Patent Application
20060265143
The technical field relates to analyzing observations of a sample to determine the concentration of a substance within the sample, and more particularly to determining a concentration via a sigmoid curve representing observations for the sample.
One way of determining the concentration of a substance in a sample is by performing serial dilution on the sample. Serial dilution techniques collect a finite number of data points for the sample by taking one or more observations (e.g., indicating optical density) of various dilutions (e.g., dilutions formed by adding various quantity of diluent to the sample). For example, dilutions of 10%, 1%, 0.1%, etc. can be measured for optical density.
The results can then be used to determine a concentration of the substance in the sample via reference to a sigmoid curve representing serial-dilution observations for a sample having a known concentration of the substance (sometimes called a “standard” or “characteristic” sigmoid curve).
The parameters of Formula (1) can be chosen so that the function ƒ(x) calculates the optical density based on a particular dilution x. Given an optical density 130 for a sample having an unknown concentration of the substance and the degree of dilution associated with the sample, the concentration of the substance can be back-calculated. In practice, plural observations of the optical density can be taken for plural degrees of dilution and applied to the standard curve.
Various techniques have been used to define the curve, analyze the observations, and calculate a concentration. One method is described by O'Connell, et al., “Calibration and assay development using the four-parameter logistic model,” Chemometrics and Intelligent Laboratory Systems, 20 (1993) 97-114, Elsevier Science Publishers B.V., Amsterdam (“O'Connell”), which is hereby incorporated herein by reference. The O'Connell approach describes determining a minimum detectable concentration (MDC) and a reliable detection limit (RDL).
The O'Connell technique can produce significant variability in its results. In certain scenarios, variability is to be avoided. Therefore, there exists a need for technologies that avoid variability in their results and otherwise make better use of the sigmoid curve.
Various technologies described herein proceed by analyzing data via a sigmoid curve, such as a standard sigmoid curve representing observations of a sample having a known quantity of a substance.
Observation of a test sample can be taken (e.g., in serial dilution scenarios) for a sample, and the technologies described herein can be applied when calculating concentration of a substance within the sample via the sigmoid curve.
In some examples, derivative techniques are applied to determine various points or ranges of the curve. The points or ranges can represent the usable portion of the sigmoid curve or a threshold value reliably indicating presence of the substance.
Some of the technologies exhibit superior reduction in variability of results and are thus useful in a variety of fields, such as testing for levels of antibodies or antigens in a sample. For example, the technologies can be used when testing for titers of antibodies or antigens (e.g., in serum) via serial dilution. The technologies can also be applied with advantage to a variety of scenarios, such as bioassay, polymerase chain reactive (“PCR”) assay, radioimmuno assay (“RIA”), cell growth assay, cell death assay, enzyme-linked immunosorbent assay (“ELISA”), toxin neutralization assay (“TNA”) (e.g., scenarios involving determining the concentration of biological toxins such as anthrax), and flow cytometry scenarios, or any test in which sample results are derived from a standard sigmoid curve.
Additional features and advantages of the disclosed technologies will be made apparent from the following detailed description of illustrated embodiments, which proceeds with reference to the accompanying drawings.
In various of the examples described herein, calculations for determining concentration of a substance in a test sample are performed by identifying a usable portion of a sigmoid curve. Performing the calculations with the usable portion of the curve can improve or optimize the concentration calculation. The portion of the curve that is usable can be determined via derivatives.
At 210, the usable portion of a standard sigmoid curve is determined. For example, a range of points between two endpoints can be defined as the usable portion. The standard sigmoid curve can be generated via observations of a sample having a known amount of the concentration present Observations can be taken for different dilutions (e.g., dilution ratios). In practice, plural observations can be taken for each dilution. To facilitate calculations related to the curve, a curve represented via four parameters (e.g., such as those of Formula 1) can be fit to the observations.
