^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: AE ARV. Performed the experiments: AE MP. Analyzed the data: AE. Contributed reagents/materials/analysis tools: AE MP. Wrote the paper: AE MP ARV.

Most recent advances in fluorescence microscopy have focused on achieving spatial resolutions below the diffraction limit. However, the inherent capability of fluorescence microscopy to non-invasively resolve different biochemical or physical environments in biological samples has not yet been formally described, because an adequate and general theoretical framework is lacking. Here, we develop a mathematical characterization of the biochemical resolution in fluorescence detection with Fisher information analysis. To improve the precision and the resolution of quantitative imaging methods, we demonstrate strategies for the optimization of fluorescence lifetime, fluorescence anisotropy and hyperspectral detection, as well as different multi-dimensional techniques. We describe optimized imaging protocols, provide optimization algorithms and describe precision and resolving power in biochemical imaging thanks to the analysis of the general properties of Fisher information in fluorescence detection. These strategies enable the optimal use of the information content available within the limited photon-budget typically available in fluorescence microscopy. This theoretical foundation leads to a generalized strategy for the optimization of multi-dimensional optical detection, and demonstrates how the parallel detection of all properties of fluorescence can maximize the biochemical resolving power of fluorescence microscopy, an approach we term Hyper Dimensional Imaging Microscopy (HDIM). Our work provides a theoretical framework for the description of the biochemical resolution in fluorescence microscopy, irrespective of spatial resolution, and for the development of a new class of microscopes that exploit multi-parametric detection systems.

Fluorescence microscopy provides an invaluable tool to probe cell and tissue biochemistry. Fluorophores sensitive to the physico-chemical properties of the environment or fluorescent sensors engineered to probe biochemical reactions encode biologically relevant information into changes of their photophysical properties. The read-out of these probes is typically performed with the quantitative detection of specific photophysical properties,

Over the past decade, most academic and industrial developments in microscopy have focused on spatial super-resolution techniques

Fisher information theory has been conveniently used to study the upper boundary of signal-to-noise ratios (SNR) achievable in fluorescence detection

Furthermore, we aim to lay down the theoretical foundations (and justifications) for multi-dimensional techniques and, more specifically, for spectrally- and polarization- resolved time-correlated single-photon counting that would enable the parallel detection of all properties of light. For conciseness, this technique will be referred to as Hyper Dimensional Imaging Microscopy or HDIM

In this work, Fisher information is used in order to define and describe the physical limit in biochemical resolving power of an optical system and to characterize how a multi-channel detection system can attain the highest possible biochemical resolution from a theoretical standpoint. In this section, we provide definitions and theorems that enable the description of biochemical precision and resolution in fluorescence microscopy; the demonstrations can be found in the

A multi-channel optical instrument partitions detected photons into histograms of photon counts collected over ranges of arrival times, wavelengths and polarization states. It is possible to characterize the general properties of the Fisher information content provided by any given partition and to demonstrate that detection systems of higher channel number and dimensionality increase the information content of an experiment as stated in the “photon partitioning theorem” below.

Let π be a partition of _{i}

In analogy to literature related to information theory

Although for a finer partition of photons the Fisher information may increase, the trivial duplication of a detection channel that collects light on an identical spectral, polarization and time range, cannot provide any advantage, but may rather increase only photon-losses or noise (see practical non-trivial partitioning note and

For any given detection channel with photon counts _{i}

Here, the vector

A fundamental and direct outcome of the photon partitioning theorem is that multi-channel multi-parametric detection systems are more photon-efficient (the capability to approach relative errors limited just by Poissonian noise)

For a non-trivial partitioning of _{x}^{2}

Photon-efficiency can be described by the figure-of-merit “^{−1/2} of the detected photons were lost. Therefore, a measure of photon-efficiency is

Let consider a photo-physical system that is sensitive to physico-chemical (biochemical) properties of its environment. If _{1}_{2}_{1}_{2}_{1}_{2}_{0}

This work is concerned with the biochemical resolution of a microscope; however, when

The photon partitioning theorem and its corollaries were described in the case of an ideal system in which the addition of a detection channel can be done without any penalty. This is useful to define the physical limits in Fisher information, precision and biochemical resolution of an optical system and not just the limit imposed by current technologies. In reality, the addition of detection channels can result in photon-losses (

The non-trivial partitioning of photons with higher channel density increases the physico-chemical (or biochemical) resolution of a detection system. Practical implications of the photon partitioning theorem can be demonstrated by implementing numerical optimization of a detection system. Fast iterative algorithms can identify the best number and configuration of detection channels by the numerical maximization of _{i} = G_{i}/N

Fluorescence lifetime imaging microscopy can be accomplished with a number of detection schemes, in the time- or frequency- domain and with single photon counting or analog detection

Gerritsen _{best} = 1.25,

A set of three typical Monte Carlo simulations (

The ^{−1} times in order to recover the same SNR compared to TCSPC. Therefore, the numerical analysis of Fisher information theory permit to evaluate which is the best performing and simplest detection system that should be used for FLIM.

