Magnetic resonance imaging of blood perfusion rate based on Helmholtz decomposition of heat flux

Objective. Thermal property (TP) maps of human tissues are useful for tumor treatment and diagnosis. In particular, the blood perfusion rate is significantly different for tumors and healthy tissues. Noninvasive techniques that reconstruct TPs from the temperature measured via magnetic resonance imaging (MRI) by solving an inverse bioheat transfer problem have been developed. A few conventional methods can reconstruct spatially varying TP distributions, but they have several limitations. First, most methods require the numerical Laplacian computation of the temperature, and hence they are sensitive to noise. In addition, some methods require the division of a region of interest (ROI) into sub-regions with homogeneous TPs using prior anatomical information, and they assume an unmeasurable initial temperature distribution. We propose a novel robust reconstruction method without the division of an ROI or the assumption of an initial temperature distribution. Approach. The proposed method estimates blood perfusion rate maps from relative temperature changes. This method avoids the computation of the Laplacian by using integral representations of the Helmholtz decomposition of the heat flux. Main Result. We compare the reconstruction results of the conventional and proposed methods using numerical simulations. The results indicate the robustness of the proposed method. Significance. This study suggests the feasibility of thermal property mapping with MRI using the robust proposed method.


Introduction
Imaging the thermal properties (TPs) of human tissues, such as blood perfusion parameters and thermal conductivity, is helpful for tumor diagnosis and treatment.Images of TPs can assist in the diagnosis of breast tumors, because breast tumors and healthy breast tissues have different TPs (Duck 2013, Lozano et al 2020, Said Camilleri et al 2022).In particular, the blood perfusion rate is significantly different for cancerous and healthy breast tissues.Therefore, several studies have identified breast tumors by solving the inverse heat transfer problem for the blood perfusion rate (Mitra and Balaji 2010, Hatwar and Herman 2017, Bezerra et al 2020, Sudarsan et al 2021).In addition, high-intensity focused ultrasound (HIFU) has been developed as a thermal treatment technique that heats and destroys target tumors using focused ultrasound (Solovchuk et al 2012, Gupta and Srivastava 2019, Izadifar et al 2020).In HIFU, thermal damage to healthy tissues should be suppressed while applying sufficient heat to tumors to ablate them.An analogous issue arises in cryosurgery which is a medical technique that freezes and destroys target tumors (Ge et al 2015, Burkov et al 2020).Thus, for the evaluation and planning of HIFU and cryosurgery, simulations of heat transfer in biological tissues are inevitable (Solovchuk et al 2012, Burkov et al 2020), and such simulations require TP images of human tissues.Magnetic resonance (MR) perfusion techniques imaging blood flow, such as dynamic contrast-enhanced magnetic resonance imaging (MRI), are also used for tumor diagnosis and treatment planning (Keserci andDuc 2017, Sadick et al 2019).The blood flow map is significantly related to the blood perfusion rate map.However, it is difficult to predict temperature changes using the blood flow, whereas it can be achieved using the TPs.It is the advantage of the TP imaging in thermal treatment planning.From these reasons, methods for determining the TPs of human tissues have been developed.
Heat transfer in biological tissues is modeled using Pennes' bioheat transfer equation (Pennes 1998).The thermal properties of biological tissues can be reconstructed from measured temperature changes by solving the inverse problem of the bioheat transfer equation.Conventional studies for TP reconstructions are mainly classified into two types in terms of regions of temperature measurements.In the first type, TPs are reconstructed from the temperature measured on the surface of human bodies using infrared (IR) cameras (Mitra and Balaji 2010, Yin et al 2013, Hatwar and Herman 2017, Bezerra et al 2020, Knupp 2021, Sudarsan et al 2021).Although IR cameras can provide high-sensitivity and high-spatial-resolution surface temperature measurements, it is difficult for these methods to reconstruct high-spatial-resolution TPs in human bodies due to the ill-posedness of the inverse problem in which inner TPs are estimated from the surface temperature.In the second type, TPs are reconstructed from the temperature inside human bodies (Cheng and Plewes 2002, Huttunen et al 2006, Sumi and Yanagimura 2007, Dragonu et al 2009, Xu et al 2009, Cornelis et al 2011, Dillon et al 2016, 2018), which can be noninvasively measured with ultrasound or MRI (Rivens et al 2007).In particular, the temperature measurement technique using MRI, which is called MR thermometry (Oh et al 2014, Lee et al 2019, Blackwell et al 2022), can provide the temperature with high spatial resolution.Although these techniques have lower sensitivity to temperature changes than IR cameras, the ill-posedness of the inverse problem is relaxed in these methods, because the temperature can be measured in inner regions.
