Patent Publication Number: US-2023149932-A1

Title: Determining a bulk concentration of a target in a sample using a digital assay with compartments having nonuniform volumes

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a continuation of non-provisional U.S. patent application Ser. No. 17/689,539, filed Mar. 8, 2020, titled DETERMINING A BULK CONCENTRATION OF A TARGET IN A SAMPLE USING A DIGITAL ASSAY WITH COMPARTMENTS HAVING NONUNIFORM VOLUMES, and naming first inventor HUYNH, Toan, which is a continuation of non-provisional U.S. patent application Ser. No. 16/200,447, now granted patent Ser. No. 11,305,284. 
     The entire contents of the above-referenced applications and of all priority documents referenced in the Application Data Sheet filed herewith are hereby incorporated by reference for all purposes. 
    
    
     SUMMARY 
     Although examples of one or more embodiments, and examples of problems solved by one or more embodiments, are described with reference to sample droplets of disparate volumes suspended in a liquid, one or more embodiments relate generally to techniques for determining a bulk concentration of a target in a source from a digital assay including compartments of disparate volumes formed in a barrier phase. 
     There are situations in which it is desirable to determine a bulk concentration of a target in a source. For example, to ensure the safety of a large number of people, a government agency may want to determine a bulk concentration of a pathogen (e.g., anthrax or another infectious agent, virus, or parasite), toxin (e.g., botulinum), or other poison (e.g., heavy metals such as lead and mercury, or chemical agents such as a nerve agent) in a municipal water supply, or may want to determine a bulk concentration of an irritant (e.g., pollen or smog), in the air. The bulk concentration is typically expressed as a ratio of the number of “pieces” (e.g., particles, molecules, cells, atoms) of the target per unit volume, or as a normalized ratio of the number of units of a volume occupied by the target to a reference number of units of the volume (e.g., parts per million). 
       FIG.  1    is a diagram that illustrates an example of an analog technique for determining a bulk concentration λ T  of a target  10  in a source  12 . For example, the target  10  may be the polio virus or bacteria (e.g.,  E Coli  or other coliform bacteria) and the source  12  may be a municipal water supply. 
     Referring to  FIG.  1   , one or more samples  14  are taken from the source  12 . For example, if the source  12  is a municipal water supply, then the sample  14  may be about 10 milliliters (mL) of water from a strategically selected location  16  (e.g., in the middle of the body of water, or near an input port where the water is drawn into a water-treatment facility). 
     Next, each sample  14  is treated with a substance, such as a reagent, that causes the sample to exhibit one or more phenomena each having a respective level related to the concentration of the target  10  in the sample. For example, a reagent added to a sample  14  can bind with the molecules of the target  10 , and cause the sample to exhibit a color having an intensity, saturation, hue, or shade that is related to the concentration of the target in the sample, where the color can be caused by the bound reagent absorbing one or more wavelengths of light, luminescing one or more wavelengths of light, or absorbing one or more first wavelengths of light and luminescing one or more second wavelengths of light. 
     Then, a technician (not shown in  FIG.  1   ) illuminates the sample with a light source designed for activating the reagent to exhibit a color. 
     Next, a human technician (not shown in  FIG.  1   ) considers the one or more exhibited phenomena for one or more samples  14  and makes an estimation {circumflex over (λ)} T  of the actual bulk concentration λ T  of the target  10  in the source  12 . For example, if a sample  14  exhibits a lighter shade of green (left end of a shade chart  16 ), then the technician estimates the bulk concentration λ T  of the target  10  in the sample  12  as “low;” conversely, if the sample exhibits a darker shade of green (right end of the shade chart), then the technician estimates the bulk concentration λ T  of the target  10  in the sample  14  as “high.” The technician considers the respective shade of green of each of one or more additional samples  14 , and, based on his/her perception of the shades of green, effectively averages the shades of green for all considered samples to arrive at a final estimate of the bulk concentration λ T . For example, the technician may characterize the bulk concentration λ T  as “high,” “medium,” “low,” “dangerous,” or “safe.” 
     But a problem with this analog technique is that the technician cannot quantify, with any precision, his/her estimate {circumflex over (λ)} T  of the bulk concentration λ T  of the target  10  in the source  12 . That is, with this analog technique, the technician can provide only a coarse, or “rough,” estimate {circumflex over (λ)} T  of the bulk concentration λ T . 
     Unfortunately, a “rough” estimate {circumflex over (λ)} T  of the bulk concentration λ T  is insufficient for some applications. 
     In another analog technique, a technician (not shown in  FIG.  1   ) compares each of one or more phenomena exhibited by one or more samples  14  to a respective chart, which quantifies the bulk concentration λ T  of the target  10  in the source  12  relative to a level of a respective phenomenon, and makes an estimation {circumflex over (λ)} T  of the bulk concentration λ T  in response to the one or more charts. For example, the chart  16  includes four shades of green that are each associated with a corresponding value λ T_1 -λ T_4  of the bulk concentration λ T , where the association between a shade of green and corresponding value λ T_n  was previously determined using one or more test sources having known values of the bulk concentration λ T  of the target  10 . The technician compares the shade of each sample  14  with the four shades of green in the chart  16 . If, per the example shown in  FIG.  1   , the shade of the sample  14  is between two of the shades (here the sample is between the two leftmost shades) in the chart  16 , then, based on his/her perception of how “close” the shade of the sample is to each of the two shades, the technician interpolates the bulk concentration λ T  as having an estimated value {circumflex over (λ)} T-Interpolated  that lies between the values λ T_1  and λ T_2  associated with the two shades in the chart (this color-shade interpolation is similar to the color-shade interpolation that one performs to determine the pH and alkalinity levels of water in a swimming pool or spa). The technician may compare, to the chart  16 , the respective shade of green of each of one or more additional samples  14 , and average the interpolated values of the bulk concentration obtained from all compared samples to arrive at a final estimated value {circumflex over (λ)} T-Interpolated  of the bulk concentration λ T  of the target  10 . 
     Although the latter analog technique may provide a more accurate estimate {circumflex over (λ)} T  of the bulk concentration λ T  of the target  10  in the source  12 , this technique still depends on the abilities of a human technician to distinguish sometimes subtle differences in the shades of a color, or in the levels of one or more other phenomena. 
       FIG.  2    is a diagram that illustrates a digital technique for determining a bulk concentration λ T  of a target  10  in a source  12 . 
     A digital assay  20  is formed by dividing sample into compartments (also called droplets if the compartments are of a liquid)  22 , each of which is small enough (e.g., ≤−100 picoliters (μL)) such that some compartments include the target  10 , and some compartments do not include the target. The number of compartments  22  in the digital assay  20  can range from tens to thousands depending on the application and the amount of precision desired. 
     The technique is a digital technique because what is considered is whether a compartment  22  does include at least one target  10  (an “on” compartment) or does not include at least one target (an “off” compartment). For example, as described above in conjunction with  FIG.  1   , a reagent is added to each compartment  22 , and binds to any molecule of the target  10  in the compartment (in this example, “at least one target” means at least one molecule of the target or, said another way, means at least one target molecule). If the compartment  22  turns any shade of green (i.e., the compartment includes at least one target), then the compartment is an “on” compartment; conversely, if the compartment does not turn green (i.e., the compartment lacks any target), then the compartment is an “off” compartment. 
     Algorithms exist for generating an estimate {circumflex over (λ)} T  of the bulk concentration λ T  of the target  10  in the source  12  in response to characteristics exhibited by a digital assay, the characteristics including the number of “on” compartments  22 , the number of “off” compartments, and the aggregate volume of the compartments. 
