# Optimization of Acquisition Schemes Towards a Better Estimation of Microstructure Parameters in Multidimensional Diffusion MRI

Constance Bocquillon, Isabelle Corouge, Emmanuel Caruyer

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Constance Bocquillon, Isabelle Corouge, Emmanuel Caruyer. Optimization of Acquisition Schemes Towards a Better Estimation of Microstructure Parameters in Multidimensional Diffusion MRI. IPMI 2025 - Information Processing in Medical Imaging, May 2025, Kos Island, Greece. pp.142-153, 10.1007/978-3-031-96628-6\_10. hal-05369014

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Distributed under a Creative Commons Attribution 4.0 International License
Optimization of acquisition schemes towards a
better estimation of microstructure parameters in
multidimensional diffusion MRI

Constance Bocquillon¹, Isabelle Corouge¹, and Emmanuel Caruyer¹

¹ Univ Rennes, Inria, CNRS, Inserm, IRISA UMR 6074, Empenn ERL 1228, Campus de Beaulieu, Rennes, F-35042

**Abstract.** Unlike traditional measurements by diffusion tensor imaging, multidimensional diffusion MRI allows the estimation of additional microstructural parameters such as anisotropy, kurtosis and orientation dispersion. To properly take advantage of this imaging modality and capturing microstructure parameters accurately and efficiently, it is crucial to use a dedicated acquisition scheme. Several models and acquisition representations can be used towards this goal.

In this paper, we focused on the q-space trajectory imaging, using b-
tensor acquisition encoding and the diffusion tensor distribution (DTD)
modeling. More specifically, our goal is to develop a framework for the
optimization of acquisition scheme based on their ability to properly
estimate microstructural parameters of interest.

We generated an extensive collection of b-tensor shapes with a fixed num-
ber of directions each, from which we efficiently selected an optimized ac-
quisition scheme. In the spirit of fingerprinting, we proposed a dictionary-
based approach. The dictionary columns were carefully adapted to the
achievable resolution in the parameter space, the parameters of inter-
est being the microscopic anisotropy, the tensor size variance and the
orientation parameter. To solve the combinatorial optimization problem
of selecting the best subset of b-tensor shapes, we implemented two ap-
proximation algorithms: a greedy approach based and a permutation
strategy.

To assess the performance of our optimization procedure, we computed
the estimation error for each parameter. The signal generated from our
scheme yielded lower or comparable errors to those of a reference scheme
proposed in the literature and designed for this purpose.

**Keywords:** Acquisition · Diffusion MRI · Diffusion tensor distribution · Q-space trajectory imaging · Microstructure · Optimal experimental design · Parameter estimation.

# 1 Introduction

Multidimensional diffusion MRI captures detailed properties such as anisotropy, kurtosis and orientation dispersion. This allows for the differentiation of microstructural features like fiber crossings, cellularity, and membrane permeability. Multidimensional dMRI has emerged as a powerful tool for studying complex
biological tissues, offering valuable insights into neural connectivity, tumor microenvironments, and other structural properties in both research and clinical settings. Specifically, q-space trajectory imaging (QTI) [17] aims at reconstructing parameters of diffusion tensor distribution (DTD) using b-tensor encoded acquisitions.

Acquisition optimization aims at improving the accuracy, precision, and efficiency of microstructural parameter estimation. By carefully selecting the MRI acquisition parameters—such as b-tensors and diffusion encoding directions—we can enhance the sensitivity of the measured signals to the underlying tissue features. The Cramer-Rao lower bound has been used as a measure of reference to evaluate the estimation accuracy when optimizing acquisition schemes [1, 16]. This strategy has been recently extended to multidimensional dMRI [11].

One drawback of the Cramer-Rao lower bound is that it is computed locally, with respect to one point in the parameter space. In this paper, we propose an alternative method inspired by fingerprinting [5, 14]. This method is based on a dictionary of signals where each column corresponds to a DTD and each line to a b-tensor. From a predefined superset of acquisitions, our objective is to select the subset of b-tensors that best preserves the separability of the columns of the dictionary. We used simulations to assess the ability of the selected subset to yield accurate estimations of the microstructural parameters of interest: the microscopic anisotropy, the tensor size variance and the orientation parameter.

# 2 Theory

In this paper, we represent scalar value by a lowercase (e.g. b), vector/first-order tensor by a bold lowercase letter (e.g. **q**), second-order tensor by a bold capital letter (e.g. **B**) and fourth-order tensor by a blackboard capital letter (e.g. **C**).

