponds to a 52% reduction. Change the starting bacterial concentration to 10

^{6}/mL and over that hour, phage concentrations increase by 1300% (1.146 log

_{10}, which is from 10

^{7}to ~1.4 × 10

^{8}phages/mL) while bacterial concentrations decline by 3.834 logs (down to 1.5 × 10

^{2}/mL, or nearly a 100% decline). Thus, in this latter case, there are sufficient bacteria present to support substantial phage population growth, and this in turn results in more substantial declines in numbers of bacteria, i.e., considerably effective active treatment is occurring. The caveats, however, are that it is difficult to determine in situ phage latent periods or burst sizes as well as bacterial rates of replication, and indeed, determining in situ phage adsorption rate constants as well. Furthermore, it is difficult to assume that in situ environments are homogeneous, or necessarily well mixed, both as required implicitly by the simulation. Still, this calculator allows one to easily play a number “what if?” scenarios regarding starting phage and bacterial concentrations.

#### 2.6.4. Additional Approaches to Predicting In Situ Efficacy, from In Vitro Characteristics

## 3. Discussion

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Improving the Realism of Phage–Bacteria Chemostat Modeling

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**Figure 1.**Comparing MOI

_{input}with MOI

_{actual.}On both sides is the same MOI

_{input}, whereas MOI

_{actual}to the right is equal to two vs. equal to zero on the left. Note that, generally, more than one adsorbable bacterium would be present and phages would adsorb over a Poisson distribution (Section 2.5), i.e., with the average number of virions adsorbed per bacterium equal to MOI

_{actual}. In addition, keep in mind that the quantitative distinction between MOI

_{input}and MOI

_{actual}results from durations of adsorption periods (t) and the phage adsorption rate constant (k), the latter defined by a combination of the properties of the adsorbing phages, adsorbable bacteria, and adsorption environment. Phage (P) and bacterial (N) concentrations, however, also play important roles in determining MOI

_{actual}, as considered below especially in Equation (5).

**Figure 2.**Adsorption with and without replacement of free phages. The mathematically simplified perspective is adsorption with replacement (

**left**) since the result is a constant free phage concentration over time. Depending on circumstances, however, that assumption may or may not be realistic. It may be realistic, though, if free phage numbers are replaced as a consequence of in situ phage replication or if bacterial numbers are small, thereby resulting in few free phage losses due to adsorption. Alternatively, free phage adsorption without replacement (

**right**) explicitly takes into account free phage losses that result from bacterial adsorptions, that is, with free phage concentrations thereby declining over time.

Topic | Section | URL |
---|---|---|

Multiplicity of Infection | Section 2.1 | moi.phage.org |

Phage Adsorptions | Section 2.2 | adsorptions.phage-therapy.org |

Bacterial Half-Life | Section 2.3.1 | b-half-life.phage.org |

Decimal Reduction Time | Section 2.3.2 | decimal.phage-therapy.org |

Phage Half-Life | Section 2.3.4 | p-half-life.phage.org |

Inundative Phage Quantities | Section 2.4 | inundative.phage-therapy.org |

Poisson Frequencies | Section 2.5 | Poisson.phage.org |

Killing Titers | Section 2.5.2 | killingtiter.phage-therapy.org |

Active Phage Therapy | Section 2.6 | active.phage-therapy.org |

Abbreviation | Description | Comments |
---|---|---|

A_{c} | Bacterial probability of being adsorbed | Likelihood of an individual bacterial cell being adsorbed per unit time, e.g., 1 min; the “c” stands for “cell” |

A_{t} | Adsorptions over time | Number of phage adsorptions that occur over some interval of time, t |

B | Burst size | Number of virions produced per phage infection; might range from 10 to well in excess of 100 |

e | Base of the natural logarithm | =2.718… (a non-repeating decimal) |

I_{P}, I_{N} | Decay rate | Rates of loss of free phages (I_{P}) or bacteria (I_{N}) that occur for reasons that are independent of phage adsorption |

IPD_{min} | Inundative phage density | Minimum phage titer required to reduce a bacterial population from some starting number to some ending number over some specified interval of time, not assuming 100% phage adsorption |

IPN_{min} | Inundative phage number | Minimum phage titer to achieve the same as IPD_{min} except here assuming 100% phage adsorption |

k | Adsorption rate constant | Probability that one virion will adsorb one bacterium as suspended in a unit volume of fluid (e.g., 1 mL) over the course of some unit time (e.g., 1 min), hence, e.g., mL^{−1} min^{−1} units, though often expressed instead as mL min^{−1} |

L | Latent period | Measure of the length of infection by a phage a bacterium |

ln | Natural logarithm | For example, ln(2) = 0.69 = −ln(0.5) = −ln(1/2); ln(e) = 1 |

