Sensitivity analysis for biomedical models

Towards a Reduced In Silico Model Predicting Biochemical Recurrence After Radiotherapy in Prostate Cancer

OBJECTIVE

Purposes of this work were i) to develop an in silico model of tumor response to radiotherapy, ii) to perform an exhaustive sensitivity analysis in order to iii) propose a simplified version and iv) to predict biochemical recurrence with both the comprehensive and the reduced model.

METHODS

A multiscale computational model of tumor response to radiotherapy was developed. It integrated the following radiobiological mechanisms: oxygenation, including hypoxic death; division of tumor cells; VEGF diffusion driving angiogenesis; division of healthy cells and oxygen-dependent response to irradiation, considering, cycle arrest and mitotic catastrophe. A thorough sensitivity analysis using the Morris screening method was performed on 21 prostate computational tissues. Tumor control probability (TCP) curves of the comprehensive model and 15 reduced versions were compared. Logistic regression was performed to predict biochemical recurrence after radiotherapy on 76 localized prostate cancer patients using an output of the comprehensive and the reduced models.

RESULTS

No significant difference was found between the TCP curves of the comprehensive and a simplified version which only considered oxygenation, division of tumor cells and their response to irradiation. Biochemical recurrence predictions using the comprehensive and the reduced models improved those made from pre-treatment imaging parameters (AUC = 0.81 ± 0.02 and 0.82 ± 0.02 vs. 0.75 ± 0.03, respectively).

CONCLUSION

A reduced model of tumor response to radiotherapy able to predict biochemical recurrence in prostate cancer was obtained.

SIGNIFICANCE

This reduced model may be used in the future to optimize personalized fractionation schedules.

Influence of Resting Venous Blood Volume Fraction on Dynamic Causal Modeling and System Identifiability

Changes in BOLD signals are sensitive to the regional blood content associated with the vasculature, which is known as V₀ in hemodynamic models. In previous studies involving dynamic causal modeling (DCM) which embodies the hemodynamic model to invert the functional magnetic resonance imaging signals into neuronal activity, V₀ was arbitrarily set to a physiolog-ically plausible value to overcome the ill-posedness of the inverse problem. It is interesting to investigate how the V₀ value influences DCM. In this study we addressed this issue by using both synthetic and real experiments. The results show that the ability of DCM analysis to reveal information about brain causality depends critically on the assumed V₀ value used in the analysis procedure. The choice of V₀ value not only directly affects the strength of system connections, but more importantly also affects the inferences about the network architecture. Our analyses speak to a possible refinement of how the hemody-namic process is parameterized (i.e., by making V₀ a free parameter); however, the conditional dependencies induced by a more complex model may create more problems than they solve. Obtaining more realistic V₀ information in DCM can improve the identifiability of the system and would provide more reliable inferences about the properties of brain connectivity.

Nonlinear estimation of BOLD signals with the aid of cerebral blood volume imaging

BACKGROUND

The hemodynamic balloon model describes the change in coupling from underlying neural activity to observed blood oxygen level dependent (BOLD) response. It plays an increasing important role in brain research using magnetic resonance imaging (MRI) techniques. However, changes in the BOLD signal are sensitive to the resting blood volume fraction (i.e., [Formula: see text]) associated with the regional vasculature. In previous studies the value was arbitrarily set to a physiologically plausible value to circumvent the ill-posedness of the inverse problem. These approaches fail to explore actual [Formula: see text] value and could yield inaccurate model estimation.

METHODS

The present study represents the first empiric attempt to derive the actual [Formula: see text] from data obtained using cerebral blood volume imaging, with the aim of augmenting the existing estimation schemes. Bimanual finger tapping experiments were performed to determine how [Formula: see text] influences the model estimation of BOLD signals within a single-region and multiple-regions (i.e., dynamic causal modeling). In order to show the significance of applying the true [Formula: see text], we have presented the different results obtained when using the real [Formula: see text] and assumed [Formula: see text] in terms of single-region model estimation and dynamic causal modeling.

RESULTS

The results show that [Formula: see text] significantly influences the estimation results within a single-region and multiple-regions. Using the actual [Formula: see text] might yield more realistic and physiologically meaningful model estimation results.

CONCLUSION

Incorporating regional venous information in the analysis of the hemodynamic model can provide more reliable and accurate parameter estimations and model predictions, and improve the inference about brain connectivity based on fMRI data.

Investigating Human Neurovascular Coupling Using Functional Neuroimaging: A Critical Review of Dynamic Models

The mechanisms that link a transient neural activity to the corresponding increase of cerebral blood flow (CBF) are termed neurovascular coupling (NVC). They are possibly impaired at early stages of small vessel or neurodegenerative diseases. Investigation of NVC in humans has been made possible with the development of various neuroimaging techniques based on variations of local hemodynamics during neural activity. Specific dynamic models are currently used for interpreting these data that can include biophysical parameters related to NVC. After a brief review of the current knowledge about possible mechanisms acting in NVC we selected seven models with explicit integration of NVC found in the literature. All these models were described using the same procedure. We compared their physiological assumptions, mathematical formalism, and validation. In particular, we pointed out their strong differences in terms of complexity. Finally, we discussed their validity and their potential applications. These models may provide key information to investigate various aspects of NVC in human pathology.