A method for estimating TPs from the temperature measured by MR thermometry with ultrasound-induced heating was initially proposed by Cheng and Plewes (Cheng and Plewes 2002).In this method, the temperature change was measured around a point heated by focused ultrasound, and then TPs at the focal point were estimated by fitting the temperature data to a function satisfying the bioheat transfer equation under the assumption of local homogeneity of TPs.Subsequently, methods with analytical solutions using focused ultrasound heating were proposed (Dragonu et al 2009, Dillon et al 2016) and applied to several in-vivo experiments (Cornelis et al 2011, Dillon et al 2018).However, they also assumed local homogeneity of TPs in the heated region and thus could not reconstruct spatially varying TP distributions.
A few methods were proposed to reconstruct inhomogeneous TP distributions by solving the bioheat transfer equation from measured spatio-temporal temperature data (Huttunen et al 2006, Sumi and Yanagimura 2007, Xu et al 2009).A method proposed by Huttunen et al estimated the values of the blood perfusion rate and thermal conductivity in sub-regions of a region of interest (ROI) by the finite element method (FEM) under the assumption that TPs are homogeneous in each sub-region (Huttunen et al 2006).The method could reconstruct heterogeneous TPs including several different tissues.However, it required to divide the ROI into the sub-regions with homogeneous TPs using prior anatomical information before the reconstruction process.Sumi et al proposed a method for reconstructing the thermal conductivity map, which did not require to assume sub-regions with homogeneous TPs (Sumi and Yanagimura 2007).However, in their method, the blood perfusion rate was ignored and could not be obtained.Moreover, their method was sensitive to noise because the computation of the numerical Laplacian of the temperature required in the bioheat transfer equation amplified the noise.Furthermore, although the above methods solve the equation for the absolute temperature, MR thermometry can measure only the relative temperature change from the initial time.Thus, these methods required an assumption of the initial temperature distribution to obtain the absolute temperature.Xu et al proposed a method for reconstructing the blood perfusion rate from the relative temperature, assuming that the thermal conductivity is homogeneous and known (Xu et al 2009).However, this method also required the computation of the Laplacian of the measured relative temperature.
In this study, to overcome these limitations in the conventional methods, we use an analogy between TP reconstructions and electrical property (EP) reconstructions that can also be used for tumor diagnosis.In the socalled magnetic resonance electrical properties tomography (MREPT) (Katscher et al 2022), EPs such as electrical conductivity and permittivity are reconstructed from the magnetic field measured using MRI by solving Maxwell's equations.We have previously proposed a method (Eda et al 2021) for MREPT based on Helmholtz decomposition of the electric field without the Laplacian computation of the magnetic field or the need to assume EP distributions.We extend this method for EP reconstructions to TP reconstructions.The key idea is that the bioheat transfer equation for the relative temperature can be split by introducing the heat flux as an intermediate variable into the simultaneous partial differential equations which have the same form as Maxwell's equation.As a result, an integral equation for TPs is derived from the Helmholtz decomposition of the heat flux.The blood perfusion rate can then be estimated from this integral equation using the measured relative temperature without Laplacian computation or division of an ROI.Furthermore, beyond a simple correspondence between the EP and TP reconstructions, it is clarified that time-series data of the relative temperature enhances the stability of TP reconstructions, in contrast to EP reconstructions with time-harmonic magnetic field data.
The remainder of this paper is organized as follows.In section 2.1, the bioheat transfer equation for the relative temperature is separated into two governing equations by introducing the heat flux.In section 2.2, a novel reconstruction method based on Helmholtz decomposition of the heat flux is proposed.The settings for numerical simulations are described in section 3.In section 4, we show the reconstruction results of the proposed method and the method proposed by Xu et al (Xu et al 2009), and discuss the features and limitations of the proposed method.Finally, this paper is concluded in section 5.