Spectral detection is the detection of fluorescence over a number of spectral channels. Detection can be achieved by

Spectra representative of fluorophores like EGFP and mCherry (

True color images of cells expressing EGFP-Actin (

The photon partitioning theorem and its corollaries permit us to predict that multi-parametric detection schemes that exploit the various properties of light can attain higher signal to noise ratios compared to detection schemes based on fewer detection channels. This prediction was tested with Monte Carlo simulations for hyper-spectral imaging (SPEC), for the combined detection of photon arrival times and wavelength that can be achieved by spectrally resolved FLIM

HDIM data was also analyzed with a generalized phasor transform (Hyper-dimensional Phasors, or HDPH). A linear gradient (

HDIM data analysis was carried out both over all 2,048 detection channels per pixel and with a data reduction algorithm based on a generalization of phasor-based data analysis. For this type of analysis, discrete cosine (DCT) and sine (DST) transformations were computed over only 2 spectral bands (440–540 nm and 540–630 nm) similarly to Digman

Data analysis was then carried out by spectral un-mixing. The spectral un-mixing problem is described by Eq. (1) for the case of an HDIM image (

Similar equations can be written for SLIM, SPEC, and HDPH. By the inversion of Eq. (1) in the presence of Poissonian noise (250, 1000 and 10000 photons per pixel), we estimated the accuracy and precision of these various techniques (see

The validity of the photon partition theorem is illustrated by the increasing precision at higher channel number. The addition of detection channels along gradients established by the random variables the experiment attempt to estimate increases the information content of the measurements; therefore, the acquisition with a higher number of detection channels results in better signal to noise ratios.

We have shown that increasing the number of detection channels also across different photophysical parameters results in increased precision. Unmixing was used to demonstrate the possibility to separate the contribution of two different characteristic HDSS representative of two fluorophores or two biochemical/photophysical environments. In analogy to the spatial resolution in fluorescence microscopy, the resolution of biochemical techniques is also limited by photon-statistics. We developed a simple algorithm (see Supporting Material) that can determine the ^{2}·400∼7,100. In the same conditions, spectral imaging resolves the mixture with ∼3,300 photons and HDIM, making use of all available information, with just 1,900 photons. Although we have demonstrated that increasing the number of channels always increases the biochemical resolving power in fluorescence microscopy, there are instances where simpler techniques can achieve optimal results already. For instance, if two biochemical environments exhibit a spectral shift of 20 nm, spectral imaging (

Panels ^{2} times more photons (∼3,300) to achieve the same result.

Therefore, the numerical analysis of Fisher information and the study of its general properties provide a useful tool to maximize the biochemical resolving power in fluorescence microscopy and to select the best (and simplest) detection system to achieve this limit efficiently.

In this work, we demonstrate and exemplify the “

We have dedicated sections of this work to the description of practical constrains on achieving the theoretical maxima in Fisher information. In practice, the addition of a detection channel is carried out at a certain “cost”. This cost can be represented by photon-losses in the coupling optics, design complexity, additional read-out noise, or even budget. However, it is important to distinguish between the theoretical limits imposed by physics (described here) and the practical limitations of state-of-the-art technologies. Current HDIM detection systems do suffer losses, for instance, from the coupling and from the masks between adjacent anodes. However, single photon detection

In this work, we have been concerned with the description of theoretical aspects of multi-channel and multi-parametric detection and with the validation of the theory by Monte Carlo simulations. Furthermore, we have demonstrated the practical application of the theory for the particular case of spectral imaging, a technique nowadays available and used in many laboratories. In order to make the advantages of these theoretical frameworks accessible, software can be found in the supporting information. These algorithms can be used to find the minimal number of detection channels that optimize fluorescence detection for FLIM or spectral imaging. Furthermore, provided that HDSSs are known experimentally or can be modeled theoretically, the best detection scheme including spectral, lifetime and anisotropy resolution can be identified. In general, Hyper-dimensional imaging microscopy will achieve the best results; however, whenever the gradient

Over the last decade, much emphasis has been placed on methodologies and technologies that increase the spatial resolution of fluorescence microscopy. With this work, we aim to highlight that affordable implementations of novel detection schemes can boost the resolving power of modern microscopes in terms of physico-chemical or biochemical detection as well. The increased resolving power and capability to discern biochemical environments with spatio-temporal resolution together promise fundamental tools for applications in biophotonics, biophysics, systems biology and biomedical research.

HeLa cells were transfected with EGFP-actin and EYFP-tubulin with Effectene (Qiagen) according to the protocol provided by the suppliers. Cells were fixed after 24–48 hrs with 5% PFA and kept in PBS. Spectral imaging was performed with a Leica SP5 equipped with a hybrid PMT operated in single photon counting mode.