At 220, the concentration (e.g., amount or quantity) of a substance in a test sample can be determined via the usable portion of the curve. For example, given a set of observations of the test sample, some of the observations may relate to portions of the curve outside of the usable area. Such observations need not be included in concentration calculation. Instead, a subset of the observations (e.g., those associated with the usable portion of the curve) can be used in the calculations to determine concentration.
The usable portion of a sigmoid curve (e.g., a standard sigmoid curve), can be determined via derivatives of the sigmoid curve. For example, a maximum and minimum of the second derivative can be used.
At 310, a first bound (e.g, endpoint) of the range is found via the second derivative of the sigmoid curve (e.g., by finding a local minimum or local maximum of the second derivative of a representation of the curve, such as the four-parameter logistic function). At 320, the other bound (e.g., endpoint) of the range is found via the second derivative of the sigmoid curve (e.g., by finding a local maximum or local maximum, inflection of the second derivative of a representation of the curve, such as the four-parameter logistic function).
The usable portion, then, can be determined as the portion of the sigmoid curve between the two bounds (e.g., endpoints).
A first point 430 on the second derivative 420 indicates a point on the curve 410 designated as a bound (e.g., a minimum of the second derivative 420 in the example). A second point 440 on the second derivative 430 indicates another point on the curve 410 designated as a bound (e.g., a maximum of the second derivative 420 in the example).
The range 460 of the curve 410 between the two bounds 430, 440 is sometimes called the “quantification range” because it is the range of the curve that is considered usable when determining the concentration of a substance in a test sample.
The point value 450 associated with the point 440 is sometimes called the “quantification titer” because it indicates a value beyond which the curve 440 is no longer useful (e.g., a lower bound of concentration reliably calculable via the curve).
A threshold value useful in calculating concentration of a substance in a test sample can be determined in a variety of ways. One way involves finding a threshold value at which a first derivative for the curve reaches a benchmark value. Such a benchmark value can be empirically determined (e.g., chosen by a researcher based on evaluation of a number of standard sigmoid curves) and then used uniformly throughout In one implementation, the benchmark value is designated as thirty percent (30%) of the maximum of the first derivative; however, a value approximating thirty percent or a different value may be useful in other scenarios. Any of the methods can be implemented in software.
The point at which the first derivative is equal to the threshold is sometimes called the “threshold titer” because it indicates a minimum detectable concentration according to the technique. Thus, if an observation of a test sample indicates a value above the threshold titer, presence of the substance is indicated, even if the exact concentration may not be reliably determined. Under some scenarios (e.g., detection of toxins), such a threshold titer can be used to advantage.
In the example, a benchmark value 540 is used to determine a point 530 on the first derivative 520 at which the standard curve 510 crosses the benchmark value 540. From the point 530, a threshold titer value 550 can be determined. If a sample has an observation above the threshold titer value 550, presence of the substance being measured is indicated.
In the example, the first derivative 520 has two points at which the curve 510 crosses the benchmark value 540. The point related to the lowest concentration (e.g., the lower value of the curve 510) is used.
The threshold titer value 550 can be used in a method to determine whether the substance of interest is present. For example, a sample for which at least one observation indicates at least such a value can be designated as containing the substance of interest Although the example describes a threshold titer, the technology can be applied to metrics other than titers.
If desired, another threshold at 560 can be found via the first derivative and used in certain scenarios.
At 610, a usable portion of a sigmoid curve is determined (e.g., via any of the technologies described herein).
At 620, a threshold value (e.g., threshold titer) is determined (e.g., via any of the technologies described herein).
At 630, for one or more observations of a sample having unknown substance concentration, the observations are compared to the threshold and a concentration is calculated for them via the usable portion of the curve. Those observations associated with a non-usable portion of the curve are not included. However, an observation above the threshold can indicate presence of the substance, even if a concentration cannot be calculated for the observation.
In practice, 610 and 620 may be performed once for a standard curve, and 630 may be performed plural times for plural samples (e.g., at a different time, location, or by a different computer user at a different computer). Alternatively, in other scenarios, 610 and 620 may be performed plural times for the standard curve (e.g., if the standard test serum is included plural times on plural respective plates).