     Because the volumes of the compartments  22  are relatively small, low-cost, portable equipment often generates the compartments having significantly different volumes, where the largest compartment volume in the digital assay  20  is, for example, approximately ten or more times the smallest compartment volume. A digital assay having compartments with such disparate volumes is called a “polydisperse digital assay.” 
     Unfortunately, the accuracy of existing algorithms decreases dramatically as the uniformity of the compartment volumes decreases. Said another way, as the disparity among the compartment volumes increases, the accuracy of the estimated bulk concentration {circumflex over (λ)} T  determined by existing algorithms decreases. 
     Consequently, for many applications, the disparity in the volumes of the compartments  22  generated by low-cost equipment is so large that existing algorithms cannot yield sufficiently accurate values of {circumflex over (λ)} T . 
     Still referring to  FIG.  2   , to increase the accuracy of existing algorithms, specialized equipment has been developed to generate digital assays having compartments with more uniform volumes. 
     For example, a digital assay  30  has compartments  32  with approximately equal volumes, and such a digital assay is called a “monodisperse digital assay.” 
     But although a monodisperse digital assay, such as the digital assay  30 , significantly increases the accuracy with which existing algorithms can estimate the bulk concentration λ T  of the target  10  in the source  12 , the generation of a monodisperse digital assay is often beset by a number of problems. 
     For example, equipment for generating a monodisperse digital assay, such as the digital assay  30 , can be expensive, bulky, and slow. Such equipment can cost US$150,000 or more; therefore, such equipment is often unattainable by charitable and other organizations with limited funds. Consequently, such organizations send out their samples to a lab for analysis, and typically wait a significant amount of time (e.g., a few weeks to a few months) for an estimate {circumflex over (λ)} T  of the bulk concentration λ T . Furthermore, such equipment can be on the order of 6 feet×6 feet×2 feet; therefore, it is often unsuitable for on-site applications (e.g., on the bank of a reservoir, at a well for drinking water). Consequently, even if an organization owns, or otherwise has access to, such equipment, transporting the sample from the source to the equipment increases the time for, and the cost of, obtaining an estimate λ T  of the bulk concentration λ T . Moreover, such equipment can take a relatively long time, e.g., on the order of one minute, to generate each compartment of a monodisperse digital assay. Consequently, because a monodisperse digital assay may include tens, hundreds, or even thousands of compartments, the throughput of such equipment is limited, and, therefore, increases the time for, and the cost of, obtaining an estimate {circumflex over (λ)} T  of the bulk concentration λ T . 
     Therefore, a need has arisen for a system that is smaller, less expensive, and faster than existing equipment, yet that is at least as accurate as existing systems. 
     In an embodiment, such a system includes a compartment-generating device, a compartment detector, and electronic computing circuitry. The device is configured to generate compartments of a digital assay, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments. The detector is configured to determine a number of the compartments each having a respective concentration of a target that is greater than a threshold concentration. And the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to the number. 
     Compared to equipment for generating a monodisperse digital assay, such a system can be portable, lower-cost, and faster, yet can yield similar accuracy. In an embodiment, these improvements flow from the system being configured to implement an algorithm that allows for accurately estimating the bulk concentration λ T  of a target in a source from a polydisperse digital assay. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG.  1    is a diagram that illustrates an analog technique for determining a bulk concentration of a target in a source. 
         FIG.  2    is diagram that illustrates two digital techniques for determining a bulk concentration of a target in a source. 
         FIG.  3    is a diagram that illustrates a digital technique for determining a bulk concentration of a target in a source, according to an embodiment. 
         FIG.  4    is a diagram of a system configured to implement the digital technique illustrated by  FIG.  3   , according to an embodiment. 
         FIG.  5    is a flow chart of the digital technique illustrated by  FIG.  3    and implemented by the system of  FIG.  4   , according to an embodiment. 
         FIG.  6    is a flow chart of the digital technique illustrated by  FIG.  3    and implemented by the system of  FIG.  4   , according to another embodiment. 
         FIG.  7    is a flow chart of a technique for characterizing the polydisperse digital assay of  FIGS.  3 - 4   , according to an embodiment. 
         FIG.  8    is a flow chart of the digital technique illustrated by  FIG.  3    and implemented by  FIG.  4   , according to yet another embodiment. 
         FIG.  9    is a diagram of a polydisperse digital assay and a residual barrier phase, according to an embodiment. 
         FIG.  10    is a diagram of a portable kit for determining a bulk concentration of a target in a source using the digital technique illustrated by  FIG.  3   , according to an embodiment. 
         FIG.  11    is a diagram of the computer of  FIG.  4   , according to an embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The words “approximately,” “substantially,” “about,” and similar words and phrases, are used below to indicate that a quantity can be in range of ±10% of a value given for the quantity, and that two or more quantities can be exactly equal, or can be within ±10% of each other. Furthermore, use of such a word to describe a range b to c indicates a range of b−10% [c−b] to c+10%·[c−b]. 
       FIG.  3    is a diagram that illustrates a digital technique for determining a bulk concentration λ T  of a target  10  in a source  12 , according to an embodiment. For example, one can use the digital technique to obtain an accurate estimate {circumflex over (λ)} T  of the bulk concentration λ T . 
     A container  40 , such as a clear test tube of glass or plastic, holds a polydisperse digital assay  42  of compartments  44  suspended in a barrier phase  46 , according to an embodiment in which a dark compartment is “on” and a light compartment is “off.” In the described example, each compartment  44  is a respective droplet of a liquid, such as water, and the barrier phase  46  is another liquid, such as an oil, in which the droplets are suspended. The combination of the droplets  44  and the liquid barrier phase  46  is an emulsion. 
       FIG.  4    is a diagram of the container  40 , the polydisperse digital assay  42  and the barrier phase  46  within the container, and a system  50  configured to determine the bulk concentration λ T  of the target  10  ( FIG.  3   ) in the source  12  ( FIG.  3   ) in response to at least some of the droplets  44  of the polydisperse digital assay, according to an embodiment. 
     The system  50  includes a droplet analyzer  52  and a computing device  54 . Both the droplet analyzer  52  and computing device  54  include electronic circuitry that is hardwired, or is configured by software or firmware, to perform the respective functions and operates described below. 
     The droplet analyzer  52  includes an “on”-droplet detector and counter  56 , a droplet counter  58 , and a droplet-volume determiner  60 . The “on”-droplet detector and counter  56  includes electronic circuitry and one more optical sensors configured to detect, and to determine the number of, “on” droplets  44  in the polydisperse digital assay  42 . The droplet counter  58  includes electronic circuitry and one or more optical sensors (one or more of which may be shared with the on-droplet detector and counter  56 ) configured to determine the total number of “on” and “off” droplets  44  in the polydisperse digital assay  42 . And the droplet-volume detector  60  includes electronic circuitry and one or more optical sensors (one or more of which may be shared with the “on”-droplet detector and counter  56  or the droplet counter  58 ) configured to measure, or otherwise to determine, the respective volume of each of the droplets  44 . 
     The computing device  54  can be any suitable computer, such a laptop, a tablet, or a smart phone, that includes one or more microprocessors or microcontrollers. 
       FIG.  5    is a flow chart  70  of an algorithm for determining a bulk concentration λ T  of a target  10  in a source  12 , according to an embodiment. 
     Referring to  FIGS.  3 - 5   , the algorithm represented by the flow chart  70 , and operation of the system  50  while implementing the algorithm, are described, according to an embodiment in which both the compartments  44  and the barrier phase  46  are liquids such that the compartments  44  are droplets suspended in the liquid barrier phase. 