MOI_{actual}, n | Actual multiplicity of infection | Number of adsorbed phages divided by the number of adsorbable bacteria; equivalent to n as used in Poisson calculations |

MOI_{input} orMOI _{addition} | Input multiplicity of infection | Number of phages added to targeted bacteria divided by the number of those bacteria |

Μ | Malthusian parameter | A measure of bacterial population growth rate in per time units |

N, N_{0}, N_{t} | Bacterial concentrations | Subscript 0 refers to initial concentrations, though in many cases this is implied so the subscript is not always present; subscript t refers to the concentration of unadsorbed bacteria following a previous time interval, t |

N_{F}, N_{T} | Bacterial numbers | Subscript F refers to a “Final” number of unadsorbed bacteria; subscript T refers to “Total” and is used instead of N_{0} to distinguish starting bacterial concentration (N_{0}) from starting bacterial numbers (N_{T}) |

p | Probability | This is lower-case “p” without italicization |

P, P_{0}, P_{F}, P_{t} | Phage titer | Subscripts are equivalent to those of N_{0}, N_{F}, N_{t}, with P in all cases referring to phage concentrations, i.e., phage titers |

P_{adsorbed} | Prior titer of adsorbed virions | Number of previously free phages that have now adsorbed, divided by volume, as to be distinguished from P_{0} |

P_{K} | Killing titer | Titer of phages required to reduce a bacterial population from a given starting number to a given ending number, assuming 100% adsorption |

r | Poisson category | Here, e.g., 0 phages adsorbed, 1 phage adsorbed, etc., all per bacterium |

r! | r factorial | For example, 3! = 1 × 2 × 3; 2! = 1 × 2; 1! = 1; 0! = 1 |

t | Time | Generally, here, this is an interval over which adsorption occurs |

t_{0.1}, t_{0.01} | Decimal reduction time(s) | Time it takes for 90% of unadsorbed bacteria to become adsorbed (t_{0.1}) or 99% (t_{0.01}) |

t_{0.5} | Bacterial half-life | Time it takes for one-half of unadsorbed bacteria to become adsorbed |

t_{MFT} | Mean free time | Average length of time it takes for a bacterium to become phage-adsorbed |

V | Volume | Volume that targeted bacteria and targeting phages are suspended in during phage treatments |

x | Fraction bacteria | As surviving following phage exposure (=N_{F}/N_{T}) |

N_{T} → | 10^{10} | 10^{9} | 10^{8} | 10^{7} | 10^{6} | 10^{5} | 10^{4} | 10^{3} | 10^{2} | |
---|---|---|---|---|---|---|---|---|---|---|

VN_{T} → | 10^{12} | 10^{11} | 10^{10} | 10^{9} | 10^{8} | 10^{7} | 10^{6} | 10^{5} | 10^{4} | |

N_{F} ↓ | ||||||||||

10^{−3} | 2.3 × 10^{8} | 2.1 × 10^{8} | 2.0 × 10^{8} | 1.8 × 10^{8} | 1.7 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | Eq. (15) |

10^{−3} | 3.5 × 10^{11} | 3.2 × 10^{10} | 3.0 × 10^{9} | 3.6 × 10^{8} | 1.8 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | Eq. (16) |

10^{−3} | 3.0 × 10^{11} | 2.8 × 10^{10} | 2.5 × 10^{9} | 2.3 × 10^{8} | 2.1 × 10^{7} | 1.8 × 10^{6} | 1.6 × 10^{5} | 1.4 × 10^{4} | 1.2 × 10^{3} | Eq. (18) |

10^{−2} | 2.1 × 10^{8} | 2.0 × 10^{8} | 1.8 × 10^{8} | 1.7 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | Eq. (15) |

10^{−2} | 3.2 × 10^{11} | 3.0 × 10^{10} | 2.8 × 10^{9} | 3.3 × 10^{8} | 1.7 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | Eq. (16) |

10^{−2} | 2.8 × 10^{11} | 2.5 × 10^{10} | 2.3 × 10^{9} | 2.1 × 10^{8} | 1.8 × 10^{7} | 1.6 × 10^{6} | 1.4 × 10^{5} | 1.2 × 10^{4} | 9.2 × 10^{2} | Eq. (18) |

10^{−1} | 2.0 × 10^{8} | 1.8 × 10^{8} | 1.7 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | Eq. (15) |

10^{−1} | 3.0 × 10^{11} | 2.8 × 10^{10} | 2.5 × 10^{9} | 3.0 × 10^{8} | 1.5 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | Eq. (16) |

10^{−1} | 2.5 × 10^{11} | 2.3 × 10^{10} | 2.1 × 10^{9} | 1.8 × 10^{8} | 1.6 × 10^{7} | 1.4 × 10^{6} | 1.2 × 10^{5} | 9.2 × 10^{3} | 6.9 × 10^{2} | Eq. (18) |