In the example, a usable portion of the curve has been determined. For observations along the test-sample curve, concentrations are found. Certain observations (e.g., OD7) have values not within the usable portion of the curve; therefore they are not used (e.g., are discarded). Thus, a subset of the observations (e.g., those associated with the usable portion of the standard curve) is used to calculate concentration.
In any of the examples described herein, the sigmoid curve can represent observed optical density for a plurality of serial dilutions. For example, a reference sample having a known concentration of a substance can be observed by measuring the optical density of a plurality of observations for a plurality of dilutions of the sample. The sample can be subjected to various conditions before or after dilution.
In some scenarios, the dilution lessens the amount of protective (e.g., cell-saving or antibody) agent, resulting in increased cell death. Optical density measurements at an appropriate wavelength can be used to measure the number of living cells.
In any of the examples described herein, the technologies can be applied to measure the concentration of antibody in a sample. For example, the sample can be diluted and added to a set of cells in the presence of a cell toxin. The presence of antibody will protect the cells from the toxin; the concentration of living cells thus indicates concentration of the antibody. Alternative techniques, such as measuring binding (e.g., via a label or other marker) can be used.
Presence of antibody can indicate presence of a toxin. For example, in one implementation, the concentration of anthrax is indirectly determined by measuring the presence of anti-PA IgG. When constructing a standard curve for such a scenario, dilution can reduce the amount of protective serum (e.g., a sample containing anti-PA IgG) that tends to preserve the life of cells. A low optical density thus indicates more dead cells; a high density indicates more living cells. Less protective serum tends to result in more dead cells and thus a lower optical density.
A sample having low optical density below the threshold titer is considered to test negative for anthrax (e.g., there was no anti-PA IgG to protect the cells). Various other procedures can be combined into a standard protocol to promote uniformity of results across test samples.
Any of the technologies described herein can be used to test for the concentration or presence of antibody. For example, a method for testing a blood serum sample for the concentration of antibody in the blood sample can proceed as follows:
(1) measuring the optical density of a test sample, wherein the test sample is generated by adding the serum to cells and a toxin neutralized by the antibody;
(2) determining whether a measurement indicating concentration of live cells (e.g., the optical density) falls within a usable portion of a standard sigmoid curve representing observations taken of a sample having a known concentration of antibody; and
(3) responsive to determining the measurement indicating concentration of live cells (e.g., optical density) falls within the usable portion, calculating a concentration via the standard sigmoid curve.
In any of the examples, the antibody can be anti-PA IgG, which indicates infection by the anthrax toxin.
During the method, one or more observations having an optical density outside the usable portion of the standard sigmoid curve can be discarded. The usable portion of the sigmoid curve can be determined via a second derivative of the sigmoid curve as described in any of the examples.
The technologies described herein can be used in conjunction with any dilution-range-type micro-biological assay, including but not limited to enzyme-linked immunosorbent assay (“ELISA”), toxin neutralization/neutralizing assay (“INA”), and flow opsono techniques.
The technologies described herein can be applied in a variety of other bioassay analysis scenarios to advantage. For example, the technologies can also be applied with advantage to polymerase chain reactive (“PCR”) assay, radioimmuno assay (“RIA”), cell growth assay, cell death assay, and flow cytometry scenarios, or any test in which sample results are derived from a standard sigmoid curve.
Observations can be taken in a variety of ways to determine concentration of a substance. For example, in biological scenarios, the number of living cells can be measured (e.g., by measuring optical density at an appropriate wavelength).
Alternatively, a binding agent can be used. Such a binding agent may include a marker (e.g., chemical, fluorescent, radioactive, and the like). The marker can assist in detection. After binding, an observation can be taken by measuring the resulting bound substance (e.g., by measuring optical density).
