The extraction of quantitative plant transport parameters from the sparse time activity curve (TAC) signals measured with positron emission tomography (PET) techniques represents one of the frontiers of plant digital imaging (Hubeau and Steppe, 2015; Galieni et al., 2021; Mincke et al., 2021a; Antonecchia et al., 2022), with a strong impact in early stress assessment (Tsukamoto et al., 2008; Yoshihara et al., 2014; Partelová et al., 2017), yield improvement (Yamazaki et al., 2015; Hubeau et al., 2019b; Mincke et al., 2020a), sustainable agriculture (Karve et al., 2015; Kuritaa et al., 2020) and climate change studies (Hubeau et al., 2019a).

The plant PET imaging technique is schematically illustrated in Figures 1A–F (Galieni et al., 2021). A ligand, generally H₂O, CO₂, or 2-Deoxyglucose (2-DG), is introduced in the plant and is transported in the vascular system (Figure 1A). It is labeled with a β⁺ emitter. Two collinear 511 keV γ rays emerge from the annihilation point of the β⁺ within the plant tissue, are detected in an array of sensors (Figure 1B), and provide tomographic information, which is used to reproduce a time-dependent 3-dimensional map of the ligand displacement in the vascular system (Figures 1C,D).

PET data are extremely sparse in space and time. Intensity maps are sampled with a voxel size Δx_exp ranging between 0.25 and 1 mm (Kuritaa et al., 2020; Antonecchia et al., 2022). As the minimal spatial resolution of a PET system is approximately 0.54 mm (FWHM) with a ¹⁸F label in water (Moses, 2011), it is impossible to identify the microscopic capillaries composing the xylem and the phloem. Therefore, voxels are grouped in larger regions of interest (ROI). For instance, a typical ROI structure along the stem of a plant is illustrated in Figure 1E (Mincke et al., 2021a). ROIs with a minimal thickness of one voxel are displaced adjacent to each other. The total activity ρ(xi,tJ) measured in the i-th ROI along the stem direction x at the time t^J is the TAC signal. It is sampled with a typical time step Δt_exp of few minutes (Figure 1F).

The time behavior of TAC signals is governed by fluid dynamics (FD) processes (Figure 1A). The measured TAC signal ρ(xi,tJ) is the sum of tracer fractions either transported through the xylem ρ1(xi,tJ) with a velocity u(xi,tJ) and a diffusion coefficient D(xi,tJ), or diffused through surrounding parenchyma ρ2(xi,tJ) with a rate a₁, or locally assimilated and stored ρ3(xi,tJ) with a rate a₂, or diffused through the atmosphere with an efflux rate a₃ (Mincke et al., 2020b).

However, the task of computing the complete fluid state XiJ=[ρ1,ρ2,ρ3,u;D,a1,a2,a3]ij from the TAC signal is affected by the sparse nature of the space-time sampling. A simple threshold-based approach illustrates the problem. The apoplastic velocity is responsible of the rising signal front. The average speed u_exp between two adjacent ROIs is u_exp = Δx_exp/(t₂−t₁), where t₁ and t₂ are the times at which the TAC crosses a certain threshold (Figure 1F). The maximal measurable average speed is limited by the sparse sampling, as uexpmax≈Δxexp/Δtexp, with a consequent bias in the determination of the other unknown variables.

To mitigate this intrinsic limitation of TAC signals, data-driven model-free techniques consider only physical parameters with already known validity ranges (Keutgen et al., 2002, 2005; Minchin and Thorpe, 2003; Matsuhashi et al., 2010; Ferrieri et al., 2012) and compartmental modeling estimates only quantities averaged in space and time (Bühler et al., 2011; Hubeau et al., 2018; Mincke et al., 2021a). These approaches cannot be extended to a complete estimation of the FD processes occurring inside the plant vascular system, which remain largely unexplored (Jensen et al., 2012).

Sequential extended Kalman Filter (EKF) data assimilation (DA), based on Bayesian interference, may seem an ideal time-stepping technique for TAC signal processing (Kalman, 1960; Suzuki et al., 2010; Suzuki, 2012; Kato et al., 2015; Suzuki and Yamamoto, 2015; Wang et al., 2017). However, EKF requires a linear implementation of the dynamic model for the prediction of the state X~J+1 at time J+1 based on the state X^J at time J:

Moreover, the physical and numerical errors are modeled with a covariance matrix P and an error covariance matrix Q (Wang et al., 2017). The predicted covariance matrix P~J+1 at time J+1 is evolved from the value P^J at time J with the linearized relationship:

Conventional CFD methods cannot be adapted easily to the form of Equations (1) and (2) as they are based on a finite volume discretization combining central schemes and Riemann solvers for the viscous and inviscid flows, respectively (Issa, 1986; Meldi and Poux, 2017; Qu et al., 2019). Approximated solutions based either on structural similarities between solvers and DA approaches, such as reduced order Kalman filtering (Suzuki, 2012), or on statistical ensemble determination of P have been proposed (Evensen, 2009).