Theory
2.1.Governing equations for relative temperature and heat flux in biological tissues Let W Ì  3 be a bounded domain representing an ROI in a human body and let ∂Ω be its boundary.Assuming that no external heat source exists in Ω, the temperature T(r, t) at r ä Ω and time t in biological tissues obeys Pennes' bioheat transfer equation where ω b is the blood perfusion rate; k, c and ρ are the thermal conductivity, heat capacity and density of tissues, respectively; Q meta is the inner heat source produced by tissue metabolism; and ρ b , c b and T b are the density, heat capacity and temperature of arterial blood, respectively.We assume that k, c and ρ are constant and known as in the conventional method proposed by Xu et al (2009).Additionally, ρ b , c b and T b are also assumed to be constant.
In equation (1), all quantities except T are assumed to be independent of time.Here, MR thermometry can only measure the relative temperature change δT(r, t) ≔ T(r, t) − T(r, t 0 ), where t 0 is the initial measurement time point.Xu et al transformed the bioheat transfer equation (1) for T(r, t) into the equation for δT(r, t) as Under the assumption that k, c and ρ were constant and known, they estimated ω b from equation (2) as where á ñ • is defined as the parameter obtained by the time-averaging operation.However, this method is sensitive to noise because of the Laplacian computation of δT in equation (3).
In the present study, we introduce the relative heat flux δJ and split equation (2) into two equations: Equations (4) and (5) represent Fourier's law and the law of conservation of energy for δT, respectively.We reconstruct ω b from the measured δT without a Laplacian computation.The Laplacian computation in equation (2) originates from the elimination of δJ in equations (4) and (5); in fact, substitution of the divergence of δJ in equation (4) into equation (5) removes δJ while generating the Laplacian of δT.In contrast, we solve the governing equations (4) and (5) for the blood perfusion rate using the integral representations of δJ.The assumption that k, c and ρ are constant and known is a limitation of the proposed method.Reconstruction errors are numerically evaluated in section 4 under the assumption of incorrect thermal parameters, k, c and ρ.

Reconstruction of blood perfusion rate based on Helmholtz decomposition of heat flux
In order to derive the integral representations of the relative heat flux δJ(r, t), we first decompose δJ(r, t) by Helmholtz decomposition (Kustepeli 2016, Freeden andGerhards 2019) in Ω given by δJ(r, t) = δJ sol (r, t) + δJ irr (r, t), where δJ sol and δJ irr are the solenoidal and irrotational components of δJ, respectively.It is shown in (Eda et al 2021) that δJ irr (r, t) can be expressed by the following two different integral representations: irr where and n is the outward unit normal to ∂Ω.Here, using Fourier's law (4), equation ( 6) is rewritten as irr Additionally, substituting the governing equations (4) and (5) into equation (7) leads to Then, equations (8) and (9) give an integral equation for the TPs in Ω: Thus, the integral equation for the blood perfusion rate to be solved in the proposed method is which can be calculated from the measured δT under the assumption that k, c and ρ are known.Setting regular grid points r i (i ä {1, 2,...,N}), equation ( 11) is a matrix obtained by the discretization of the integral operation on the left-hand side of equation (11).Because equation ( 12) is satisfied at all measurement times t j ( j ä {1, 2,...,M}), we solve the system of equation (12) for all t j : When solving the linear equation (13), we impose L 2 total variation regularization for ω b and minimize the cost function given by is the discretized gradient matrix, and λ is a regularization parameter.To effectively solve ∇Φ = 0, we combine the conjugate-gradient (CG) method and the fast Fourier transform (FFT).The CG method (Hestenes and Stiefel 1952) is well-known for quickly solving a large system of linear equations.Additionally, the matrix operation A(t)W b can be quickly calculated using the FFT, because the operation expresses a discretized convolution.In the calculation of the convolution, the singularity of the kernel