## 2.1 B-tensor encoded acquisitions

In [6, 17], the authors generalized the concept of b-tensors using q-space trajectories. In its general form, it can be defined as:

$$ \mathbf{B} = \int_{0}^{\tau} \mathbf{q}(t)^{\otimes 2} dt \qquad (1) $$

where $\tau$ is the echo time, $\mathbf{q}(t)^{\otimes 2}$ is the outer product of $\mathbf{q}(t)$ with itself and $\mathbf{q}(t)$ is a time dependent trajectory in q-space associated with a gradient waveform $\mathbf{g}(t)$ as:

$$ \mathbf{q}(t) = \gamma \int_{0}^{\tau} \mathbf{g}(t') dt' \qquad (2) $$

where $\gamma$ is the gyromagnetic ratio.

In this work, we focused on cylindrical b-tensors. They can be described by their diagonal form in their principal axis system by:
$$ \mathbf{B} = \begin{pmatrix} b_{\parallel} & 0 & 0 \\ 0 & b_{\perp} & 0 \\ 0 & 0 & b_{\perp} \end{pmatrix} \qquad (3) $$

where $b_{\parallel}$ is the axial eigenvalue of the b-tensor and $b_{\perp}$ is the radial eigenvalue. Cylindrical b-tensors shapes are described as planar, oblate, spherical, prolate or linear depending on the relative values of $b_{\parallel}$ and $b_{\perp}$ as shown in Figure 1. The b-value is defined as the trace of the b-tensor and is given by:

$$ b_{val} = b_{\parallel} + 2b_{\perp} \qquad (4) $$

<figure><img src="image_2.png" /><figcaption>Fig. 1. Cylindrical b-tensor shapes based on their eigenvalues.</figcaption></figure>

## 2.2 Q-space trajectory imaging

Q-space trajectory imaging [17] considers that the signal originates from a collection of microenvironments in which diffusion is assumed to be Gaussian.

$$ s(\mathbf{B}) = s_0 \int p(\mathbf{D}) \exp(-\mathbf{B} : \mathbf{D}) d\mathbf{D} = s_0 \langle \exp(-\langle \mathbf{B} : \mathbf{D} \rangle) \rangle \qquad (5) $$

where $s_0$ is the non-diffusion weighted signal, $p(\mathbf{D})$ is the diffusion tensor distribution, $\mathbf{B} : \mathbf{D}$ is the inner product of $\mathbf{B}$ with $\mathbf{D}$, $\mathbf{D}$ is the diffusion tensor and $\langle \cdot \rangle$ represents the averaging over the distribution.

The cumulant expansion of $s$ gives:

$$ s(\mathbf{B}) \approx s_0 \exp(-\langle \mathbf{B} : \langle \mathbf{D} \rangle \rangle + \frac{1}{2} \langle \mathbf{B} : \mathbf{C} \rangle) \qquad (6) $$

where $\mathbf{B}$ is the outer product of $\mathbf{B}$ with itself, $\mathbf{C}$ is the covariance matrix of $\mathbf{D}$ given by:

$$ \mathbf{C} = \langle \mathbf{D}^{\otimes 2} \rangle - \langle \mathbf{D} \rangle^{\otimes 2} \qquad (7) $$

This framework was further improved by adding positivity constraints on $\mathbf{D}$ and $\mathbf{C}$ in [8].
## 2.3 The diffusion tensor distribution (DTD) parameters

The DTD introduced in Equation 5 gives several microstructure parameters that can be defined through the moments of the distribution [17]. Here, we focused on the microscopic anisotropy, the tensor size variance and the orientation parameter. Their effect is illustrated in Figure 2.

<figure><img src="image_3.png" /><figcaption>Fig. 2. Collection of DTDs with several values for each parameter of interest.</figcaption></figure>

**Tensor-size variance (C<sub>MD</sub>)** The tensor size variance describes the variance of the mean diffusivity of the microscopic tensors. It starts at 0 where all the microscopic tensors have the same size and increases with the size disparity. It can be computed by:

$$C_{MD} = \frac{\langle C, E_{bulk} \rangle}{\langle \mathbf{D}^{\otimes 2} \rangle, E_{bulk}} \quad (8)$$

