Quantitative prediction of contrast enhancement from test bolus data in cardiac MSCT

Abstract

The purpose of this study was to evaluate a new algorithm for the prediction of contrast enhancement from test bolus data in cardiac multislice spiral computed tomography (MSCT). An algorithm for the prediction of contrast enhancement using test bolus data was developed. A total of 30 consecutive patients (15 male, 69.5 ± 9.6 years) underwent cardiac MSCT (12 × 0.75 mm, 120 kV, 500 mAs_eff.) with a biphasic contrast material injection protocol. Contrast timing was derived from a standard 20 ml test bolus injection. Based on the test bolus time attenuation curves, expected enhancement values were computed for the ascending and descending aorta and the pulmonary trunk and compared with measured data from the cardiac CT scan. At the level of the test bolus measurement in the ascending aorta, the corresponding attenuation values were 309.4 ± 49.6 Hounsfield Units (HU) for the predicted and 285.6 ± 42.6 HU for the measured attenuation, respectively. The mean deviation between predicted and measured CT values was 32.8 ± 48.2 HU (upper and lower limits of agreement 101.4/−53.8 HU), indicating a slight systematic tendency for overestimation. For 80% of the patients the prediction error was less than 50 HU. Prediction of contrast enhancement in cardiac MSCT from test bolus data is feasible with a relatively small mean deviation; 80% of the predictions were within a range that might be acceptable for routine clinical application.

Appendix

Defining the problem

A short test bolus of contrast agent is injected with constant flow during a certain time and the resulting enhancement curve is measured. From this we want to predict the resulting enhancement curve after having injected the actual examination bolus. The prediction of the expected contrast agent flow is made using the theory of time-invariant linear systems and its main assumptions of time invariance, superposition and homogeneity. As in CT measured enhancement is proportional to the tracer concentration, we do not need to explicitly consider enhancement, concentration, contrast amount or the type of contrast agent used as long as we measure both test bolus and final injection data under the same conditions.

Fourier approach

The problem can in principle be solved using a Fourier transformation; this approach was successfully used for the optimization of aortic CT angiography studies with scan times in the range of 30 s [24, 25]. It requires test bolus data with high temporal resolution and over a sufficiently long time span that also includes recirculation. For lower quality data and few data points, this approach is numerically not very stable. If the test bolus data are corrupt, e.g. if the scan started too late or was stopped too early, a prediction using a Fourier transform becomes even more unreliable.

Modelling

A test bolus b _T (t) of contrast agent is injected with a constant injection rate F _T from time t _0T until t _FT . With Θ(t) denoting the step (or Heaviside) function, this can be written as

$$ b_{T} {\left( t \right)} = F_{T} \Theta {\left( {t - t_{{0T}} } \right)}\Theta {\left( {t_{{FT}} - t} \right)}. $$

The enhancement curve $\widetilde{c}_{T} {\left( t \right)}$ after injecting this bolus is measured. We want to predict the enhancement curve $\widetilde{c}_{R} {\left( t \right)}$that results after having injected the examination bolus

$$ b_{R} {\left( t \right)} = F_{R} \Theta {\left( {t - t_{{0R}} } \right)}\Theta {\left( {t_{{FR}} - t} \right)}, $$

with constant flow F _R from time t _0R until t _FR .

Generalized summation

For a linear system the dependence of cause (injection of contrast agent) and effect (enhancement) can be mathematically expressed as

$$\widetilde{c}{\left( t \right)} = {\int\limits_{ - \infty }^\infty {dt\prime } }k{\left( {t - t\prime } \right)}b{\left( {t\prime } \right)},$$

with k(t) being the patient and ROI specific response function. In Fourier space this can be simplified to

$$\widetilde{C}{\left( \xi \right)} = K{\left( \xi \right)}B{\left( \xi \right)}.$$

If we use the test bolus data to determine

$$ K{\left( \xi \right)} = \frac{{C_{T} {\left( \xi \right)}}} {{B_{T} {\left( \xi \right)}}}, $$

we can calculate the desired $\widetilde{c}_{R} {\left( t \right)}$ as the inverse Fourier transform of

$$\widetilde{C}_{R} {\left( \xi \right)} = \frac{{B_{R} {\left( \xi \right)}}}{{B_{T} {\left( \xi \right)}}}C_{T} {\left( \xi \right)}.$$

The Fourier transform of the bolus function can be analytically determined as

$$ B{\left( \xi \right)} = F\frac{1} {\xi }{\left( {\exp {\left( {i\xi t_{F} } \right)} - \exp {\left( {i\xi t_{0} } \right)}} \right)} $$

Therefore the desired enhancement curve results as

$$\widetilde{c}_{R} {\left( t \right)} = \frac{1}{{2\pi }}\frac{{F_{R} }}{{F_{T} }}{\int\limits_{ - \infty }^\infty {d\xi \exp {\left( { - i\xi t} \right)}} }\frac{{\exp {\left( {i\xi t_{{FR}} } \right)} - \exp {\left( {i\xi t_{{0R}} } \right)}}}{{\exp {\left( {i\xi t_{{FT}} } \right)} - \exp {\left( {i\xi t_{{0T}} } \right)}}}\widetilde{C}_{T} {\left( \xi \right)}.$$