Numerical experiment
We validated the effectiveness of the proposed method by numerical simulations.We reconstructed ω b and compared the reconstruction results with those obtained with the conventional method using equation (3).
We examined a hemispherical simulation model (radius: 72 mm) representing the shape of a breast, referring to the model used by Hatwar and Herman (Hatwar and Herman 2017).As shown in figure 1(a), the simulation model consisted of three spheres representing inclusions (radii: 10, 7.5 and 5 mm; depth from skin surface: 30 mm) inside the gland region.The thermal parameters for the tissues in the simulation are summarized in table 1.
Temperature data were computed using FEM-based software (COMSOL Multiphysics, COMSOL Inc.).We fixed the temperature on the bottom surface of the muscle region to the core body temperature of 37 °C, assuming that the muscle touches the chest wall.Additionally, we assumed the skin surface to be exposed to ambient air and imposed the convective boundary condition on the surface as n s • J = h(T − T air ), where n s is the outward unit normal to the skin surface, h = 10 W/(K•m 2 ) is the convective heat transfer coefficient and T air = 21 °C is the temperature of ambient air.The parameters of arterial blood were set to ρ b = 1060 kg m −3 , c b = 3770 J (kg•K) −1 and T b = 37 °C.To investigate the dependence of the conditions of the inverse problem on the initial temperature distribution, we simulated two cases given two different initial temperature distributions at t = 0 s as shown in figure 1(b), hereinafter referred to as Case 1 and Case 2. In both cases, the initial temperature distributions produced by ultrasound-induced heating were represented by Gaussian functions plus a constant equal to the core body temperature.The parameters for the Gaussian functions are summarized in table 2. Although the highest temperatures in both cases were nearly identical, the temperature distribution was more focused in Case 1 than in Case 2. During thermal treatments, the increase in the temperature of healthy tissues should be kept below 4 °C in order to avoid tissue damage (Huttunen et al 2006), which was satisfied in the entire breast in both cases.The voxel size of the simulated temperature data was set to 1.4 mm × 1.4 mm × 5 mm and the temporal resolution of the data was set to 30 s in both cases.Although most MR thermometry techniques used for HIFU treatments have a spatial resolution of a few mm, temporal resolution of a few seconds and temperature resolution of about 1 °C (Blackwell et al 2022), Fagan et al proposed an MR thermometry technique with a low temporal resolution of 20 s but with a high temperature resolution of 0.1 °C using a 7 T MRI to measure small temperature changes of about 1 °C compared to those in the HIFU Table 1.Thermal parameters used in simulations.treatment (Fagan et al 2021).The increase in the temperature was also about 1 °C in this study, and hence we set the lower temporal resolution than that of a typical MR thermometry technique.The ROIs for both cases were 60 mm × 35 mm × 40 mm cuboid regions inside the gland region (with a matrix size of 42 × 24 × 8), as shown in figure 1(b), and temporal temperature data from 150 to 750 s (21 epochs) were used.The temperature data was given by the relative temperature δT(r, t) = T(r, t) − T(r, t 0 ) where t 0 = 150 s.Gaussian noise (standard deviation: 0.1, 0.3 and 0.5 °C) was added to the relative temperature data.We calculated the spatial and temporal derivatives of the temperature by applying a four-dimensional Savitzky-Golay (SG) filter (Press and Teukolsky 1990) with a size of 3 × 3 × 3 × 3 to both the noiseless and noisy data.Furthermore, in the noisy case, we also reconstructed the blood perfusion rate with the conventional method using a 9 × 9 × 3 × 3 SG filter, in addition to the reconstruction using the 3 × 3 × 3 × 3 SG filter.When reconstructing the blood perfusion rate by the conventional and proposed methods, k, c and ρ in the ROI were assumed to be the same constants as those of the gland indicated in table 1.Then, in order to discuss the influence of providing incorrect thermal parameters, we changed the assumed values of k and c in the reconstruction of the proposed method while maintaining the value of ρ fixed at that of the gland, because the product of ρ and c influences bioheat transfer in the bioheat equation (1).The regularization parameter λ in the cost function ( 14) was fixed heuristically at λ = 10 −12 for all conditions.
The estimated blood perfusion rate was evaluated using the normalized root mean square errors (NRMSEs) defined as where x e and x t are the estimated and true scalar fields.