A Sensitivity Analysis of fMRI Balloon Model

Functional magnetic resonance imaging (fMRI) allows the mapping of the brain activation through measurements of the Blood Oxygenation Level Dependent (BOLD) contrast. The characterization of the pathway from the input stimulus to the output BOLD signal requires the selection of an adequate hemodynamic model and the satisfaction of some specific conditions while conducting the experiment and calibrating the model. This paper, focuses on the identifiability of the Balloon hemodynamic model. By identifiability, we mean the ability to estimate accurately the model parameters given the input and the output measurement. Previous studies of the Balloon model have somehow added knowledge either by choosing prior distributions for the parameters, freezing some of them, or looking for the solution as a projection on a natural basis of some vector space. In these studies, the identification was generally assessed using event-related paradigms. This paper justifies the reasons behind the need of adding knowledge, choosing certain paradigms, and completing the few existing identifiability studies through a global sensitivity analysis of the Balloon model in the case of blocked design experiment.

Nonlinear Bayesian estimation of BOLD signal under non-Gaussian noise

Modeling the blood oxygenation level dependent (BOLD) signal has been a subject of study for over a decade in the neuroimaging community. Inspired from fluid dynamics, the hemodynamic model provides a plausible yet convincing interpretation of the BOLD signal by amalgamating effects of dynamic physiological changes in blood oxygenation, cerebral blood flow and volume. The nonautonomous, nonlinear set of differential equations of the hemodynamic model constitutes the process model while the weighted nonlinear sum of the physiological variables forms the measurement model. Plagued by various noise sources, the time series fMRI measurement data is mostly assumed to be affected by additive Gaussian noise. Though more feasible, the assumption may cause the designed filter to perform poorly if made to work under non-Gaussian environment. In this paper, we present a data assimilation scheme that assumes additive non-Gaussian noise, namely, the e-mixture noise, affecting the measurements. The proposed filter MAGSF and the celebrated EKF are put to test by performing joint optimal Bayesian filtering to estimate both the states and parameters governing the hemodynamic model under non-Gaussian environment. Analyses using both the synthetic and real data reveal superior performance of the MAGSF as compared to EKF.

What can we learn from global sensitivity analysis of biochemical systems?

Most biological models of intermediate size, and probably all large models, need to cope with the fact that many of their parameter values are unknown. In addition, it may not be possible to identify these values unambiguously on the basis of experimental data. This poses the question how reliable predictions made using such models are. Sensitivity analysis is commonly used to measure the impact of each model parameter on its variables. However, the results of such analyses can be dependent on an exact set of parameter values due to nonlinearity. To mitigate this problem, global sensitivity analysis techniques are used to calculate parameter sensitivities in a wider parameter space. We applied global sensitivity analysis to a selection of five signalling and metabolic models, several of which incorporate experimentally well-determined parameters. Assuming these models represent physiological reality, we explored how the results could change under increasing amounts of parameter uncertainty. Our results show that parameter sensitivities calculated with the physiological parameter values are not necessarily the most frequently observed under random sampling, even in a small interval around the physiological values. Often multimodal distributions were observed. Unsurprisingly, the range of possible sensitivity coefficient values increased with the level of parameter uncertainty, though the amount of parameter uncertainty at which the pattern of control was able to change differed among the models analysed. We suggest that this level of uncertainty can be used as a global measure of model robustness. Finally a comparison of different global sensitivity analysis techniques shows that, if high-throughput computing resources are available, then random sampling may actually be the most suitable technique.

Reliable and efficient approach of BOLD signal with dual Kalman filtering

By introducing the conflicting effects of dynamic changes in blood flow, volume, and blood oxygenation, Balloon model provides a biomechanical compelling interpretation of the BOLD signal. In order to obtain optimal estimates for both the states and parameters involved in this model, a joint filtering (estimate) method has been widely used. However, it is flawed in several aspects (i) Correlation or interaction between the states and parameters is incorporated despite its nonexistence in biophysical reality. (ii) A joint representation for states and parameters necessarily means the large dimension of state space and will in turn lead to huge numerical cost in implementation. Given this knowledge, a dual filtering approach is proposed and demonstrated in this paper as a highly competent alternative, which can not only provide more reliable estimates, but also in a more efficient way. The two approaches in our discussion will be based on unscented Kalman filter, which has become the algorithm of choice in numerous nonlinear estimation and machine learning applications.