Serum samples were analyzed during 2001, 2002, and 2003 according to a protocol for a TNA for anti-PA antibodies. The protocol is described in Quinn et al., “Specific, sensitive, and quantitative enzyme-linked immunosorbent assay for human immunoglobulin G antibodies to anthrax toxin protective antigen”, Emerging Infectious Diseases, 8: 10, Centers for Disease Control, October 2002, (available at http://www.cdc.gov/ncidod/EID/vol8no10/02-0380.htm) which is hereby incorporated herein by reference. A standard human serum was tested for neutralization of PA toxin separately on 103 plates.
Four samples including a serum standard (AVR414) sample were analyzed on each plate. Each sample consisted of a seven-point dilution with triplicates at each dilution. For samples which met primary quality control (“QC”), four-parameter logistic curves (see Formula 1) were fitted to the raw data, using the NLIN procedure of SAS software (SAS Institute, Cary N.C., USA).
The statistical observations and analytical conclusions are based on empirical data developed by Center for Disease Control's anthrax serology lab over the last two years. Numeric results apply specifically to subsets of approximately 1500 anthrax TNA plates which have been analyzed in the lab and whose lab results have been analyzed statistically.
Quantification Titer, and Threshold Titer involved less variability as measured by Coefficient of Variation (“CV”) and by range of values over the 103 plates. The results are shown in Table 1. Titer indicates some measure of concentration. For the nominal lower limit, MDC ranged from 1600 to 11,650 with an overall CV of 41%, whereas the Threshold Titer ranged from 1690 to 8,810 with a CV of 37%. For the reliable lower limit, RDL ranged from 1180 to 7,660 with a CV of 38%, while the Quantification Titer ranged from 1130 to 3,960 with a CV of 29%.
The described technologies can take advantage of the fact that the knees (i.e., the upper and lower bends) of a symmetric sigmoid curve (e.g., the four-parameter logistic-log curve) can be identified directly through the use of the first and/or second derivatives. Such an approach can be superior to the O'Connell approach, which evaluates error in the inverse function.
The described Quantification Titer and Threshold Titer are especially precise for assays whose curve-fit is especially good (e.g., curves whose R-squares [R-squared for the non-linear curve fit is defined as the difference between the sum of squares for the observed values and the sum of squares for the residuals divided by the sum of squares for the observed values: (SSQobserved-SSQresidual)/SSQobserved] are in the range of 0.99 to 1.00).
In general, titers are computed by identifying some response threshold. Then, the fitted curve is used to translate that threshold from the response axis (y or vertical axis) to the dilution axis (x or horizontal axis). An example is shown in
Methods of determining that response threshold are described herein. Also, the four-parameter logistic (“4PL”) curve was used to “back-compute” (e.g., apply the inverse 4PL function to compute) sample concentrations, by applying the relationship between response variable (optical density) and dilution from the standard-serum curve to the observed sample curve or raw data points. An example is shown in
The described technologies can be incorporated into a TNA system. The system can compute the 4PL curve, the ED50, two implicit titers (e.g., threshold and quantification), the usable portion of the curve or quantification range, and the implicit concentrations of test samples. The method system can identify the “knees” of the 4PL curve directly through the use of the derivatives.
The Threshold Titer can be defined using an empirically derived threshold in the first derivative to define the point on the curve, thereby identifying a titer point on the dilution scale (e.g., as shown in
The fitted curve for the standard serum, having passed primary and secondary quality control (“QC”), can establish the normalized relationship between optical density (“OD”) and dilution for a specific plate. Mathematically, this establishes OD as a function of dilution, i.e. OD=ƒ(DIL), where ƒ is the four-parameter logistic function. This is called the standard curve or characteristic curve for the plate.
Further, the usable portion of the standard curve is that domain of dilutions or that corresponding range of OD's over which the inverse function can be used reliably, where “reliably” is defined by a specified level of precision, typically stated as a specified percent-CV across the implicit concentrations either across each and every dilution (“within-dilution”) or across all dilutions (“within assay.”) That is, the inverse function, DIL=ƒ−1(OD) can be and is applied to the OD's of the test samples to “back-calculate” the dilutions and therefore the initial sample concentrations, since the starting titration dilutions are known.