Results and discussion
Figure 2 shows the reconstructed blood perfusion rate in Case 1 without noise.Both the conventional and proposed methods achieved the reconstructions with great accuracy in the absence of noise and identified all three small inclusions at a depth of 30 mm from the skin surface.The conventional study (Hatwar and Herman 2017) reported that it is difficult for the method with surface temperature measurements to estimate the sizes, locations and blood perfusion rates of deep and small tumors, even when reducing unknown parameters by imposing constraints on their geometries, due to the ill-posedness of the inverse problem.The results in figure 2 suggest that methods with MR thermometry have the potential to identify deeper and smaller tumors and retrieve their blood perfusion rate values without the assumptions of their geometries, compared to the methods with surface temperature measurements.Figure 3 and 4 show the reconstructed blood perfusion rates in Case 1 and Case 2, respectively, with Gaussian noise.As shown in figure 3(a), high-frequency artifacts appear when using the conventional method with a 3 × 3 × 3 × 3 SG filter due to the numerical Laplacian calculation.
In addition, as shown in figure 3(b), although the conventional method using a large-sized SG filter suppressed the high-frequency artifacts, the boundary between the inclusion and healthy tissue was smoothed, which is also indicated in the line profile in figure 3(d).This is a crucial issue for the diagnosis of small tumors.Conversely, the results of the proposed method are robust to noise without the large-sized filter, as in figure 3(c) and line profile in figure 3(d), because the computation of the second-order derivative is not necessary.This is also indicated by the NRMSEs in table 3. Furthermore, as shown in table 3, the reconstruction errors of both methods are smaller in Case 2 than in Case 1.This is because the conditions of the inverse problem depend on the initial temperature distribution.Figure 5 shows relative temperature distributions of two cases in the ROI in the final epoch.As shown in the left panel, in Case 1, the relative temperature was small in the region |x| > 20 mm, meaning that the temperature was maintained near the core body temperature.This is because the initial increase from the core body temperature was focused, as shown in figure 1(b).As a result, the SNR of the relative temperature was low so that the reconstruction for Case 1 in this region was ill-posed.In contrast, as shown in the right panel of figure 5, a significant temperature decrease occurred in the entire ROI in Case 2, because a large increase relative to the core body temperature was initially given throughout the ROI, as shown in figure 1(b).Thus, the SNR of the relative temperature was uniformly high in the ROI, and hence the reconstruction condition was improved in Case 2. In actual setup, the initial temperature is determined by the intensity and extent of heat sources.Additionally, in clinical applications, such as tumor detection, the initial temperature in tumors is significantly related to the performance of the reconstruction method, which is affected by the sizes and positions of tumors compared to those of the heat sources.In both cases, the standard deviation of the Gaussian function that represented the temperature distributions produced by the ultrasound-induced heating was 18 mm and larger than the radii of the inclusions ( 10 mm), and the area of two standard deviations from the center of the Gaussian function covered the inclusions.Thus, a sufficient initial temperature change occurred in the inclusions to reconstruct the blood perfusion rate.Furthermore, in Case 2, the initial temperature was represented by two Gaussian functions and the centers of the Gaussian functions were close to the inclusions.As a result, the SNR of the relative temperature in the inclusions was higher in Case 2 than in Case 1, as in figure 5, and hence the reconstruction condition in the inclusions was also improved.