     First, at a step  72 , a technician (not shown in  FIGS.  3 - 5   ) generates the droplets  44  of different volumes (e.g., in a range of approximately 1 pL-100s pL) from a sample of the source  12  to form the polydisperse digital assay  42 . For example, the technician adds an indicated volume of the barrier-phase liquid  46 , such as an oil, to the container  40 , adds an indicated sample volume of the source  10  to the barrier-phase liquid in the container, plugs the top of the container, and shakes the container to form an emulsion of the droplets  44  suspended in the barrier-phase liquid. Further in example, the container  40  includes a measurement line (not shown in  FIGS.  3 - 5   ) to indicate the volume of the barrier-phase liquid to be added, and the technician uses a measurement device, such as an “eye” dropper, to obtain and measure the volume of the sample. Alternatively, the container  40  may include another measurement line (not shown in  FIGS.  3 - 5   ), higher up on the container than the barrier-phase measurement line, to indicate the volume of the sample to be added after the barrier-phase liquid is added. Still further in example, the technician adds to the container  40 , before the technician shakes it, a reagent for rendering target-carrying droplets  44  luminescent as described above in conjunction with  FIGS.  1 - 2   . Using this “shake-and-bake” technique, the technician can generate the polydisperse digital assay  42  in a matter of a few ones to tens of seconds, and in no more than a few minutes even if the time for setting up the system  50  and obtaining the sample is included. 
     Next, at a step  74 , the “on”-droplet detector and counter  56  determines the number a of “on” droplets  44  in the polydisperse digital assay  42 . For example, the counter  56  can include a combination illumination device and image-capture device, such as a light source and a small camera, which the technician holds up near, or against, the container  40 . The illumination device illuminates the droplets  44  so that droplets including the target luminesce a color having shades respectively corresponding to the concentrations of the target in the droplets. The technician then presses a button on the device, or a virtual button displayed by the computer  54 , to capture an image of the droplets  44  in the container while the droplets including the target are luminescing. Using conventional image-processing techniques, the computer  54  analyzes the image, detects the droplets  44 , and determines whether each detected droplet  44  is “on” or “off” by determining, for each droplet, whether the number of targets (or, said another way, the number of the target) within the droplet exceeds a threshold number (e.g., one target molecule, five target molecules, ten target molecules). For example, an optical signal that the target luminesces has a property (e.g., intensity, color, color shade) indicative of the number of targets within a droplet  44 , and the computer  54  determines whether the number of targets within the droplet exceeds the threshold number by determining whether the property of the target-related optical signal exceeds (or is below) a signal-property threshold. Further in example, the computer  54  compares a shade of the color (e.g., green), or an opacity, of a droplet  44  to a threshold shade or opacity, determines that the droplet is “on” if the level of the shade or opacity is greater than or equal to the threshold, and determine that the droplet is “off” if the level of the shade or opacity is less than the threshold. Alternatively, the technician uses the “on”-droplet detector and counter  56  to capture multiple images of the droplets  44  from different orientations relative to the container  40  so that the computer  54  is able to detect droplets that might otherwise be obscured by other droplets in a single image. 
     Then, at a step  76 , the computer  54  generates an estimate {circumflex over (λ)} T  of the bulk concentration λ T  of the target  10  in the source  12  in response to the number a of “on” droplets  44  in the container  40 . For example, as described below in conjunction with  FIG.  6   , the computer  54  executes one or more equations to solve for {circumflex over (λ)} T  in response to a. 
     The system  50 , and the algorithm that the system implements, provide one or more advantages over existing systems and techniques for determining a bulk concentration of a target in a source. For example, the container  40  and the barrier phase  46  are configured to provide inexpensive, on-site, and fast generation of the digital assay  42 . Furthermore, the system  50  is configured to provide inexpensive, on-site, and fast estimation of the bulk concentration λ T  of the target  10  in the source  12  even in response to a polydisperse digital assay  42  having compartments  44  of disparate volumes. 
     Still referring to  FIGS.  3 - 5   , alternate embodiments of the system  50  and of the above-described algorithm are contemplated. For example, one or more steps can be added to the algorithm, and one or more of the above-described steps can be omitted from the algorithm. Furthermore, one or more components can be added to the system  50 , and one or more of the above-described components can be omitted from the system. Moreover, use of a reagent may be omitted if, for example, the target luminesces without the reagent, or if the system  50  can determine, in a manner that does not involve use of a reagent, whether the number of targets in a compartment  44  exceeds the threshold number by determining whether a property of a target-related signal (e.g., color, color shade) exceeds a signal-property threshold. In addition, the computer  54  may perform one or more functions and operations attributed to the droplet analyzer  52 , and the droplet analyzer may perform one or more functions and operations attributed to the computer. Furthermore, a cloud server may complement, or replace, the computer  54 . Moreover, the system  50  and the above-described algorithm yield similar results and advantages for a monodisperse digital assay. In addition, embodiments described below in conjunction with  FIGS.  6 - 10    may be applicable to the system  50  and to the above-described algorithm. 
       FIG.  6    is a flow chart  80  of a digital-variable-volume (DVV) algorithm for determining a bulk concentration λ T  of a target  10  ( FIG.  3   ) in a source  12  (FIG.  3 ), according to an embodiment. The DVV algorithm is suitable for situations in which the system  50  includes the droplet-volume determiner  60 , or in which the respective volumes of the “on” droplets  44 , and the aggregate volume of all the droplets, otherwise can be determined. 
     Referring to  FIGS.  3 - 4  and  6   , the DVV algorithm represented by the flow chart  80 , and operation of the system  50  while implementing the algorithm, are described according to an embodiment in which both the compartments  44  and the barrier phase  46  are liquids such that the compartments  44  are droplets suspended in the barrier phase to form an emulsion. 
     First, at a step  82 , a technician (not shown in  FIGS.  3 - 4  and  6   ) generates the droplets  44  of disparate volumes (e.g., in a range of approximately 1 pL-100s pL) from a sample of the source  12  to form the polydisperse digital assay  42 . For example, the technician may form the polydisperse digital assay  42  using a method that is the same as, or that is similar to, the “shake-and-bake” method described above in conjunction with step  72  of  FIG.  5   . 
     Next, at a step  84 , the “on”-droplet detector and counter  56  determines the number a of “on” droplets  44  in the polydisperse digital assay  42 . For example, the “on”-droplet detector and counter  56  may determine the number a using a method that is the same as, or that is similar to, the method described above in conjunction with step  74  of  FIG.  5   . 
     Then, at a step  86 , the droplet-volume determiner  60  determines the respective volume v i  of each of the detected “on” droplets  44 , and, if necessary, determines the aggregate volume V Total  of the droplets  44  by summing the respective volumes of the detected “on” and “off” droplets. For example, to determine the respective volume v i  of each of the α “on” droplets  44 , the determiner  60  analyzes the one or more images that the system  50  captured at the step  84  using a conventional droplet-volume-determining algorithm. To determine the aggregate volume V Total , the determiner  60  also determines the volumes of the “off” droplets  44  in the same way that the determiner determines the volumes v i  of the “on” droplets, sums the volumes of the “off” droplets with the volumes v i  of the “on” droplets, and sets V Total  equal to the determined sum. Alternatively, the droplet-volume determiner  60  is configured to operate as described above but is part of, or is otherwise included in, the computer  54  instead of the droplet analyzer  52 . In yet another alternative, because the aggregate volume of the sample is the same as the aggregate volume V Total  of the droplets  44 , and because the volume of the sample is known per step  82 , the technician enters into the computer  54  the volume of the sample, and the computer sets V Total  equal to the entered sample volume. 
     Next, at a step  88 , the computer  54  solves for the estimated bulk concentration {circumflex over (λ)} T  of the target  10  in the source  12  according to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       a 
                     