In this paper a novel procedure is proposed consisting of assimilating the sparsely sampled TAC signal with computational fluid dynamics (CFD) data simulated at a fine time sampling Δt_sim. A first high resolution predictor stage integrates the FD model forwards in time and, when the experimental data are available, a second correction stage adjusts the model parameters before continuing to the next cycle (Figure 1G). A key aspect of this study is the adoption of a novel set of quasi gas dynamic (QGD) equations for plant PET TAC signals, which can be reduced directly in the form of Equation (1). On this basis a prediction-correction sequential time stepping data assimilation procedure has been developed with the unique feature of a direct computation of the time evolution of the covariance matrix as in Equation (2) without statistical approximations. With respect to existing digital signal processing approaches to plant PET TAC signals, the novel kinetically consistent data assimilation (KC-DA) procedure estimates the complete FD state profile along the vascular system of the plant, controlling explicitly the interplay between physical FD-related mechanisms and computational numerical errors caused by the sparsity of the signal (Figure 1G). It is the first time that Kalman filtering is used in combination with kinetically consistent algorithms for data-driven modeling in plant science. The validity of KC-DA is experimentally demonstrated with an example of zucchini sprouts measurement.

The findings presented above approach the problem of a quantitative interpretation of the dynamic plant PET data from a physical and computational point of view. The key advance of KC-DA with respect to the state of the art is the possibility of the extraction of a continuous profile of the FD variables along the plant stem based on a realistic dynamic physical model, as shown in Figure 6. The relevance of novel mathematical modeling based on biophysical mechanisms has been recently emphasized (Tredenick and Farquhar, 2021) and models which describe the mechanistic properties of water movement in the different parts of plants became fundamental for the interpretation of measurements using nuclear science approaches, such as PET. Although these models are based on fluid dynamics and are extended with compartmental modeling for exchange rates between different functional compartments in plant tissues (Bühler et al., 2011; Mincke et al., 2021b), they are able to extract only an averaged quantities along small sections of the stem. Similarly, model-free techniques such as input-output approaches (Matsuhashi et al., 2010) can calculate only the mean speed of tracer transport and the proportion of tracer moved between specified image positions by means of transfer function analysis. For instance, it has been observed with these techniques that local variations of the average transport speed of water between 0.7 and 1.8 cm/min occurs at different parts of the stem of Sorghum (Keutgen et al., 2005). It is interesting to note that, from the difference of tracer arrival times at three equally spaced points along the stem of a soybean plant, it has been estimated that water transport occurs with an approximate constant speed of 4 mm/s in a total length of 30 mm (Ohya et al., 2008). These two apparently contradictory results are well representative of the need of a precise estimation of the continuous profiles along longer segments of the stem, as proposed in this paper (Figure 6), which will support agronomists in the estimation of such interesting and still unexplored quantitative feature of the plant, which are otherwise not directly accessible with other experimental techniques.

From a mathematical point of view, KC-DA approaches the problem of a reliable estimation of the correlation matrix between the parameters. While typical approaches include either a sensitivity analysis or a Monte Carlo based error estimation (Bühler et al., 2011; Mincke et al., 2021b), KC-DA has the distinctive feature of a direct computation of the correlation matrix P as in Equation (2), which takes also into account the numerical errors due to the discretization of the computational mesh used in the predictor and to the intrinsic voxel size of the measured PET data (Figure 5D). The results in Figure 2 show that the method exhibits a good numerical stability and precision.

A limit of KC-DA is that the convergence improves with time as shown in Figure 3A, and the initial frames of the tracer dynamic cannot be estimated with precision. This feature has a direct effect on the estimated parameters. While the viscosity coefficient and the exchange rates are found in an expected range (Bühler et al., 2011; Mincke et al., 2020b), the velocity profile appears almost constant and with an average value low in comparison with the above mentioned results. As observed in Figures 6A,B, this occurs as, while apoplastic flow dominates at the initial stages of the tracer immersion, at later times the diffusive flow is dominant. Such effect suggests that the dynamic model in Equation (8) could be extended by including a time-dependent velocity, allowing therefore to precisely capture the tracer dynamics at early stages. The next step of this study will be to demonstrate the validity of the KC-DA approach with more conservation equations and ligands used in plant science, toward an increasingly complex modeling of plant PET TAC signals.

A key feature of KC-DA is the ability of calculating the continuous profile of kinetic variables associated to the FD flow in plant transport. This makes KC-DA particularly suited to the quantification of plant vascular flow. KC-DA has an explicit computational implementation thanks to the analytical form of the evolution and noise correlation matrices. The direct calculation of the numerical dissipative terms plays a pivotal role in the stabilization of the time stepping procedure and helps the stability of the estimation of the transport parameters in the plant vascular system.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

ND'A ideated the data-driven kinetically consistent approach and wrote the analysis code. ND'A and MP analyzed the data and wrote the manuscript. EA, MC, GP, and MP designed the possible experiments and applications to agronomy. ND'A, MP, and QX supervised the research focus and designed the scientific background. All authors contributed to the article and approved the submitted version.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.