The necessity for assuming the thermal conductivity, heat capacity and density is a limitation of the proposed method.In the above reconstructions, the thermal parameters of the gland indicated in table 1 were provided.To investigate the influence of providing incorrect thermal parameters, we assumed thermal conductivity values of 0.3 or 0.7 W (m•K) −1 for all tissues, although the true values vary depending on the tissues.The reconstruction results for Case 1 without noise are compared in figure 6.As shown in the line profile of figure 6(c), the reconstruction errors are sufficiently small to not interfere with detecting the inclusions, even when the assumed thermal conductivity is incorrect.The results indicate that the blood perfusion term in equation (2) causes the temperature changes in the simulation more dominantly than the heat transfer term.Next, the effect of an erroneous assumption of the heat capacity on the blood perfusion rate reconstruction was examined.Assuming c = 2500 or 4500 J (kg•K) −1 for all tissues, the blood perfusion rate was reconstructed for Case 1 without noise.The results are compared in figure 7.As shown in the line profile of figure 7(c), the reconstructed blood perfusion rate is approximately proportional to the assumed heat capacity value.This is because the heat transfer term in equation (2) is negligible compared to the perfusion term.However, as shown in figure 7, the contrast between the blood perfusion rate in the inclusions and that in the healthy tissues was still observable with c = 2500 J (kg•K) −1 .Therefore, even with incorrect thermal parameters, the proposed method can retrieve the contrast of the blood perfusion rate and identify the inclusions.This implies the feasibility of the proposed method in clinical applications under the assumption that the thermal conductivity, heat capacity and density are constant and known.Furthermore, the proposed method has the potential to estimate the thermal parameters assumed in this study as additional unknown parameters in the integral equation (10), although the integral equation may be ill-conditioned due to the increase in the number of unknown parameters.Further analysis is necessary.
The TP reconstruction problem is analogous to MREPT, as described in section 1.In MREPT, substitution of the derivatives of the electric field in Ampere's law into Faraday's law eliminates the electric field while generating the Laplacian of the measured magnetic field.Therefore, in TP reconstructions, the measured temperature and the introduced heat flux respectively correspond to the magnetic and electric fields in MREPT.Based on this analogy, we extended our previously reported technique for MREPT (Eda et al 2021) in this study to TP reconstructions.However, there are two differences between TP and EP reconstructions.First, the governing equations (4) and (5) depend on time, although Maxwell's equations in MREPT are time-harmonic.In the present study, temporal temperature data were used to derive a system of integral equations, which would improve the ill-posedness of the inverse problem.Additionally, it is possible to change the spatial and temporal distributions of the heat sources to improve the conditions of the inverse problem in TP reconstructions, as indicated by the results given two different initial temperature distributions in figures 3 and 4, whereas changing the electromagnetic field source in MREPT is difficult.This provides another control parameter for TP reconstructions.Optimizing the design of heat sources is left for further studies.
In this study, the blood perfusion rate was assumed to be independent of time.However, blood perfusion typically increases with increasing temperature in healthy tissues more than in tumors (Bianchi et al 2022).In this case, the proposed method probably estimates the temporal average of the blood perfusion rate, and hence   the contrast between the blood perfusion rate in tumors and that in healthy tissues may be retrieved but less observable.Furthermore, blood temperature also increases with increasing temperature in tissues (Laakso and Hirata 2011), although it was assumed to be constant in this study.The blood temperature vanishes in the perfusion term of equation (2) under the assumption.If it increases, its variation remains in the perfusion term, which leads to the reconstruction error in the proposed method.However, the influences of the temporal dependence of blood perfusion and the increase in blood temperature may be sufficiently small to reconstruct the blood perfusion rate, because the simulated temperature change is small compared to that in the thermal treatments.Further verification in realistic simulations considering the influences will be necessary.