                     
                       
                         v 
                         i 
                       
                       
                         1 
                         - 
                         
                           e 
                           
                             
                               - 
                               
                                 v 
                                 i 
                               
                             
                             ⁢ 
                             
                               
                                 λ 
                                 ^ 
                               
                               T 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     V 
                     Total 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     A derivation and explanation of equation (1) is included below. 
     In an ideal example, each “on” droplet  44  would contain one and only one of the target such that the computer  54  could determine the estimated bulk concentration {circumflex over (λ)} T  from the number a of “on” droplets  44  divided by the volume V Total  of the sample  14 , where a would also equal the number of targets in the sample. 
     But because in an actual, non-ideal, example each “on” droplet  44  may contain more than one of the target  10 , determining the estimated bulk concentration {circumflex over (λ)} T  from α/V Total  may lead to an error caused by an undercounting of the number of the target in the sample  14 . 
     To reduce or eliminate such an undercounting error where the respective number of the target in one or more of the “on” droplets  44  of the sample  14  is unknown, the computer  54  is configured to use equation (1) to estimate the bulk concentration λ T  of the target in the source  12 . 
     For each “on” droplet  44 , equation (1) includes a respective expression for the probability that the droplet includes at least one of the target, the probability being dependent on, and, therefore, the respective expression including, the respective volume of the droplet. 
     Consequently, equation (1) not only effectively accounts for the possibility that each of one or more of the “on” droplets  44  contains more than one of the target  10 , equation (1) also effectively accounts for a larger “on” droplet  44  being more likely than a smaller “on” droplet to contain more than one of the target. 
     The system  50 , and the DVV algorithm that the system  50  is configured to implement, provide one or more advantages over existing systems and techniques. For example, the container  40  and binary phase  46  provide for inexpensive, on-site, and fast generation of the digital assay  42 . Furthermore, the system  50  provides for inexpensive, on-site, fast, and accurate estimation of a bulk concentration λ T  of a target  10  in a source  12  even in response to a polydisperse digital assay  42  having droplets  44  of disparate volumes. 
     Still referring to  FIGS.  3 - 4  and  6   , alternate embodiments of the system  50  and the DVV algorithm are contemplated. For example, instead of the “on”-droplet detector and counter  56  determining the number a of “on” droplets  44 , the technician may count the number a of “on” droplets and enter the number a into the computer  54 . Moreover, instead of the droplet-volume determiner  60  determining the volumes v; of the “on” droplets  44 , the technician may use a device, such as a ruler or microscope, to estimate the volumes v i , and then enter these volumes into the computer  54 . Moreover, the system  50  and the above-described algorithm yield similar results and advantages for a monodisperse digital assay. In addition, alternate embodiments described above in conjunction with  FIGS.  3 - 5    and below in conjunction with  FIGS.  7 - 11    may be applicable to the system  50  and the DVV algorithm. 
       FIG.  7    is a flow chart  100  of an algorithm for characterizing the volumes of the compartments (e.g., droplets)  44  generated with the container  40  and barrier phase  46  of  FIG.  4   , according to an embodiment. 
       FIG.  8    is a flow chart  110  of a digital-variable-volume-approximation (DVVA) algorithm for determining a bulk concentration λ T  of a target  10  ( FIG.  3   ) in a source  12  ( FIG.  3   ), according to an embodiment. The DVVA algorithm is suitable for situations in which the system  50  lacks the droplet-volume determiner  60 , or in which the respective volumes of the “on” droplets  44 , and the aggregate volume of all the droplets, are otherwise unknown. 
     Referring to  FIGS.  3 - 4  and  7   , the compartment-volume characterizing algorithm is described, according to an embodiment in which the compartments  44  are droplets. 
     At a step  102 , a technician (not shown in  FIG.  3 - 4  or  7   ) generates a set of test droplets of different volumes (e.g., in a range of approximately 1 pL-100 pL) from a test sample that is similar to a sample of the source  12  of  FIG.  3    to form a test polydisperse digital assay; for example, if the intended source  12  is a body of water, then the technician may use a sample of water. The technician adds an indicated volume of the barrier-phase liquid  46 , such as an oil, to the container  40 , adds an indicated volume of the test sample to the barrier-phase liquid in the container, plugs the top of the container, and shakes the container to form an emulsion of test droplets suspended in the barrier-phase liquid. The container  40  may include a measurement line (not shown in  FIG.  3 - 4  or  7   ) to indicate the volume of the barrier-phase liquid to be added, and the technician may use a measurement device, such as an “eye” dropper, to add an indicated volume of the test sample. Alternatively, the container may include another measurement line (not shown in  FIG.  3 - 4  or  7   ), higher up on the container than the barrier-phase measurement line, to indicate the volume of the test sample to be added after the barrier-phase liquid is added. 
     Next, at a step  104 , the droplet counter  58 , or a similar counter, determines the number m of test droplets, and at a step  106 , the droplet-volume determiner  60 , or a similar determiner, determines a respective volume V i_characterized  for each of the m test droplets. 
     Then, at a step  108 , the computer  54  stores the number m of test droplets, and stores the volumes v i_characterized  of the test droplets, in a memory (not shown in  FIG.  3   ). Alternatively, the number m and the corresponding volumes V i_characterized  can be stored in another memory from which the computer  54  is configured to download the values of m and V i_characterized . 
     The theory behind generating the characterized number m and the characterized volumes v i_characterized  is that similar containers, sample substances, and barrier phases will generate similar values for m and V i_characterized  such that the values of m and V i_characterized  can be used to determine a bulk concentration λ T  of the target  10  ( FIG.  3   ) in the sample  12  ( FIG.  3   ) in situations where the volumes v i  of the actual droplets  44  cannot be determined or are otherwise unknown. That is, the statistical dependence between the actual droplet volumes v i  and the characterized droplet volumes V i_characterized  is high enough that, as described below, the number m and volumes V i_characterized  of the test droplets can be used to determine the actual bulk concentration λ T  of the target  10  in the source  12  in a situation where the volumes v i  of the actual droplets  44  are unknown. 
     Still referring to  FIG.  7   , alternative embodiments of the droplet-characterization algorithm are contemplated. For example, the steps  102 - 106  can be repeated any suitable number of times to generate sets of test values for m and V i_characterized , and the computer  54 , or a similar computer, can calculate the final values of m and V i_characterized  by interpolating values from one or more of the sets of test values. 
     Referring to  FIGS.  3 - 4  and  8   , the DVVA algorithm represented by the flow chart  110 , and operation of the system  50  while implementing the DVVA algorithm, are described according to an embodiment in which both the compartments  44  and the barrier phase  46  are liquids such that the compartments  44  are droplets suspended in the barrier phase to form an emulsion. 
     First, at a step  112 , a technician (not shown in  FIGS.  3 - 4  and  8   ) generates the droplets  44  of disparate volumes (e.g., in a range of approximately 1 pL-100 pL) from a sample of the source  12  to form the polydisperse digital assay  42 . For example, the technician may form the polydisperse digital assay  42  using a method that is the same as, or similar to, the “shake-and-bake” method described above in conjunction with step  72  of  FIG.  5   . 
     Next, at a step  114 , the “on”-droplet detector and counter  56  determines the number a of “on” droplets  44  in the polydisperse digital assay  42 . For example, the on-droplet detector and counter  56  may determine the number a using a method that is the same as, or similar to, the method described above in conjunction with step  74  of  FIG.  5   . 
     Then, at a step  116 , the droplet counter  58  determines the number n of all droplets  44  (i.e., the sum of the “on” and “off” droplets) in the polydisperse digital assay  42  in the container  40 . For example, to determine the number n of all droplets  44 , the droplet counter  58  analyzes the one or more images that the system  50  captured at the step  114  using a conventional droplet-counting algorithm. Alternatively, the droplet counter  58  counts only the number b of “off” droplets  44 , and adds this number to the number a of “on” droplets, to generate n=a+b. Alternatively, the droplet counter  58  operates in a similar manner but is part of, or is included in, the computer  54  instead of being part of, or included in, the droplet analyzer  52 . 
     Next, at a step  118 , the computer  54  solves for the estimated bulk concentration λ T  of the target  10  in the source  12  according to the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       
                         1 
                         m 
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             m 
                           
                           
                             
                               e 
                               
                                 - 
                                 v 
                               
                             
                             
                               
                                 
                                   i 
                                   ⁢ 
                                   _ 
                                   ⁢ 
                                   characterized 
                                 
                                 
                                   
                                     λ 
                                     ^ 
                                   
                                   T 
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                     - 
                     1 
                     + 
                     
                       a 
                       n 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
     where m is the characterized number of droplets and each V i_characterized  is the characterized volume of a respective one of the m droplets as described above in conjunction with  FIG.  7   . 
     A derivation and explanation of equation (2) is included below. 
     The system  50 , and the DVVA algorithm that the system is configured to implement, provide one or more advantages over existing systems and techniques. For example, the container  40  and barrier phase  46  provide for inexpensive, on-site, and fast generation of the digital assay  42 . Furthermore, the system  50  provides for inexpensive, on-site, and fast estimation of a bulk concentration λ T  of the target  10  in the source  12  even in response to a polydisperse digital assay  42  having droplets  44  of disparate volumes that are unknown. 
     Still referring to  FIGS.  3 - 4  and  8   , alternate embodiments of the system  50  and the DVVA algorithm are contemplated. For example, from the number m and volumes V i_characterized  of the test droplets described above in conjunction with  FIG.  7   , one may determine the probability density function ƒ(v) of the volumes of the test droplets, and solve for the estimated bulk concentration {circumflex over (λ)} T  of the target  10  in the source  12  according to the following equation: 
     
       
         
           
             
               
                 
                   
                     a 
                     n 
                   
                   = 
                   
                     1 
                     - 
                     
                       
                         ∫ 
                         
                           - 
                           ∞ 
                         
                         ∞ 
                       
                       
                         
                           e 
                           
                             
                               - 
                               
                                 
                                   λ 
                                   ^ 
                                 
                                 T 
                               
                             
                             ⁢ 
                             v 
                           
                         
                         ⁢ 
                         
                           f 
                           ⁡ 
                           ( 
                           v 
                           ) 
                         