10^{0} | 1.8 × 10^{8} | 1.7 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | Eq. (15) |

10^{0} | 2.8 × 10^{11} | 2.5 × 10^{10} | 2.3 × 10^{9} | 2.7 × 10^{8} | 1.3 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | Eq. (16) |

10^{0} | 2.3 × 10^{11} | 2.1 × 10^{10} | 1.8 × 10^{9} | 1.6 × 10^{8} | 1.4 × 10^{7} | 1.2 × 10^{6} | 9.2 × 10^{4} | 6.9 × 10^{3} | 4.6 × 10^{2} | Eq. (18) |

10^{1} | 1.7 × 10^{8} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | Eq. (15) |

10^{1} | 2.5 × 10^{11} | 2.3 × 10^{10} | 2.1 × 10^{9} | 2.4 × 10^{8} | 1.2 × 10^{8} | 9.3 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | Eq. (16) |

10^{1} | 2.1 × 10^{11} | 1.8 × 10^{10} | 1.6 × 10^{9} | 1.4 × 10^{8} | 1.2 × 10^{7} | 9.2 × 10^{5} | 6.9 × 10^{4} | 4.6 × 10^{3} | 2.3 × 10^{2} | Eq. (18) |

10^{2} | 1.5 × 10^{8} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | Eq. (15) |

10^{2} | 2.3 × 10^{11} | 2.1 × 10^{10} | 1.8 × 10^{9} | 2.1 × 10^{8} | 9.9 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | Eq. (16) |

10^{2} | 1.8 × 10^{11} | 1.6 × 10^{10} | 1.4 × 10^{9} | 1.2 × 10^{8} | 9.2 × 10^{6} | 6.9 × 10^{5} | 4.6 × 10^{4} | 2.3 × 10^{3} | Eq. (18) | |

10^{3} | 1.4 × 10^{8} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (15) |

10^{3} | 2.1 × 10^{11} | 1.8 × 10^{10} | 1.6 × 10^{9} | 1.8 × 10^{8} | 8.3 × 10^{7} | 6.2 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (16) |

10^{3} | 1.6 × 10^{11} | 1.4 × 10^{10} | 1.2 × 10^{9} | 9.2 × 10^{7} | 6.9 × 10^{6} | 4.6 × 10^{5} | 2.3 × 10^{4} | Eq. (18) | ||

10^{4} | 1.2 × 10^{8} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (15) | |

10^{4} | 1.8 × 10^{11} | 1.6 × 10^{10} | 1.4 × 10^{9} | 1.5 × 10^{8} | 6.6 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (16) | |

10^{4} | 1.4 × 10^{11} | 1.2 × 10^{10} | 9.2 × 10^{8} | 6.9 × 10^{7} | 4.6 × 10^{6} | 2.3 × 10^{5} | Eq. (18) | |||

10^{5} | 1.1 × 10^{8} | 9.2 × 10^{7} | 7.7 × 10^{7} | 6.1 × 10^{7} | 4.6 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (15) | ||

10^{5} | 1.6 × 10^{11} | 1.4 × 10^{10} | 1.2 × 10^{9} | 1.2 × 10^{8} | 5.0 × 10^{7} | 3.1 × 10^{7} | 1.5 × 10^{7} | Eq. (16) | ||

10^{5} | 1.2 × 10^{11} | 9.2 × 10^{9} | 6.9 × 10^{8} | 4.6 × 10^{7} | 2.3 × 10^{6} | Eq. (18) |

_{T}refers to starting bacterial numbers within 1 mL, VN

_{T}refers to starting bacterial numbers here within 100 mL, and N

_{F}refers to ending bacterial numbers, with the value N

_{T}in its two instances being used equivalent to N

_{0}. Stacked quantities from top to bottom are IPD

_{min}assuming constant phage titers over time (Equation (15)), IPD

_{min}not assuming constant phage titers over time (Equation (16)), and IPN

_{min}(Equation (18)). “Equation” in the last column has been abbreviated as “Eq.”

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**MDPI and ACS Style**

Abedon, S.T.
Automating Predictive Phage Therapy Pharmacology. *Antibiotics* **2023**, *12*, 1423.
https://doi.org/10.3390/antibiotics12091423

**AMA Style**

Abedon ST.
Automating Predictive Phage Therapy Pharmacology. *Antibiotics*. 2023; 12(9):1423.
https://doi.org/10.3390/antibiotics12091423

**Chicago/Turabian Style**

Abedon, Stephen T.
2023. "Automating Predictive Phage Therapy Pharmacology" *Antibiotics* 12, no. 9: 1423.
https://doi.org/10.3390/antibiotics12091423