Conclusion
We proposed a novel reconstruction method for blood perfusion rate distributions from the relative temperature under the assumption that the thermal conductivity, heat capacity and density are constant and known.The numerical results showed that the proposed method was robust to noise due to the elimination of the Laplacian computation by using integral representations of the Helmholtz decomposition of the heat flux.
The results also indicated that the proposed method could retrieve the contrast between the blood perfusion rates in inclusions and healthy tissues even when the assumed thermal parameters were incorrect.The estimation of the parameters assumed in this study in addition to the blood perfusion rate will be the subject of future studies.
r r is avoided by using a kernel truncated by a sphere with a radius large enough to encompass Ω(Vico et al 2016, Eda et al 2021).

Figure 1 .
Figure 1.Simulation model.(a) Geometry of simulation model.The model includes four healthy tissues (Dermis, Fat, Gland and Muscle) and three inclusions.The temperature on the bottom surface that touches the chest wall is fixed to the core body temperature.The convective boundary condition is imposed on the skin surface.(b) Initial temperature distributions in two cases.The left and right panels show the distributions in Case 1 and Case 2, respectively.The dash-dotted rectangles represent ROIs.

Figure 2 .
Figure 2. Reconstructed blood perfusion rate in Case 1 without noise.(a) True blood perfusion rate map.(b) Reconstruction results for conventional method using 3 × 3 × 3 × 3 SG filter.(c) Reconstruction results for proposed method using 3 × 3 × 3 × 3 SG filter.The upper and lower rows in (a)-(c) show the results for planes at z = −2.5 mm and at y = 44.3mm, respectively.(d) Line profile of reconstructed blood perfusion rate along dashed line in (a).

Figure 3 .
Figure 3. Reconstructed blood perfusion rate in Case 1 with Gaussian noise.(a) Reconstruction results for conventional method using 3 × 3 × 3 × 3 SG filter.(b) Reconstruction results for conventional method using 9 × 9 × 3 × 3 SG filter.(c) Reconstruction results for proposed method using 3 × 3 × 3 × 3 SG filter.The left, middle and right columns in (a)-(c) show the results for the data added Gaussian noise with standard deviations of 0.1, 0.3 and 0.5 °C, respectively.(d) Line profile of reconstructed blood perfusion rate in Case 1 with Gaussian noise (standard deviation is 0.1 °C) along dashed line in figure 2(a).

Figure 4 .
Figure 4. Reconstructed blood perfusion rate in Case 2 with Gaussian noise.The left, middle and right columns show the results for the data added Gaussian noise with standard deviations of 0.1, 0.3 and 0.5 °C, respectively.(a) Reconstruction results for conventional method using 3 × 3 × 3 × 3 SG filter.(b) Reconstruction results for proposed method using 3 × 3 × 3 × 3 SG filter.

Figure 5 .
Figure 5. Relative temperature distributions at t = 750 s in both cases.The left and right panels show the distributions in Case 1 and Case 2, respectively.

Figure 6 .
Figure 6.Reconstructed blood perfusion rate using proposed method in Case 1 without noise assuming incorrect thermal conductivities.(a) Reconstruction results assuming thermal conductivities of 0.3 W (m•K) −1 for all tissues.(b) Reconstruction results assuming thermal conductivities of 0.7 W (m•K) −1 for all tissues.(c) Line profile of reconstructed blood perfusion rate along dashed line in figure 2 (a).

Figure 7 .
Figure 7. Reconstructed blood perfusion rate using proposed method in Case 1 without noise assuming incorrect heat capacities.(a) Reconstruction results assuming heat capacities of 2500 J (kg•K) −1 for all tissues.(b) Reconstruction results assuming heat capacities of 4500 J (kg•K) −1 for all tissues.(c) Line profile of reconstructed blood perfusion rate along dashed line in figure 2 (a).

Table 2 .
Gaussian function parameters in initial temperature distributions in two cases.

Table 3 .
Normalized root mean square errors averaged over 10 reconstructions of both methods using a 3 × 3 × 3 × 3 SG filter for Case 1 and Case 2 without and with Gaussian noise.