                         ⁢ 
                         dv 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     Moreover, alternate embodiments described above in conjunction with  FIGS.  3 - 7    and below in conjunction with  FIGS.  9 - 11    may be applicable to the system  50  and the DVVA algorithm. 
       FIG.  9    is a diagram of the polydisperse digital assay  42  in the container  40 , according another embodiment. 
     If the barrier phase  46 , e.g., an oil, has a higher density than the droplets  44 , then the droplets may “float” over a residual region  120  of the barrier phase that is devoid of droplets as shown in  FIG.  9   . 
     Conversely, if the barrier phase  46 , e.g., an oil, has a lower density than the droplets  44 , then the droplets may “sink” to the bottom of the container  40  such that the residual region  120  of the barrier phase that is devoid of droplets “floats” over the droplets (the residual region “floating” over the droplets is not shown in  FIG.  9   ). 
       FIG.  10    is a diagram of a digital-assay-generator-and-analyzer kit  130 , according to an embodiment. The combination of the kit  130 , a portable computer (e.g., laptop, tablet, smart phone), and one or more of the algorithms described above in conjunction with  FIGS.  5 - 8    provides for inexpensive, on-site, and fast generation of the digital assay  42 , and provides for inexpensive, on-site, and fast estimation of a bulk concentration λ T  of a target  10  ( FIG.  3   ) in a source  12  ( FIG.  3   ) in spite of the polydisperse digital assay having droplets  44  of disparate volumes that may be unknown. For example, the kit  130  may cost approximately $30-$100, the amount of the barrier phase  46  required to generate each digital assay  42  may cost approximately $1 or less, the combined weight of the kit  130  and the computer  54  may be approximately 3 pounds (lbs.) to 10 lbs., and the total test time (from collection of a sample  14  to the computer  54  rendering an estimated bulk concentration {circumflex over (λ)} T ) may be approximately 20 minutes to 80 minutes. 
     In addition to the container  40  and the droplet analyzer  52 , the kit  130  includes a container stopper  132 , a re-openable and re-closable package (e.g., a screw-top bottle)  134  of the barrier phase  46 , a re-openable and re-closable optional package (e.g., a screw-top bottle)  136  of a reagent, a dropper  138 , and a non-transitory computer-readable medium  140 . The stopper  132  is configured to form a liquid-tight seal at the opening of the container  40  to allow shaking of the container to form the polydisperse digital assay  42 . The dropper  138  allows a technician to transfer the barrier phase  46  and the reagent from their respective packages  134  and  136  to the container  40 , and allows a technician to obtain a liquid sample (e.g., water) from a source (e.g., reservoir) and to transfer the sample to the container. And the computer-readable medium is a suitable non-volatile memory that stores program instructions that, when executed by a portable computer, cause the computer to implement one of the algorithms described above in conjunction with  FIGS.  5 - 8   . 
     Still referring to  FIG.  10   , alternate embodiments of the kit  130  are contemplated. For example, the kit  130  may include a carrying case in which all of the other system components may be stored and carried. Furthermore, embodiments describe above in conjunction with  FIGS.  1 - 9    and below in conjunction with  FIG.  11    may be applicable to the kit  130 . 
       FIG.  11    is a block diagram of the computer  54  of  FIG.  4   , according to an embodiment. 
     The computer  54  includes computing circuitry  150 , one or more input devices  152 , one or more output devices  154 , and one or more data-storage devices  156 . 
     The computing circuitry  150  includes circuitry that is configured to perform various functions and operations, such as the functions and operations described above in conjunction with  FIGS.  3 - 8  and  10   . For example, the computing circuitry  150  includes a microprocessor or microcontroller that is hardwired or configured with firmware, or that executes software, to perform the above-described functions and operations. 
     The one or more input devices  152  are configured to allow an operator or device to provide data or other information or signals to the computer  54 . Examples of an input device  152  include a keyboard, mouse, touch screen, audible or voice-recognition component, the droplet analyzer  52  ( FIG.  4   ), and so on 
     The one or more output devices  154  are configured to provide data from the computing circuitry  150  to an operator or device in a suitable form, or to perform a function or operation under control of the computing circuitry  150 . Examples of an output device  154  include a printer, video display, audio output components, the droplet analyzer  52  ( FIG.  4   ), and so on. 
     The one or more data-storage devices  156  are configured to store data on or to retrieve data from volatile or non-volatile storage media (not shown). Examples of a data-storage device  156  include a magnetic disk, a FLASH memory, other types of solid state memory such as a random-access memory (RAM, SRAM, DRAM, USB “stick”), a ferro-electric memory, a tape drive, an optical disk like a compact disk and a digital versatile disk (DVDs), and so on. 
     Still referring to  FIG.  11   , alternate embodiments of the computer  54  are contemplated. For example, the computer  54  may omit one or more of the above-described devices, and may include one or more other devices. 
     Derivation and Explanation of Equations (1) and (2) 
     General Definitions and Assumptions 
     In digital assays, the targets (molecules, cells, etc.) in the bulk sample are randomly distributed into many compartments. A compartment with one or more targets gives a signal (e.g., fluorescent intensity after nucleic acid amplification), and is called an “on” compartment. A compartment without targets does not provide a signal, and is called an “off” compartment. Targets are distributed into compartments following the Poisson distribution. An assay system, such as the system  50  of  FIG.  4   , can detect and count the number of “on” compartments (but not the number of targets per compartment). 
     For each assay, the bulk concentration needs to be calculated using a certain inference method. The digital-variable-volume (DVV) and digital-variable-volume-approximation (DVVA) methods are based on maximum likelihood estimation; the concentration estimate is the one that maximizes the likelihood of observing a certain experimental result. The choice of maximum likelihood estimation was inspired by its use in multivolume digital PCR (where each assay utilizes a handful of predetermined, precisely controlled volumes), which has been inspired by limiting dilution assays for microorganism counting. In particular, an important feature is that results from different volumes are readily combined by way of multiplying the likelihoods. Below, are derived the expressions used to calculate the concentration estimates and the standard errors using the maximum likelihood framework. The terms relevant to the descriptions of the DVV and DVVA methods are described in Table 1. 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Definitions of mathematical symbols. 
               
            
           
           
               
               
            
               
                 Symbol 
                 Definition 
               
               
                   
               
               
                 {circumflex over (x)} 
                 Estimator of a particular parameter 
               
               
                   
                 denoted as x 
               
               
                     x    
                 Expectation of the quantity x over a 
               
               
                   
                 certain distribution 
               
               
                 λ T   
                 Bulk concentration (number of 
               
               
                   
                 targets/unit volume) 
               
               
                 {circumflex over (λ)} T   
                 Estimator of λ T  (inferred from the 
               
               
                   
                 assay result) 
               
               
                 Λ ≡ ln(λ T ) 
                 Natural log of bulk concentration 
               
               
                 {circumflex over (Λ)} 
                 Estimator of Λ (inferred from the 
               
               
                   
                 assay result) 
               
               
                   
               
               
                 
                   
                     
                       
                         σ 
                         
                           Λ 
                           ^ 
                         
                       
                     
                   
                 
                 Standard error of {circumflex over (Λ)} 
               
               
                   
               
               
                 Λ 0   
                 
                   
                     
                       
                         Λ 
                         ⁢ 
                             
                         with 
                         ⁢ 
                             
                         smallest 
                         ⁢ 
                             
                         
                           σ 
                           
                             Λ 
                             ^ 
                           
                         
                       
                     
                   
                 
               
               
                   
               
               
                 N 
                 Total number of compartments 
               
               
                 V total   
                 Total volume of compartments 
               
               
                 A 
                 Number of ON compartments 
               
               
                 A ≡ {v 1 , v 2 , . . . , v a } 
                 Set of volumes of ON compartments 
               
               
                 b ≡ n − a 
                 Number of OFF compartments 
               
               
                   
               
               
                 
                   
                     
                       
                         
                           V 
                           
                             
                                 
                               b 
                             
                             
                               
                                 def 
                                 _ 
                               
                               _ 
                             
                           
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             b 
                           
                             
                           
                             v 
                             i 
                           
                         
                       
                     
                   
                 
                 Total Volume of OFF compartments 
               
               
                   
               
               
                 M 
                 Number of pre-measured (test or 
               
               
                   
                 characterization) volumes 
               
               
                 M ≡ {v 1 , v 2 , . . . , v m } 
                 Set of pre-measured (test or 
               
               
                   
                 characterization) volumes 
               
               
                 f(v) 
                 Volume probability density function 
               
               
                 μ v   
                 Mean volume 
               
               
                 σ v   
                 Standard deviation of volume 
               
               
                 μ ln V 
                 Geometric mean of volume 
               
               
                 W 
                 Product logarithm function (also 
               
               
                   
                 known as Lambert W function) 
               
               
                   
               
            
           
         
       
     
     Begin by calculating the probability that a particular compartment turns “on” given the volume and bulk concentration (equation (a)). It is the same as the probability of having more than one target in the compartment, based on the Poisson distribution with the mean of Vλ T . This probability is useful in subsequent derivation steps. 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       each 
                     
                     ( 
                     
                       
                         λ 
                         T 
                       
                       , 
                       v 
                     
                     ) 
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         Prob 
                         ⁡ 
                         ( 
                         notargets 
                         ) 
                       
                     
                     = 
                     
                       
                         1 
                         - 
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     v 
                                     ⁢ 
                                     
                                       λ 
                                       T 
                                     
                                   
                                   ) 
                                 
                                 k 
                               
                               ⁢ 
                               
                                 e 
                                 
                                   
                                     - 
                                     v 
                                   
                                   ⁢ 
                                   
                                     λ 
                                     T 
                                   
                                 
                               
                             
                             
                               k 
                               ! 
                             
                           
                           
                             
                               ❘ 
                               &#34;\[LeftBracketingBar]&#34; 
                             
                             
                               k 
                               = 
                               0 
                             
                           
                         
                       
                       = 
                       
                         1 
                         - 
                         
                           e 
                           
                             
                               - 
                               v 
                             
                             ⁢ 
                             
                               λ 
                               T 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   a 
                   ) 
                 
               
             
           
         
       
     
     Digital Variable Volume (DVV) 
     The likelihood l(λ T ) of observing a certain assay result, i.e., particular numbers of “on” and “off” compartments (a and b, respectively) with the associated volumes is the product of individual likelihoods calculated using equation (a). 
       Π i=1   a Peach(λ T   ,V   i )Π i=1   b Peach[1−Peach(λ T   ,V   i )]=Π 1=1   a (1− e   −v   i   λT )Π 1   b   e   −v   i   λT   (b)
 
     The value of λ T  that maximizes l(λ T ) is then found. Use the natural logarithm of the concentration (Λ≡In(λ T )) and the loglikelihood function (L(Λ) ≡In(λ T )) to conveniently calculate the standard errors and enforce the requirement for positive concentrations. The calculation of the standard error is also more appropriate for Λ than for λ T  because the distribution of Λ is less skewed. Therefore, the goal is now finding the Λ value that maximizes L(Λ). The expression for L(Λ) and the first and second derivatives are shown below. 
     