**Microscopic anisotropy (μFA)** The microscopic anisotropy ranges from 0 to 1 where 0 corresponds to spherical microscopic tensors, and 1 corresponds to a distribution of sticks-like tensors. It can be computed by:

$$C_{\mu} = \frac{\langle \mathbf{D}^{\otimes 2} \rangle, E_{shear} >}{\langle \mathbf{D}^{\otimes 2} \rangle, E_{iso} >} = \mu FA^2 \quad (9)$$
**Orientation parameter (OP)** The orientation parameter ranges from 0 to 1 where 0 corresponds to fully randomly oriented tensors and 1 to fully aligned ones. It can be computed by:

$$OP^2 = \frac{\langle \mathbf{D} \rangle^{\otimes 2}, \mathbb{E}_{shear} >}{\langle \mathbf{D}^{\otimes 2} \rangle, \mathbb{E}_{shear} >} \qquad (10)$$

To compute these parameters, the isotropic 4th order isotropic tensor $\mathbb{E}_{iso}$ and the bulk modulus $\mathbb{E}_{bulk}$ are defined as

$$\mathbb{E}_{iso} = \frac{1}{3} \begin{pmatrix} 1 & 0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{pmatrix}; \mathbb{E}_{bulk} = \mathbb{E}_{iso}^{\otimes 2} = \frac{1}{9} \begin{pmatrix} 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix} \qquad (11)$$

and the shear modulus as

$$\mathbb{E}_{shear} = \mathbb{E}_{iso} - \mathbb{E}_{bulk} \qquad (12)$$

# 3 Methods

In order to optimize the sensitivity of the acquisition to microstructural parameters of interest, we designed an approach in several steps. First, we generated a superset of acquisitions from which we will select the best subset of b-tensors. Then, we generated a collection of DTD parameterized by OP, μFA and $C_{MD}$ which will become the columns of our dictionary. Next, we optimized the choice of a subset of acquisitions to maximize the separability of the columns in our dictionary. Finally, we evaluated the accuracy of the reconstructed parameters from the acquisitions associated with the selected subset.

## 3.1 Acquisition superset

We constructed a superset of b-tensors, $S$, containing every axisymmetric b-tensor shape with $b_\parallel$ and $b_\perp$ in the range [0, 2000] s/mm² divided into 60 intervals. We kept only those with a b-value under 2000 s/mm² and ensuring a step of 100 in b-values. The principal axis of each b-tensor shape was rotated into 30 directions generated from [9]. This results in a superset, $S$, containing $n = 331$ unique b-tensor shapes, each rotated into 30 directions.

## 3.2 Fingerprinting the parameter space

We generated a dictionary of signals, inspired by fingerprinting methods [14], where each column corresponds to a DTD and each row corresponds to a b-tensor $\mathbf{B} \in S$. Each DTD can be parameterized by three parameters: μFA, OP
and C<sub>MD</sub>. The range of values is [0, 1] for µFA and OP, and [0, 0.3] for C<sub>MD</sub>. We restricted ourselves to DTD solely containing axisymmetric micro-tensors. We sampled 10000 micro-tensors for each DTD; the shape, size and orientation of each micro-tensor are sampled independently. The corresponding signal is simply computed according to Equation 5.

Instead of dividing the parameter space on a regular grid, such as commonly done in fingerprinting [5,14], we locally adapted the granularity of the subdivision to the resolution permitted by the superset S. More precisely, in a construction inspired by octrees, we started from the initial cube representing the whole space of parameters. At each step, and for each parameter, we tried to divide the cube into two half-cubes; the division was only kept if it could discriminate the pair of newly created DTDs using our superset of acquisitions, S.

For a given acquisition scheme, S, we consider that two DTDs can be discriminated if:

$$ \max_{\mathbf{B} \in S} (|s_{\text{DTD1}}(\mathbf{B}) - s_{\text{DTD2}}(\mathbf{B})|) \geq \frac{1}{\text{SNR}} \quad (13) $$

where B is a b-tensor in the acquisition scheme, s<sub>DTD1</sub>(B) (resp. s<sub>DTD2</sub>(B)) is the signal generated from DTD 1 (resp. DTD 2) for the b-tensor B.

## 3.3 Selection of acquisition schemes

Among the superset S, the objective is to select a small subset of k b-tensor shapes, such that the number of pairs of DTD that can be distinguished is maximized. The number of possible subsets, C<sub>n</sub><sup>k</sup>, grows very quickly with n and k which renders the brute-force approach impractical.

The criterion in equation 13 ensures that if we take a subset S<sub>∞</sub> ⊂ S, the number of pairs of DTD that can be discriminated is at most the same. To compare the shapes, we assigned a score to each. This score is the number of pairs of DTD that can be discriminated using a scheme with this b-tensor shape on every direction considered. This way, the stronger the shape, the more DTD can be discriminated, the weaker the shape, the less DTD can be discriminated. We focused on two approaches to select the acquisition subsets: a greedy algorithm (see Algorithm 1) and a permutation algorithm (see Algorithm 2).