Simulations were performed with Matlab (MathWorks, Cambridge, UK) on a workstation equipped with an Intel Xeon X5647 CPU and 24 GB of RAM. Images of 256 by 256 pixels were generated with photons partitioned (

The typical solution of the unmixing problem (see Eq. (1)) was achieved by unconstrained least squares and non-negative least squares, although at a higher dimensionality compared to hyper-spectral imaging. Results from unconstrained least squares are shown because the latter did not show better performances on synthetic data. In Eq. (1),

A figure of merit for accuracy (

Because fluorescence emission is a Poissonian process, if _{i}_{i}_{i}

_{i}_{i}

Here,

Under basic assumptions of regularity and differentiability, the analytical representation of the Fisher information can be obtained computing the negative of the expectation of the second derivative of the log-likelihood that describes the experiment:

Functions that describe the fluorescence emission of fluorophores at room temperature are regular and smooth, thus this identity can be considered always true. Therefore, it is simple to show that:

From this follows that:

Because the expectation of

In order to show that the Fisher information increases with the addition of photon counting channels, let π’ be a finer partition of

Therefore, the Fisher information carried by the two new detection channels will be equal to:

Therefore, inasmuch as the second addendum in Eq. (10) is always positive or null, we have proven that partitioning the

Theoretically, detection channels with different (hyper) spectral characteristics can be combined to generate a large number of possible partitions of photons collected in any given experiment. The trivial duplication of a channel will not increase the Fisher information. A finer partition

We note that the mere duplication of a channel is trivial by definition and exhibit null derivative of the partitioning function therefore resulting in no gain of information.

Although the partitioning function has such a fundamental role in the optimization of detection schemes, we have demonstrated that optimization can be achieved without analytical solutions, but simply applying the general principles described by the photon partitioning theorem and its corollaries. However, it is instructive to analyze the partitioning functions for the simple case of fluorescence lifetime detection in the case that the random variable _{1}_{2}_{3}_{2}_{1}_{2}_{1}_{3}

The derivative of _{3}>_{1}, conditions always true. In fluorescence detection at room temperature, spectra, anisotropy and lifetime decays are always represented by smooth and continue functions and, therefore, partitioning functions are always defined.

The Fisher information is the inverse of the Rao-Cramer limit on the variance of an unbiased estimator. It is therefore rather trivial to demonstrate that for any non-trivial partitioning the following inequality holds:

This implies that with an increasing number of independent non-trivial noiseless channels the estimate of the unknown will be measured with decreasing uncertainty. We can therefore generalize the concept of photon-economy (or photon-efficiency) that has been described for fluorescence lifetime imaging microscopy

It is difficult to establish a general criterion for the resolution of biochemical techniques. In practical cases, however, biochemical systems are characterized by a parameter _{1}_{2}

For non-trivial partitions of higher channel densities, both the uncertainties _{1} and σ_{2}, therefore, the separability of the detection system tend to increase, i.e., _{1}_{2}_{0} can be evaluated by substituting in Eq. (15) _{1}_{2}_{,} σ_{1} = σ_{2} = σ_{0} (assumption inherent in the Rayleigh criterion) with _{0}_{0}

At the net of its spatial resolution, the biochemical/photophysical resolving power (_{0}

As we have demonstrated that for a non-trivial partitioning of channels increases the photon-efficiency of a detection system (

Li

Therefore, by a simple derivation of Eq. (18), it is possible to prove that the best

Mono-exponential fluorescence decays with τ = 2 ns and 250, 1000 and 10000 photon/pixel were simulated, in the presence of Poissonian noise, to find the optimal configuration of a two time gates based system with uneven gate width. The time range between 0–50 ns was used in order to include all photons emitted by the simulated fluorophores. Simulated photon counts were binned into two gates and the F-value was estimated using 15,000 replicates of the simulation in order to estimate the precision of the technique. The beginning of the second time gate was scanned from 0 to 10 ns in 0.05 ns steps. The best F-value for all three photon-count levels was 1.24 at 1.59τ.

In order to optimize Fisher information, it is possible to implement numerical iterative strategies that maximize the gain of Fisher information when creating a denser partition of detection channels. The Rao-Cramer lower bound for the precision of a measurement for a given partition is _{i}^{−1}_{i}_{i}N_{i}

For instance, in the case of spectral imaging, the relative fraction of one fluorophore depends on the abundances of both fluorophores. In this case, the Fisher information is a matrix and its numerical evaluation can be carried out as:

Inverting the Fisher information matrix permits to find the covariance matrix:

Many figure of merits can be defined to evaluate the combined error on the estimation of the relative abundances; for the sake of example, the software we provide uses the average of relative errors:

Spectral unmixing of spectrally overlapping fluorescent proteins. Spectral unmixing on images of cells expressing EGFP-Actin and EYFP-Tubulin is described in the main text and in

(DOCX)

Hyper Dimensional Spectral Signatures of two synthetic fluorophores peaked at 540 nm, 560 nm and their sum. These HDSS have been used to test unmixing algorithms (see

(DOCX)

Costs and photon-losses: case studies. A review of photon losses in optical relay systems.

(DOCX)

Biochemical Resolving Power. The definition of biochemical resolving power.

(DOCX)

Hyper Dimensional Phasors (HDPH). The definition of phasors used to analyse the data.

(DOCX)

Matlab code for Fisher information analysis. The zipped folders contain three subfolders with software designed to optimize Fisher information numerically, making use of the photon partitioning theorem and its corollaries: •

(ZIP)

We would like to thank Emma Kay Richardson for supplying the HeLa cells expressing EGFP-actin and EYFP-tubulin.