       
         
           
             
               
                 
                   
                     
                       L 
                       ⁡ 
                       ( 
                       Λ 
                       ) 
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           a 
                         
                         
                           ln 
                           ⁡ 
                           ( 
                           
                             1 
                             - 
                             
                               e 
                               
                                 
                                   - 
                                   
                                     v 
                                     i 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   Λ 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           e 
                           Λ 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             b 
                           
                             
                           
                             v 
                             i 
                           
                         
                       
                     
                   
                   ⁢ 
                     
                   
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           a 
                         
                         
                           ln 
                           ⁡ 
                           ( 
                           
                             1 
                             - 
                             
                               e 
                               
                                 
                                   - 
                                   
                                     v 
                                     i 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   Λ 
                                 
                               
                             
                           
                           ) 
                         
                       
                       - 
                       
                         
                           e 
                           Λ 
                         
                         ( 
                         
                           
                             V 
                             total 
                           
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               a 
                             
                             
                               v 
                               i 
                             
                           
                         
                         ) 
                       
                     
                   
                   ⁢ 
                     
                   
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           a 
                         
                         
                           [ 
                           
                             
                               ln 
                               ⁡ 
                               ( 
                               
                                 1 
                                 - 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                   
                                 
                               
                               ) 
                             
                             + 
                             
                               
                                 v 
                                 i 
                               
                               ⁢ 
                               
                                 e 
                                 Λ 
                               
                             
                           
                           ] 
                         
                       
                       - 
                       
                         V 
                         
                           total 
                           
                             e 
                             Λ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   c 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                   
                   
                     
                       
                         L 
                         ′ 
                       
                       ( 
                       Λ 
                       ) 
                     
                     = 
                     
                       
                         e 
                         Λ 
                       
                       ( 
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             a 
                           
                           
                             
                               v 
                               i 
                             
                             
                               1 
                               - 
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     Λ 
                                   
                                 
                               
                             
                           
                         
                         - 
                         
                           V 
                           total 
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   d 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       L 
                       ″ 
                     
                     ( 
                     Λ 
                     ) 
                   
                   = 
                   
                     
                       
                         e 
                         Λ 
                       
                       ( 
                       
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             a 
                           
                           
                             
                               v 
                               i 
                             
                             
                               1 
                               - 
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     Λ 
                                   
                                 
                               
                             
                           
                         
                         - 
                         
                           V 
                           total 
                         
                       
                       ) 
                     
                     - 
                     
                       
                         e 
                         
                           2 
                           ⁢ 
                           Λ 
                         
                       
                       ⁢ 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           a 
                         
                         
                           
                             
                               v 
                               i 
                               2 
                             
                             ⁢ 
                             
                               e 
                               
                                 
                                   - 
                                   
                                     v 
                                     i 
                                   
                                 
                                 ⁢ 
                                 
                                   e 
                                   Λ 
                                 
                               
                             
                           
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   e 
                   ) 
                 
               
             
           
         
       
     
     To calculate {circumflex over (Λ)}, the root of the first derivative (equation (d)) is determined, i.e., equation (1), which is repeated below, is solved. 
     
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         i 
                         = 
                         1 
                       
                       a 
                     
                     
                       
                         v 
                         i 
                       
                       
                         1 
                         - 
                         
                           e 
                           
                             
                               - 
                               
                                 v 
                                 i 
                               
                             
                             ⁢ 
                             
                               
                                 λ 
                                 ^ 
                               
                               T 
                             
                           
                         
                       
                     
                   
                   = 
                   
                     V 
                     Total 
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     Plugging L′(Λ)=0 into equation (e) gives L″(Λ)&lt;0. So the Λ value found using equation (1) indeed maximizes L(Λ). Also, using the derivatives at Λ, the standard error of Λ also can be calculated using the observed Fisher information L″({circumflex over (Λ)}). 
     
       
         
           
             
               
                 
                   
                     σ 
                     
                       Λ 
                       ^ 
                     
                   
                   = 
                   
                     
                       variance 
                     
                     = 
                     
                       
                         
                           1 
                           
                             
                               - 
                               L 
                             
                             ⁢ 
                             
                               ″ 
                               ⁡ 
                               ( 
                               
                                 Λ 
                                 ^ 
                               
                               ) 
                             
                           
                         
                       
                       = 
                       
                         1 
                         / 
                         
                           
                             
                               e 
                               
                                 2 
                                 ⁢ 
                                 
                                   Λ 
                                   ^ 
                                 
                               
                             
                             · 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 a 
                               
                               
                                 
                                   
                                     v 
                                     i 
                                     2 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         - 
                                         
                                           v 
                                           i 
                                         
                                       
                                       ⁢ 
                                       
                                         Λ 
                                         ^ 
                                       
                                     
                                   
                                 
                                 
                                   
                                     ( 
                                     
                                       1 
                                       - 
                                       
                                         e 
                                         
                                           
                                             - 
                                             
                                               v 
                                               i 
                                             
                                           
                                           ⁢ 
                                           
                                             Λ 
                                             ^ 
                                           
                                         
                                       
                                     
                                     ) 
                                   
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   f 
                   ) 
                 
               
             
           
         
       
     
     This σ {circumflex over (Λ)}  can be used to calculate the confidence interval. Calculating σ {circumflex over (Λ)}  using the expected Fisher information is not feasible because the volume distribution is unknown. In fact, to implement the DVV technique, the volume distribution is not required and need not be the same from one experiment to another. 
     Digital Variable Volume Approximation (DVVA) 
     In general, the probability a compartment turns ON can be calculated using the volume distribution (specified by the probability density function ƒ(v)). 
         Pon (λ T )=∫ƒ( v )PeaCh(λ T   ,v ) dV =∫ƒ( v )(1− e   −vλT ) dv= 1−∫ƒ( v ) e   vλT   dv   (g)
 
     Previously, ƒ(v) has been chosen to follow the gamma distribution or truncated normal distribution. However, in practice, ƒ(v) may not be described by a simple function. And even when that is true, a set of pre-measured volumes (M as in Table 1) still needs to be experimentally obtained to characterize ƒ(v). Therefore, for the DVVA technique, a set of separately measured volumes, M, is used instead of ƒ(v). 
     
       
         
           
             
               
                 
                   
                     
                       p 
                       on 
                     
                     ( 
                     λ 
                     ) 
                   
                   = 
                   
                     
                       1 
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             1 
                           
                           m 
                         
                         
                           
                             1 
                             m 
                           
                           ⁢ 
                           
                             ( 
                             
                               e 
                               
                                 
                                   - 
                                   
                                     v 
                                     i 
                                   
                                 
                                 ⁢ 
                                 
                                   λ 
                                   T 
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       1 
                       - 
                       
                         
                           1 
                           m 
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 1 
                               
                               m 
                             
                             
                               e 
                               
                                 
                                   - 
                                   
                                     v 
                                     i 
                                   
                                 
                                 ⁢ 
                                 
                                   λ 
                                   T 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   h 
                   ) 
                 
               
             
           
         
       
     
     The likelihood function can then be obtained using the binomial distribution (for the case of a ON compartments out of n compartments with the probability of  pon (λ). 
     
       
         
           
             
               
                 
                   
                     l 
                     ⁡ 
                     ( 
                     
                       λ 
                       T 
                     
                     ) 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             
                               n 
                             
                           
                           
                             
                               a 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           
                             p 
                             ON 
                             a 
                           
                           ( 
                           
                             1 
                             - 
                             
                               p 
                               ON 
                             
                           
                           ) 
                         
                         
                           n 
                           - 
                           a 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             
                               ( 
                               
                                 
                                   
                                     n 
                                   
                                 
                                 
                                   
                                     a 
                                   
                                 
                               
                               ) 
                             
                             [ 
                             
                               1 
                               - 
                               
                                 
                                   1 
                                   m 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       m 
                                     
                                     
                                       e 
                                       
                                         
                                           - 
                                           
                                             v 
                                             i 
                                           
                                         
                                         ⁢ 
                                         
                                           λ 
                                           T 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             ] 
                           
                           a 
                         
                         [ 
                         
                           
                             1 
                             m 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     λ 
                                     T 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         ] 
                       
                       
                         n 
                         - 
                         a 
                       
                     
                   
                 
               
               
                 
                   ( 
                   i 
                   ) 
                 
               
             
           
         
       
     
     As motivated above, the loglikelihood function can be calculated with the change of variable Λ≡In(λ T ), and subsequently, its first and second derivatives. 
     