Any of the examples described herein can be used in any number of protocols. For example, a certain number of observations falling within the usable portion of the curve can be required, various quality assurance (“QA”) tests can be required, etc.
In any of the examples herein, a number of metrics can be used to determine the concentration of a substance. Some of the examples use a metric of optical density, but the technologies described herein can be applied to other metrics.
In any of the examples herein, the technologies can measure the concentration of one substance to indicate the concentration of another substance. Thus, measurement of a concentration can be achieved indirectly. For example, for an antigen such as a toxin, the concentration of antibody present in a sample taken from a subject (e.g., a human patient) can indicate the concentration of the toxin in the subject's body, for example the concentration in a body fluid, such as blood, for example, blood serum. Further, the concentration of live cells in the presence of a toxin can indicate the presence of an antibody neutralizing the toxin. Thus, the concentration of live cells in a toxin neutralization assay can be used to indirectly indicate the concentration of the toxin.
For any of the technologies described herein, a plurality of points can be used to generate a test (e.g., for a sample) sigmoid curve (e.g., according to the 4PL formula). The sample curve can then be used to calculate concentration in light of a usable portion of a standard sigmoid curve (e.g., determined via any of the technologies described herein). Such an approach can be used instead of or in addition to using points to calculate concentration.
Although examples herein describe arrangements in which the usable portion of the standard curve is determined, an approach could alternatively or in addition determine the usable portion of a test curve.
When determining concentration in light of the two curves, calculations can proceed by considering a number of points on the test curve within the usable portion of the standard curve (e.g., using a particular point resolution).
Even though a curve can be generated from test points falling outside the usable portion of the standard curve, it may be desirable to require one or more test points to fall within the usable portion of the standard curve (e.g., for quality control purposes).
Various advantages can be gained by using the examples described herein. The examples described herein may benefit from zero or more of the following advantages.
When the described technology is used across plates, it establishes very specific quantitative characteristics of a given material such as a serum standard.
The described technology allows great flexibility in determining if and how a reagent fits the expected model and therefore provides more robust computation of endpoints.
The described technology provides a strong quality control test for known material or substance.
The described technology increases accuracy.
The described technology increases repeatability.
Given an assay which is sufficiently precise biologically, the described technology presented improves on the reproducibility and therefore the accuracy and precision of prior computational methods.
For example, when the described technologies relating to first and second derivatives are applied to an anthrax TNA scenario, the long-run effect is that the stability of the described technology is higher (e.g., as measured by higher variance) than O'Connell's MDC and RDL.
The described technology provides a continuous, as opposed to discreet, endpoint determination and is therefore more precise. For purposes of illustration, assume a compared % CV technique can only use only specific, observed dilutions of the test-sample for end-points (e.g., of the usable portion of the curve). In a worst-case example, if the true (but yet-unknown) usable portion of a curve covered the domain from a dilution of for example, 500 to 1500, and the dilution sequence for the test sample were 200, 400, 800, 1600 . . . then the compared % CV approach would fail at 400 and fail at 1600, leaving only the test triplicate at 800 for the back-calculation. By many protocols, use of a single triplicate would be insufficient, so in this specific example, there would be no reportable test result.
By contrast, the described technology can avoid the above outcome if the test-sample itself produced an acceptable sigmoid curve. In such a case, additional OD points on the test curve between dilutions of 500 and 799 and between 801 and 1500 can be imputed, using knowledge of the test curve gained from the dilutions 400, 1600 and beyond, even though these data are outside the usable portion of the standard curve. These additional points can be used to back-calculate additional concentrations and produce reportable results. These results rely in a predictable way on information gained from the test curve and from the standard curve without constraint to the two or three test dilutions points (one point in the above example) which might happen to fall in the usable domain of the standard curve.