We chose the number of selected b-tensor shapes so as to be comparable to the reference scheme, Q3. The scheme Q3, described in Table 1, is optimized using the Cramér-Rao lower bound trace of the mean diffusivity, µFA, the isotropic kurtosis and the anisotropic kurtosis described in [11]. Since Q3 uses 120 acquisitions, we selected 4 shapes of b-tensor acquired on 30 directions each. The 30 directions were generated analytically following Koay's method [10]. The upper b-value in Q3 is 2000 s/mm², which is the same as what we retained for our superset.

## 3.4 Parameter estimation

We evaluated the accuracy of the parameters of interest, using signal simulated following each of the optimized acquisition design. In order to do that, we es-
**Algorithm 1:** Greedy algorithm

**Input:** Distance matrix, nb\_shape
**Output:** Acquisition subset
subset ← [];
scores ← Array of size nb\_shape, containing the number of pair of DTDs that
can be discriminated by each shape;
**for** i ← 1 **to** nb\_shape **do**
    best\_shape ← Best shape available outside of subset;
    subset ← subset+best\_shape;
    scores ← Number of new pairs that can be discriminated using each shape;
**end**
**return** subset

**Algorithm 2:** Permutation algorithm

**Input:** Distance matrix, nb\_shape, nb\_rep
**Output:** Best acquisition subset generated
best\_subset\_seen ← [];
best\_nb\_dtds ← 0;
**for** i ← 1 **to** nb\_rep **do**
    subset ← random selection of nb\_shape shapes;
    changed ← True;
    **while** changed **do**
        weakest\_shape ← Weakest shape of subset;
        nb\_weakest ← number of pairs for the subset with the weakest shape;
        next\_best\_shape ← Best shape available outside of the subset;
        nb\_next\_best ← number of pairs for the subset with the
            next\_best\_shape instead;
        **if** nb\_weakest < nb\_next\_best **then**
            Switching the weakest shape with the next best one in subset;
        **end**
        **else**
            changed ← False
        **end**
    **end**
    nb\_dtds\_subset ← number of pairs of DTDs that can be discriminated;
    **if** nb\_dtds\_subset > best\_nb\_dtds **then**
        best\_nb\_dtds gets nb\_dtds\_subset;
        best\_subset\_seen ← subset;
    **end**
**end**
**return** best\_subset\_seen

timated each parameter using QTI+ constraints [8] and its implementation in
Dipy [7].

The acquisition scheme that we use as the reference is Q3 [11]. We focused
more specifically on the estimation of OP and the estimation of µFA. For a given
OP (resp. µFA), we generated 11 DTDs with 10000 tensors and a $C_{MD}$ at 0.1. The 11 DTDs were generated using the full range of the varying parameter: 0 to 1 with a step of 0.1 between values.

Then, for each of these DTDs, we generated the signals from each acquisition scheme that we did or did not corrupt with Rician noise. In the case of a corruption with noise, we fixed the SNR at 30 and generated 100 signals corrupted with noise for each DTD.

# 4 Results

The hierarchical division of the parameter space yields a total of 124 cubes when we set SNR = 25, out of a total of $2^{3 \times 6} = 262144$ possible cubes, since we used 6 recursive levels. On the resulting subdivision, on Fig. 3, we can see that the $C_{MD}$ value has no significant impact on signals as only one division was made within the range. Also, we clearly have very few subdivisions in the parameter space when the µFA is small. This can be explained as it is hardly possible to find direction information within sphere-like tensors.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Division of the parameter space in boxes of DTD with generated signal according to the achievable resolution.</figcaption></figure>
<table><caption>**Table 1.** Repartition of the b-tensor encoded acquisitions. Q3 is the reference scheme, PTE corresponds to planar b-tensors, OTE to oblate b-tensors, STE to spherical b-tensors and LTE to linear b-tensors.</caption><thead><tr><th rowspan="2">Acquisition scheme<br>b (s/mm²)</th><th colspan="3">Q3</th><th colspan="3">Greedy algorithm</th><th colspan="3">Permutation algorithm</th></tr><tr><th>PTE</th><th>OTE</th><th>STE</th><th>PTE</th><th>OTE</th><th>STE</th><th>PTE</th><th>OTE</th><th>STE</th><th>LTE</th></tr></thead><tbody><tr><td>100</td><td>7</td><td>-</td><td>-</td><td>9</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>800</td><td>9</td><td>-</td><td>-</td><td>50</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1000</td><td>-</td><td>-</td><td>-</td><td>-</td><td>30</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>1100</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>30</td><td>-</td></tr><tr><td>1200</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>30</td><td>-</td><td>-</td><td>30</td><td>-</td></tr><tr><td>1300</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>-</td><td>30</td><td>-</td><td>-</td><td>30</td></tr><tr><td>2000</td><td>-</td><td>-</td><td>30</td><td>15</td><td>-</td><td>-</td><td>-</td><td>30</td><td>-</td><td>-</td></tr></tbody></table>