       
         
           
             
               
                 
                                   
                   
                     
                       L 
                       ⁡ 
                       ( 
                       Λ 
                       ) 
                     
                     = 
                     
                       
                         
                           ( 
                           a 
                           ) 
                         
                         ⁢ 
                         
                           ln 
                           ⁡ 
                           ( 
                           
                             1 
                             - 
                             
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                   
                                 
                               
                               m 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           ( 
                           
                             n 
                             - 
                             a 
                           
                           ) 
                         
                         ⁢ 
                         
                           ln 
                           ⁡ 
                           ( 
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     e 
                                     Λ 
                                   
                                 
                               
                             
                             m 
                           
                           ) 
                         
                       
                       + 
                       
                         ln 
                         ⁡ 
                         ( 
                         
                           
                             
                               n 
                             
                           
                           
                             
                               a 
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   j 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       L 
                       ′ 
                     
                     ( 
                     Λ 
                     ) 
                   
                   = 
                   
                     
                       
                         
                           ( 
                           
                             
                               a 
                               n 
                             
                             - 
                             1 
                             + 
                             
                               
                                 1 
                                 m 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           n 
                           m 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             m 
                           
                           
                             
                               v 
                               i 
                             
                             ⁢ 
                             
                               e 
                               
                                 Λ 
                                 - 
                                 
                                   
                                     v 
                                     i 
                                   
                                   ⁢ 
                                   
                                     e 
                                     Λ 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 1 
                                 m 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         
                           1 
                           m 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             m 
                           
                           
                             e 
                             
                               
                                 - 
                                 
                                   v 
                                   i 
                                 
                               
                               ⁢ 
                               
                                 e 
                                 Λ 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           [ 
                           
                             
                               a 
                               n 
                             
                             - 
                             
                               
                                 p 
                                 ON 
                               
                               ( 
                               
                                 e 
                                 Λ 
                               
                               ) 
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           n 
                           m 
                         
                         ⁢ 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               1 
                             
                             m 
                           
                           
                             
                               v 
                               i 
                             
                             ⁢ 
                             
                               e 
                               
                                 Λ 
                                 - 
                                 
                                   
                                     v 
                                     i 
                                   
                                   ⁢ 
                                   
                                     e 
                                     Λ 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             p 
                             ON 
                           
                           ( 
                           
                             e 
                             Λ 
                           
                           ) 
                         
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                               ON 
                             
                             ( 
                             
                               e 
                               Λ 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   k 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     L 
                     ″ 
                   
                   = 
                   
                     
                       
                         
                           L 
                           ′ 
                         
                         ( 
                         Λ 
                         ) 
                       
                       ⁢ 
                       
                         
                           e 
                           Λ 
                         
                         [ 
                         
                           1 
                           - 
                           
                             
                               
                                 1 
                                 m 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 
                                   
                                     v 
                                     i 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         - 
                                         
                                           v 
                                           i 
                                         
                                       
                                       ⁢ 
                                       
                                         e 
                                         Λ 
                                       
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   p 
                                   ON 
                                 
                                 ( 
                                 
                                   e 
                                   Λ 
                                 
                                 ) 
                               
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     p 
                                     ON 
                                   
                                   ( 
                                   
                                     e 
                                     Λ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           - 
                           
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 v 
                                 i 
                                 2 
                               
                             
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 
                                   v 
                                   i 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     v 
                                     i 
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                     
                     - 
                     
                       
                         
                           
                             ne 
                             
                               2 
                               ⁢ 
                               Λ 
                             
                           
                           ( 
                           
                             
                               1 
                               m 
                             
                             ⁢ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 
                                   v 
                                   i 
                                 
                                 ⁢ 
                                 
                                   e 
                                   
                                     
                                       - 
                                       
                                         v 
                                         i 
                                       
                                     
                                     ⁢ 
                                     
                                       e 
                                       Λ 
                                     
                                       
                                   
                                 
                               
                             
                           
                           ) 
                         
                         2 
                       
                       
                         
                           
                             p 
                             ON 
                           
                           ( 
                           
                             e 
                             Λ 
                           
                           ) 
                         
                         [ 
                         
                           1 
                           - 
                           
                             
                               p 
                               ON 
                             
                             ( 
                             
                               e 
                               Λ 
                             
                             ) 
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   l 
                   ) 
                 
               
             
           
         
       
     
     To maximize L(Λ), the root of L′(Λ) is found (equation (2), which is repeated below), and it is verified that it corresponds to a maximum by checking the sign of the second derivative (equation (m)). An interesting observation is that equation (2) can be obtained by using a/n to estimate  PoN  (λ T ) 
     
       
         
           
             
               
                 
                   0 
                   = 
                   
                     
                       
                         L 
                         ′ 
                       
                       ( 
                       Λ 
                       ) 
                     
                     = 
                     
                       
                         
                           a 
                           n 
                         
                         - 
                         1 
                         + 
                         
                           
                             1 
                             m 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   1 
                                 
                                 m 
                               
                               
                                 e 
                                 
                                   
                                     - 
                                     
                                       v 
                                       i 
                                     
                                   
                                   ⁢ 
                                   
                                     λ 
                                     T 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       = 
                       
                         
                           
                             a 
                             n 
                           
                           - 
                           
                             
                               p 
                               ON 
                             
                             ( 
                             
                               e 
                               Λ 
                             
                             ) 
                           
                         
                         = 
                         
                           
                             a 
                             n 
                           
                           - 
                           
                             
                               p 
                               ON 
                             
                             ( 
                             
                               λ 
                               T 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
       
         
           
             
               
                 
                                   
                   
                     
                       
                         
                           L 
                           ″ 
                         
                         ( 
                         Λ 
                         ) 
                       
                       
                         
                           ❘ 
                           &#34;\[LeftBracketingBar]&#34; 
                         
                         
                           Λ 
                           = 
                           
                             Λ 
                             ^ 
                           
                         
                       
                     
                     = 
                     
                       
                         
                           
                             L 
                             ″ 
                           
                           ( 
                           Λ 
                           ) 
                         
                         
                           
                             ❘ 
                             &#34;\[LeftBracketingBar]&#34; 
                           
                           
                             
                               L 
                               ⁢ 
                               
                                 ′ 
                                 ⁡ 
                                 ( 
                                 Λ 
                                 ) 
                               
                             
                             = 
                             0 
                           
                         
                       
                       = 
                       
                         
                           0 
                           - 
                           
                             
                               
                                 
                                   ne 
                                   
                                     2 
                                     ⁢ 
                                     Λ 
                                   
                                 
                                 ( 
                                 
                                   
                                     1 
                                     m 
                                   
                                   ⁢ 
                                   
                                     
                                       ∑ 
                                       
                                         i 
                                         = 
                                         1 
                                       
                                       m 
                                     
                                     
                                       
                                         v 
                                         i 
                                       
                                       ⁢ 
                                       
                                         e 
                                         
                                           
                                             - 
                                             
                                               v 
                                               i 
                                             
                                           
                                           ⁢ 
                                           
                                             e 
                                             Λ 
                                           
                                             
                                         
                                       
                                     
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               
                                 
                                   p 
                                   ON 
                                 
                                 ( 
                                 
                                   e 
                                   Λ 
                                 
                                 ) 
                               
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     p 
                                     ON 
                                   
                                   ( 
                                   
                                     e 
                                     Λ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                         
                         &lt; 
                         0 
                       
                     
                   
                 
               
               
                 
                   ( 
                   m 
                   ) 
                 
               
             
           
         
       
     
     Then σ {circumflex over (Λ)}  is calculated using the expected Fisher information, − L″(Λ) . The second derivative, L″(Λ), is a linear function of 
     
       
         
           
             
               a 
               n 
             
             - 
             
               
                 p 
                 ON 
               
               ( 
               
                 e 
                 Λ 
               
               ) 
             
           
         
       
     
     ((equation 2)). 
     
       
         
           
             
               〈 
               
                 
                   a 
                   n 
                 
                 - 
                 
                   
                     p 
                     ON 
                   
                   ( 
                   
                     e 
                     Λ 
                   
                   ) 
                 
               
               〉 
             
             = 
             0 
           
         
       
     
     can be plugged into equation (k), and the subsequent result can be plugged into equation (l) to obtain the following expression for σ {circumflex over (Λ)} . 
     