The described technology can define the usable portion of the curve before the application of the curve to the test-sample results, rather than after the application of the inverse function. In a comparison technique, the measure of the usable portion of the standard curve is whether or not it did work as evidenced by post-use error, (i.e. % CV in the implicit test-sample concentrations). By contrast, the described technology can directly measures the slope of the curve and the second-derivative characteristics of that slope, thereby allowing direct identification of the knees of the curve and the domain between the knees. Thus, the usable portion of the curve can be defined. The described technology can exploit the characteristics of the standard curve itself, rather than implicit characteristics based on any particular test sample.
Although the described technologies can be used with any concentration units, exemplary units used in association with determining concentration include titers, grams/liter (e.g., micrograms/milliliter), or other measurements of concentration.
Any of the methods described herein can be implemented in software. For any of the methods implemented in software, the depicted actions can be achieved via computer-executable instructions carrying out the depicted actions. Such instructions can be encoded on one or more computer-readable media (e.g., RAM, ROM, CD-ROM, DVD, hard disk, removable media, flash media, and the like).
When implemented in software, the various inputs to the methods (e.g., observations of a test sample) can be provided via a user interface or sent directly via test equipment to the computer system. The outputs can be provided via user interface, sent directly to output, or communicated via a network to another computer system, as desired.
A comparer 1140 is operable to determine whether an observation of 1130 is within the usable portion of the sigmoid curve as indicated by the representation 1120. For example, a function or other logic can make such a determination.
A calculator 1150 is operable to calculate a concentration for the test sample via the observations 1130 and the sigmoid curve representation 1110. The calculator 1150 can be configured so that calculation is performed responsive to determining that the observation is within the usable portion of the sigmoid curve. Observations determined to be outside the usable portion of the sigmoid curve can be rejected.
The usable portion of the sigmoid curve as represented by the representation 1120 can be calculated via a second derivative of the sigmoid curve as described in any of the examples herein. The system 1100 can further include means for calculating such usable portion (e.g., by determining endpoints).
The pictured system 1100 can be implemented in software stored on one or more computer-readable media and can be implemented in any number of computer languages.
In the example, analysis has been performed for six (6) samples. The window 1230 indicates the sample identifier, a concentration of a substance within the sample, an indication of whether the substance was present in the sample, the number of observations for the sample, and the number of observations discarded (e.g., for falling outside of the usable portion of the standard sigmoid curve).
Additional, fewer, or different elements can be added to the user interface 1200. For example, a representation (e.g., file name or list of parameters) of the sigmoid curve can be presented. Also, other discarded observations (e.g., failing to meet QC) can be noted. The presence of the substance can be alternatively indicated (e.g., by color, flashing text, or icon). Similarly, the other results can be alternatively indicated.
Any combination of the user interface described herein can be used in software. Additional, fewer, or different elements can be used as an alternative to those pictured.
For those actions specified as computer-executable, such actions can be performed fully-automatically (e.g., without human intervention) or semi-automatically (e.g., with assistance from a human operator). One or more computer-readable media can comprise the instructions described as computer-executable.
Various implementations of the technologies described herein can be called the “Taylor method,” named after inventor Thomas H. Taylor, Jr.
In view of the many possible embodiments to which the principles of the invention may be applied, it should be recognized that the illustrated embodiments are examples of the invention, and should not be taken as a limitation on the scope of the invention. Rather, the scope of the invention includes what is covered by the following claims. I therefore claim as my invention all that comes within the scope and spirit of these claims.
This application claims the benefit of U.S. Provisional Patent Application No. 60/456,613 filed Mar. 21, 2003, which is incorporated herein by reference.
| Filing Document | Filing Date | Country | Kind | 371c Date |
|---|---|---|---|---|
| PCT/US04/08566 | 3/19/2004 | WO | 9/20/2005 |
| Number | Date | Country | |
|---|---|---|---|
| 60456613 | Mar 2003 | US |