This can be integrated without explicit knowledge of $\widetilde{C}_{T} {\left( \xi \right)}$ by developing the denominator in a geometrical series and then using an inverse Fourier transform

$$\widetilde{c}_{R} {\left( t \right)} = \frac{{F_{R} }}{{F_{T} }}{\sum\nolimits_{n = 0}^\infty {{\left( {\widetilde{c}_{T} {\left( {t - t_{{0R}} + t_{{0T}} - n\Delta _{T} } \right)} - \widetilde{c}_{T} {\left( {t - t_{{FR}} + t_{{0T}} - n\Delta _{T} } \right)}} \right)}} },$$

where Δ _T  = t _FT –t _0T . This way the actual numerical Fourier transformation of the enhancement data can be avoided. The infinite sum causes no problems, because $\widetilde{c}_{T} {\left( t \right)}$ vanishes before and after certain time points (i.e. before the bolus has been injected and after the injection has passed sufficiently long). Therefore, in practical cases only a finite number of terms have to be considered. In the special case that ${{\left( {{\text{t}}_{{{\text{FR}}}} {\text{ - t}}_{{{\text{0R}}}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\text{t}}_{{{\text{FR}}}} {\text{ - t}}_{{{\text{0R}}}} } \right)}} {\Delta _{T} }}} \right. \kern-\nulldelimiterspace} {\Delta _{T} }$ is an integer the sum reduces to a simple finite shifted adding up of the test bolus data. For arbitrary values of ${{\left( {{\text{t}}_{{{\text{FR}}}} {\text{ - t}}_{{{\text{0R}}}} } \right)}} \mathord{\left/ {\vphantom {{{\left( {{\text{t}}_{{{\text{FR}}}} {\text{ - t}}_{{{\text{0R}}}} } \right)}} {\Delta _{T} }}} \right. \kern-\nulldelimiterspace} {\Delta _{T} }$ it can considered a generalization of this concept. In any case the resulting enhancement can be determined as a sum of terms of the test bolus data, not requiring the Fourier transformation. This has the advantage of being both numerically robust and simple. The disadvantage lies in the fact that it can only make predictions for times where the test bolus has been measured or can safely be assumed to vanish. If the test bolus data are corrupt, i.e. the scan started too late or was broken off too early, the prediction becomes unreliable.

Physiological modelling

In this approach a simple physiological model to describe the flow of the contrast agent is used [26]. The contrast agent is injected (re the source term on the right hand side) and assumed to be transported through the body with a constant mean velocity (represented by the left hand side). By adding a diffusion term we can allow for the tracer to have different velocities or take different paths within the system. For a flow rate of F we get a function depending on velocity v and diffusion constant D

$$ \frac{\partial } {{\partial t}}c{\left( {x,t} \right)} + v\frac{\partial } {{\partial x}}c{\left( {x,t} \right)} - D\frac{{\partial ^{2} }} {{\partial x^{2} }}c{\left( {x,t} \right)} = F\delta ^{{(1)}} {\left( x \right)}\Theta {\left( {t - t_{0} } \right)}\Theta {\left( {t_{F} - t} \right)}. $$

where t ₀ denotes the start and t _F the end time of the injection. As this is the well-known one-dimensional inhomogeneous heat equation with an additional drift term, its solution is straightforward and approximately given as

$$ b{\left( {x,t} \right)} = \frac{1} {{2v}}\Theta {\left( {t - t_{0} } \right)}{\left( {\Theta {\left( {t_{F} - t} \right)}{\left( {1 - Erf{\left( {\frac{{x - v{\left( {t - t_{0} } \right)}}} {{{\sqrt {4D{\left( {t - t_{0} } \right)}} }}}} \right)}} \right)} + \Theta {\left( {t - t_{F} } \right)}{\left( {Erf{\left( {\frac{{x - v{\left( {t - t_{F} } \right)}}} {{{\sqrt {4D{\left( {t - t_{F} } \right)}} }}}} \right)} - Erf{\left( {\frac{{x - v{\left( {t - t_{0} } \right)}}} {{{\sqrt {4D{\left( {t - t_{0} } \right)}} }}}} \right)}} \right)}} \right)} $$

in terms of the error function

$$ Erf{\left( x \right)} = \frac{2} {{{\sqrt \pi }}}{\int\limits_{ - \infty }^x {dt\exp {\left( { - t^{2} } \right)}} }, $$

where we have used its property of being unequal to 1 or −1 only around the zeros of its argument and assumed both x and v to be positive.

The fit of the test bolus data provides the patient (e.g. cardiac output) and ROI (e.g. transit time to this specific ROI) dependent parameters and allows predicting the expected behaviour by changing the source term to the contrast agent protocol to be predicted. This approach has the advantage of being able to account for partially missing data. Furthermore, the recirculation effects may be described by these solutions similar to a gamma variate function commonly used to correct for recirculation [27].

Combined approach

In practice a combination of both methods was used in our implementation. The generalized summation approach has the advantage of being both numerically robust and simple. Its drawback is that it can only make predictions for times where the test bolus has either been measured or can safely be assumed to vanish. Therefore, if enough data for the prediction are available, or in other words, if the test bolus curve with its recirculation is complete, only generalized summation is used. If the test bolus data are incomplete the second approach is used, which allows one to account for partially missing data. If there are not enough data, the physiological model is used to expand the test bolus data, such that we end up with a complete test bolus curve. The expanded data are then used as input for the generalized summation.