       
         
           
             
               
                 
                   
                     σ 
                     
                       Λ 
                       ^ 
                     
                   
                   = 
                   
                     
                       variance 
                     
                     = 
                     
                       
                         
                           
                             1 
                             
                               - 
                               
                                 〈 
                                 
                                   L 
                                   ⁢ 
                                   
                                     ″ 
                                     ⁡ 
                                     ( 
                                     Λ 
                                     ) 
                                   
                                 
                                 〉 
                               
                             
                           
                           = 
                         
                       
                       ⁢ 
                       
                         1 
                         
                           
                             ( 
                             
                               e 
                               Λ 
                             
                             ) 
                           
                           ⁢ 
                           
                             ( 
                             
                               
                                 1 
                                 m 
                               
                               ⁢ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     1 
                                   
                                   m 
                                 
                                 
                                   
                                     v 
                                     i 
                                   
                                   ⁢ 
                                   
                                     e 
                                     
                                       
                                         - 
                                         
                                           v 
                                           i 
                                         
                                       
                                       ⁢ 
                                       
                                         e 
                                         Λ 
                                       
                                     
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         
                           
                             
                               
                                 p 
                                 on 
                               
                               ( 
                               
                                 e 
                                 Λ 
                               
                               ) 
                             
                             · 
                             
                               [ 
                               
                                 1 
                                 - 
                                 
                                   
                                     p 
                                     on 
                                   
                                   ( 
                                   
                                     e 
                                     Λ 
                                   
                                   ) 
                                 
                               
                               ] 
                             
                           
                           n 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   n 
                   ) 
                 
               
             
           
         
       
     
     In this particular case, the standard error calculated using the observed Fisher information, −L″({circumflex over (Λ)}), is also the same as equation (n) evaluated at Λ={circumflex over (Λ)}. This can be verified by plugging L′({circumflex over (Λ)})=0 into L″({circumflex over (Λ)}) (equation (0). 
     EXAMPLE EMBODIMENTS 
     Example 1 includes a system, comprising: a device configured to generate compartments of a sample, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments; a detector configured to determine a number of the compartments each having a respective number of a target that is greater than a threshold number of the target; and electronic circuitry configured to determine a bulk concentration of the target in a source of the sample in response to the determined number of the compartments. 
     Example 2 includes the system of Example 1 wherein the device includes a container configured to generate the compartments of the sample in a barrier phase in response to the container moving. 
     Example 3 includes the system of any of Examples 1-2 wherein the device includes a container configured to generate the compartments of the sample in a liquid in response to a shaking of the container. 
     Example 4 includes the system of any of Examples 1-3 wherein the device includes a container configured to generate the compartments of the sample as droplets of the sample in an oil in response to a shaking of the container. 
     Example 5 includes the system of any of Examples 1-4 wherein the device includes a container configured to generate the compartments of the sample as droplets of the sample in a barrier phase in response to a shaking of the container, the droplets each have a viscosity, the barrier phase having a viscosity that is greater than the viscosity of the droplets. 
     Example 6 includes the system of any of Examples 1-5 wherein the detector is configured to determine the number of the compartments each having a respective number of the target that is greater than the threshold number of the target in response to a wavelength of electromagnetic energy at which each of the number of the compartments luminesces. 
     Example 7 includes the system of any of Examples 1-6 wherein the detector is configured to determine the number of the compartments each having a respective number of the target that is greater than the threshold number of the target in response to a wavelength of electromagnetic energy that each of the number of the compartments absorbs. 
     Example 8 includes the system of any of Examples 1-7 wherein the detector is configured to determine the number of the compartments each having a respective number of the target that is greater than the threshold number of the target in response to a wavelength of electromagnetic energy that each of the number of the compartments passes. 
     Example 9 includes the system of any of Examples 1-8 wherein the detector is configured to determine the number of the compartments each having a respective number of the target that is greater than the threshold number of the target in response to a wavelength of electromagnetic energy that each of the number of the compartments blocks. 
     Example 10 includes the system of any of Examples 1-9 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a respective measured volume of each of the number of compartments. 
     Example 11 includes the system of any of Examples 1-10 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a sum of a respective measured volume of each of the compartments. 
     Example 12 includes the system of any of Examples 1-11 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a number of the compartments. 
     Example 13 includes the system of any of Examples 1-12 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a number of other compartments. 
     Example 14 includes the system of any of Examples 1-13 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a respective measured volume of each of other compartments. 
     Example 15 includes the system of any of Examples 1-14 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a number of other compartments and a respective measured volume of each of the other compartments. 
     Example 16 includes the system of any of Examples 1-15 wherein the electronic circuitry is configured to determine a bulk concentration of the target in a source of the sample in response to a probability density function of compartment volume. 
     Example 17 includes a system, comprising: a barrier-phase liquid; a container configured to receive the barrier-phase liquid, to receive a sample including a target, and to generate compartments of the sample suspended in the barrier-phase liquid in response to a shaking of the container, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments; and a detector configured to determine a number of the compartments each having a respective number of the target that is greater than a threshold number of the target. 
     Example 18 includes the system of Example 17 wherein the barrier-phase liquid includes an oil. 
     Example 19 includes the system of any of Examples 17-18 wherein the container includes a clear tube. 
     Example 20 includes the system of any of Examples 17-19 wherein the detector includes an electronic detector. 
     Example 21 includes the system of any of Examples 17-20 wherein the detector is configured to determine a number of the compartments. 
     Example 22 includes the system of any of Examples 17-21, further comprising an apparatus configured to obtain the sample from a source including the target. 
     Example 23 includes the system of any of Examples 17-22, further comprising a computer-readable medium storing instructions that, when executed by a computing circuit, cause the computing circuit to determine a bulk concentration of the target in a source of the sample in response to the number of the compartments each having a respective number of the target that is greater than a threshold number of the target. 
     Example 24 includes a method, comprising: generating compartments of a sample, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments; determining a number of the compartments each having a respective number of a target that is greater than a threshold number of the target; and determining a bulk concentration of the target in a source of the sample in response to the number of the compartments. 
     Example 25 includes the method of Example 24 wherein generating the compartments includes generating the compartments suspended in a barrier phase by shaking a container that includes the sample and the barrier phase. 
     Example 26 includes the method of any of Examples 24-25 wherein generating the compartments includes generating droplets suspended in a liquid by shaking a container that includes the sample and the liquid. 
     Example 27 includes the method of any of Examples 24-26 wherein determining the number of compartments each having a respective number of the target that is greater than the threshold number of the target includes determining the number of compartments in response to a wavelength of electromagnetic energy at which each of the number of the compartments luminesces. 
     Example 28 includes the method of any of Examples 24-27 wherein determining the number of the compartments each having a respective number of the target that is greater than the threshold number of the target includes determining the number of compartments in response to a wavelength of electromagnetic energy that each of the number of the compartments absorbs. 
     Example 29 includes the method of any of Examples 24-28 wherein determining the number of the compartments each having a respective number of the target that is greater than the threshold number of the target includes determining the number of compartments in response to a wavelength of electromagnetic energy that each of the number of the compartments passes. 
     Example 30 includes the method of any of Examples 24-29 wherein determining the number of the compartments each having a respective number of the target that is greater than the threshold number of the target includes determining the number of compartments in response to a wavelength of electromagnetic energy that each of the number of the compartments blocks. 
     Example 31 includes the method of any of Examples 24-30 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a respective measured volume of each of the number of compartments. 
     Example 32 includes the method of any of Examples 24-31 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a sum of a respective measured volume of each of the compartments. 
     Example 33 includes the method of any of Examples 24-32 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a number of the compartments. 
     Example 34 includes the method of any of Examples 24-33 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a number of other compartments. 
     Example 35 includes the method of any of Examples 24-34 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a respective measured volume of each of other compartments. 
     Example 36 includes the method of any of Examples 24-35 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a number of other compartments and a respective measured volume of each of the other compartments. 
     Example 37 includes the method of any of Examples 24-36 wherein determining the bulk concentration of the target in the source of the sample includes determining the bulk concentration in response to a probability density function of compartment volume. 
     Example 38 includes a tangible non-transitory computer-readable medium storing instructions that, when executed by a computing circuit, cause the computing circuit: to determine a number of compartments of a sample each having a respective number of a target that is greater than a threshold number of the target, at least one of the compartments having a respective volume that is different from a respective volume of each of at least another one of the compartments; and to determine a bulk concentration of the target in a source of the sample in response to the determined number of compartments. 
     From the foregoing it will be appreciated that, although specific embodiments have been described herein for purposes of illustration, various modifications may be made without deviating from the spirit and scope of the disclosure. Furthermore, where an alternative is disclosed for a particular embodiment, this alternative may also apply to other embodiments even if not specifically stated. In addition, any described component or operation may be implemented/performed in hardware, software, firmware, or a combination of any two or more of hardware, software, and firmware. For example, any of one, more, or all of the above-described operations and functions can be performed by electronic circuitry that is hardwire configured to perform one or more operations or functions, that is configured to execute program instructions to perform one or more operations or functions, that is configured with firmware, or otherwise configured, to perform one or more operations or functions, or that is configured with a combination of two or more of the aforementioned configurations. For example, one or more of the components of the computer  54  of  FIG.  11    can include such electronic circuitry. Furthermore, one or more components of a described apparatus or system may have been omitted from the description for clarity or another reason. Moreover, one or more components of a described apparatus or system that have been included in the description may be omitted from the apparatus or system. In addition, one or more steps of a described method may have been omitted from the description for clarity or another reason. Moreover, one or more steps of a described methods that have been included in the description may be omitted from the method.