Concurrent bias correction in hemodynamic data assimilation

Low-frequency drift in fMRI datasets can be caused by various sources and are generally not of interest in a conventional task-based fMRI experiment. This feature complicates the assimilation approach that is always under specific assumption on statistics of system uncertainties. In this paper, we present a novel approach to the assimilation of nonlinear hemodynamic system with stochastic biased noise. By treating the drift variation as a random-walk process, the assimilation problem was translated into the identification of a nonlinear system in the presence of time-varying bias. We developed a bias aware unscented Kalman estimator to efficiently handle this problem. In this framework, the estimates of bias-free states and drift are separately carried out in two parallel filters, the optimal estimates of the system states then are corrected from bias-free states with drift estimates. The approach can simultaneously deal with the fMRI responses and drift in an assimilation cycle in an on-line fashion. It makes no assumptions of the structure and statistics of the drift, thereby is particularly suited for fMRI imaging where the formulation of real drift remains difficult to acquire. Experiments with synthetic data and real fMRI data are performed to demonstrate feasibility of our approach and to explore its potential advantages over classic polynomial approach. Moreover, we include the comparison of the variability of observables from the scanner and of normalized signal used in assimilation procedure in Appendix.

Quantitative evaluation of activation state in functional brain imaging

Neuronal activity can evoke the hemodynamic change that gives rise to the observed functional magnetic resonance imaging (fMRI) signal. These increases are also regulated by the resting blood volume fraction (V (0)) associated with regional vasculature. The activation locus detected by means of the change in the blood-oxygen-level-dependent (BOLD) signal intensity thereby may deviate from the actual active site due to varied vascular density in the cortex. Furthermore, conventional detection techniques evaluate the statistical significance of the hemodynamic observations. In this sense, the significance level relies not only upon the intensity of the BOLD signal change, but also upon the spatially inhomogeneous fMRI noise distribution that complicates the expression of the results. In this paper, we propose a quantitative strategy for the calibration of activation states to address these challenging problems. The quantitative assessment is based on the estimated neuronal efficacy parameter [Formula: see text] of the hemodynamic model in a voxel-by-voxel way. It is partly immune to the inhomogeneous fMRI noise by virtue of the strength of the optimization strategy. Moreover, it is easy to incorporate regional vascular information into the activation detection procedure. By combining MR angiography images, this approach can remove large vessel contamination in fMRI signals, and provide more accurate functional localization than classical statistical techniques for clinical applications. It is also helpful to investigate the nonlinear nature of the coupling between synaptic activity and the evoked BOLD response. The proposed method might be considered as a potentially useful complement to existing statistical approaches.

Exploiting magnetic resonance angiography imaging improves model estimation of BOLD signal

The change of BOLD signal relies heavily upon the resting blood volume fraction ([Formula: see text]) associated with regional vasculature. However, existing hemodynamic data assimilation studies pretermit such concern. They simply assign the value in a physiologically plausible range to get over ill-conditioning of the assimilation problem and fail to explore actual [Formula: see text]. Such performance might lead to unreliable model estimation. In this work, we present the first exploration of the influence of [Formula: see text] on fMRI data assimilation, where actual [Formula: see text] within a given cortical area was calibrated by an MR angiography experiment and then was augmented into the assimilation scheme. We have investigated the impact of [Formula: see text] on single-region data assimilation and multi-region data assimilation (dynamic cause modeling, DCM) in a classical flashing checkerboard experiment. Results show that the employment of an assumed [Formula: see text] in fMRI data assimilation is only suitable for fMRI signal reconstruction and activation detection grounded on this signal, and not suitable for estimation of unobserved states and effective connectivity study. We thereby argue that introducing physically realistic [Formula: see text] in the assimilation process may provide more reliable estimation of physiological information, which contributes to a better understanding of the underlying hemodynamic processes. Such an effort is valuable and should be well appreciated.

Detrend-free hemodynamic data assimilation of two-stage Kalman estimator

Spurious temporal drift is abundant in fMRI data, and its removal is a critical preprocessing step in fMRI data assimilation due to the sparse nature and the complexity of the data. Conventional data-driven approaches rest upon specific assumptions of the drift structure and signal statistics, and may lead to inaccurate results. In this paper we present an approach to the assimilation of nonlinear hemodynamic system, with special attention on drift. By treating the drift variation as a random-walk process, the assimilation problem was translated into the identification of a nonlinear system in the presence of time varying bias. We developed two-stage unscented Kalman filter (UKF) to efficiently handle this problem. In this framework the assimilation can implement with original fMRI data without detrending preprocessing. The fMRI responses and drift were estimated simultaneously in an assimilation cycle. The efficacy of this approach is demonstrated in synthetic and real fMRI experiments. Results show that the joint estimation strategy produces more accurate estimation of physiological states, fMRI response and drift than separate processing due to no assumption of structure of the drift that is not available in fMRI data.