Table 1 details the repartition of the b-tensor encoded acquisitions for the three acquisition schemes we compared. The first one is Q3 [11] and includes planar, spherical and linear b-tensors. The second one is the scheme generated from the greedy algorithm. The b-values selected are between 1000 and 2000 s/mm² and includes planar, oblate and linear b-tensors but no spherical. The last one is the scheme generated from the permutation algorithm with 100 random initializations. Selected b-values are 1000, 1100 and 1200 s/mm² with some planar, oblate and linear b-tensor selected.

Figure 4 shows the error on the estimation of OP over its full range for two values of µFA, without noise or with Rician noise with an SNR of 30. On the top left, we have the comparison of the estimation error for our three schemes at SNR 30 with a µFA of 0.6. In this case, similar results were achieved for the reference scheme Q3 and the scheme selected through the greedy algorithm. The permutation algorithm gives a more important error. It is interesting to note that the error is fairly high for every scheme considered here. When we look at the top right figure, we keep the same value of the noise but we increase the µFA. While the scheme from the permutation algorithm still gives the least interesting results, the scheme generated from the greedy algorithm gives an error that is decreased and better than the reference scheme. This pattern is also present in the bottom row where the signal was not corrupted with noise.

Figure 5 shows the error on the estimation of µFA over its full range for two values of OP, with a Rician noise at an SNR of 30 or without noise. The top row presents the estimation with Rician noise and the bottom row the estimation without noise. The first column gives the estimation for an OP of 0.4 while the second gives it for an OP of 0.8. Overall, the estimation error is smaller than it is for the estimation of OP. In all four cases, Q3 and the greedy algorithm scheme perform similarly. In presence of noise, the permutation algorithm generated the most errors. However, without noise, it generates the best results.
<figure><img src="image_5.png" /><figcaption>**Fig. 4.** OP estimation error on its full range. In blue, the reference scheme; in orange, the greedy algorithm result; in green, the permutation algorithm result. (top row) The signal was corrupted with a Rician noise at an SNR 30. (bottom row) The signal was not corrupted with any noise. (first column) µFA was set at a medium value. (second column) µFA was set at a high value.</figcaption></figure>

# 5 Discussion

We proposed a framework aiming to improve the accuracy of DTD parameters estimation. We found that the b-tensor encoded acquisition scheme generated by the greedy approach provided the best results in terms of estimation compared to the previously proposed scheme we used as a reference.

Our approach is promising and can be further improved in several ways. One improvement that needs to be implemented is to adapt the number of directions based on the b-value and the shape of the b-tensor. That can be achieved by computing the band limit in spherical harmonic for each b-tensor shape [4]. Then, we can select a number of direction based on the band-limit using spherical codes [12]. Another direction for improvement is to take into consideration the impact of the shape of a b-tensor and its b-value on the echo time (TE), directly related to the waveform [15]. That way, we can not only select an upper bound in the number of acquisitions selected, but also guide the choice towards b-tensor shapes with shorter TE, leading to an overall improvement in SNR.
<figure><img src="image_6.png" /><figcaption>**Fig. 5.** µFA estimation error on its full range. In blue, the reference scheme; in orange, the greedy algorithm result; in green, the permutation algorithm result. (top row) The signal was corrupted with a Rician noise at an SNR 30. (bottom row) The signal was not corrupted with any noise. (first column) OP was set at a medium value. (second column) OP was set at a high value.</figcaption></figure>

In this work, we focused on the DTD framework but this could easily be adapted to work with biophysical models such as AxCaliber [3], CHARMED [2], NODDI [18] or SANDI [13] as they have a limited number of parameters of interest.

# 6 Conclusion

The estimation of microstructure parameter is greatly influenced by the acquisition scheme. Our approach allows us to generate acquisition schemes optimized towards the reconstruction of selected parameters.

While the current results can be further improved, we generated acquisition schemes with similar or better estimation errors while ensuring its feasibility on clinical MR scanner.
Our approach could also be implemented for other models with a reasonable number of parameters of interest in order to generate acquisition schemes tailored specifically for their reconstruction.

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